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Mathematical Principles

of

Natural Philosophy

Translated into English by Andrew Motte

Text and images from Newton's Principia: The Mathematical Principles of Natural Philosophy ; to which is added, Newton's System of the World / translated by Andrew Motte [first American edition; New York: Daniel Adee, c1846] and based on the transcription done by Wikisource, with images for mathematical expressions replaced by text.

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- Of the method of first and last ratios of quantities.
- Of the Invention of Centripetal Forces.
- Of the motion of bodies in eccentric conic sections.
- Of the finding of elliptic, parabolic, and hyperbolic orbits, from the focus given.
- How the orbits are to be found when neither focus is given.
- How the motions are to be found in given orbits.
- Concerning the rectilinear ascent and descent of bodies.
- Of the invention of orbits wherein bodies will revolve, being acted upon by any sort of centripetal force.
- Of the motion of bodies in moveable orbits; and of the motion of the apsides.
- Of the motion of bodies in given superficies, and of the reciprocal motion of funependulous bodies.
- Of the motions of bodies tending to each other with centripetal forces.
- Of the attractive forces of sphaerical bodies.
- Of the attractive forces of bodies which are not of a sphaerical figure.
- Of the motion of very small bodies when agitated by centripetal forces tending to the several parts of any very great body.

- Of the motion of bodies that are resisted in the ratio of the velocity.
- Of the motion of bodies that are resisted in the duplicate ratio of their velocities.
- Of the motions of bodies which are resisted partly in the ratio of the velocities, and partly in the duplicate of the same ratio.
- Of the circular motion of bodies in resisting mediums.
- Of the density and compression of fluids; and of hydrostatics.
- Of the motion and resistance of funependulous bodies.
- Of the motion of fluids, and the resistance made to projected bodies.
- Of motion propagated through fluids.
- Of the circular motion of fluids.

SINCE the ancients (as we are told by Pappus), made great account of the science of mechanics in the investigation of natural things : and the moderns, laying aside substantial forms and occult qualities, have endeavoured to subject the phenomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics so far as it regards philosophy. The ancients considered mechanics in a twofold respect ; as rational, which proceeds accurately by demonstration ; and practical. To practical mechanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accurate is called geometrical , what is less so, is called mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic ; and if any could work with perfect accuracy, he would be the most perfect mechanic of all ; for the description if right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn ; for it requires that the learner should first be taught to describe these accurately, before he enters upon geometry ; then it shows how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from mechanics ; and by geometry the use of them, when so solved, is shown ; and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things. Therefore geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that geometry is commonly referred to their magnitudes, and mechanics to their motion. In this sense rational mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any motions, accurately proposed and demonstrated. This part of mechanics was cultivated by the ancients in the five powers which relate to manual arts, who considered gravity (it not being a manual power), ho Otherwise than as it moved weights by those powers. Our design not respecting arts, but philosophy, and our subject not manual but natural powers, we consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces, whether attractive or impulsive ; and therefore we offer this work as the mathematical principles of philosophy ; for all the difficulty of philosophy seems to consist in this from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena ; and to this end the general propositions in the first and second book are directed. In the third book we give an example of this in the explication of the System of the World : for by the propositions mathematically demonstrated in the former books, we in the third derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces, by other propositions which are also mathematical, we deduce the motions of the planets, the comets, the moon, and the sea. I wish we could derive the rest of the phenomena of nature by the same kind of reasoning from mechanical principles; for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto at tempted the search of nature in vain ; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy. In the publication of this work the most acute and universally learned Mr. Edmund Halley not only assisted me with his pains in correcting the press and taking care of the schemes, but it was to his solicitations that its becoming public is owing ; for when he had obtained of me my demonstrations of the figure of the celestial orbits, he continually pressed me to communicate the same to the Royal Society, who afterwards, by their kind encouragement and entreaties, engaged me to think of publishing them. But after I had begun to consider the inequalities of the lunar motions, and had entered upon some other things relating to the laws and measures of gravity, and other forces ; and the figures that would be described by bodies attracted according to given laws ; and the motion of several bodies moving among themselves; the motion of bodies in resisting mediums; the forces, densities, and motions, of mediums ; the orbits of the comets, and such like ; deferred that publication till I had made a search into those matters, and could put forth the whole together. What relates to the lunar motions (being imperfect), I have put all together in the corollaries of Prop. 66, to avoid being obliged to propose and distinctly demonstrate the several things there contained in a method more prolix than the subject deserved, and interrupt the series of the several propositions. Some things, found out after the rest, I chose to insert in places less suitable, rather than change the number of the propositions and the citations. I heartily beg that what I have here done may be read with candour; and that the defects in a subject so difficult be not so much reprehended as kindly supplied, and investigated by new endeavours of my readers.

Isaac Newton.

Cambridge, Trinity College May 8, 1688.

In the second edition the second section of the first book was enlarged. In the seventh section of the second book the theory of the resistances of fluids was more accurately investigated, and confirmed by new experiments. In the third book the moon's theory and the praecession of the equinoxes were more fully deduced from their principles ; and the theory of the comets was confirmed by more examples of the calculation of their orbits, done also with greater accuracy.

In this third edition the resistance of mediums is somewhat more largely handled than before; and new experiments of the resistance of heavy bodies falling in air are added. In the third book, the argument to prove that the moon is retained in its orbit by the force of gravity is enlarged on ; and there are added new observations of Mr. Pound's of the proportion of the diameters of Jupiter to each other : there are, besides, added Mr. Kirk's observations of the comet in 1680 ; the orbit of that comet computed in an ellipsis by Dr. Halley ; and the orbit of the comet in 1723 computed by Mr. Bradley.

*The quantity of matter is the measure of the same, arising
from its density and bulk conjunctly.*

THUS air of a double density, in a double space, is quadruple in quantity ; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body ; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.

*The quantity of motion is the measure of the same, arising
from the velocity and quantity of matter conjunctly.*

The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double ; with twice the velocity, it is quadruple,

The *vis insita*, or innate force of matter,
is a power of resisting, by which every body, as much as in it
lies, endeavours to persevere in its present stale, whether it be
of rest, or of moving uniformly forward in a right line.

This force is ever proportional to the body whose force it is ; and
differs nothing from the inactivity of the mass, but in our manner
of conceiving it. A body, from the
inactivity of matter, is not without difficulty put out of its state
of rest or motion. Upon which account, this *vis insita*,
may, by a most significant name, be called *vis inertia*, or
force of inactivity. But a body exerts this force only, when another
force, impressed upon it, endeavours to change its condition ; and
the exercise of this force may be considered both as resistance and
impulse ; it is resistance, in so far as the body, for maintaining
its present state, withstands the force impressed; it is impulse, in
so far as the body, by not easily giving way to the impressed force
of another, endeavours to change the state of that other. Resistance
is usually ascribed to bodies at rest, and impulse to those in
motion; but motion and rest, as commonly conceived, are only
relatively distinguished ; nor are those bodies always truly at
rest, which commonly are taken to be so.

*An impressed force is an action exerted upon a body, in order
to change its state, either of rest, or of moving uniformly
forward in a right line.*

This force consists in the action only; and remains no longer in
the body, when the action is over. For a body maintains every new
state it acquires, by its *vis inertiae* only. Impressed
forces are of different origins as from percussion, from pressure,
from centripetal force.

*A centripetal force is that by which bodies are drawn or
impelled, or any way tend, towards a point as to a centre.*

Of this sort is gravity, by which bodies tend to the centre of the earth magnetism, by which iron tends to the loadstone ; and that force, what ever it is, by which the planets are perpetually drawn aside from the rectilinear motions, which otherwise they would pursue, and made to revolve in curvilinear orbits. A stone, whirled about in a sling, endeavours to recede from the hand that turns it ; and by that endeavour, distends the sling, and that with so much the greater force, as it is revolved with the greater velocity, and as soon as ever it is let go, flies away. That force which opposes itself to this endeavour, and by which the sling perpetually draws back the stone towards the hand, and retains it in its orbit, because it is directed to the hand as the centre of the orbit, I call the centripetal force. And the same thing is to be understood of all bodies, revolved in any orbits. They all endeavour to recede from the centres of their orbits ; and wore it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call centripetal, would fly off in right lines, with an uniform motion. A projectile, if it was not for the force of gravity, would not deviate towards the earth, but would go off from it in a right line, and that with an uniform motion, if the resistance of the air was taken away. It is by its gravity that it is drawn aside perpetually from its rectilinear course, and made to deviate towards the earth, more or less, according to the force of its gravity, and the velocity of its motion. The less its gravity is, for the quantity of its matter, or the greater the velocity with which it is projected, the less will it deviate from a rectilinear course, and the farther it will go. If a leaden ball projected from the top of a mountain by the force of gunpowder with a given velocity, and in a direction parallel to the horizon, is carried in a curve line to the distance of two miles before it falls to the ground ; the same, if the resistance of the air were taken away, with a double or decuple velocity, would fly twice or ten times as far. And by increasing the velocity, we may at pleasure increase the distance to which it might be projected, and diminish the curvature of the line, which it might describe, till at last it should fall at the distance of 10, 30, or 90 degrees, or even might go quite round the whole earth before it falls ; or lastly, so that it might never fall to the earth, but go forward into the celestial spaces, and proceed in its motion in infinitum. And after the same manner that a projectile, by the force of gravity, may be made to revolve in an orbit, and go round the whole earth, the moon also, either by the force of gravity, if it is endued with gravity, or by any other force, that impels it towards the earth, may be perpetually drawn aside towards the earth, out of the rectilinear way, which by its innate force it would pursue; and would be made to revolve in the orbit which it now describes ; nor could the moon with out some such force, be retained in its orbit. If this force was too small, it would not sufficiently turn the moon out of a rectilinear course : if it was too great, it would turn it too much, and draw down the moon from its orbit towards the earth. It is necessary, that the force be of a just quantity, and it belongs to the mathematicians to find the force, that may serve exactly to retain a body in a given orbit, with a given velocity ; and vice versa, to determine the curvilinear way, into which a body projected from a given place, with a given velocity, may be made to deviate from its natural rectilinear way, by means of a given force.

The quantity of any centripetal force may be considered as of three kinds; absolute, accelerative, and motive.

*The absolute quantity of a centripetal force is the measure
of the same proportional to the efficacy of the cause that
propagates it from the centre, through the spaces round about.*

Thus the magnetic force is greater in one load-stone and less in another according to their sizes and strength of intensity.

*The accelerative quantity of a centripetal force is the
measure, of the same, proportional to the velocity which it
generates in a given time.*

Thus the force of the same load-stone is greater at a less distance, and less at a greater : also the force of gravity is greater in valleys, less on tops of exceeding high mountains ; and yet less (as shall hereafter be shown), at greater distances from the body of the earth ; but at equal distances, it is the same everywhere ; because (taking away, or allowing for, the resistance of the air), it equally accelerates all falling bodies, whether heavy or light, great or small.

*The motive quantity of a centripetal force, is the measure of
the same proportional to the motion which it generates in a
given time.*

Thus the weight is greater in a greater body, less in a less body ; and in the same body, it is greater near to the earth, and less at remoter distances. This sort of quantity is the centripetency, or propension of the whole body towards the centre, or, as I may say, its weight ; and it is always known by the quantity of an equal and contrary force just sufficient to hinder the descent of the body.

These quantities of forces, we may, for brevity's sake, call by the names of motive, accelerative, and absolute forces ; and, for distinction's sake, con sider them, with respect to the bodies that tend to the centre ; to the places of those bodies ; and to the centre of force towards which they tend ; that is to say, I refer the motive force to the body as an endeavour and propensity of the whole towards a centre, arising from the propensities of the several parts taken together ; the accelerative force to the place of the body, as a certain power or energy diffused from the centre to all places around to move the bodies that are in them : and the absolute force to the centre, as endued with some cause, without which those motive forces would not be propagated through the spaces round about ; whether that cause be some central body (such as is the load-stone, in the centre of the magnetic force, or the earth in the centre of the gravitating force), or anything else that does not yet appear. For I here design only to give a mathematical notion of those forces, without considering their physical causes and seats.

Wherefore the accelerative force will stand in the same relation to the motive, as celerity does to motion. For the quantity of motion arises from the celerity drawn into the quantity of matter : and the motive force arises from the accelerative force drawn into the same quantity of matter. For the sum of the actions of the accelerative force, upon the several ; articles of the body, is the motive force of the whole. Hence it is, that near the surface of the earth, where the accelerative gravity, or force productive of gravity, in all bodies is the same, the motive gravity or the weight is as the body : but if we should ascend to higher regions, where the accelerative gravity is less, the weight would be equally diminished, and would always be as the product of the body, by the accelerative gravity. So in those regions, where the accelerative gravity is diminished into one half, the weight of a body two or three times less, will be four or six times less.

I likewise call attractions and impulses, in the same sense, accelerative, and motive ; and use the words attraction, impulse or propensity of any sort towards a centre, promiscuously, and indifferently, one for another ; considering those forces not physically, but mathematically : wherefore, the reader is not to imagine, that by those words, I anywhere take upon me to define the kind, or the manner of any action, the causes or the physical reason thereof, or that I attribute forces, in a true and physical sense, to certain centres (which are only mathematical points) ; when at any time I happen to speak of centres as attracting, or as endued with attractive powers.

Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be under stood in the following discourse. I do not define time, space, place and motion, as being well known to all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which, it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common.

I. Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration : relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time ; such as an hour, a day, a month, a year.

II. Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces ; which our senses determine by its position to bodies ; and which is vulgarly taken for immovable space ; such is the dimension of a subterraneous, an aereal, or celestial space, determined by its position in respect of the earth. Absolute and relative space, are the same in figure and magnitude ; but they do not remain always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains always the same, will at one time be one part of the absolute space into which the air passes ; at another time it will be another part of the same, and so, absolutely understood, it will be perpetually mutable.

III. Place is a part of space which a body takes up, and is according to the space, either absolute or relative. I say, a part of space ; not the situation, nor the external surface of the body. For the places of equal solids are always equal ; but their superfices, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same thing with the sum of the motions of the parts ; that is, the translation of the whole, out of its place, is the same thing with the sum of the translations of the parts out of their places ; and therefore the place of the whole is the same thing with the sum of the places of the parts, and for that reason, it is internal, and in the whole body.

IV. Absolute motion is the translation of a body from one absolute place into another ; and relative motion, the translation from one relative place into another. Thus in a ship under sail, the relative place of a body is that part of the ship which the body possesses ; or that part of its cavity which the body fills, and which therefore moves together with the ship : and relative rest is the continuance of the body in the same part of the ship, or of its cavity. But real, absolute rest, is the continuance of the body in the same part of that immovable space, in which the ship itself, its cavity, and all that it contains, is moved. Wherefore, if the earth is really at rest, the body, which relatively rests in the ship, will really and absolutely move with the same velocity which the ship has on the earth. But if the earth also moves, the true and absolute motion of the body will arise, partly from the true motion of the earth, in immovable space ; partly from the relative motion of the ship on the earth ; and if the body moves also relatively in the ship ; its true motion will arise, partly from the true motion of the earth, in immovable space, and partly from the relative motions as well of the ship on the earth, as of the body in the ship ; and from these relative motions will arise the relative motion of the body on the earth. As if that part of the earth, where the ship is, was truly moved toward the east, with a velocity of 10010 parts; while the ship itself, with a fresh gale, and full sails, is carried towards the west, with a velocity expressed by 10 of those parts ; but a sailor walks in the ship towards the east, with 1 part of the said velocity ; then the sailor will be moved truly in immovable space towards the east, with a velocity of 10001 parts, and relatively on the earth towards the west, with a velocity of 9 of those parts.

Absolute time, in astronomy, is distinguished from relative, by the equation or correction of the vulgar time. For the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time ; astronomers correct this inequality for their more accurate deducing of the celestial motions. It may be, that there is no such thing as an equable motion, whereby time may H accurately measured. All motions may be accelerated and retarded; but the true, or equable, progress of absolute time is liable to no change. The duration or perseverance of the existence of things remains the same, whether the motions are swift or slow, or none at all : and therefore it ought to be distinguished from what are only sensible measures thereof ; and out of which we collect it, by means of the astronomical equation. The necessity of which equation, for deter mining the times of a phaenomenon, is evinced as well from the experiments of the pendulum clock, as by eclipses of the satellites of Jupiter.

As the order of the parts of time is immutable, so also is the order of the parts of space. Suppose those parts to be moved out of their places, and they will be moved (if the expression may be allowed) out of themselves. For times and spaces are, as it were, the places as well of themselves as of all other things. All things are placed in time as to order of succession ; and in space as to order of situation. It is from their essence or nature that they are places ; and that the primary places of things should be moveable, is absurd. These are therefore the absolute places ; and translations out of those places, are the only absolute motions.

But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body considered as immovable, we define all places ; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs ; but in philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred.

But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes and effects. It is a property of rest, that bodies really at rest do rest in respect to one another. And therefore as it is possible, that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest ; but impossible to know, from the position of bodies to one another in our regions whether any of these do keep the same position to that remote body; it follows that absolute rest cannot be determined from the position of bodies in our regions.

It is a property of motion, that the parts, which retain given positions to their wholes, do partake of the motions of those wholes. For all the parts of revolving bodies endeavour to recede from the axis of motion ; and the impetus of bodies moving forward, arises from the joint impetus of all the parts. Therefore, if surrounding bodies are moved, those that are relatively at rest within them, will partake of their motion. Upon which account, the true and absolute motion of a body cannot be determined by the translation of it from those which only seem to rest ; for the external bodies ought not only to appear at rest, but to be really at rest. For otherwise, all included bodies, beside their translation from near the surrounding ones, partake likewise of their true motions ; and though that translation were not made they would not be really at rest, but only seem to be so. For the surrounding bodies stand in the like relation to the surrounded as the exterior part of a whole does to the interior, or as the shell does to the kernel ; but, if the shell moves, the kernel will also move, as being part of the whole, without any removal from near the shell.

A property, near akin to the preceding, is this, that if a place is moved, whatever is placed therein moves along with it ; and therefore a body, which is moved from a place in motion, partakes also of the motion of its place. Upon which account, all motions, from places in motion, are no other than parts of entire and absolute motions ; and every entire motion is composed of the motion of the body out of its first place, and the motion of this place out of its place ; and so on, until we come to some immovable place, as in the before-mentioned example of the sailor. Where fore, entire and absolute motions can be no otherwise determined than by immovable places : and for that reason I did before refer those absolute motions to immovable places, but relative ones to movable places. Now no other places are immovable but those that, from infinity to infinity, do all retain the same given position one to another ; and upon this account must ever remain unmoved ; and do thereby constitute immovable space.

The causes by which true and relative motions are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed upon the body moved : but relative motion may be generated or altered without any force impressed upon the body. For it is sufficient only to impress some force on other bodies with which the former is compared, that by their giving way, that relation may be changed, in which the relative rest or motion of this other body did consist. Again, true motion suffers always some change from any force impressed upon the moving body ; but relative motion docs not necessarily undergo any change by such forces. For if the same forces are likewise impressed on those other bodies, with which the comparison is made, that the relative position may be pre served, then that condition will be preserved in which the relative motion consists. And therefore any relative motion may be changed when the true motion remains unaltered, and the relative may be preserved when the true suffers some change. Upon which accounts; true motion does by no means consist in such relations.

The effects which distinguish absolute from relative motion arc, the forces of receding from the axis of circular motion. For there are no such forces in a circular motion purely relative, but in a true and absolute circular motion., they are greater or less, according t the quantity of the motion. If a vessel, hung: by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and held at rest together with the water ; after, by the sudden action of another force, it is whirled about the contrary way, and while the cord is untwisting itself, the vessel continues for some time in this motion ; the surface of the water will at first be plain, as before the vessel began to move : but the vessel; by gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming itself into a concave figure (as I have experienced), and the swifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. This ascent of the water shows its endeavour to recede from the axis of its motion ; and the true and absolute circular motion of the water, which is here directly contrary to the relative, discovers itself, and may be measured by this endeavour. At first, when the relative motion of the water in the vessel was greatest, it produced no endeavour to recede from the axis ; the water showed no tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its endeavour to recede from the axis ; and this endeavour showed the real circular motion of the water perpetually increasing, till it had acquired its greatest quantity, when the water rested relatively in the vessel. And therefore this endeavour does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be defined by such translation. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavouring to recede from its axis of motion, as its proper and adequate effect ; but relative motions, in one and the same body, are innumerable, according to the various relations it bears to external bodies, and like other relations, are altogether destitute of any real effect, any otherwise than they may perhaps partake of that one only true motion. And therefore in their system who suppose that our heavens, revolving below the sphere of the fixed stars, carry the planets along with them ; the several parts of those heavens, and the planets, which are indeed relatively at rest in their heavens, do yet really move. For they change their position one to another (which never happens to bodies truly at rest), and being carried together with their heavens, partake of their motions, and as parts of revolving wholes, endeavour to recede from the axis of their motions.

Wherefore relative quantities are not the quantities themselves, whose names they bear, but those sensible measures of them (either accurate or inaccurate), which are commonly used instead of the measured quantities themselves. And if the meaning of words is to he determined by their use, then by the names time, space, place and motion, their measures are properly to be understood ; and the expression will be unusual, and purely mathematical, if the measured quantities themselves are meant. Upon which account, they do strain the sacred writings, who there interpret those words for the measured quantities. Nor do those less defile the purity of mathematical and philosophical truths, who confound real quantities themselves with their relations and vulgar measures.

It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent ; be cause the parts of that immovable space, in which those motions are performed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate : for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions ; partly from the forces, which are the causes and effects of the true motions. For instance, if two globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common centre of gravity, we might, from the tension of the cord, discover the endeavour of the globes to recede from the axis of their motion, and from thence we might compute the quantity of their circular motions. And then if any equal forces should be impressed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decrease of the tension of the cord, we might infer the increment or decrement of their motions : and thence would be found on what faces those forces ought to be impressed, that the motions of the globes might be most augmented ; that is, we might discover their hinder-most faces, or those which, in the circular motion, do follow. But the faces which follow being known, and consequently the opposite ones that precede, we should likewise know the determination of their motions. And thus we might find both the quantity and the determination of this circular motion, even in an immense vacuum, where there was nothing external or sensible with which the globes could be compared. But now, if in that space some remote bodies were placed that kept always a given position one to another, as the fixed stars do in our regions, we could not indeed determine from the relative translation of the globes among those bodies, whether the motion did belong to the globes or to the bodies. But if we observed the cord, and found that its tension was that very tension which the motions of the globes required, we might conclude the motion to be in the globes, and the bodies to be at rest ; and then, lastly, from the translation of the globes among the bodies, we should find the determination of their motions. But how we are to collect the true motions from their causes, effects, and apparent differences ; and, vice versa, how from the motions, either true or apparent, we may come to the knowledge of their causes and effects, shall be explained more at large in the following tract. For to this end it was that I composed it.

*Every body perseveres in its state of rest, or of uniform
motion in a right line, unless it is compelled to change that
state by forces impressed thereon.*

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.

*The alteration of motion is ever proportional to the motive
force impressed; and is made in the direction of the right line
in which that force is impressed.*

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

*To every action there is always opposed an equal reaction: or
the mutual actions of two bodies upon each other are always
equal, and directed to contrary parts.*

Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinge upon another, and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.

*A body by two forces conjoined will describe the diagonal of
a parallelogram, in the same time that it would describe the
sides, by those forces apart.*

If a body in a given time, by the force M impressed apart in the place A, should with an uniform motion be carried from A to B; and by the force N impressed apart in the same place, should be carried from A to C; complete the parallelogram ABCD, and, by both forces acting together, it will in the same time be carried in the diagonal from A to D. For since the force N acts in the direction of the line AC, parallel to BD, this force (by the second law) will not at all alter the velocity generated by the other force M, by which the body is carried towards the line BD. The body therefore will arrive at the line BD in the same time, whether the force N be impressed or not; and therefore at the end of that time it will be found somewhere in the line BD. By the same argument, at the end of the same time it will be found somewhere in the line CD. Therefore it will be found in the point D, where both lines meet. But it will move in a right line from A to D, by Law I.

*And hence is explained the composition of any one direct
force AD, out of any two oblique forces AC and CD; and, on the
contrary, the resolution of any one direct force AD into two
oblique forces AC and CD: which composition and resolution are
abundantly confirmed from mechanics.*

As if the unequal radii OM and ON drawn from the centre O of any wheel, should sustain the weights A and P by the cords MA and NP; and the forces of those weights to move the wheel were required. Through the centre O draw the right line KOL, meeting the cords perpendicularly in K and L; and from the centre O, with OL the greater of the distances OK and OL, describe a circle, meeting the cord MA in D: and drawing OD, make AC parallel and DC perpendicular thereto. Now, it being indifferent whether the points K, L, D, of the cords be fixed to the plane of the wheel or not, the weights will have the same effect whether they are suspended from the points K and L, or from D and L. Let the whole force of the weight A be represented by the line AD, and let it be resolved into the forces AC and CD; of which the force AC, drawing the radius OD directly from the centre, will have no effect to move the wheel: but the other force DC, drawing the radius DO perpendicularly, will have the same effect as if it drew perpendicularly the radius OL equal to OD; that is, it will have the same effect as the weight P, if that weight is to the weight A as the force DC is to the force DA; that is (because of the similar triangles ADC, DOK), as OK to OD or OL. Therefore the weights A and P, which are reciprocally as the radii OK and OL that lie in the same right line, will be equipollent, and so remain in equilibrio; which is the well known property of the balance, the lever, and the wheel. If either weight is greater than in this ratio, its force to move the wheel will be so much greater.

If the weight *p*, equal to the weight P, is partly
suspended by the cord N*p*, partly sustained by the oblique
plane *p*G; draw *p*H, NH, the former perpendicular
to the horizon, the latter to the plane *p*G; and if the
force of the weight *p* tending downwards is represented by
the line *p*H, it may be resolved into the forces *p*N,
HN. If there was any plane *p*Q, perpendicular to the cord *p*N,
cutting the other plane *p*G in a line parallel to the
horizon, and the weight *p* was supported only by those
planes *p*Q, *p*G, it would press those planes
perpendicularly with the forces *p*N; HN; to wit, the plane
*p*Q with the force *p*N, and the plane *p*G
with the force HN. And therefore if the plane *p*Q was taken
away, so that the weight might stretch the cord, because the cord,
now sustaining the weight, supplies the place of the plane that was
removed, it will be strained by the same force *p*N which
pressed upon the plane before. Therefore, the tension of this
oblique cord *p*N will be to that of the other perpendicular
cord PN as *p*N to *p*H. And therefore if the weight
*p* is to the weight A in a ratio compounded of the
reciprocal ratio of the least distances of the cords PN, AM, from
the centre of the wheel, and of the direct ratio of *p*H to
*p*N, the weights will have the same effect towards moving
the wheel, and will therefore sustain each other; as any one may
find by experiment.

But the weight *p* pressing upon those two oblique planes,
may be considered as a wedge between the two internal surfaces of a
body split by it; and hence the forces of the wedge and the mallet
may be determined; for because the force
with which the weight *p* presses the plane *p*Q is
to the force with which the same, whether by its own gravity, or by
the blow of a mallet, is impelled in the direction of the line *p*H
towards both the planes, as *p*N to *p*H; and to the
force with which it presses the other plane *p*G, as *p*N
to NH. And thus the force of the screw may be deduced from a like
resolution of forces; it being no other than a wedge impelled with
the force of a lever. Therefore the use of this Corollary spreads
far and wide, and by that diffusive extent the truth thereof is
farther confirmed. For on what has been said depends the whole
doctrine of mechanics variously demonstrated by different authors.
For from hence are easily deduced the forces of machines, which are
compounded of wheels, pullies, levers, cords, and weights, ascending
directly or obliquely, and other mechanical powers; as also the
force of the tendons to move the bones of animals.

*The quantity of motion, which is collected by taking the sum
of the motions directed towards the same parts, and the
difference of those that are directed to contrary parts, suffers
no change from the action of bodies among themselves.*

For action and its opposite re-action are equal, by Law III, and therefore, by Law II, they produce in the motions equal changes towards opposite parts. Therefore if the motions are directed towards the same parts, whatever is added to the motion of the preceding body will be subducted from the motion of that which follows; so that the sum will be the same as before. If the bodies meet, with contrary motions, there will be an equal deduction from the motions of both; and therefore the difference of the motions directed towards opposite parts will remain the same.

Thus if a spherical body A with two parts of velocity is triple of a spherical body B which follows in the same right line with ten parts of velocity, the motion of A will be to that of B as 6 to 10. Suppose, then, their motions to be of 6 parts and of 10 parts, and the sum will be 16 parts. Therefore, upon the meeting of the bodies, if A acquire 3, 4, or 5 parts of motion, B will lose as many; and therefore after reflexion A will proceed with 9, 10, or 11 parts, and B with 7, 6, or 5 parts; the sum remaining always of 16 parts as before. If the body A acquire 9, 10, 11, or 12 parts of motion, and therefore after meeting proceed with 15, 16, 17, or 18 parts, the body B, losing so many parts as A has got, will either proceed with 1 part, having lost 9, or stop and remain at rest, as having lost its whole progressive motion of 10 parts; or it will go back with 1 part, having not only lost its whole motion, but (if I may so say) one part more; or it will go back with 2 parts, because a progressive motion of 12 parts is taken off. And so the sums of the conspiring motions 15+1, or 16+0, and the differences of the contrary motions 17−1 and 18−2, will always be equal to 16 parts, as they were before the meeting and reflexion of the bodies. But, the motions being known with which the bodies proceed after reflexion, the velocity of either will be also known, by taking the velocity after to the velocity before reflexion, as the motion after is to the motion before. As in the last case, where the motion of the body A was of 6 parts before reflexion and of 18 parts after, and the velocity was of 2 parts before reflexion, the velocity thereof after reflexion will be found to be of 6 parts; by saying, as the 6 parts of motion before to 18 parts after, so are 2 parts of velocity before reflexion to 6 parts after.

But if the bodies are either not spherical, or, moving in different right lines, impinge obliquely one upon the other, and their motions after reflexion are required, in those cases we are first to determine the position of the plane that touches the concurring bodies in the point of concourse, then the motion of each body (by Corol. II) is to be resolved into two, one perpendicular to that plane, and the other parallel to it. This done, because the bodies act upon each other in the direction of a line perpendicular to this plane, the parallel motions are to be retained the same after reflexion as before; and to the perpendicular motions we are to assign equal changes towards the contrary parts; in such manner that the sum of the conspiring and the difference of the contrary motions may remain the same as before. From such kind of reflexions also sometimes arise the circular motions of bodies about their own centres. But these are cases which I do not consider in what follows; and it would be too tedious to demonstrate every particular that relates to this subject.

*The common centre of gravity of two or more bodies does not
alter its state of motion or rest by the actions of the bodies
among themselves; and therefore the common centre of gravity of
all bodies acting upon each other (excluding outward actions and
impediments) is either at rest, or moves uniformly in a right line.*

For if two points proceed with an uniform motion in right lines,
and their distance be divided in a given ratio, the dividing point
will be either at rest, or proceed uniformly in a right line. This
is demonstrated hereafter in Lem. XXIII and its Corol., when the
points are moved in the same plane; and by a like way of arguing, it
may be demonstrated when the points are not moved in the same plane.
Therefore if any number of bodies move uniformly in right lines, the
common centre of gravity of any two of them is either at rest, or
proceeds uniformly in a right line; because the line which connects
the centres of those two bodies so moving is divided at that common
centre in a given ratio. In like manner the common centre of those
two and that of a third body will be either at rest or moving
uniformly in a right line because at that centre the distance
between the common centre of the two bodies,
and the centre of this last, is divided in a given ratio. In like
manner the common centre of these three, and of a fourth body, is
either at rest, or moves uniformly in a right line; because the
distance between the common centre of the three bodies, and the
centre of the fourth is there also divided in a given ratio, and so
on *in infinitum*. Therefore, in a system of bodies where
there is neither any mutual action among themselves, nor any foreign
force impressed upon them from without, and which consequently move
uniformly in right lines, the common centre of gravity of them all
is either at rest or moves uniformly forward in a right line.

Moreover, in a system of two bodies mutually acting upon each other, since the distances between their centres and the common centre of gravity of both arc reciprocally as the bodies, the relative motions of those bodies, whether of approaching to or of receding from that centre, will be equal among themselves. Therefore since the changes which happen to motions are equal and directed to contrary parts, the common centre of those bodies, by their mutual action between themselves, is neither promoted nor retarded, nor suffers any change as to its state of motion or rest. But in a system of several bodies, because the common centre of gravity of any two acting mutually upon each other suffers no change in its state by that action: and much less the common centre of gravity of the others with which that action does not intervene; but the distance between those two centres is divided by the common centre of gravity of all the bodies into parts reciprocally proportional to the total sums of those bodies whose centres they are: and therefore while those two centres retain their state of motion or rest, the common centre of all does also retain its state: it is manifest that the common centre of all never suffers any change in the state of its motion or rest from the actions of any two bodies between themselves. But in such a system all the actions of the bodies among themselves either happen between two bodies, or are composed of actions interchanged between some two bodies; and therefore they do never produce any alteration in the common centre of all as to its state of motion or rest. Wherefore since that centre, when the bodies do not act mutually one upon another, either is at rest or moves uniformly forward in some right line, it will, notwithstanding the mutual actions of the bodies among themselves, always persevere in its state, either of rest, or of proceeding uniformly in a right line, unless it is forced out of this state by the action of some power impressed from without upon the whole system. And therefore the same law takes place in a system consisting of many bodies as in one single body, with regard to their persevering in their state of motion or of rest. For the progressive motion, whether of one single body, or of a whole system of bodies, is always to be estimated from the motion of the centre of gravity.

The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion.

For the differences of the motions tending towards the same parts, and the sums of those that tend towards contrary parts, are, at first (by supposition), in both cases the same; and it is from those sums and differences that the collisions and impulses do arise with which the bodies mutually impinge one upon another. Wherefore (by Law II), the effects of those collisions will be equal in both cases; and therefore the mutual motions of the bodies among themselves in the one case will remain equal to the mutual motions of the bodies among themselves in the other. A clear proof of which we have from the experiment of a ship; where all motions happen after the same manner, whether the ship is at rest, or is carried uniformly forwards in a right line.

If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same, manner as if they had been urged by no such forces.

For these forces acting equally (with respect to the quantities of the bodies to be moved), and in the direction of parallel lines, will (by Law II) move all the bodies equally (as to velocity), and therefore will never produce any change in the positions or motions of the bodies among themselves.

Hitherto I have laid down such principles as have been received by
mathematicians, and are confirmed by abundance of experiments. By
the first two Laws and the first two Corollaries, Galileo discovered
that the descent of bodies observed the duplicate ratio of the time,
and that the motion of projectiles was in the curve of a parabola;
experience agreeing with both, unless so far as these motions are a
little retarded by the resistance of the air. When a body is
falling, the uniform force of its gravity acting equally, impresses,
in equal particles of time, equal forces upon that body, and
therefore generates equal velocities; and in the whole time
impresses a whole force, and generates a whole velocity proportional
to the time. And the spaces described in proportional times are as
the velocities and the times conjunctly; that is, in a duplicate
ratio of the times. And when a body is thrown upwards, its uniform
gravity impresses forces and takes off velocities proportional to
the times; and the times of ascending to the greatest heights are as
the velocities to be taken off, and those heights are as the
velocities and the times conjunctly, or in the duplicate ratio of
the velocities. And if a body be projected in any direction, the
motion arising from its projection is compounded with the
motion
arising from its gravity. As if the body A by its motion of
projection alone could describe in a given time the right line AB,
and with its motion of falling alone could
describe in the same time the altitude AC; complete the
paralellogram
ABDC, and the body by that compounded motion will at the end of the
time be found in the place D; and the curve line AED, which that
body describes, will be a parabola, to which the right line AB will
be a tangent in A; and whose ordinate BD will be as the square of
the line AB. On the same Laws and Corollaries depend those things
which have been demonstrated concerning the times of the vibration
of pendulums, and are confirmed by the daily experiments of pendulum
clocks. By the same, together with the third Law, Sir Christ. Wren,
Dr. Wallis, and Mr. Huygens, the greatest geometers of our times,
did severally determine the rules of the congress and reflexion of
hard bodies, and much about the same time communicated their
discoveries to the Royal Society, exactly agreeing among themselves
as to those rules. Dr. Wallis, indeed, was something more early in
the publication; then followed Sir Christopher Wren, and, lastly,
Mr. Huygens. But Sir Christopher Wren confirmed the truth of the
thing before the Royal Society by the experiment of pendulums, which
Mr. Mariotte soon after thought fit to explain in a treatise
entirely upon that subject. But to bring this experiment to an
accurate agreement with the theory, we are to have a due regard as
well to the resistance of the air as to the elastic force of the
concurring bodies. Let the spherical bodies A, B be suspended by the
parallel and
equal strings AC, BD, from the
centres C, D. About these centres, with those intervals, describe
the semicircles EAF, GBH, bisected by the radii CA, DB. Bring the
body A to any point R of the arc EAF, and (withdrawing the body B)
let it go from thence, and after one oscillation suppose it to
return to the point V: then RV will be the retardation arising from
the resistance of the air. Of this RV let ST be a fourth part,
situated in the middle, to wit, so as RS and TV may be equal, and RS
may be to ST as 3 to 2, then will ST represent very nearly the
retardation during the descent from S to A. Restore the body B to
its place: and, supposing the body A to be let fall from the point
S, the velocity thereof in the place of reflexion A, without
sensible error, will be the same as if it had descended *in
vacuo* from the point T. Upon which account this velocity may
be represented by the chord of the arc TA. For it is a proposition
well known to geometers, that the velocity of a pendulous body in
the lowest point is as the chord of the arc which it has described
in its descent. After reflexion, suppose the
body A comes to the place *s*, and the body B to the place *k*.
Withdraw the body B, and find the place *v*, from which if
the body A, being let go, should after one oscillation return to the
place *r*, *st* may be a fourth part of *rv*,
so placed in the middle thereof as to leave *rs* equal to *tv*,
and let the chord of the arc *t*A. represent the velocity
which the body A had in the place A immediately after reflexion. For
*t* will be the true and correct place to which the body A
should have ascended, if the resistance of the air had been taken
off. In the same way we are to correct the place *k* to
which the body B ascends, by finding the place *l* to which
it should have ascended *in vacuo*. And thus everything may
be subjected to experiment, in the same manner as if we were really
placed *in vacuo*. These things being done, we are to take
the product (if I may so say) of the body A, by the chord of the arc
TA (which represents its velocity), that we may have its motion in
the place A immediately before reflexion; and then by the chord of
the arc *t*A, that we may have its motion in the place A
immediately after reflexion. And so we are to take the product of
the body B by the chord of the arc B*l*, that we may have the
motion of the same immediately after reflexion. And in like manner,
when two bodies are let go together from different places, we are to
find the motion of each, as well before as after reflexion; and then
we may compare the motions between themselves, and collect the
effects of the reflexion. Thus trying the thing with pendulums of
ten feet, in unequal as well as equal bodies, and making the bodies
to concur after a descent through large spaces, as of 8, 12, or 16
feet, I found always, without an error of 3 inches, that when the
bodies concurred together directly, equal changes towards the
contrary parts were produced in their motions, and, of consequence,
that the action and reaction were always equal. As if the body A
impinged upon the body B at rest with 9 parts of motion, and losing
7, proceeded after reflexion with 2, the body B was carried
backwards with those 7 parts. If the bodies concurred with contrary
motions, A with twelve parts of motion, and B with six, then if A
receded with 2, B receded with 8; to wit, with a deduction of 14
parts of motion on each side. For from the motion of A subducting
twelve parts, nothing will remain; but subducting 2 parts more, a
motion will be generated of 2 parts towards the contrary way; and
so, from the motion of the body B of 6 parts, subducting 14 parts, a
motion is generated of 8 parts towards the contrary way. But if the
bodies were made both to move towards the same way, A, the swifter,
with 14 parts of motion, B, the slower, with 5, and after reflexion
A went on with 5, B likewise went on with 14 parts; 9 parts being
transferred from A to B. And so in other cases. By the congress and
collision of bodies, the quantity of motion, collected from the sum
of the motions directed towards the same way, or from the difference
of those that were directed towards contrary ways, was never
changed. For the error of an inch or two in measures may be easily
ascribed to the difficulty of executing
everything with accuracy. It was not easy to let go the two
pendulums so exactly together that the bodies should impinge one
upon the other in the lowermost place AB; nor to mark the places *s*,
and *k*, to which the bodies ascended after congress. Nay,
and some errors, too, might have happened from the unequal density
of the parts of the pendulous bodies themselves, and from the
irregularity of the texture proceeding from other causes.

But to prevent an objection that may perhaps be alledged against the rule, for the proof of which this experiment was made, as if this rule did suppose that the bodies were either absolutely hard, or at least perfectly elastic (whereas no such bodies are to be found in nature), I must add, that the experiments we have been describing, by no means depending upon that quality of hardness, do succeed as well in soft as in hard bodies. For if the rule is to be tried in bodies not perfectly hard, we are only to diminish the reflexion in such a certain proportion as the quantity of the elastic force requires. By the theory of Wren and Huygens, bodies absolutely hard return one from another with the same velocity with which they meet. But this may be affirmed with more certainty of bodies perfectly elastic. In bodies imperfectly elastic the velocity of the return is to be diminished together with the elastic force; because that force (except when the parts of bodies are bruised by their congress, or suffer some such extension as happens under the strokes of a hammer) is (as far as I can perceive) certain and determined, and makes the bodies to return one from the other with a relative velocity, which is in a given ratio to that relative velocity with which they met. This I tried in balls of wool, made up tightly, and strongly compressed. For, first, by letting go the pendulous bodies, and measuring their reflexion, I determined the quantity of their elastic force; and then, according to this force, estimated the reflexions that ought to happen in other cases of congress. And with this computation other experiments made afterwards did accordingly agree; the balls always receding one from the other with a relative velocity, which was to the relative velocity with which they met as about 5 to 9. Balls of steel returned with almost the same velocity: those of cork with a velocity something less; but in balls of glass the proportion was as about 15 to 16. And thus the third Law, so far as it regards percussions and reflexions, is proved by a theory exactly agreeing with experience.

In attractions, I briefly demonstrate the thing after this manner.
Suppose an obstacle is interposed to hinder the congress of any two
bodies A, B, mutually attracting one the other: then if either body,
as A, is more attracted towards the other body B, than that other
body B is towards the first body A, the obstacle will be more
strongly urged by the pressure of the body A than by the pressure of
the body B, and therefore will not remain in equilibrio: but the
stronger pressure will prevail, and will make the system of the two
bodies, together with the obstacle, to move directly towards
the parts on which B lies; and in free spaces, to go forward *in
infinitum* with a motion perpetually accelerated; which is
absurd and contrary to the first Law. For, by the first Law, the
system ought to persevere in its state of rest, or of moving
uniformly forward in a right line: and therefore the bodies must
equally press the obstacle, and be equally attracted one by the
other. I made the experiment on the loadstone and iron. If these,
placed apart in proper vessels, are made to float by one another in
standing water, neither of them will propel the other; but, by being
equally attracted, they will sustain each other's pressure, and rest
at last in an equilibrium.

So the gravitation betwixt the earth and its parts is mutual. Let
the earth FI be cut by any plane EG into two parts EGF and EGI, and
their
weights one towards the other
will be mutually equal. For if by another plane HK, parallel to the
former EG, the greater part EGI is cut into two parts EGKH and HKI,
whereof HKI is equal to the part EFG, first cut off, it is evident
that the middle part EGKH, will have no propension by its proper
weight towards either side, but will hang as it were, and rest in an
equilibrium betwixt both. But the one extreme part HKI will with its
whole weight bear upon and press the middle part towards the other
extreme part EGF; and therefore the force with which EGI, the sum of
the parts HKI and EGKH, tends towards the third part EGF, is equal
to the weight of the part HKI, that is, to the weight of the third
part EGF. And therefore the weights of the two parts EGI and EGF,
one towards the other, are equal, as I was to prove. And indeed if
those weights were not equal, the whole earth floating in the
non-resisting aether would give way to the greater weight, and,
retiring from it, would be carried off *in infinitum*.

And as those bodies are equipollent in the congress and reflexion, whose velocities are reciprocally as their innate forces, so in the use of mechanic instruments those agents are equipollent, and mutually sustain each the contrary pressure of the other, whose velocities, estimated according to the determination of the forces, are reciprocally as the forces.

So those weights are of equal force to move the arms of a balance; which during the play of the balance are reciprocally as their velocities upwards and downwards; that is, if the ascent or descent is direct, those weights are of equal force, which are reciprocally as the distances of the points at which they are suspended from the axis of the balance; but if they are turned aside by the interposition of oblique planes, or other obstacles, and made to ascend or descend obliquely, those bodies will be equipollent, which are reciprocally as the heights of their ascent and descent taken according to the perpendicular; and that on account of the determination of gravity downwards.

And in like manner in the pully, or in a combination of pullies, the force of a hand drawing the rope directly, which is to the weight, whether ascending directly or obliquely, as the velocity of the perpendicular ascent of the weight to the velocity of the hand that draws the rope, will sustain the weight.

In clocks and such like instruments, made up from a combination of wheels, the contrary forces that promote and impede the motion of the wheels, if they are reciprocally as the velocities of the parts of the wheel on which they are impressed, will mutually sustain the one the other.

The force of the screw to press a body is to the force of the hand that turns the handles by which it is moved as the circular velocity of the handle in that part where it is impelled by the hand is to the progressive velocity of the screw towards the pressed body.

The forces by which the wedge presses or drives the two parts of the wood it cleaves are to the force of the mallet upon the wedge as the progress of the wedge in the direction of the force impressed upon it by the mallet is to the velocity with which the parts of the wood yield to the wedge, in the direction of lines perpendicular to the sides of the wedge. And the like account is to be given of all machines.

The power and use of machines consist only in this, that by
diminishing the velocity we may augment the force, and the contrary:
from whence in all sorts of proper machines, we have the solution of
this problem; *To move a given weight with a given power*,
or with a given force to overcome any other given resistance. For if
machines are so contrived that the velocities of the agent and
resistant are reciprocally as their forces, the agent will just
sustain the resistant, but with a greater disparity of velocity will
overcome it. So that if the disparity of velocities is so great as
to overcome all that resistance which commonly arises either from
the attrition of contiguous bodies as they slide by one another, or
from the cohesion of continuous bodies that are to be separated, or
from the weights of bodies to be raised, the excess of the force
remaining, after all those resistances are overcome, will produce an
acceleration of motion proportional thereto, as well in the parts of
the machine as in the resisting body. But to treat of mechanics is
not my present business. I was only willing to show by those
examples the great extent and certainty of the third Law of motion.
For if we estimate the action of the agent from its force and
velocity conjunctly, and likewise the reaction of the impediment
conjunctly from the velocities of its several parts, and from the
forces of resistance arising from the attrition, cohesion, weight,
and acceleration of those parts, the action and reaction in the use
of all sorts of machines will be found always equal to one another.
And so far as the action is propagated by the intervening
instruments, and at last impressed upon the resisting body, the
ultimate determination of the action will be always contrary to the
determination of the reaction.

*Quantities, and the ratios of quantities, which in any finite
time converge continually to equality, and before the end of
that time approach nearer the one to the other than by any given
difference, become ultimately equal.*

If you deny it, suppose them to be ultimately unequal, and let D be their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference D; which is against the supposition.

*If in any figure* AacE*, terminated by the right lines*
Aa, AE, *and the curve* acE*, there be inscribed any
number of parallelograms* Ab, Be, Cd, *&c.,
comprehended under equal bases* AB, BC, CD, *&c.,
and the sides,* Bb, Cc, Dd, *&c., parallel to one
side* Aa *of the figure; and the parallelograms*
aKbl, bLcm, cMdn, *&c., are completed. Then if the breadth
of those parallelograms be supposed to be diminished, and their
number to be augmented* in infinitum*; I say, that the
ultimate ratios which the inscribed figure* AKbLcMdD*,
the circumscribed figure* AalbmcndoE*, and curvilinear
figure* AabcdE*, will have to one another, are ratios of
equality.*

For the difference of the inscribed and circumscribed figures is
the sum of the parallelograms K*l*, L*m*, M*u*,
D*o*, that is (from the equality of all their bases), the
rectangle under one of their bases K*b* and the sum of their
altitudes A*a*, that is, the rectangle AB*la*. But
this rectangle, because its breadth AB is
supposed diminished *in infinitum*, becomes less than any
given space. And therefore (by Lem. I) the figures inscribed and
circumscribed become ultimately equal one to the other; and much
more will the intermediate curvilinear figure be ultimately equal to
either. Q.E.D.

*The same ultimate ratios are also ratios of equality, when the, breadths,*
AB, BC, DC, *&c., of the parallelograms are unequal, and are all diminished*
in infinitum*.*

For suppose AF equal to the greatest breadth, and complete the
parallelogram FA*af*. This parallelogram will be greater than
the difference of the inscribed and circumscribed figures; but,
because its breadth AF is diminished *in infinitum*, it will
be come less than any given rectangle. Q.E.D.

Cor. 1. Hence the ultimate sum of those evanescent parallelograms will in all parts coincide with the curvilinear figure.

Cor. 2. Much more will the rectilinear
figure comprehended under the chords of the evanescent arcs *ab,
bc, cd,* &c., ultimately coincide with the curvilinear
figure.

Cor. 3. And also the circumscribed rectilinear figure comprehended under the tangents of the same arcs.

Cor. 4 And therefore these ultimate figures
(as to their perimeters *ac*E) are not rectilinear, but
curvilinear limits of rectilinear figures.

*If in two figures* AacE, PprT*,
you inscribe (as before) two ranks of parallelograms, an equal number in each
rank, and, when their breadths are diminished* in infinitum*,
the ultimate ratios of the parallelograms in one figure to those
in the other, each to each respectively, are the same; I say,
that those two figures* AacE, PprT*, are to one another in that same ratio.*

For as the parallelograms in the one are severally to the parallelograms in the other, so (by composition) is the sum of all in the one to the sum of all in the other; and so is the one figure to the other; because (by Lem. III) the former figure to the former sum, and the latter figure to the latter sum, are both in the ratio of equality. Q.E.D.

Cor. Hence if two quantities of any kind
are any how divided into an equal number of parts, and those
parts, when their number is augmented, and their
magnitude diminished *in infinitum*, have a given ratio one
to the other, the first to the first, the second to the second, and
so on in order, the whole quantities will be one to the other in
that same given ratio. For if, in the figures of this Lemma, the
parallelograms are taken one to the other in the ratio of the parts,
the sum of the parts will always be as the sum of the
parallelograms; and therefore supposing the number of the
parallelograms and parts to be augmented, and their magnitudes
diminished *in infinitum*, those sums will be in the
ultimate ratio of the parallelogram in the one figure to the
correspondent parallelogram in the other; that is (by the
supposition), in the ultimate ratio of any part of the one quantity
to the correspondent part of the other.

*In similar figures, all sorts of homologous sides, whether
curvilinear or rectilinear, are proportional; and the areas are
in the duplicate ratio of the homologous sides.*

*If any arc* ACB*, given in position is subtended by
its chord* AB*, and in any point* A*, in the
middle of the continued curvature, is touched by a right line*
AD*, produced both ways; then if the points A and B approach
one another and meet, I say, the angle* BAD*, contained
between, the chord and the tangent, will be diminished* in
infinitum*, and ultimately will vanish.*

For if that angle does not vanish, the arc ACB will contain with the tangent AD an angle equal to a rectilinear angle; and therefore the curvature at the point A will not be continued, which is against the supposition.

*The same things being supposed, I say that the ultimate ratio
of the arc, chord, and tangent, any one to any other, is the
ratio of equality.*

For while the point B approaches towards the point A, consider
always AB and AD as produced to the remote points *b* and *d*,
and parallel to the secant BD draw *bd*: and let the arc A*cb*
be always similar to the arc ACB. Then, supposing the points A and B
to coincide, the angle *d*A*b* will vanish, by the
preceding Lemma; and therefore the right lines A*b*, A*d*
(which are always finite), and the intermediate arc A*cb*,
will coincide, and become equal among themselves. Wherefore, the
right lines AB, AD, and the intermediate arc
ACB (which are always proportional to the former), will vanish, and
ultimately acquire the ratio of equality. Q.E.D.

Cor. 1. Whence if through B we draw BF parallel to the tangent, always cutting any right line AF passing through A in F, this line BF will be ultimately in the ratio of equality with the evanescent arc ACB; because, completing the parallelogram AFBD, it is always in a ratio of equality with AD.

Cor. 2. And if through B and A more right lines are drawn, as BE, BD, AF, AG, cutting the tangent AD and its parallel BF; the ultimate ratio of all the abscissas AD, AE, BF, BG, and of the chord and arc AB, any one to any other, will be the ratio of equality.

Cor. 3. And therefore in all our reasoning about ultimate ratios, we may freely use any one of those lines for any other.

*If the right lines* AR, BR*, with the arc* ACB*,
the chord* AB*, and the tangent* AD*, constitute
three triangles* RAB, RACB, RAD*, and the points* A *and*
B *approach and meet: I say, that the ultimate form of these
evanescent triangles is that of similitude, and their ultimate
ratio that of equality.*

For while the point B approaches towards the point A, consider
always AB, AD, AR, as produced to the remote points *b, d,*
and *r*, and *rbd* as drawn parallel to RD, and let
the arc A*cb* be always similar to the arc ACB. Then
supposing the points A and B to coincide, the angle *b*A*d*
will vanish; and therefore the three triangles *r*A*b*,
*r*A*cb*, *r*A*d* (which are always
finite), will coincide, and on that account become both similar and
equal. And therefore the triangles RAB, RACB, RAD, which are always
similar and proportional to these, will ultimately be come both
similar and equal among themselves. Q.E.D.

Cor. And hence in all reasonings about ultimate ratios, we may indifferently use any one of those triangles for any other.

*If a right line* AE*, and a curve Line* ABC*,
both given by position, cut each other in a given angle,* A*;
and to that right line, in another given angle,* BD, CE *are
ordinately applied, meeting the curve in* B, C*; and the
points* B *and* C *together approach towards and
meet in the point* A*: I say, that the areas of the
triangles* ABD, ACE*, will ultimately be one to the other
in the duplicate ratio of the sides.*

For while the points B, C, approach towards the point A, suppose
always AD to be produced to the remote points *d* and *e*,
so as A*d*, A*e* may be proportional to AD, AE; and
the ordinates *db*, *ec*, to be drawn parallel to
the ordinates DB and EC, and meeting AB and AC produced in *b*
and *c*. Let the curve A*bc* be similar to the curve
ABC, and draw the right line A*g* so as to touch both curves
in A, and cut the ordinates DB, EC, *db, ec*, in F, G, *f,
g*. Then, supposing the length A*e* to remain the same,
let the points B and C meet in the point A; and the angle *c*A*g*
vanishing, the curvilinear areas A*bd*, A*ce* will
coincide with the rectilinear areas A*fd*, A*ge*; and
therefore (by Lem. V) will be one to the other in the duplicate
ratio of the sides A*d*, A*e*. But the areas ABD, ACE
are always proportional to these areas; and so the sides AD, AE are
to these sides. And therefore the areas ABD, ACE are ultimately one
to the other in the duplicate ratio of the sides AD, AE.
Q.E.D.

*The spaces which a body describes by any finite force urging
it, whether that force is determined and immutable, or is
continually augmented or continually diminished, are in the very
beginning of the motion one to the other in the duplicate ratio
of the times.*

Let the times be represented by the lines AD, AE, and the velocities generated in those times by the ordinates DB, EC. The spaces described with these velocities will be as the areas ABD, ACE, described by those ordinates, that is, at the very beginning of the motion (by Lem. IX), in the duplicate ratio of the times AD, AE. Q.E.D.

Cor. 1. And hence one may easily infer, that the errors of bodies describing similar parts of similar figures in proportional times, are nearly as the squares of the times in which they are generated; if so be these errors are generated by any equal forces similarly applied to the bodies, and measured by the distances of the bodies from those places of the similar figures, at which, without the action of those forces, the bodies would have arrived in those proportional times.

Cor. 2. But the errors that are generated by proportional forces, similarly applied to the bodies at similar parts of the similar figures, are as the forces and the squares of the times conjunctly.

Cor. 3. The same thing is to be understood of any spaces whatsoever described by bodies urged with different forces; all which, in the very beginning of the motion, are as the forces and the squares of the times conjunctly.

Cor. 4. And therefore the forces are as the spaces described in the very beginning of the motion directly, and the squares of the times inversely.

Cor. 5. And the squares of the times are as the spaces described directly, and the forces inversely.

If in comparing indetermined quantities of different sorts one with
another, any one is said to be as any other directly or inversely,
the meaning is, that the former is augmented or diminished in the
same ratio with the latter, or with its reciprocal. And if any one
is said to be as any other two or more directly or inversely, the
meaning is, that the first is augmented or diminished in the ratio
compounded of the ratios in which the others, or the reciprocals of
the others, are augmented or diminished. As if A is said to be as B
directly, and C directly, and D inversely, the meaning is, that A is
augmented or diminished in the same ratio with B
× C × 1

D, that is to say, that A
and BC

D are one to the other in a given ratio.

*The evanescent subtense of the angle of contact, in
all curves which at the point of contact have a finite
curvature, is ultimately in the duplicate ratio of the subtense
of the conterminate arc.*

Case 1. Let AB be that arc, AD its tangent,
BD the subtense of the angle of contact perpendicular on the
tangent, AB the subtense of the arc. Draw BG perpendicular to the
subtense AB, and AG to the tangent AD, meeting in G; then let the
points D, B, and G, approach to the points *d, b,* and *g*,
and suppose J to be the ultimate intersection of the lines BG, AG,
when the points D, B, have come to A. It is evident that the
distance GJ may be less than any assignable. But (from the nature of
the circles passing through the points A, B, G, A, *b, g*)
AB^{2} = AG × BD, and Ab^{2}
= Ag × bd; and therefore the ratio of AB² to A*b*²
is compounded of the ratios of AG to A*g*, and of B*d*
to *bd*. But because GJ may be assumed of less length than
any assignable, the ratio of AG to A*g* may be such as to
differ from the ratio of equality by less than any assignable
difference; and therefore the ratio of AB² to A*b*² may be
such as to differ from the ratio of BD to *bd* by less than
any assignable difference. There fore, by Lem. I, the ultimate ratio
of AB² to A*b*² is the same with the ultimate ratio of BD to
*bd*. Q.E.D.

Case 2. Now let BD be inclined to AD in any
given angle, and the ultimate ratio of BD to *bd* will
always be the same as before, and therefore the same with the ratio
of AB² to A*b*². Q.E.D.

Case 3. And if we
suppose the angle D not to be given, but that the right line BD
converges to a given point, or is determined by any other condition
whatever; nevertheless the angles D, *d*, being determined
by the same law, will always draw nearer to equality, and approach
nearer to each other than by any assigned difference, and therefore,
by Lem. I, will at last be equal; and therefore the lines BD, *bd*
are in the same ratio to each other as before. Q.E.D.

Cor. 1. Therefore since the tangents AD, A*d*,
the arcs AB, A*b*, and their sines, BC, *bc*, become
ultimately equal to the chords AB, A*b*, their squares will
ultimately become as the subtenses BD, *bd*.

Cor. 2. Their squares are also ultimately
as the versed sines of the arcs, bisecting the chords, and
converging to a given point. For those versed sines are as the
subtenses BD, *bd*.

Cor. 3. And therefore the versed sine is in the duplicate ratio of the time in which a body will describe the arc with a given velocity.

Cor. 4. The rectilinear triangles ADB, A*db*
are ultimately in the triplicate ratio of the sides AD, A*d*,
and in a sesquiplicate ratio of the sides DB, *db*; as being
in the ratio compounded of the sides AD to DB, and of A*d* to
*db*. So also the triangles ABC, A*bc* are ultimately
in the triplicate ratio of the sides BC, *bc*. What I call
the sesquiplicate ratio is the subduplicate of the triplicate, as
being compounded of the simple and subduplicate ratio.

Cor. 5. And because DB, *db* are
ultimately parallel and in the duplicate ratio of the lines AD, A*d*,
the ultimate curvilinear areas ADB, A*db* will be (by the
nature of the parabola) two thirds of the rectilinear triangles ADB,
A*db* and the segments AB, A*b* will be one third of
the same triangles. And thence those areas and those segments will
be in the triplicate ratio as well of the tangents AD, A*d*,
as of the chords and arcs AB, AB.

But we have all along supposed the angle of contact to be neither
infinitely greater nor infinitely less than the angles of contact
made by circles and their tangents; that is, that the curvature at
the point A is neither infinitely small nor infinitely great, or
that the interval AJ is of a finite magnitude. For DB may be taken
as AD³: in which case no circle can be drawn through the point A,
between the tangent AD and the curve AB, and therefore the angle of
contact will be infinitely less than those of circles. And by a like
reasoning, if DB be made successfully as AD^{4}, AD^{5},
AD^{6}, AD^{7}, &c., we shall have a series of
angles of contact, proceeding *in infinitum*, wherein every
succeeding term is infinitely less than the preceding. And
if DB be made successively as AD^{2}; AD^{3/2},
AD^{4/3}, AD^{5/4}, AD^{6/5},
AD^{7/6}, &c., we shall have another infinite
series of angles of contact, the first of which is of the same sort
with those of circles, the second infinitely greater, and every
succeeding one infinitely greater than the preceding. But between
any two of these angles another series of intermediate angles of
contact may be interposed, proceeding both ways *in infinitum*,
wherein every succeeding angle shall be infinitely greater or
infinitely less than the preceding. As if between the terms AD^{2}
and AD^{3} there were interposed the series AD^{13/6},
AD^{11/5}, AD^{9/4}, AD^{7/3},
AD^{5/2}, AD^{8/3}, AD^{11/4},
AD^{14/5}, AD^{17/6} &c.
And again, between any two angles of this series, a new series of
intermediate angles may be interposed, differing from one another by
infinite intervals. Nor is nature confined to any bounds.

Those things which have been demonstrated of curve lines, and the
superfices which they comprehend, may be easily applied to the curve
superfices and contents of solids. These Lemmas are premised to
avoid the tediousness of deducing perplexed demonstrations *ad
absurdum*, according to the method of the ancient geometers.
For demonstrations are more contracted by the method of
indivisibles: but because the hypothesis of indivisibles seems
somewhat harsh, and therefore that method is reckoned less
geometrical, I chose rather to reduce the demonstrations of the
following propositions to the first and last sums and ratios of
nascent and evanescent quantities, that is, to the limits of those
sums and ratios; and so to premise, as short as I could, the
demonstrations of those limits. For hereby the same thing is
performed as by the method of indivisibles; and now those principles
being demonstrated, we may use them with more safety. Therefore if
hereafter I should happen to consider quantities as made up of
particles, or should use little curve lines for right ones, I would
not be understood to mean indivisibles, but evanescent divisible
quantities: not the sums and ratios of determinate parts, but always
the limits of sums and ratios; and that the force of such
demonstrations always depends on the method laid down in the
foregoing Lemmas.

Perhaps it may be objected, that there is no ultimate proportion, of evanescent quantities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument, it may be alledged, that a body arriving at a certain place, and there stopping, has no ultimate velocity: because the velocity, before the body comes to the place, is not its ultimate velocity; when it has arrived, is none. But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of nascent quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished). There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be. And since such limits are certain and definite, to determine the same is a problem strictly geometrical. But whatever is geometrical we may be allowed to use in determining and demonstrating any other thing that is likewise geometrical.

It may also be objected, that if the ultimate ratios of evanescent
quantities are given, their ultimate magnitudes will be also given:
and so all quantities will consist of indivisibles, which is
contrary to what Euclid has demonstrated concerning
incommensurables, in the 10th Book of his Elements. But this
objection is founded on a false supposition. For those ultimate
ratios with which quantities vanish are not truly the ratios of
ultimate quantities, but limits towards which the ratios of
quantities decreasing without limit do always converge; and to which
they approach nearer than by any given difference, but never go
beyond, nor in effect attain to, till the quantities are diminished
*in infinitum*. This thing will appear more evident in
quantities infinitely great. If two quantities, whose difference is
given, be augmented *in infinitum*, the ultimate ratio of
these quantities will be given, to wit, the ratio of equality; but
it does not from thence follow, that the ultimate or greatest
quantities themselves, whose ratio that is, will be given. Therefore
if in what follows, for the sake of being more easily understood, I
should happen to mention quantities as least, or evanescent, or
ultimate, you are not to suppose that quantities of any determinate
magnitude are meant, but such as are conceived to be always
diminished without end.

*The areas, which revolving bodies describe by radii drawn to an
immovable centre of force do lie in the same immovable planes, and
are proportional to the times in which they are described.*

For suppose the time to be divided into equal parts, and in the first
part of that time let the body by its innate force describe the right
line AB In the second part of that time, the same would (by Law I.),
if not hindered, proceed directly to *c*, along the line B*c*
equal to AB; so that by the radii AS, BS, *c*S, drawn to the
centre, the equal areas ASB, BS*c*, would be described.
But when the body is arrived at
B, suppose that a centripetal force acts at once with a great impulse;
and, turning aside the body from the right line B*c*, compels
it afterwards to continue its motion along the right line BC. Draw *c*C
parallel to BS meeting BC in C; and at the end of the second part of
the time, the body (by Cor. I. of the Laws) will be found in C, in the
same plane with the triangle ASB. Join SC, and, because SB and C*c*
are parallel, the triangle SBC will be equal to the triangle SB*c*,
and therefore also to the triangle SAB. By the like argument, if the
centripetal force acts successively in C, D, E. &c.; and makes the
body, in each single particle of time, to describe the right lines CD,
DE, EF, &c., they will all lie in the same plane; and the triangle
SCD will be equal to the triangle SBC, and SDE to SCD, and SEF to SDE.
And therefore, in equal times, equal areas are described in one
immovable plane: and, by composition, any sums SADS, SAFS, of those
areas, are one to the other as the times in which they are described.
Now let the number of those triangles be augmented, and their breadth
diminished *in infinitum*; and (by Cor. 4, Lem. III.) their
ultimate perimeter ADF will be a curve line: and therefore the
centripetal force, by which the body is perpetually drawn back from
the tangent of this curve, will act continually; and any described
areas SADS, SAFS, which are always proportional to the times of
description, will, in this case also, be proportional to those times.
Q.E.D.

Cor. 1. The velocity of a body attracted towards an immovable centre, in spaces void of resistance, is reciprocally as the perpendicular let fall from that centre on the right line that touches the orbit. For the velocities in those places A, B, C, D, E, are as the bases AB, BC, CD, DE, EF, of equal triangles; and these bases are reciprocally as the perpendiculars let fall upon them.

Cor. 2. If the chords AB, BC of two arcs,
successively described in equal times by the same body, in spaces void
of resistance, are completed into a parallelogram ABCV, and the
diagonal BV of this parallelogram; in the position which it ultimately
acquires when those arcs are diminished *in infinitum*, is
produced both ways, it will pass through the centre of force.

Cor. 3. If the chords AB, BC, and DE, EF, of
arcs described in equal times, in spaces void
of resistance, are completed into the parallelograms ABCV, DEFZ; the
forces in B and E are one to the other in the ultimate ratio of the
diagonals BV, EZ, when those arcs are diminished in infinitum. For the
motions BC and EF of the body (by Cor. 1 of the Laws) are compounded
of the motions B*c*, BV, and E*f*, EZ: but BV and EZ,
which are equal to C*c* and F*f*, in the demonstration
of this Proposition, were generated by the impulses of the centripetal
force in B and E, and are therefore proportional to those impulses.

Cor. 4. The forces by which bodies, in spaces void of resistance, are drawn back from rectilinear motions, and turned into curvilinear orbits, are one to another as the versed sines of arcs described in equal times; which versed sines tend to the centre of force, and bisect the chords when those arcs are diminished to infinity. For such versed sines are the halves of the diagonals mentioned in Cor. 3.

Cor. 5. And therefore those forces are to the force of gravity as the said versed sines to the versed sines perpendicular to the horizon of those parabolic arcs which projectiles describe in the same time.

Cor. 6. And the same things do all hold good (by Cor. 5 of the Laws), when the planes in which the bodies are moved, together with the centres of force which are placed in those planes, are not at rest, but move uniformly forward in right lines.

*Every body that moves in any curve line described in a plane,
and by a radius, drawn to a point either immovable, or moving
forward with an uniform rectilinear motion, describes about that
point areas proportional to the times, is urged by a centripetal
force directed to that point.*

Case. 1. For every body that moves in a curve
line, is (by Law 1) turned aside from its rectilinear course by the
action of some force that impels it. And that force by which the body
is turned off from its rectilinear course, and is made to describe, in
equal times, the equal least triangles SAB, SBC, SCD, &c., about
the immovable point S (by Prop. XL. Book 1, Elem. and Law II), acts in
the place B, according to the direction of a line parallel to
*c*C, that is, in the direction of the line BS, and in the
place C, according to the direction of a line parallel to *d*D,
that is, in the direction of the line CS, &c.; and therefore acts
always in the direction of lines tending to the immovable point S.
Q.E.D.

Case. 2. And (by Cor. 5 of the Laws) it is indifferent whether the superfices in which a body describes a curvilinear figure be quiescent, or moves together with the body, the figure described, and its point S, uniformly forward in right lines.

Cor. 1. In non-resisting spaces or mediums,
if the areas are not proportional to the times, the forces are not
directed to the point in which the radii meet; but deviate therefrom *in
consequentia*, or towards the parts to which the motion is
directed, if the description of the areas is accelerated; but *in
antecedentia*, if retarded.

Cor. 2. And even in resisting mediums, if the description of the areas is accelerated, the directions of the forces deviate from the point in which the radii meet; towards the parts to which the motion tends.

A body may be urged by a centripetal force compounded of several forces; in which case the meaning of the Proposition is, that the force which results out of all tends to the point S. But if any force acts perpetually in the direction of lines perpendicular to the described surface, this force will make the body to deviate from the plane of its motion: but will neither augment nor diminish the quantity of the described surface, and is therefore to be neglected in the composition of forces.

*Every body, that by a radius drawn to the centre of another
body, how soever moved, describes areas about that centre
proportional to the times, is urged by a force compounded out of
the centripetal force tending to that other body, and of all the
accelerative force by which that other body is impelled.*

Let L represent the one, and T the other body; and (by Cor. 6 of the Laws) if both bodies are urged in the direction of parallel lines, by a new force equal and contrary to that by which the second body T is urged, the first body L will go on to describe about the other body T the same areas as before: but the force by which that other body T was urged will be now destroyed by an equal and contrary force; and therefore (by Law I.) that other body T, now left to itself, will either rest, or move uniformly forward in a right line: and the first body L impelled by the difference of the forces, that is, by the force remaining, will go on to describe about the other body T areas proportional to the times. And therefore (by Theor. II.) the difference of the forces is directed to the other body T as its centre. Q.E.D

Cor. 1. Hence if the one body L, by a radius drawn to the other body T, describes areas proportional to the times; and from the whole force, by which the first body L is urged (whether that force is simple, or, according to Cor. 2 of the Laws, compounded out of several forces), we subduct (by the same Cor.) that whole accelerative force by which the other body is urged; the whole remaining force by which the first body is urged will tend to the other body T, as its centre.

Cor. 2. And, if these areas are proportional to the times nearly, the remaining force will tend to the other body T nearly.

Cor. 3. And *vice versa*, if the
remaining force tends nearly to the other body T, those areas will be
nearly proportional to the times.

Cor. 4. If the body L, by a radius drawn to the other body T, describes areas, which, compared with the times, are very unequal; and that other body T be either at rest, or moves uniformly forward in a right line: the action of the centripetal force tending to that other body T is either none at all, or it is mixed and compounded with very powerful actions of other forces: and the whole force compounded of them all, if they are many, is directed to another (immovable or moveable) centre. The same thing obtains, when the other body is moved by any motion whatsoever; provided that centripetal force is taken, which remains after subducting that whole force acting upon that other body T.

Because the equable description of areas indicates that a centre is respected by that force with which the body is most affected, and by which it is drawn back from its rectilinear motion, and retained in its orbit; why may we not be allowed, in the following discourse, to use the equable description of areas as an indication of a centre, about which all circular motion is performed in free spaces?

*The centripetal forces of bodies, which by equable motions
describe different circles, tend to the centres of the same
circles; and are one to the other as the squares of the arcs
described in equal times applied to the radii of the circles.*

These forces tend to the centres of the circles (by Prop. II., and Cor. 2, Prop. I.), and are one to another as the versed sines of the least arcs described in equal times (by Cor. 4, Prop. I.); that is, as the squares of the same arcs applied to the diameters of the circles (by Lem. VII.); and therefore since those arcs are as arcs described in any equal times, and the diameters are as the radii, the forces will be as the squares of any arcs described in the same time applied to the radii of the circles. Q.E.D.

Cor. 1. Therefore, since those arcs are as the velocities of the bodies the centripetal forces are in a ratio compounded of the duplicate ratio of the velocities directly, and of the simple ratio of the radii inversely.

Cor. 2. And since the periodic times are in a ratio compounded of the ratio of the radii directly, and the ratio of the velocities inversely, the centripetal forces, are in a ratio compounded of the ratio of the radii directly, and the duplicate ratio of the periodic times inversely.

Cor. 3. Whence if the periodic times are equal, and the velocities therefore as the radii, the centripetal forces will be also as the radii; and the contrary.

Cor. 4. If the periodic times and the velocities are both in the subduplicate ratio of the radii, the centripetal forces will be equal among themselves; and the contrary.

Cor. 5. If the periodic times are as the radii, and therefore the velocities equal, the centripetal forces will be reciprocally as the radii; and the contrary.

Cor. 6. If the periodic times are in the sesquiplicate ratio of the radii, and therefore the velocities reciprocally in the subduplicate ratio of the radii, the centripetal forces will be in the duplicate ratio of the radii inversely; and the contrary.

Cor. 7. And universally, if the periodic time
is as any power R^{n} of the radius R, and therefore the
velocity reciprocally as the power R^{n−1} of the radius, the
centripetal force will be reciprocally as the power R^{2n−1}
of the radius; and the contrary.

Cor. 8. The same things all hold concerning the times, the velocities, and forces by which bodies describe the similar parts of any similar figures that have their centres in a similar position with those figures; as appears by applying the demonstration of the preceding cases to those. And the application is easy, by only substituting the equable description of areas in the place of equable motion, and using the distances of the bodies from the centres instead of the radii.

Cor. 9. From the same demonstration it likewise follows, that the arc which a body, uniformly revolving in a circle by means of a given centripetal force, describes in any time, is a mean proportional between the diameter of the circle, and the space which the same body falling by the same given force would descend through in the same given time.

The case of the 6th Corollary obtains in the celestial bodies (as Sir Christopher Wren, Dr. Hooke, and Dr. Halley have severally observed); and therefore in what follows, I intend to treat more at large of those things which relate to centripetal force decreasing in a duplicate ratio of the distances from the centres.

Moreover, by means of the preceding Proposition and its Corollaries,
we may discover the proportion of a
centripetal force to any other known force, such as that of gravity.
For if a body by means of its gravity revolves in a circle concentric
to the earth, this gravity is the centripetal force of that body. But
from the descent of heavy bodies, the time of one entire revolution,
as well as the arc described in any given time, is given (by Cor. 9 of
this Prop.). And by such propositions, Mr. Huygens, in his excellent
book *De Horologio Oscillatorio*, has compared the force of
gravity with the centrifugal forces of revolving bodies.

The preceding Proposition may be likewise demonstrated after this
manner. In any circle suppose a polygon to be inscribed of any number
of sides. And if a body, moved with a given velocity along the sides
of the polygon, is reflected from the circle at the several angular
points, the force, with which at every reflection it strikes the
circle, will be as its velocity: and therefore the sum of the forces,
in a given time, will be as that velocity and the number of
reflections conjunctly: that is (if the species of the polygon be
given), as the length described in that given time, and increased or
diminished in the ratio of the same length to the radius of the
circle; that is, as the square of that length applied to the radius;
and therefore the polygon, by having its sides diminished *in
infinitum*, coincides with the circle, as the square of the arc
described in a given time applied to the radius. This is the
centrifugal force, with which the body impels the circle; and to which
the contrary force, wherewith the circle continually repels the body
towards the centre, is equal.

*There being given, in any places, the velocity with which a
body describes a given figure, by means of forces directed to some
common centre: to find that centre.*

Let the three right lines PT, TQV, VR touch the figure described in as many points, P, Q, R, and meet in T and V. On the tangents erect the perpendiculars PA, QB, RC, reciprocally proportional to the velocities of the body in the points P, Q, R, from which the perpendiculars were raised; that is, so that PA may be to QB as the velocity in Q, to the velocity in P, and QB to RC as the velocity in R to the velocity in Q. Through the ends A, B, C, of the perpendiculars draw AD, DBE, EC, at right angles, meeting in D and E: and the right lines TD, VE produced, will meet in S, the centre required.

For the perpendiculars let fall from the centre S on the tangents PT, QT, are reciprocally as the velocities of the bodies in the points P and Q (by Cor. 1, Prop. I.), and therefore, by construction, as the perpendiculars AP, BQ directly; that is, as the perpendiculars let fall from the point D on the tangents. Whence it is easy to infer that the points S, D, T, are in one right line. And by the like argument the points S, E, V are also in one right line; and therefore the centre S is in the point where the right lines TD, VE meet. Q.E.D.

*In a space void of resistance, if a body revolves in any orbit
about an immovable centre, and in the least time describes any arc
just then nascent; and the versed sine of that arc is supposed to
be drawn bisecting the chord, and produced passing through the
centre of force: the centripetal force in the middle of the arc
will be as the versed sine directly and the square of the time
inversely.*

For the versed sine in a given time is as the force (by Cor. 4, Prop. 1); and augmenting the time in any ratio, because the arc will be augmented in the same ratio, the versed sine will be augmented in the duplicate of that ratio (by Cor. 2 and 3, Lem. XI.), and therefore is as the force and the square of the time. Subduct on both sides the duplicate ratio of the time, and the force will be as the versed sine directly, and the square of the time inversely. Q.E.D.

And the same thing may also be easily demonstrated by Corol. 4, Lem. X.

Cor. 1. If a body P revolving about the
centre S describes a curve line APQ, which a right line ZPR touches in
any point P; and from any other point Q of the curve, QR is drawn
parallel to the distance SP, meeting the tangent in R; and QT is drawn
perpendicular to the distance SP; the centripetal force will be
reciprocally as the solid SP^{2}
× QT^{2}

QR, if the solid be taken of that magnitude which it
ultimately acquires when the points P and Q coincide. For QR is equal
to the versed sine of double the arc QP, whose middle is P: and double
the triangle SQP, or SP × QT is
proportional to the time in which that double arc is described; and
therefore may be used for the exponent of the time.

Cor. 2. By a like reasoning, the centripetal
force is reciprocally as the solid SY^{2}
× QP^{2}

QR; if SY is a perpendicular from the centre of force on PR
the tangent of the orbit. For the rectangles SY ×
QP and SP × QT are equal.

Cor. 3. If the orbit
is either a circle, or touches or cuts a circle concentrically, that
is, contains with a circle the least angle of contact or section,
having the same curvature and the same radius of curvature at the
point P; and if PV be a chord of this circle, drawn from the body
through the centre of force; the centripetal force will be
reciprocally as the solid SP^{2} × PV.
For PV is QP^{2}

QR.

Cor. 4. The same things being supposed, the centripetal force is as the square of the velocity directly, and that chord inversely. For the velocity is reciprocally as the perpendicular SY, by Cor. 1. Prop. I.

Cor. 5. Hence if any curvilinear figure APQ
is given, and therein a point S is also given, to which a centripetal
force is perpetually directed, that law of centripetal force may be
found, by which the body P will be continually drawn back from a
rectilinear course, and being detained in the perimeter of that
figure, will describe the same by a perpetual revolution. That is, we
are to find, by computation, either the solid SP^{2} × QT^{2}

QR or the solid SP^{2} × PV,
reciprocally proportional to this force. Examples of this we shall
give in the following Problems.

*If a body revolves in the circumference of a circle; it is
proposed to find the law of centripetal force directed to any given point.*

Let VQPA be the circumference of the circle; S the given point to
which as to a centre the force tends; P the body moving in the
circumference; Q the next place into which it is to move; and PRZ the
tangent of the circle at the preceding place. Through the point S draw
the chord PV, and the diameter VA of the circle: join AP, and draw QT
perpendicular to SP, which produced, may meet the tangent PR in Z; and
lastly, through the point Q, draw LR parallel to SP, meeting the
circle in L, and the tangent PZ in R. And, because of the similar
triangles ZQR, ZTP, VPA, we shall have RP², that is, QRL to QT² as AV²
to PV². And therefore
QRL × SP^{2}

AV^{2} is equal to QT². Multiply those equals by
SP^{2}

QR,
and the points P and Q coinciding, for RL write PV; then we shall have
SP^{2}
× PV^{3}

AV^{2} = SP^{2}
× QT^{2}

QR. And therefore (by Cor 1 and 5, Prop. VI.)
the centripetal force is reciprocally as SP^{2} × PV^{3}

AV^{2}; that is (because
AV² is given), reciprocally as the square of the distance or altitude
SP, and the cube of the chord PV conjunctly. Q.E.I.

On the tangent PR produced let fall the perpendicular SY; and
(because of the similar triangles SYP, VPA), we shall have AV to PV as
SP to SY, and therefore SP
× PV

AV = SY, and SP^{2} × PV^{3}

AV^{2} = SY^{2}
× PV. And therefore (by Corol. 3 and 5, Prop. VI),
the centripetal force is reciprocally as SP^{2}
× PV^{3}

AV^{2}; that is (because AV
is given), reciprocally as SP^{2} × PV^{3}.
Q.E.I.

Cor. 1. Hence if the given point S, to which the centripetal force always tends, is placed in the circumference of the circle, as at V, the centripetal force will be reciprocally as the quadrato-cube (or fifth power) of the altitude SP.

Cor. 2. The force by which the body P in the
circle APTV revolves about the centre of force S is to the force by
which the same body P may revolve in the same circle, and in the same
periodic time, about any other centre of force R, as RP^{2}
× SP to the cube of the right line SG, which, from the first
centre of force S is drawn parallel to the distance PR of the body
from the second centre of force R, meeting the tangent PG of the orbit
in G. For by the construction of this Proposition, the former force is
to the latter as RP^{2} × PT^{3}
to SP^{2} × PV^{3} ; that
is, as SP × RP^{2} to SP^{3} × PV^{3}

PT^{3}; or (because of the
similar triangles PSG, TPV) to SG³.

Cor. 3. The force by which the body P in any
orbit revolves about the centre of force S, is to the force by which
the same body may revolve in the same orbit, and the same periodic
time, about any other centre of force R, as the solid SP
× RP^{2}, contained under the distance of the body
from the first centre of force S, and the square of its distance from
the second centre of force R, to the cube of the right line SG, drawn
from the first centre of the force S, parallel to the distance RP of
the body from the second centre of force R, meeting the tangent PG of
the orbit in G. For the force in this orbit at any point P is the same
as in a circle of the same curvature.

*If a body moves in the semi-circumference* PQA*; it is
proposed to find the law of the centripetal force tending to a
point* S*, so remote, that all the lines* PS, RS *drawn
thereto, may be taken for parallels.*

From C, the centre of the semi-circle, let the semi-diameter CA he
drawn, cutting the parallels at right angles in M and N, and join CP.
Because of the similar triangles CPM, PZT, and RZQ, we shall have CP²
to PM² as PR² to QT²; and, from the nature of the circle, PR² is equal
to the rectangle QR × (RN + QN), or, the
points P, Q, coinciding, to the rectangle QR × 2PM.
Therefore CP² is to PM² as QR × 2PM to QT²;
and QT^{2}

QR = 2PM^{3}

CP^{2}, and QT^{2} × SP^{2}

QR = 2PM^{3} ×
SP^{2}

CP^{2}. And therefore (by Corol. 1 and 5;
Prop. VI.), the centripetal force is reciprocally as 2PM^{3} × SP^{2}

CP^{2}; that is (neglecting the given ratio 2SP^{2}

CP^{2} ), reciprocally as PM³. Q.E.I.

And the same thing is likewise easily inferred from the preceding Proposition.

And by a like reasoning, a body will be moved in an ellipsis, or even in an hyperbola, or parabola, by a centripetal force which is reciprocally ae the cube of the ordinate directed to an infinitely remote centre of force.

*If a body revolves in a spiral* PQS*, cutting all the
radii* SP, SQ, *&c., in a given angle; it is proposed
to find the law of the centripetal force tending to the centre of
that spiral.*

Suppose the indefinitely small angle PSQ to be given; because, then,
all the angles are given, the figure SPRQT will be given in specie.
Therefore the ratio QT

QR is also given, and QT^{2}

QR is as QT, that is (because the figure is given in specie),
as SP. But if the angle PSQ is any way changed, the right line QR,
subtending the angle of contact QPR (by Lemma
XI) will be changed in the duplicate ratio of PR or QT. Therefore the
ratio QT^{2}

QR remains the same as before, that is, as SP. And QT^{2} × SP^{2}

QR is as SP³, and therefore (by Corol. 1 and 5, Prop. VI) the
centripetal force is reciprocally as the cube of the distance SP.
Q.E.I.

The perpendicular SY let fall upon the tangent, and the chord PV of the circle concentrically cutting the spiral, are in given ratios to the height SP; and therefore SP³ is as SY² × PV, that is (by Corol. 3 and 5, Prop. VI) reciprocally as the centripetal force.

*All parallelograms circumscribed about any conjugate diameters
of a given ellipsis or hyperbola are equal among themselves.*

This is demonstrated by the writers on the conic sections.

*If a body revolves in an ellipsis; it is proposed to find the
law of the centripetal force tending to the centre of the ellipsis.*

Suppose CA, CB to be semi-axes of the ellipsis; GP, DK, conjugate
diameters; PF, QT perpendiculars to those diameters; Q*v* an
ordinate to the diameter GP; and if the parallelogram Q*v*PR be
completed, then (by the properties of the conic sections) the
rectangle P*v*G will be to Q*v*² as PC² to CD²; and
(because of the similar triangles Q*v*T, PCF), Q*v*² to
QT² as PC² to PF²; and, by composition, the ratio of P*v*G to
QT² is compounded of the ratio of PC² to CD², and of the ratio of PC²
to PF², that is, *v*G to QT^{2}

Pv as PC² to CD^{2} × PF^{2}

PC^{2}. Put QR for P*v*,
and (by Lem. XII) BC × CA for CD × PF; also (the points P and Q
coinciding) 2PC for *v*G; and multiplying the
extremes and means together, we shall have QT^{2}
× PC^{2}

QR equal to 2BC^{2}
× CA^{2}

PC. Therefore (by Cor. 5, Prop. VI),
the centripetal force is reciprocally as 2BC^{2}
× CA^{2}

PC; that is (because 2BC^{2}
× CA^{2} is given), reciprocally as 1

PC; that is, directly as the distance
PC. QEI.

In the right line PG on the other side of the point T, take the point
*u* so that T*u* may be equal to T*v*; then take
*u*V, such as shall be to *v*G as DC² to PC². And
because Q*v*² is to P*v*G as DC² to PC² (by the conic
sections), we shall have QV^{2}=Pv × uV.
Add the rectangle *u*P*v* to both sides, and the square
of the chord of the arc PQ will be equal to the rectangle VP*v*;
and therefore a circle which touches the conic section in P, and
passes through the point Q, will pass also through the point V. Now
let the points P and Q meet, and the ratio of *u*V to *v*G,
which is the same with the ratio of DC² to PC², will become the ratio
of PV to PG, or PV to 2PC; and therefore PV will be equal to 2DC^{2}

PC. And therefore the force by which
the body P revolves in the ellipsis will be reciprocally as 2DC^{2}

PC × PF^{2} (by Cor. 3, Prop
VI); that is (because 2DC² × PF² is given) directly as PC.
Q.E.I.

Cor. 1. And therefore the force is as the
distance of the body from the centre of the ellipsis; and, *vice
versa*, if the force is as the distance, the body will move in
an ellipsis whose centre coincides with the centre of force, or
perhaps in a circle into which the ellipsis may degenerate.

Cor. 2. And the periodic times of the revolutions made in all ellipses whatsoever about the same centre will be equal. For those times in similar ellipses will be equal (by Corol. 3 and 8, Prop. IV); but in ellipses that have their greater axis common, they are one to another as the whole areas of the ellipses directly, and the parts of the areas described in the same time inversely; that is, as the lesser axes directly, and the velocities of the bodies in their principal vertices inversely; that is, as those lesser axes directly, and the ordinates to the same point of the common axes inversely; and therefore (because of the equality of the direct and inverse ratios) in the ratio of equality.

If the ellipsis, by having its centre removed to an infinite
distance, de generates into a parabola, the body will move in this
parabola; and the force, now tending to a
centre infinitely remote, will become equable. Which is *Galileo's*
theorem. And if the parabolic section of the cone (by changing the
inclination of the cutting plane to the cone) degenerates into an
hyperbola, the body will move in the perimeter of this hyperbola,
having its centripetal force changed into a centrifugal force. And in
like manner as in the circle, or in the ellipsis, if the forces are
directed to the centre of the figure placed in the abscissa, those
forces by increasing or diminishing the ordinates in any given ratio;
or even by changing the angle of the inclination of the ordinates to
the abscissa, are always augmented or diminished in the ratio of the
distances from the centre; provided the periodic times remain equal;
so also in all figures whatsoever, if the ordinates are augmented or
diminished in any given ratio, or their inclination is any way
changed, the periodic time remaining the same, the forces directed to
any centre placed in the abscissa are in the several ordinates
augmented or diminished in the ratio of the distances from the centre.

*If a body revolves in an ellipsis; it is required to find
the law of the centripetal force tending to the focus of the ellipsis.*

Let S be the focus of the ellipsis. Draw SP cutting the diameter DK
of the ellipsis in E, and the ordinate Q*v* in *x*; and
complete the parallelogram Q*x*PR. It is evident that EP is
equal to the greater semi-axis AC: for drawing HI from the other focus
H of the ellipsis parallel to EC, because CS, CH are equal, ES, EI
will be also equal; so that EP is the half sum of PS, PI, that is
(because of the parallels HI, PR, and the equal angles IPR, HPZ), of
PS, PH, which taken together are equal to the whole axis 2AC. Draw QT
perpendicular to SP, and putting L for the principal latus rectum of
the ellipsis (or for 2BC^{2}

AC ), we shall have L × QR to L × P*v*
as QR to P*v*, that is, as PE or AC to PC; and L × P*v*
to G*v*P as L to G*v*; and *Gv*P to Q*v*²
as PC² to CD²; and by (Corol. 2, Lem. VII) the points Q and P
coinciding, Q*v*² is to Q*x*² in the ratio of equality;
and Q*x*² or Q*v*² is to QT² as EP² to PF², that is, as
CA² to PF², or (by Lem. XII) as CD² to CB². And compounding all those
ratios together, we shall have L × QR to
QT² as AC × L × PC² × CD², or 2CB²
× PC² × CD² to PC × Gv × CD² × CB²,
or as 2PC to G*v*. But the points Q and P coinciding, 2PC and G*v*
are equal. And therefore the quantities L × QR
and QT², proportional to these, will be also equal. Let those equals
be drawn into SP^{2}

QR, and L × SP² will become equal to SP^{2} × QT^{2}

QR. And therefore (by Corol. 1 and 5, Prop. VI) the
centripetal force is reciprocally as L × SP², that is, reciprocally in
the duplicate ratio of the distance SP. Q.E.I.

Since the force tending to the centre of the ellipsis, by which the
body P may revolve in that ellipsis, is (by Corol. 1, Prop. X.) as the
distance CP of the body from the centre C of the ellipsis; let CE be
drawn parallel to the tangent PR of the ellipsis; and the force by
which the same body P may revolve about any other point's of the
ellipsis, if CE and PS intersect in E, will be as PE^{3}

SP^{2} (by Cor. 3, Prop. VII.);
that is, if the point S is the focus of the ellipsis, and therefore PE
be given as SP² reciprocally. Q.E.I.

With the same brevity with which we reduced the fifth Problem to the parabola, and hyperbola, we might do the like here: but because of the dignity of the Problem and its use in what follows. I shall confirm the other cases by particular demonstrations.

*Suppose a body to move in an hyperbola; it is required to
find the law of the centripetal force tending to the focus of that
figure.*

Let CA, CB be the semi-axes of the hyperbola; PG, KD other conjugate
diameters; PF a perpendicular to the diameter KD; and Q*v* an
ordinate to the diameter GP. Draw SP cutting the diameter DK in E, and
the ordinate Q*v* in *x*, and complete the
parallelogram QRP*x*. It is evident that EP is equal to the
semi-transverse axis AC; for drawing HI, from the other focus H of the
hyperbola, parallel to EC, because CS, CH are equal, ES, EI will be
also equal; so that EP is the half difference
of PS, PI; that is (because of the parallels
IH, PR, and the equal angles IPR, HPZ), of PS, PH, the difference of
which is equal to the whole axis 2AC. Draw QT perpendicular to SP; and
putting L for the principal latus rectum of the hyperbola (that is,
for 2BC^{2}

AC, we shall have L × QR to L × P*v*
as QR to P*v*, or P*x* to P*v*, that is (because
of the similar triangles P*xv*, PEC), as PE to PC, or AC to PC.
And L × P*v* will be to G*v* × P*v* as L to G*v*;
and (by the properties of the conic sections) the rectangle G*v*P
is to Q*v*² as PC² to CD²; and by (Cor. 2, Lem. VII.), Q*v*²
to Q*x*² the points Q and P coinciding, becomes a ratio of
equality; and Q*x*² or Q*v*² is to QT² as EP² to PF²,
that is, as CA² to PF², or (by Lem. XII.) as CD² to CB²: and,
compounding all those ratios together, we shall have L × QR to QT² as
AC × L × PC² × CD², or 2CB² × PC² × CD² to PC × G*v* × CD² ×
CB², or as 2PC to G*v*. But the points P and Q coinciding, 2PC
and G*v* are equal. And therefore the quantities L × QR and
QT², proportional to them, will be also equal. Let those equals be
drawn into SP^{2}

QR, and we shall have L × SP² equal to SP^{2} × QT^{2}

QR. And therefore (by Cor. I and 5, Prop. VI.) the
centripetal force is reciprocally as L × SP², that is, reciprocally in
the duplicate ratio of the distance SP. Q.E.I.

Find out the force tending from the centre C of the hyperbola. This
will be proportional to the distance CP. But from thence (by Cor. 3,
Prop. VII.) the force tending to the focus S will be as PE^{3}

SP^{2}, that is, because PE is
given reciprocally as SP². Q.E.I.

And the same way may it be demonstrated, that the body having its centripetal changed into a centrifugal force, will move in the conjugate hyperbola.

*The latus rectum of a parabola belonging to any vertex is
quadruple the distance of that vertex from the focus of the
figure.*

This is demonstrated by the writers on the conic sections.

*The perpendicular, let fall from the focus of a parabola on its
tangent, is a mean proportional between the distances of the focus
from the point of contact, and from the principal vertex of the figure.*

For, let AP be the parabola, S its focus, A its principal vertex, P the point of contact, PO an ordinate to the principal diameter, PM the tangent meeting the principal diameter in M, and SN the perpendicular from the focus on the tangent: join AN, and because of the equal lines MS and SP, MN and NP, MA and AO, the right lines AN, OP, will be parallel; and thence the triangle SAN will be right-angled at A, and similar to the equal triangles SNM, SNP; therefore PS is to SN as SN to SA. Q.E.D.

Cor. 1. PS² is to SN² as PS to SA.

Cor. 2. And because SA is given, SN² will be as PS.

Cor. 3. And the concourse of any tangent PM, with the right line SN. drawn from the focus perpendicular on the tangent, falls in the right line AN that touches the parabola in the principal vertex.

*If a body moves in the perimeter of a parabola; it is required
to find the law of the centripetal force tending to the focus of that figure.*

Retaining the construction of the preceding Lemma, let P be the body
in the perimeter of the parabola; and from the place Q, into which it
is next to succeed, draw QR parallel and QT perpendicular to SP, as
also Q*v* parallel to the tangent, and meeting the diameter PG
in *v*, and the distance SP in *x*.
Now, because of the similar triangles P*xv*, SPM, and of the
equal sides SP, SM of the one, the sides P*x* or QR and P*v*
of the other will be also equal. But (by the conic sections) the
square of the ordinate Q*v* is equal to the rectangle under the
latus rectum and the segment P*v* of the diameter; that is (by
Lem. XIII.), to the rectangle 4PS × P*v*, or 4PS × QR; and the
points P and Q coinciding, the ratio of Q*v* to Q*x* (by
Cor. 2, Lem. VII.,) becomes a ratio of equality. And therefore Q*x*²,
in this case, becomes equal to the rectangle 4PS × QR. But (because of
the similar triangles Q*x*T, SPN), Q*x*² is to QT² as
PS² to SN², that is (by Cor. 1, Lem. XIV.), as PS to SA; that is, as
4PS × QR to 4SA × QR, and therefore (by Prop. IX. Lib. V., Elem.) QT²
and 4SA × QR are equal. Multiply these equals by SP^{2}

QR, and SP^{2} ×
QT^{2}

QR will become equal to SP² × 4SA: and therefore (by Cor. 1
and 5, Prop. VI.), the centripetal force is reciprocally as SP² × 4SA;
that is, because 4SA is given; reciprocally in the duplicate ratio of
the distance SP. Q.E.I.

Cor. 1. From the three last Propositions it follows, that if any body P goes from the place P with any velocity in the direction of any right line PR, and at the same time is urged by the action of a centripetal force that is reciprocally proportional to the square of the distance of the places from the centre, the body will move in one of the conic sections, having its focus in the centre of force; and the contrary. For the focus, the point of contact, and the position of the tangent, being given, a conic section may be described, which at that point shall have a given curvature. But the curvature is given from the centripetal force and velocity of the body being given; and two orbits, mutually touching one the other, cannot be described by the same centripetal force and the same velocity.

Cor. 2. If the velocity with which the body
goes from its place P is such, that in any infinitely small moment of
time the lineola PR may be thereby described; and the centripetal
force such as in the same time to move the same body through the space
QR; the body will move in one of the conic sections, whose principal
latus rectum is the quantity QT^{2}

QR in its ultimate state, when the
lineolae PR, QR are diminished *in infinitum*. In these
Corollaries I consider the circle as an ellipsis; and I except the
case where the body descends to the centre in a right line.

*If several bodies revolve about one common centre, and the
centripetal force is reciprocally in the duplicate ratio of the
distance of places from the centre; I say, that the principal
latera recta of their orbits are in the duplicate ratio of the areas,
which the bodies by radii drawn to the centre describe in the same time.*

For (by Cor. 2, Prop. XIII) the latus rectum L is equal to the
quantity QT^{2}

QR in its ultimate state when the
points P and Q coincide. But the lineola QR in a given time is as the
generating centripetal force; that is (by supposition), reciprocally
as SP² . And therefore QT^{2}

QR is as QT² × SP²; that is, the latus
rectum L is in the duplicate ratio of the area QT × SP.
Q.E.D.

Cor. Hence the whole area of the ellipsis, and the rectangle under the axes, which is proportional to it, is in the ratio compounded of the subduplicate ratio of the latus rectum, and the ratio of the periodic time. For the whole area is as the area QT × SP, described in a given time, multiplied by the periodic time.

*The same things being supposed, I say, that the periodic times
in ellipses are in the sesquiplicate ratio of their greater axes.*

For the lesser axis is a mean proportional between the greater axis and the latus rectum; and, therefore, the rectangle under the axes is in the ratio compounded of the subduplicate ratio of the latus rectum and the sesquiplicate ratio of the greater axis. But this rectangle (by Cor. 3. Prop. XIV) is in a ratio compounded of the subduplicate ratio of the latus rectum, and the ratio of the periodic time. Subduct from both sides the subduplicate ratio of the latus rectum, and there will remain the sesquiplicate ratio of the greater axis, equal to the ratio of the periodic time. Q.E.D.

Cor. Therefore the periodic times in ellipses are the same as in circles whose diameters are equal to the greater axes of the ellipses.

*The same things being supposed, and right lines being drawn to
the bodies that shall touch the orbits, and perpendiculars being
let fall on those tangents from the common focus; I say, that the
velocities of the bodies are in a ratio compounded of the ratio of
the perpendiculars inversely, and the subduplicate ratio of the
principal latera recta directly.*

From the focus S draw SY perpendicular to the tangent PR, and the
velocity of the body P will be reciprocally in the subduplicate ratio
of the quantity SY^{2}

L. For that velocity is as the
infinitely small arc PQ described
in a given moment of time, that is (by Lem. VII), as the tangent PR;
that is (because of the proportionals PR to QT, and SP to SY), as
SP × QT

SY; or as SY reciprocally, and SP × QT
directly; but SP × QT is as the area described in the given time, that
is (by Prop. XIV), in the subduplicate ratio of the latus rectum.
Q.E.D.

Cor. 1. The principal latera recta are in a ratio compounded of the duplicate ratio of the perpendiculars and the duplicate ratio of the velocities.

Cor. 2. The velocities of bodies, in their greatest and least distances from the common focus, are in the ratio compounded of the ratio of the distances inversely, and the subduplicate ratio of the principal latera recta directly. For those perpendiculars are now the distances.

Cor. 3. And therefore the velocity in a conic section, at its greatest or least distance from the focus, is to the velocity in a circle, at the same distance from the centre, in the subduplicate ratio of the principal latus rectum to the double of that distance.

Cor. 4. The velocities of the bodies revolving in ellipses, at their mean distances from the common focus, are the same as those of bodies revolving in circles, at the same distances; that is (by Cor. 6, Prop. IV), reciprocally in the subduplicate ratio of the distances. For the perpendiculars are now the lesser semi-axes, and these are as mean proportionals between the distances and the latera recta. Let this ratio inversely be compounded with the subduplicate ratio of the latera recta directly, and we shall have the subduplicate ratio of the distance inversely.

Cor. 5. In the same figure, or even in different figures, whose principal latera recta are equal, the velocity of a body is reciprocally as the perpendicular let fall from the focus on the tangent.

Cor. 6. In a parabola, the velocity is reciprocally in the subduplicate ratio of the distance of the body from the focus of the figure; it is more variable in the ellipsis, and less in the hyperbola, than according to this ratio. For (by Cor. 2, Lem. XIV) the perpendicular let fall from the focus on the tangent of a parabola is in the subduplicate ratio of the distance. In the hyperbola the perpendicular is less variable; in the ellipsis more.

Cor. 7. In a parabola, the velocity of a body at any distance from the focus is to the velocity of a body revolving in a circle, at the same distance from the centre, in the subduplicate ratio of the number 2 to 1; in the ellipsis it is less, and in the hyperbola greater, than according to this ratio, (by Cor. 2 of this Prop.) the velocity at the vertex of a parabola is in this ratio, and (by Cor. 6 of this Prop. and Prop. IV) the same proportion holds in all distances. And hence, also, in a parabola, the velocity is everywhere equal to the velocity of a body revolving in a circle at half the distance; in the ellipsis it is less, and in the hyperbola greater.

Cor. 8. The velocity of a body revolving in any conic section is to the velocity of a body revolving in a circle, at the distance of half the principal latus rectum of the section, as that distance to the perpendicular let fall from the focus on the tangent of the section. This appears from Cor. 5.

Cor. 9. Wherefore since (by Cor. 6, Prop.
IV), the velocity of a body revolving in this circle is to the
velocity of another body revolving in any other circle reciprocally in
the subduplicate ratio of the distances; therefore, *ex aequo*,
the velocity of a body revolving in a conic section will be to the
velocity of a body revolving in a circle at the same distance as a
mean proportional between that common distance, and half the principal
latus rectum of the section, to the perpendicular let fall from the
common focus upon the tangent of the section.

*Supposing the centripetal force to be reciprocally proportional
to the squares of the distances of places from the centre, and
that the absolute quantity of that force is known; it is required
to determine the line which a body will describe that is let go
from a given place with a given velocity in, the direction of a given right line.*

Let the centripetal force tending to the point S be such as will make
the body *p* revolve in any given orbit *pq*; and
suppose the velocity of this body in the place *p* is known.
Then from the place P suppose the body P to be let with a given
velocity in the direction of the line PR; but by virtue of a
centripetal force to be immediately turned aside from that right line
into the conic section PQ. This, the right line PR will therefore
touch in P. Suppose likewise that the right line *pr* touches
the orbit *pq* in *p*, and if from S you suppose
perpendiculars let fall on those tangents, the principal latus rectum
of the conic section (by Cor. 1, Prop. XVI) will be to the principal
latus rectum of that orbit in a ratio compounded of the duplicate
ratio of the perpendiculars, and the duplicate ratio of the
velocities; and is therefore given. Let this latus rectum be L; the
focus S of the conic section is also given.
Let the angle RPH be the complement of the angle RPS to two right; and
the line PH, in which the other focus H is placed, is given by
position. Let fall SK perpendicular on PH, and erect the conjugate
semi-axis BC; this done, we shall have SP^{2}
− 2KPH + PH^{2} = SH^{2}
= 4CH^{2} = 4BH^{2}
− 4BC^{2} = (SP + PH^{2})
− L × (SP + PH) = SP^{2} + 2SPH +
PH^{2} − L × (SP + PH). Add on both sides 2KPH
− SP^{2} − PH^{2} + L × (SP + PH), and we
shall have L × (SP + PH) = 2SPH + 2KPH, or
SP + PH to PH, as 2SP + 2KP to L. Whence PH is given both in length
and position. That is, if the velocity of the body in P is such that
the latus rectum L is less than 2SP + 2KP, PH will lie on the same
side of the tangent PR with the line SP; and therefore the figure will
be an ellipsis, which from the given foci S, H, and the principal axis
SP + PH, is given also. But if the velocity of the body is so great,
that the latus rectum L becomes equal to 2SP + 2KP, the length PH will
be infinite; and therefore, the figure will be a parabola, which has
its axis SH parallel to the line PK, and is thence given. But if the
body goes from its place P with a yet greater velocity, the length PH
is to be taken on the other side the tangent; and so the tangent
passing between the foci, the figure will be an hyperbola having its
principal axis equal to the difference of the lines SP and PH, and
thence is given. For if the body, in these cases, revolves in a conic
section so found, it is demonstrated in Prop. XI, XII, and XIII, that
the centripetal force will be reciprocally as the square of the
distance of the body from the centre of force S; and therefore we have
rightly determined the line PQ, which a body let go from a given place
P with a given velocity, and in the direction of the right line PR
given by position, would describe with such a force.
Q.E.F.

Cor. 1. Hence in every conic section, from the principal vertex D, the latus rectum L, and the focus S given, the other focus H is given, by taking DH to DS as the latus rectum to the difference between the latus rectum and 4DS. For the proportion, SP + PH to PH as 2SP + 2KP to L, becomes, in the case of this Corollary, DS + DH to DH as 4DS to L, and by division DS to DH as 4DS − L to L.

Cor. 2. Whence if the velocity of a body in the principal vertex D is given, the orbit may be readily found; to wit, by taking its latus rectum to twice the distance DS, in the duplicate ratio of this given velocity to the velocity of a body revolving in a circle at the distance DS (by Cor. 3, Prop. XVI.), and then taking DH to DS as the latus rectum to the difference between the latus rectum and 4DS.

Cor. 3. Hence also if a body move in any conic section, and is forced out of its orbit by any impulse, you may discover the orbit in which it will afterwards pursue its course. For by compounding the proper motion of the body with that motion, which the impulse alone would generate, you will have the motion with which the body will go off from a given place of impulse in the direction of a right line given in position.

Cor. 4. And if that body is continually disturbed by the action of some foreign force, we may nearly know its course, by collecting the changes which that force introduces in some points, and estimating the continual changes it will undergo in the intermediate places, from the analogy that appears in the progress of the series.

If a body P, by means of a centripetal force tending to any given
point R, move in the perimeter of any given conic section whose centre
is C; and the law of the centripetal force is required: draw CG
parallel to the radius RP, and meeting the tangent PG of the orbit in
G; and the force required (by Cor. 1, and Schol. Prop. X., and Cor. 3,
Prop. VII.) will be as CG^{3}

RP^{2}.

*If from the two foci* S, H*, of any ellipsis or
hyberbola, we draw to any third point* V *the right lines*
SV, HV, *whereof one* HV *is equal to the principal axis
of the figure, that is, to the axis in which the foci are
situated, the other,* SV*, is bisected in* T *by
the perpendicular* TR *let fall upon it; that
perpendicular* TR *will somewhere touch the conic section:
and,* vice versa*, if it does touch it,* HV *will
be equal to the principal axis of the figure.*

For, let the perpendicular TR cut the right line HV, produced, if need be, in R; and join SR. Because TS, TV are equal, therefore the right lines SR, VR, as well as the angles TRS, TRV, will be also equal. Whence the point R will be in the conic section, and the perpendicular TR, will touch the same; and the contrary. Q.E.D.

*From a focus and the principal axes given, to describe elliptic
and hyperbolic trajectories, which shall pass through given
points, and touch right lines given by position.*

Let S be the common focus of the figures; AB the length of the
principal axis of any trajectory; P a point through which the
trajectory should pass; and TR a right line which it should touch.
About the centre P, with the interval AB − SP, if the orbit is an
ellipsis, or AB + SP, if the orbit is an hyperbola, describe the
circle HG. On the tangent TR let fall the perpendicular ST, and
produce the same to V, so that TV may be equal to ST; and about V as a
centre with the interval AB describe the circle FH. In this manner,
whether two points P, *p*, are given, or two tangents TR, *tr*,
or a point P and a tangent TR, we are to describe two circles. Let H
be their common intersection, and from the foci S, H, with the given
axis describe the trajectory: I say, the thing is done. For (be cause
PH + SP in the ellipsis, and PH − SP in the hyperbola, is equal to the
axis) the described trajectory will pass through the point P, and (by
the preceding Lemma) will touch the right line TR. And by the same
argument it will either pass through the two points P, *p*, or
touch the two right lines TR, *tr*. Q.E.F.

*About a given focus, to describe a parabolic trajectory, which
shall pass through given points, and touch right lines given by position.*

Let S be the focus, P a point, and TR a tangent of the trajectory to
be described. About P as a centre, with the interval PS, describe the
circle FG. From the focus let fall ST perpendicular on the tangent,
and produce the same to V, so as TV may be equal to ST. After the same
manner another circle *fg* is to be described, if another
point *p* is given; or another point *v* is to be
found, if another tangent *tr* is given; then draw the right
line IF, which shall touch the two circles FG, *fg*, if two
points P, *p* are given; or pass through the two points V, *v*,
if two tangents TR, *tr*, are given: or touch the circle FG,
and pass through the point V, if the point P and the tangent TR are
given. On FI let fall the perpendicular SI, and bisect the same in K;
and with the axis SK and principal vertex K describe a parabola: I say
the thing is done. For this parabola (because SK is equal to IK, and
SP to FP) will pass through the point P; and (by
Cor. 3, Lem. XIV) because ST is equal to TV, and STR a right angle, it
will touch the right line TR. Q.E.F.

*About a given focus to describe any trajectory given in specie
which shall pass through given points, and touch right lines given by position.*

Case 1. About the focus S it is required to
describe a trajectory ABC, passing through two points B, C. Because
the trajectory is given in specie, the ratio of the principal axis to
the distance of the foci will be given. In that ratio take KB to BS,
and LC to CS. About the centres B, C, with the intervals BK, CL,
describe two circles; and on the right line KL, that touches the same
in K and L, let fall the perpendicular SG; which cut in A and *a*,
so that GA may be to AS, and G*a* to *a*S, as KB to BS;
and with the axis A*a*, and vertices A, *a*, describe a
trajectory: I say the thing is done. For let H be the other focus of
the described figure, and seeing GA is to AS as G*a* to *a*S,
then by division we shall have G*a* − GA, or A*a* to *a*S
− AS, or SH in the same ratio, and therefore in the ratio which the
principal axis of the figure to be described has to the distance of
its foci; and therefore the described figure is of the same species
with the figure which was to be described. And since KB to BS, and LC
to CS, are in the same ratio, this figure will pass through the points
B, C, as is manifest from the conic sections.

Case 2. About the focus S it is required to
describe a trajectory which shall somewhere touch two right lines TR,
*tr*. From the focus on those tangents let fall the
perpendiculars ST, S*t*, which produce to V, *v*, so
that TV, *tv* may be equal to TS, *t*S. Bisect V*v*
in O, and erect the indefinite perpendicular OH, and cut the right
line VS infinitely produced in K and *k*, so that VK be to KS,
and V*k* to *k*S, as the principal axis of the
trajectory to be described is to the distance of its foci. On the
diameter K*k* describe a circle cutting OH in H; and with the
foci S, H, and principal axis equal to VH, describe a trajectory: I
say, the thing is done. For bisecting K*k* in X, and joining
HX, HS, HV, H*v*, because VK is to KS as V*k* to *k*S;
and by composition, as VK + V*k* to KS + *k*S; and by
division, as V*k* − VK to *k*S − KS, that is, as 2VX to
2KX, and 2KX to 2SX, and therefore as VX to HX and HX to SX, the
triangles VXH, HXS will be similar; therefore VH will be to SH as VX
to XH; and therefore as VK to KS. Wherefore VH, the principal axis of
the described trajectory, has the same ratio to SH, the distance of
the foci, as the principal axis of the
trajectory which was to be described has to the distance of its foci;
and is therefore of the same species. And seeing VH, *v*H are
equal to the principal axis, and VS, *v*S are perpendicularly
bisected by the right lines TR, *tr*, it is evident (by Lem.
XV) that those right lines touch the described trajectory.
Q.E.F.

Case. 3. About the focus S it is required to
describe a trajectory, which shall touch a right line TR in a given
Point R. On the right line TR let fall the perpendicular ST, which
produce to V, so that TV may be equal to ST; join VR, and cut the
right line VS indefinitely produced in K and *k*, so that VK
may be to SK, and V*k* to S*k*, as the principal axis of
the ellipsis to be described to the distance of its foci; and on the
diameter K*k* describing a circle, cut the right line VR
produced in H; then with the foci S, H, and principal axis equal to
VH, describe a trajectory: I say, the thing is done. For VH is to SH
as VK to SK, and therefore as the principal axis of the trajectory
which was to be described to the distance of its foci (as appears from
what we have demonstrated in Case 2); and therefore the described
trajectory is of the same species with that which was to be described;
but that the right line TR, by which the angle VRS is bisected,
touches the trajectory in the point R, is certain from the properties
of the conic sections. Q.E.F.

Case 4. About the focus S it is required to
describe a trajectory APB that shall touch a right line TR, and pass
through any given point P without the tangent, and shall be similar to
the figure *apb*, described with the principal axis *ab*,
and foci *s, h.* On the tangent TR let fall the perpendicular
ST, which produce to V, so that TV may be equal to ST; and making the
angles *hsq, shq*, equal to the angles VSP, SVP, about *q*
as a centre, and with an interval which shall be to *ab* as SP
to VS, describe a circle cutting the figure *apb* in *p*:
join *sp*, and draw SH such that it
may be to *sh* as SP is to *sp*, and may make the
angle PSH equal to the angle *psh*, and the angle VSH equal to
the angle *psq*. Then with the foci S, H, and principal axis
AB, equal to the distance VH, describe a conic section: I say, the
thing is done; for if *sv* is drawn so that it shall be to
*sp* as *sh* is to *sq*,
and shall make the angle *vsp* equal to the angle *hsq*,
and the angle *vsh* equal to the angle *psq*, the
triangles *svh, spq*, will be similar, and therefore *vh*
will be to *pq* as *sh* is to *sq*; that is
(because of the similar triangles VSP, *hsq*), as VS is to SP,
or as *ab* to *pq*. Wherefore *vh* and *ab*
are equal. But, because of the similar triangles VSH, *vsh*,
VH is to SH as *vh* to *sh*; that is, the axis of the
conic section now described is to the distance of its foci as the axis
*ab* to the distance of the foci *sh*; and therefore
the figure now described is similar to the figure *aph*. But,
because the triangle PSH is similar to the triangle *psh*,
this figure passes through the point P; and because VH is equal to its
axis, and VS is perpendicularly bisected by the right line TR, the
said figure touches the right line TR. Q.E.F

*From three given points to draw to a fourth point that is not
given three right lines whose differences shall be either given, or none at all.*

Case 1. Let the given points be A, B, C, and Z the fourth point which we are to find; because of the given difference of the lines AZ, BZ, the locus of the point Z will be an hyperbola whose foci are A and B, and whose principal axis is the given difference. Let that axis be MN. Taking PM to MA as MN is to AB, erect PR perpendicular to AB, and let fall ZR perpendicular to PR; then from the nature of the hyperbola, ZR will be to AZ as MN is to AB. And by the like argument, the locus of the point Z will be another hyperbola, whose foci are A, C, and whose principal axis is the difference between AZ and CZ; and QS a perpendicular on AC may be drawn, to which (QS) if from any point Z of this hyperbola a perpendicular ZS is let fall (this ZS), shall be to AZ as the difference between AZ and CZ is to AC. Wherefore the ratios of ZR and ZS to AZ are given, and consequently the ratio of ZR to ZS one to the other; and therefore if the right lines RP, SQ, meet in T, and TZ and TA are drawn, the figure TRZS will be given in specie, and the right line TZ, in which the point Z is somewhere placed, will be given in position. There will be given also the right line TA, and the angle ATZ; and because the ratios of AZ and TZ to ZS are given, their ratio to each other is given also; and thence will be given likewise the triangle ATZ, whose vertex is the point Z. Q.E.I.

Case 2. If two of the three lines, for example AZ and BZ, are equal, draw the right line TZ so as to bisect the right line AB; then find the triangle ATZ as above. Q.E.I.

Case 3. If all the three are equal, the point Z will be placed in the centre of a circle that passes through the points A, B, C. Q.E.I.

This problematic Lemma is likewise solved in Apollonius's Book of Tactions restored by Vieta.

*About a given focus to describe a trajectory that shall pass
through given points and touch right lines given by position.*

Let the focus S, the point P, and the tangent TR be given, and suppose that the other focus H is to be found. On the tangent let fall the perpendicular ST, which produce to Y, so that TY may be equal to ST, and YH will be equal to the principal axis. Join SP, HP, and SP will be the difference between HP and the principal axis. After this manner, if more tangents TR are given, or more points P, we shall always determine as many lines YH, or PH, drawn from the said points Y or P, to the focus H, which either shall be equal to the axes, or differ from the axes by given lengths SP; and therefore which shall either be equal among themselves, or shall have given differences; from whence (by the preceding Lemma), that other focus H is given. But having the foci and the length of the axis (which is either YH, or, if the trajectory be an ellipsis, PH + SP; or PH − SP, if it be an hyperbola), the trajectory is given. Q.E.I.

When the trajectory is an hyperbola, I do not comprehend its conjugate hyperbola under the name of this trajectory. For a body going on with a continued motion can never pass out of one hyperbola into its conjugate hyperbola.

The case when three points are given is more readily solved thus. Let
B, C, D, be the given points. Join BC, CD, and produce them to E, F,
so as EB may be to EC as SB to SC; and FC to FD as SC to SD. On EF
drawn and produced let fall the perpendiculars SG, BH, and in GS
produced indefinitely take GA to AS, and G*a* to *a*S,
as HB is to BS; then A will be the vertex, and A*a* the
principal axis of the trajectory; which, according as GA is greater
than, equal to, or less than AS. will be
either an ellipsis, a parabola, or an hyperbola; the point *a*
in the first case falling on the same side of the line GF as the point
A; in the second, going off to an infinite distance; in the third,
falling on the other side of the line GF. For if on GF the
perpendiculars CI, DK are let fall, IC will be to HB as EC to EB; that
is, as SC to SB; and by permutation, IC to SC as HB to SB, or as GA to
SA. And, by the like argument, we may prove that KD is to SD in the
same ratio. Wherefore the points B, C, D lie in a conic section
described about the focus S, in such manner that all the right lines
drawn from the focus S to the several points of the section, and the
perpendiculars let fall from the same points on the right line GF, are
in that given ratio.

That excellent geometer M. De la Hire has solved this Problem much after the same way, in his Conics, Prop. XXV., Lib. VIII.

*If from any point* P *of a given conic section, to the
four produced sides* AB, CD, AC, DB*, of any trapezium*
ABDC *inscribed in that section, as many right lines* PQ,
PR, PS, PT *are drawn in given angles, each line to each side;
the rectangle* PQ × PR *of those on the opposite sides*
AB, CD*, will be to the rectangle* PS × PT *of those on
the other two opposite sides* AC, BD*, in a given ratio.*

Case 1. Let us suppose, first, that the lines drawn to one pair of opposite sides are parallel to either of the other sides; as PQ and PR to the side AC, and PS and PT to the side AB. And farther, that one pair of the opposite sides, as AC and BD, are parallel betwixt themselves; then the right line which bisects those parallel sides will be one of the diameters of the conic section, and will likewise bisect RQ. Let O be the point in which RQ is bisected, and PO will be an ordinate to that diameter. Produce PO to K, so that OK may be equal to PO, and OK will be an ordinate on the other side of that diameter. Since, therefore, the points A, B, P and K are placed in the conic section, and PK cuts AB in a given angle, the rectangle PQK (by Prop. XVII., XIX., XXI. and XXIII., Book III., of Apollonius's Conics) will be to the rectangle AQB in a given ratio. But QK and PR are equal, as being the differences of the equal lines OK, OP, and OQ, OR; whence the rectangles PQK and PQ × PR are equal; and therefore the rectangle PQ × PR is to the rectangle A B, that is, to the rectangle PS × PT in a given ratio. Q.E.D

Case 2. Let us next suppose that the opposite
sides AC and BD of the trapezium are not parallel. Draw B*d*
parallel to AC, and meeting as well the right line ST in *t*,
as the conic section in *d*. Join C*d* cutting PQ in *r*,
and draw DM parallel to PQ, cutting C*d* in M, and AB in N.
Then (because of the similar triangles BT*t*, DBN), B*t*
or PQ is to T*t* as DN to NB. And so R*r* is to AQ or PS
as DM to AN. Wherefore, by multiplying the antecedents by the
antecedents, and the consequents by the consequents, as the rectangle
PQ × R*r* is to the rectangle PS × T*t*, so will the
rectangle NDM be to the rectangle ANB; and (by Case 1) so is the
rectangle PQ × P*r* to the rectangle PS × P*t*; and by
division, so is the rectangle PQ × PR to the rectangle PS × PT.
Q.E.D.

Case 3. Let us suppose, lastly, the four
lines PQ, PR, PS, PT, not to be parallel to the sides AC, AB, but any
way inclined to them. In their place draw P*q*, P*r*,
parallel to AC; and P*s*, P*t* parallel to AB; and
because the angles of the triangles PQ*q*, PR*r*, PS*s*,
PT*t* are given, the ratios of PQ to P*q*, PR to P*r*,
PS to P*s*, PT to P*t* will be also given; and therefore
the compounded ratios PQ × PR to P*q* × P*r*, and PS ×
PT to P*s* × P*t* are given. But from what we have
demonstrated before, the ratio of P*q* × P*r* to P*s*
× P*t* is given; and therefore also the ratio of PQ × PR to PS
× PT. Q.E.D.

*The same things supposed, if the rectangle* PQ × PR *of
the lines drawn to the two opposite sides of the trapezium is to
the rectangle* PS × PT *of those drawn to the other two
sides in a given ratio, the point* P*, from whence those
lines are drawn, will be placed in a conic section described about
the trapezium.*

Conceive a conic section to be described passing through the points *A,
B, C, D*, and any one of the infinite number of points P, as for
example *p*; I say, the point P will be always placed in this
section. If you deny the thing, join AP cutting this conic section
somewhere else, if possible, than in P, as in *b*. Therefore
if from those points *p* and *b*, in the given angles
to the sides of the trapezium, we draw the right lines *pq, pr,
ps, pt*, and *bk, bn, bf, bd*, we shall have, as *bk
× bn* to *bf × bd*, so (by Lem.
XVII) *pq × pr* to *ps × pt*; and so (by supposition)
PQ × PR to PS × PT. And because of the similar trapezia *bk*A*f*,
PQAS, as *bk* to *bf*, so PQ to PS. Wherefore by
dividing the terms of the preceding proportion by the correspondent
terms of this, we shall have *bn* to *bd* as PR to PT.
And therefore the equiangular trapezia D*nbd*, DRPT, are
similar, and consequently their diagonals D*b*, DP do coincide.
Wherefore *b* falls in the intersection of the right lines AP,
DP, and consequently coincides with the point P. And therefore the
point P, wherever it is taken, falls to be in the assigned conic
section. Q.E.D.

Cor. Hence if three right lines PQ, PR, PS, are drawn from a common point P, to as many other right lines given in position, AB, CD, AC, each to each, in as many angles respectively given, and the rectangle PQ × PR under any two of the lines drawn be to the square of the third PS in a given ratio; the point P, from which the right lines are drawn, will be placed in a conic section that touches the lines AB, CD in A and C; and the contrary. For the position of the three right lines AB, CD, AC remaining the same, let the line BD approach to and coincide with the line AC; then let the line PT come likewise to coincide with the line PS; and the rectangle PS × PT will become PS², and the right lines AB, CD, which before did cut the curve in the points A and B, C and D, can no longer cut, but only touch, the curve in those coinciding points.

In this Lemma, the name of conic section is to be understood in a
large sense, comprehending as well the rectilinear section through the
vertex of the cone, as the circular one parallel to the base. For if
the point *p* happens to be in a right line, by which the
points A and D, or C and B are joined, the conic section will be
changed into two right lines, one of which is that right line upon
which the point *p* falls, and the other is a right line that
joins the other two of the four points. If the two opposite angles of
the trapezium taken together are equal to two right angles, and if the
four lines PQ, PR, PS, PT, are drawn to the sides thereof at right
angles, or any other equal angles, and the rectangle PQ × PR under two
of the lines drawn PQ and PR, is equal to the rectangle PS × PT under
the other two PS and PT, the conic section will become a circle. And
the same thing will happen if the four lines are drawn in any angles,
and the rectangle PQ × PR, under one pair of the lines drawn, is to
the rectangle PS × PT under the other pair as the rectangle under the
sines of the angles S, T, in which the two last lines PS, PT are drawn
to the rectangle under the sines of the angles Q, R, in which the
first two PQ, PR are drawn. In all other
cases the locus of the point P will be one of the three figures which
pass commonly by the name of the conic sections. But in room of the
trapezium ABCD, we may substitute a quadrilateral figure whose two
opposite sides cross one another like diagonals. And one or two of the
four points A, B, C, D may be supposed to be removed to an infinite
distance, by which means the sides of the figure which converge to
those points, will become parallel; and in this case the conic section
will pass through the other points, and will go the same way as the
parallels *in infinitum*.

*To find a point* P *from which if four right lines*
PQ, PR, PS, PT *are drawn to as many other right lines* AB,
CD, AC, BD, *given by position, each to each, at given angles,
the rectangle* PQ × PR*, under any two of the lines drawn,
shall be to the rectangle* PS × PT*, under the other two,
in a given ratio.*

Suppose the lines AB, CD, to which the two right lines PQ, PR, containing one of the rectangles, are drawn to meet two other lines, given by position, in the points A, B, C, D. From one of those, as A, draw any right line AH, in which you would find the point P. Let this cut the opposite lines BD, CD, in H and I; and, because all the angles of the figure are given, the ratio of PQ to PA, and PA to PS, and therefore of PQ to PS, will be also given. Subducting this ratio from the given ratio of PQ × PR to PS × PT, the ratio of PR to PT will be given; and adding the given ratios of PI to PR, and PT to PH, the ratio of PI to PH, and therefore the point P will be given. Q.E.I.

Cor. 1. Hence also a tangent may be drawn to any point D of the locus of all the points P. For the chord PD, where the points P and D meet, that is, where AH is drawn through the point D, becomes a tangent. In which case the ultimate ratio of the evanescent lines IP and PH will be found as above. Therefore draw CF parallel to AD, meeting BD in F, and cut it in E in the same ultimate ratio, then DE will be the tangent; because CF and the evanescent IH are parallel, and similarly cut in E and P.

Cor. 2. Hence also the locus of all the
points P may be determined. Through any of the points A, B, C, D, as
A, draw AE touching the locus, and through any other point B parallel
to the tangent, draw BF meeting the locus in F; and find the point F
by this Lemma. Bisect BF in G, and, drawing the indefinite line AG,
this will be the position of the diameter to which BG and FG are
ordinates. Let this AG meet the locus
in H, and AH will be its diameter or latus transversum, to which the
latus rectum will be as BG² to AG × GH. If AG nowhere meets the locus,
the line AH being infinite, the locus will be a parabola; and its
latus rectum corresponding to the diameter AG will be BG^{2}

AG. But if it does meet it anywhere,
the locus will be an hyperbola, when the points A and H are placed on
the same side the point G; and an ellipsis, if the point G falls
between the points A and H; unless, perhaps, the angle AGB is a right
angle, and at the same time BG² equal to the rectangle AGH, in which
case the locus will be a circle.

And so we have given in this Corollary a solution of that famous Problem of the ancients concerning four lines, begun by Euclid, and carried on by Apollonius; and this not an analytical calculus, but a geometrical composition, such as the ancients required.

*If the two opposite angular points* A *and* P *of
any parallelogram* ASPQ *touch any conic section in the
points* A *and* P*; and the sides* AQ, AS *of
one of those angles, indefinitely produced, meet the same conic
section in* B *and* C*; and from the points of
concourse* B *and* C *to any fifth point* D *of
the conic section, two right lines* BD, CD *are drawn
meeting the two other sides* PS, PQ *of the parallelogram,
indefinitely produced in* T *and* R*; the parts*
PR *and* PT*, cut off from the sides, will always be one
to the other in a given ratio. And* vice versa*, if those
parts cut off are one to the other in a given ratio, the locus of
the point* D *will be a conic section passing through the
four points* A, B, C, P*.*

Case 1. Join BP, CP, and from the point D draw the two right lines DG, DE, of which the first DG shall be parallel to AB, and meet PB, PQ, CA in H, I, G; and the other DE shall be parallel to AC, and meet PC, PS, AB, in F, K, E; and (by Lem. XVII) the rectangle DE × DF will be to the rectangle DG × DH in a given ratio. But PQ is to DE (or IQ) as PB to HB, and consequently as PT to DH; and by permutation PQ is to PT as DE to DH. Likewise PR is to DF as RC to DC, and therefore as (IG or) PS to DG; and by permutation PR is to PS as DF to DG; and, by compounding those ratios, the rectangle PQ × PR will be to the rectangle PS × PT as the rectangle DE × DF is to the rectangle DG × DH, and consequently in a given ratio. But PQ and PS are given, and therefore the ratio of PR to PT is given. Q.E.D.

Case 2. But if PR and PT are supposed to be in a given ratio one to the other, then by going back again, by a like reasoning, it will follow that the rectangle DE × DF is to the rectangle DG × DH in a given ratio; and so the point D (by Lem. XVIII) will lie in a conic section passing through the points A, B, C, P, as its locus. Q.E.D.

Cor. 1. Hence if we draw BC cutting PQ in *r*
and in PT take P*t* to P*r* in the same ratio which PT
has to PR; then B*t* will touch the conic section in the point
B. For suppose the point D to coalesce with the point B, so that the
chord BD vanishing, BT shall become a tangent, and CD and BT will
coincide with CB and B*t*.

Cor. 2. And, vice versa, if B*t* is a
tangent, and the lines BD, CD meet in any point D of a conic section,
PR will be to PT as P*r* to P*t*. And, on the contrary,
if PR is to PT as P*r* to P*t*, then BD and CD will meet
in some point D of a conic section.

Cor. 3. One conic section cannot cut another
conic section in more than four points. For, if it is possible, let
two conic sections pass through the five points A, B, C, P, O; and let
the right line BD cut them in the points D, *d*, and the right
line C*d* cut the right line PQ in *q*. Therefore PR is
to PT as P*q* to PT: whence PR and P*q* are equal one to
the other, against the supposition.

*If two moveable and indefinite right lines* BM, CM *drawn
through given points* B, C*, as poles, do by their point of
concourse* M *describe a third right line* MN *given
by position; and other two indefinite right lines* BD, CD *are
drawn, making with the former two at those given points* B, C*,
given angles,* MBD, MCD*: I say, that those two right lines*
BD, CD *will by their point of concourse* D *describe a
conic section passing through the points* B, C*. And,*
vice versa*, if the right lines* BD, CD *do by their
point of concourse* D *describe a conic section passing
through the given points* B, C, A*, and the angle* DBM
*Is always equal to the given angle* ABC*, as well as the
angle* DCM *always equal to the given angle* ACB*,
the point* M *will lie in a right line given by position,as its locus.*

For in the right line *MN* let a point N be given, and when
the moveable point M falls on the immoveable point N. let the moveable
point D fall on an immovable point P. Join CN, BN, CP, BP, and from
the point P draw the right lines PT, PR meeting BD, CD in T and R, and
making the angle BPT equal to the given angle BNM, and the angle CPR
equal to the given angle CNM. Wherefore since (by
supposition) the angles MBD, NBP are equal, as also the angles MCD,
NCP, take away the angles NBD and NCD that are common, and there will
remain the angles NBM and PBT, NCM and PCR equal; and therefore the
triangles NBM, PBT are similar, as also the triangles NCM, PCR.
Wherefore PT is to NM as PB to NB; and PR to NM as PC to NC. But the
points, B, C, N, P are immovable: wherefore PT and PR have a given
ratio to NM, and consequently a given ratio between themselves; and
therefore, (by Lemma XX) the point D wherein the moveable right lines
BT and CR perpetually concur, will be placed in a conic section
passing through the points B, C, P. Q.E.D.

And, *vice versa*, if the moveable point D lies in a conic
section passing through the given points B, C, A; and the angle DBM is
always equal to the given angle ABC, and the angle DCM always equal to
the given angle ACB, and when the point D falls successively on any
two immovable points *p*, P, of the conic section, the
moveable point M falls successively on two immovable points *n*,
N. Through these points *n*, N, draw the right line *n*N:
this line *n*N will be the perpetual locus of that moveable
point M. For, if possible, let the point M be placed in any curve
line. Therefore the point D will be placed in a conic section passing
through the five points B, C, A, *p*, P, when the point M is
perpetually placed in a curve line. But from what was demonstrated
before, the point D will be also placed in a conic section passing
through the same five points B, C, A, *p*, when the point M is
perpetually placed in a right line. Wherefore the two conic sections
will both pass through the same five points, against Corol. 3, Lem.
XX. It is therefore absurd to suppose that the point M is placed in a
curve line. Q.E.D.

*To describe a trajectory that shall pass through five given points.*

Let the five given points be A, B, C, P, D. From any one of them, as
A, to any other two as B, C, which may be called the poles, draw the
right lines AB, AC, and parallel to those the lines TPS, PRQ, through
the fourth point P. Then from the two poles B, C, draw through the
fifth point D two indefinite lines BDT, CRD, meeting with the last
drawn lines TPS, PRQ (the former with the
former, and the latter with the latter) in T and R. Then drawing the
right line *tr* parallel to TR, cutting off from the right
lines PT, PR, any segments P*t*, P*r*, proportional to
PT, PR; and if through their extremities, *t, r*, and the
poles B, C, the right lines B*t*, C*r* are drawn,
meeting in *d*, that point *d* will be placed in the
trajectory required. For (by Lem. XX) that point *d* is placed
in a conic section passing through the four points A, B, C, P; and the
lines R*r*, T*t* vanishing, the point *d* comes
to coincide with the point D. Wherefore the conic section passes
through the five points A, B, C, P, D. Q.E.D.

Of the given points join any three, as A, B, C; and about two of them
B, C, as poles, making the angles ABC, ACB of a given magnitude to
revolve, apply the legs BA, CA, first to the point D, then to the
point P, and mark the points M, N, in which the other legs BL, CL
intersect each other in both cases. Draw the indefinite right line MN,
and let those moveable angles revolve about their poles B, C, in such
manner that the intersection, which is now supposed to be *d*,
of the legs BL, CL, or BM, CM, may always fall in that indefinite
right line MN; and the intersection, which is now supposed to be *m*,
of the legs BA, CA, or BD, CD, will describe the trajectory required,
PAD*d*B. For (by Lem. XXI) the point *d* will be placed
in a conic section passing through the points B, C; and when the point
*m* comes to coincide with the points L, M, N, the point *d*
will (by construction) come to coincide with the points A, D, P.
Wherefore a conic section will be described that shall pass through
the five points A, B. C, P, D. Q.E.F.

Cor. 1. Hence a right line may be readily
drawn which shall be a tangent to the trajectory in any given point B.
Let the point *d* come to coincide with the point B, and the
right line B*d* will become the tangent required.

Cor. 2. Hence also may be found the centres, diameters, and latera recta of the trajectories, as in Cor. 2, Lem. XIX.

The former of these constructions will become something more simple
by joining BP, and in that line, produced, if need be, taking B*p*
to BP as PR is to PT; and through *p* draw the indefinite
right line *pe* parallel to SPT, and in that line *pe*
taking always *pe* equal to P*r*, and draw the right
lines B*e*, C*r* to meet in *d*.
For since P*r* to P*t*, PR to PT, *p*B to PB, *pe*
to Pt, are all in the same ratio, *pe* and P*r* will be
always equal. After this manner the points of the trajectory are most
readily found, unless you would rather describe the curve
mechanically, as in the second construction.

*To describe a trajectory that shall pass through four given
points, and touch a right line given by position.*

Case 1. Suppose that HB is the given tangent,
B the point of contact, and C, D, P, the three other given points.
Join BC, and draw PS parallel to BH, and PQ parallel to BC; complete
the parallelogram BSPQ. Draw BD cutting SP in T, and CD cutting PQ in
R. Lastly, draw any line *tr* parallel to TR, cutting off from
PQ, PS, the segments P*r*, P*t* proportional to PR, PT
respectively; and draw C*r*, B*t* their point of
concourse *d* will (by Lem. XX) always fall on the trajectory
to be described.

Let the angle CBH of a given magnitude revolve about the pole B; as also the rectilinear radius BC, both ways produced, about the pole C. Mark the points M, N, on which the leg BC of the angle cuts that radius when BH, the other leg thereof, meets the same radius in the points P and D. Then drawing the indefinite line MN, let that radius CP or CD and the leg BC of the angle perpetually meet in this line; and the point of concourse of the other leg BH with the radius will delineate the trajectory required.

For if in the constructions of the preceding Problem the point A comes to a coincidence with the point B, the lines CA and CB will coincide, and the line AB, in its last situation, will become the tangent BH; and therefore the constructions there set down will become the same with the constructions here described. Wherefore the concourse of the leg BH with the radius will describe a conic section passing through the points C, D, P, and touching the line BH in the point B. Q.E.F.

Case 2. Suppose the four points B, C, D, P, given, being situated with out the tangent HI. Join each two by the lines BD, CP meeting in G, and cutting the tangent in H and I. Cut the tangent in A in such manner that HA may be to IA as the rectangle under a mean proportional between CG and GP, and a mean proportional between BH and HD is to a rectangle under a mean proportional between GD and GB, and a mean proportional between PI and IC, and A will be the point of contact. For if HX, a parallel to the right line PI, cuts the trajectory in any points X and Y, the point A (by the properties of the conic sections) will come to be so placed, that HA² will become to AI² in a ratio that is compounded out of the ratio of the rectangle XHY to the rectangle BHD, or of the rectangle CGP to the rectangle DGB; and the ratio of the rectangle BHD to the rectangle PIC. But after the point of contact A is found, the trajectory will be described as in the first Case. Q.E.F. But the point A may be taken either between or without the points H and I, upon which account a twofold trajectory may be described.

*To describe a trajectory that shall pass through three given
points, and touch two right lines given by position.*

Suppose HI, KL to be the given tangents and B, C, D, the given points. Through any two of those points, as B, D, draw the indefinite right line BD meeting the tangents in the points H, K. Then likewise through any other two of these points, as C, D, draw the indefinite right line CD meeting the tangents in the points I, L. Cut the lines drawn in R and S, so that HR may be to KR as the mean proportional between BH and HD is to the mean proportional between BK and KD; and IS to LS as the mean proportional between CI and ID is to the mean proportional between CL and LD. But you may cut, at pleasure, either within or between the points K and H, I and L, or without them; then draw RS cutting the tangents in A and P, and A and P will be the points of contact. For if A and P are supposed to be the points of contact, situated anywhere else in the tangents, and through any of the points H, I, K, L, as I, situated in either tangent HI, a right line IY is drawn parallel to the other tangent KL, and meeting the curve in X and Y, and in that right line there be taken IZ equal to a mean proportional between IX and IY, the rectangle XIY or IZ², will (by the properties of the conic sections) be to LP² as the rectangle CID is to the rectangle CLD, that is (by the construction), as SI is to SL², and therefore IZ is to LP as SI to SL. Wherefore the points S, P, Z, are in one right line. Moreover, since the tangents meet in G, the rectangle XIY or IZ² will (by the properties of the conic sections) be to IA² as GP² is to GA², and consequently IZ will be to IA as GP to GA. Wherefore the points P, Z, A, lie in one right line, and therefore the points S, P, and A are in one right line. And the same argument will prove that the points R, P, and A are in one right line. Wherefore the points of contact A and P lie in the right line RS. But after these points are found, the trajectory may be described, as in the first Case of the preceding Problem. Q.E.F.

In this Proposition, and Case 2 of the foregoing, the constructions are the same, whether the right line XY cut the trajectory in X and Y, or not; neither do they depend upon that section. But the constructions being demonstrated where that right line does cut the trajectory, the constructions where it does not are also known; and therefore, for brevity's sake, I omit any farther demonstration of them.

*To transform figures into other figures of the same kind.*

Suppose that any figure HGI is to be transformed. Draw, at pleasure,
two parallel lines AO, BL, cutting any third line AB, given by
position, in A and B, and from any point G of the figure, draw out any
right line GD, parallel to OA, till it meet the right line AB. Then
from any given point O in the line OA, draw to the point D the right
line OD, meeting BL in *d*; and from the point of concourse
raise the right line *dg* containing any given angle with the
right line BL, and having such ratio to O*d* as DG has to OD;
and *g* will be the point in the new figure *hgi*,
corresponding to the point G. And in like manner the several points of
the first figure will give as many correspondent points of the new
figure. If we therefore conceive the point G to be carried along by a
continual motion through all the points of the first figure, the point
*g* will be likewise carried along by a continual motion
through all the points of the new figure, and describe the same. For
distraction's sake, let us call DG the first ordinate, *dg*
the new ordinate, AD the first abscissa, *ad* the new
abscissa; O the pole, OD the abscinding radius, OA the first ordinate
radius, and O*a* (by which the parallelogram OAB*a* is
completed) the new ordinate radius.

I say, then, that if the point G is placed in a right line given by
position, the point *g* will be also placed in a right line
given by position. If the point G is placed in a conic section, the
point *g* will be likewise placed in
a conic section. And here I understand the circle as one of the conic
sections. But farther, if the point G is placed in a line of the third
analytical order, the point *g* will also be placed in a line
of the third order, and so on in curve lines of higher orders. The two
lines in which the points G, *g*, are placed, will be always
of the same analytical order. For as *ad* is to OA, so are O*d*
to OD, *dg* to DG, and AB to AD; and therefore AD is equal to
OA × AB

ad, and DG equal to OA × dg

ad. Now if the point G is placed in a
right line, and therefore, in any equation by which the relation
between the abscissa AD and the ordinate GD is expressed, those
indetermined lines AD and DG rise no higher than to one dimension, by
writing this equation OA × AB

ad in place of AD, and OA × dg

ad in place of DG, a new equation will
be produced, in which the new abscissa *ad* and new ordinate *dg*
rise only to one dimension; and which therefore must denote a right
line. But if AD and DG (or either of them) had risen to two dimensions
in the first equation, *ad* and *dg* would likewise
have risen to two dimensions in the second equation. And so on in
three or more dimensions. The indetermined lines, *ad, dg* in
the second equation, and AD, DG, in the first, will always rise to the
same number of dimensions; and therefore the lines in which the points
G, *g*, are placed are of the same analytical order.

I say farther, that if any right line touches the curve line in the
first figure, the same right line transferred the same way with the
curve into the new figure will touch that curve line in the new
figure, and *vice versa*. For if any two points of the curve
in the first figure are supposed to approach one the other till they
come to coincide, the same points transferred will approach one the
other till they come to coincide in the new figure; and therefore the
right lines with which those points are joined will be come together
tangents of the curves in both figures. I might have given
demonstrations of these assertions in a more geometrical form; but I
study to be brief.

Wherefore if one rectilinear figure is to be transformed into
another, we need only transfer the intersections of the right lines of
which the first figure consists, and through the transferred
intersections to draw right lines in the new figure. But if a
curvilinear figure is to be transformed, we must transfer the points,
the tangents, and other right lines, by means of which the curve line
is defined. This Lemma is of use in the solution of the more difficult
Problems; for thereby we may transform the proposed figures, if they
are intricate, into others that are more simple. Thus any right lines
converging to a point are transformed into parallels, by taking for
the first ordinate radius any right line that passes through the point
of concourse of the converging lines, and that because their point of
concourse is by this means made to go off *in
infinitum*; and parallel lines are such as tend to a point
infinitely remote. And after the problem is solved in the new figure,
if by the inverse operations we transform the new into the first
figure, we shall have the solution required.

This Lemma is also of use in the solution of solid problems. For as often as two conic sections occur, by the intersection of which a problem may be solved, any one of them may be transformed, if it is an hyperbola or a parabola, into an ellipsis, and then this ellipsis may be easily changed into a circle. So also a right line and a conic section, in the construction of plane problems, may be transformed into a right line and a circle

*To describe a trajectory that shall pass through two given
points, and touch three right lines given by position.*

Through the concourse of any two of the tangents one with the other,
and the concourse of the third tangent with the right line which
passes through the two given points, draw an indefinite right line;
and, taking this line for the first ordinate radius, transform the
figure by the preceding Lemma into a new figure. In this figure those
two tangents will become parallel to each other, and the third tangent
will be parallel to the right line that passes through the two given
points.
Suppose *hi, kl* to be
those two parallel tangents, *ik* the third tangent, and *hl*
a right line parallel thereto, passing through those points *a, b*,
through which the conic section ought to pass in this new figure; and
completing the parallelogram *hikl*, let the right lines *hi,
ik, kl* be so cut in *c, d, e*, that *hc* may be
to the square root of the rectangle *ahb, ic,* to *id,*
and *ke* to *kd*, as the sum of the right lines *hi*
and *kl* is to the sum of the three lines, the first whereof
is the right line *ik*, and the other two are the square roots
of the rectangles *ahb* and *alb*; and *c, d, e*,
will be the points of contact. For by the properties of the conic
sections, *hc*² to the rectangle *ahb*, and *ic²*
to *id²*, and *ke²* to *kd²*, and *el²*
to the rectangle *alb*, are all in the same ratio; and
therefore *hc* to the square root of *ahb, ic* to *id,
ke* to *kd*, and *el* to the square root of *alb*,
are in the subduplicate of that ratio; and by composition, in the
given ratio of the sum of all the antecedents *hi + kl*, to
the sum of all the consequents √(ahb)+ik+√(alb).
Wherefore from that given ratio we have the points of contact *c,
d, e*, in the new figure. By the inverted operations of the last
Lemma, let those points be transferred into the first figure, and the
trajectory will be there described by Prob. XIV. Q.E.F.
But according as the points *a, b*, fall between
the points *h, l,* or without them, the points *c, d, e,*
must be taken either between the points, *h,
i, k, l,* or without them. If one of the points *a, b*,
falls between the points *h, i*, and the other without the
points *h, l*, the Problem is impossible.

*To describe a trajectory that shall pass through a given point,
and touch four right lines given by position.*

From the common intersections, of any two of the tangents to the
common intersection of the other two, draw an indefinite right line;
and taking this line for the first ordinate radius; transform the
figure (by Lem. XXII) into a new figure, and the two pairs of
tangents, each of which before concurred in the first ordinate radius,
will now become parallel. Let *hi* and *kl*, *ik*
and *hl*, be those pairs of parallels completing the
parallelogram *hikl*. And let *p* be the point in this
new figure corresponding to the given point in the first figure.
Through O the centre of the figure draw *pq*: and O*q*
being equal to O*p*, *q* will be the other point
through which the conic section must pass in this new figure. Let this
point be transferred, by the inverse operation of Lem. XXII into the
first figure, and there we shall have the two points through which the
trajectory is to be described. But through those points that
trajectory may be described by Prop. XVII.

*If two right lines, as* AC, BD *given by position, and
terminating in given points* A, B*, are in a given ratio
one to the other, and the right line* CD*, by which the
indetermined points* C, D *are joined is cut in* K *in
a given ratio; I say, that the point* K *will be placed in
a right line given by position.*

For let the right lines AC, BD meet in E, and in BE take BG to AE as BD is to AC, and let FD be always equal to the given line EG; and, by construction, EC will be to GD, that is, to EF, as AC to BD, and therefore in a given ratio; and therefore the triangle EFC will be given in kind. Let CF be cut in L so as CL may be to CF in the ratio of CK to CD; and because that is a given ratio, the triangle EFL will be given in kind, and therefore the point L will be placed in the right line EL given by position. Join LK, and the triangles CLK, CFD will be similar; and because FD is a given line, and LK is to FD in a given ratio, LK will be also given. To this let EH be taken equal, and ELKH will be always a parallelogram. And therefore the point K is always placed in the side HK (given by position) of that parallelogram. Q.E.D.

Cor. Because the figure EFLC is given in kind, the three right lines EF, EL, and EC, that is, GD, HK, and EC, will have given ratios to each other.

*If three right lines, two whereof are parallel, and given by
position, touch any conic section; I say, that the semi-diameter
of the section which is parallel to those two is a mean
proportional between the segments of those two that are
intercepted between the points of contact and the third tangent.*

Let AF, GB be the two parallels touching the conic section ADB in A and B; EF the third right line touching the conic section in I, and meeting the two former tangents in F and G, and let CD be the semi-diameter of the figure parallel to those tangents; I say, that AF, CD, BG are continually proportional.

For if the conjugate diameters AB, DM meet the tangent FG in E and H,
and cut one the other in C, and the parallelogram IKCL be completed;
from the nature of the conic sections, EC will be to CA as CA to CL;
and so by division, EC − CA to CA − CL, or EA to AL; and by
composition, EA to EA + AL or EL, as EC to EC + CA or EB; and
therefore (because of the similitude of the triangles EAF, ELI, ECH,
EBG) AF is to LI as CH to BG. Likewise, from the nature of the conic
sections, LI (or CK) is to CD as CD to CH; and therefore (*ex aequo
perturbatè*) AF is to CD as CD to BG. Q.E.D.

Cor. 1. Hence if two tangents FG, PQ, meet
two parallel tangents AF, BG in F and G, P and Q, and cut one the
other in O; AF (*ex aequo perturbatè*) will be to BQ as AP to
BG, and by division, as FP to GQ, and therefore as FO to OG.

Cor. 2. Whence also the two right lines PG, FQ drawn through the points P and G, F and Q, will meet in the right line ACB passing through the centre of the figure and the points of contact A, B.

*If four sides of a parallelogram indefinitely produced touch
any conic section, and are cut by a fifth tangent; I say, that,
taking those segments of any two conterminous sides that terminate
in opposite angles of the parallelogram, either segment is to the
side from which it is cut off as that part of the other
conterminous side which is intercepted between the point of
contact and the third side is to the other segment.*

Let the four sides ML, IK, KL, MI, of the parallelogram MLIK touch the F conic section in A, B, C, D; and let the fifth tangent FQ cut those sides in F, Q, H, and E; and taking the segments ME, KQ of the sides MI, KI, or the segments KH, MF of the sides KL, ML; I say, that ME is to MI as BK to KQ; and KH to KL as AM to MF. For, by Cor. 1 of the preceding Lemma, ME is to EI as (AM or) BK to BQ; and, by composition, ME is to MI as BK to KQ. Q.E.D. Also KH is to HL as (BK or) AM to AF; and by division, KH to KL as AM to MF. Q.E.D.

Cor. 1. Hence if a parallelogram IKLM described about a given conic section is given, the rectangle KQ × ME, as also the rectangle KH × MF equal thereto, will be given. For, by reason of the similar triangles KQH, MFE, those rectangles are equal.

Cor. 2. And if a sixth tangent *eq*
is drawn meeting the tangents KI, MI in *q* and *e*,
the rectangle KQ × ME will be equal to the rectangle K*q* × M*e*,
and KQ will be to M*e* as K*q* to ME, and by division as
Q*q* to E*e*.

Cor. 3. Hence, also, if E*q*, *e*Q,
are joined and bisected, and a right line is drawn through the points
of bisection, this right line will pass through the centre of the
conic section. For since Q*q* is to E*e* as KQ to M*e*,
the same right line will pass through the middle of all the lines E*q*,
*e*Q, MK (by Lem. XXIII), and the middle point of the right
line MK is the centre of the section.

*To describe a trajectory that may touch five right lines given by position.*

Supposing ABG, BCF, GCD, FDE, EA to be the tangents given by position. Bisect in M and N, AF, BE, the diagonals of the quadrilateral figure ABFE contained under any four of them; and (by Cor. 3, Lem. XXV) the right line MN drawn through the points of bisection will pass through the centre of the trajectory. Again, bisect in P and Q, the diagonals (if I may so call them) BD, GF of the quadrilateral figure BGDF contained under any other four tangents, and the right line PQ, drawn through the points of bisection will pass through the centre of the trajectory; and therefore the centre will be given in the con course of the bisecting lines. Suppose it to be O. Parallel to any tangent BC draw KL at such distance that the centre O may be placed in the middle between the parallels; this KL will touch the trajectory to be described. Let this cut any other two tangents GCD, FDE, in L and K. Through the points C and K, F and L, where the tangents not parallel, GL, FK meet the parallel tangents OF, KL, draw OK, FL meeting in R; and the right line OR drawn and produced, will cut the parallel tangents CF, KL, in the points of contact. This appears from Cor. 2, Lem. XXIV. And by the same method the other points of contact may be found, and then the trajectory may be described by Prob. XIV. Q.E.F.

Under the preceding Propositions are comprehended those Problems
wherein either the centres or asymptotes of the trajectories are
given. For when points and tangents and the centre are given, as many
other points and as many other tangents are given at an equal distance
on the other side of the centre. And an asymptote is to be considered
as a tangent, and its infinitely remote extremity (if we may say so)
is a point of contact. Conceive the point of contact of any tangent
removed *in infinitum*, and the tangent will degenerate into
an asymptote, and the constructions of the preceding Problems will be
changed into the constructions of those Problems wherein the asymptote
is given.

After the trajectory is described, we may find its axes and foci in
this manner. In the construction and figure of Lem. XXI, let those
legs BP, CP, of the moveable angles PBN, PCN, by the concourse of
which the trajectory was described, be made parallel one to the other;
and retaining that position, let them revolve about their poles B, C,
in that figure. In the mean while let the other legs CN, BN, of those
angles, by their concourse K or *k*, describe the circle BKGC.
Let O be the centre of this circle; and from this centre upon the
ruler MN, wherein those legs CN, BN did concur while the trajectory
was described, let fall the perpendicular OH meeting the circle in K
and L. And when those other legs CK, BK meet in the point K that is
nearest to the ruler, the first legs CP, BP will be parallel to the
greater axis, and perpendicular on the lesser; and the contrary
will happen if those legs meet in the remotest
point L. Whence if the centre of the trajectory is given; the axes
will be given; and those being given, the foci will be readily found.

But the squares of the axes are one to the other as KH to LH, and thence it is easy to describe a trajectory given in kind through four given points. For if two of the given points are made the poles C, B, the third will give the moveable angles PCK, PBK; but those being given, the circle BGKC may be described. Then, because the trajectory is given in kind, the ratio of OH to OK, and therefore OH itself, will be given. About the centre O, with the interval OH, describe another circle, and the right line that touches this circle, and passes through the concourse of the legs CK, BK, when the first legs CK, BP meet in the fourth given point, will be the ruler MN, by means of which the trajectory may be described. Whence also on the other hand a trapezium given in kind (excepting a few cases that are impossible) may be inscribed in a given conic section.

There are also other Lemmas, by the help of which trajectories given in kind may be described through given points, and touching given lines. Of such a sort is this, that if a right line is drawn through any point given by position, that may cut a given conic section in two points, and the distance of the intersections is bisected, the point of bisection will touch another conic section of the same kind with the former, and having its axes parallel to the axes of the former. But I hasten to things of greater use.

*To place the three angles of a triangle, given both in kind and
magnitude, in respect of as many rigid lines given by position,
provided they are not all parallel among themselves, in such
manner that the several angles may touch the several lines.*

Three indefinite right lines AB, AC, BC, are given by position, and
it is required so to place the triangle DEF that its angle D may touch
the line AB, its angle E the line AC, and its angle F the line BC.
Upon DE, DF, and EF, describe three segments of circles DRE, DGF, EMF,
capable of angles equal to the angles BAC, ABC, ACB respectively. But
those segments are to be described towards such sides of the lines DE,
DF, EF, that the letters DRED may turn round
about in the same order with the letters BACB; the letters DGFD in the
same order with the letters ABCA; and the letters EMFE in the same
order with the letters ACBA; then; completing those segments into
entire circles let the two former circles cut one the other in G, and
suppose P and Q, to be their centres. Then joining GP, PQ, take G*a*
to AB as GP is to PQ; and about the centre G, with the interval G*a*,
describe a circle that may cut the first circle DGE in *a*.
Join *a*D cutting the second circle DFG in *b*, as
well as *a*E cutting the third circle EMF in *c*.
Complete the figure ABC*def* similar and equal to the figure *abc*DEF:
I say, the thing is done.

For drawing F*c* meeting *a*D in *n*, and
joining *a*G, *b*G, QG, QD, PD, by construction the
angle E*a*D is equal to the angle CAB, and the angle *ac*F
equal to the angle ACB; and therefore the triangle *anc*
equiangular to the triangle ABC. Wherefore the angle *anc* or
F*n*D is equal to the angle ABC, and consequently to the angle
F*b*D; and therefore the point *n* falls on the point *b*.
Moreover the angle GPQ, which is half the angle GPD at the centre, is
equal to the angle G*a*D at the circumference; and the angle
GQP, which is half the angle GQD at the centre, is equal to the
complement to two right angles of the angle G*b*D at the
circumference, and therefore equal to the angle G*ba*. Upon
which account the triangles GPQ, G*ab*, are similar, and G*a*
is to *ab* as GP to PQ; that is (by construction), as G*a*
to AB. Wherefore *ab* and AB are equal; and consequently the
triangles *abc*, ABC, which we have now proved to be similar,
are also equal. And therefore since the angles D, E, F, of the
triangle DEF do respectively touch the sides *ab, ac, bc* of
the triangle *abc*, the figure ABC*def* may be
completed similar and equal to the figure *abc*DEF, and by
completing it the Problem will be solved. Q.E.F.

Cor. Hence a right line may be drawn whose
parts given in length may be intercepted between three right lines
given by position. Suppose the triangle DEF, by the access of its
point D to the side EF, and by having the sides DE, DF placed *in
directum* to be changed into a right line whose given part DE is
to be interposed between the right lines AB, AC given by position; and
its given part DF is to be interposed between the right lines AB, BC,
given by position; then, by applying the preceding construction to
this case; the Problem will be solved.

*To describe a trajectory given both in kind and magnitude,
given parts of which shall be interposed between three right lines given by position.*

Suppose a trajectory is to be described that may be similar and equal to the curve line DEF, and may be cut by three right lines AB, AC, BC, given by position, into parts DE and EF, similar and equal to the given parts of this curve line.

Draw the right lines DE, EP, DF: and place the angles D, E, F, of this triangle DEF, so as to touch those right lines given by position (by Lem. XXVI). Then about the triangle describe the trajectory, similar and equal to the curve DEF. Q.E.F.

*To describe a trapezium given in kind, the angles whereof may
be so placed, in respect of four right lines given by position,
that are neither all parallel among themselves, nor converge to
one common point, that the several angles may touch the several lines.*

Let the four right lines ABC, AD, BD, CE, be given by position; the
first cutting the second in A, the third in B, and the fourth in C;
and suppose a trapezium *fghi* is to be described that may be
similar to the trapezium FGHI, and whose angle *f*, equal to
the given angle F, may touch the right line ABC; and the other angles
*g, h, i,* equal to the other given angles, G, H, I, may touch
the other lines AD, BD, CE, respectively. Join FH, and upon FG, FH, FI
describe as many segments of circles FSG, FTH, FVI, the first of which
FSG may be capable of an angle equal to the angle BAD; the second FTH
capable of an angle equal to the angle CBD; and the third FVI of an
angle equal to the angle ACE. But the segments are to be described
towards those sides of the lines FG, FH, FI, that the circular order
of the letters FSGF may be the same as of the letters BADB, and that
the letters FTHF may turn about in the same order as the letters CBDC
and the letters FVIF in the game order as the letters ACEA. Complete
the segments into entire circles, and let P be the centre of the first
circle FSG, Q the centre of the second FTH. Join and produce both ways
the line PQ, and in it take QR in the same ratio to PQ as BC has to
AB. But QR is to be taken towards that side of the point Q, that the
order of the letters P, Q, R
may be the same as of the letters A, B, C; and about the centre R with
the interval RF describe a fourth circle FN*c* cutting the
third circle FVI in *c*. Join F*c* cutting the first
circle in *a*, and the second in *b*. Draw *a*G,
*b*H, *c*I, and let the figure ABC*fghi* be made
similar to the figure *abc*FGHI; and the trapezium *fghi*
will be that which was required to be described.

For let the two first circles FSG, FTH cut one the other in K; join
PK, QK, RK, *a*K, *b*K, *c*K, and produce QP
to L. The angles F*a*K, F*b*K, F*c*K at the
circumferences are the halves of the angles FPK, FQK, FRK, at the
centres, and therefore equal to LPK, LQK, LRK, the halves of those
angles. Wherefore the figure PQRK is equiangular and similar to the
figure *abc*K, and consequently *ab* is to *bc*
as PQ to QR, that is, as AB to BC. But by construction, the angles *f*A*g*,
*f*B*h*, *f*C*i*, are equal to the angles
F*a*G, F*b*H, F*c*I. And therefore the figure ABC*fghi*
may be completed similar to the figure *abc*FGHI. Which done a
trapezium *fghi* will be constructed similar to the trapezium
FGHI, and which by its angles *f, g, h, i* will touch the
right lines ABC, AD, BD, CE. Q.E.F.

Cor. Hence a right line may be drawn whose
parts intercepted in a given order, between four right lines given by
position, shall have a given proportion among themselves. Let the
angles FGH, GHI, be so far increased that the right lines FG, GH, HI,
may lie *in directum*; and by constructing the Problem in this
case, a right line *fghi* will be drawn, whose parts *fg,
gh, hi*, intercepted between the four right lines given by
position, AB and AD, AD and BD, BD and CE, will be one to another as
the lines FG, GH, HI, and will observe the same order among them
selves. But the same thing may be more readily done in this manner.

Produce AB to K and BD to L, so as BK may be to AB as HI to GH; and
DL to BD as GI to FG; and join KL meeting the right line CE in *i*.
Produce *i*L to M, so as LM may be to *i*L as GH to
HI; then draw MQ parallel to LB, and meeting the right line AD in *g*,
and join *gi* cutting AB, BD in *f, h*; I say, the
thing is done.

For let M*g* cut the right line AB in Q, and AD the right line
KL in S, and draw AP parallel to BD, and
meeting *i*L in P, and *g*M to L*h* (*gi*
to *hi*, M*i* to L*i*, GI to HI, AK to BK) and
AP to BL, will be in the same ratio. Cut DL in R, so as DL to RL may
be in that same ratio; and because *g*S to *g*M, AS to
AP, and DS to DL are proportional; therefore (*ex aequo*) as *g*S
to L*h*, so will AS be to BL, and DS to RL; and mixtly, BL − RL
to L*h* − BL, as AS − DS to *g*S − AS. That is, BR is
to B*h* as AD is to A*g*, and therefore as BD to *g*Q.
And alternately BR is to BD as B*h* to *g*Q, or as *fh*
to *fg*. But by construction the line BL was cut in D and R in
the same ratio as the line FI in G and H; and therefore BR is to BD as
FH to FG. Wherefore *fh* is to *fg* as FH to FG.
Since, therefore, *gi* to *hi* likewise is as M*i*
to L*i*, that is, as GI to HI, it is manifest that the lines
FI, *fi*, are similarly cut in G and H, *g* and *h*.
Q.E.F.

In the construction of this Corollary, after the line LK is drawn
cutting CE in *i*, we may produce *i*E to V, so as EV
may be to E*i* as FH to HI, and then draw V*f* parallel
to BD. It will come to the same, if about the centre *i* with
an interval IH, we describe a circle cutting BD in X, and produce *i*X
to Y so as *i*Y may be equal to IF, and then draw Y*f*
parallel to BD.

Sir Christopher Wren and Dr. Wallis have long ago given other solutions of this Problem.

*To describe a trajectory given in kind, that may be cut by four
right lines given by position, into parts given in order, kind, and proportion.*

Suppose a trajectory is to be described that may be similar to the
curve line FGHI, and whose parts, similar and proportional to the
parts FG, GH, HI of the other, may be intercepted between the right
lines AB and AD, AD, and BD, BD and CE given by position, viz., the
first between the first pair of those lines, the second between the
second, and the third between the third. Draw the right lines FG, GH,
HI, FI; and (by Lem. XXVII) describe a trapezium *fghi* that
may be similar to the trapezium FGHI, and whose angles *f, g, h, i*,
may touch the right lines given by position AB, AD, BD, CE, severally
according to their order. And then about this trapezium describe a
trajectory, that trajectory will be similar to the curve line FGHI.

This problem may be likewise constructed in the following manner.
Joining FG, GH, HI, FI, produce GF to V, and join FH, IG, and make
the angles CAK, DAL equal to the
angles FGH, VFH. Let AK, AL meet the right line BD in K and L, and
thence draw KM, LN, of which let KM make the angle AKM equal to the
angle GHI, and be itself to AK as HI is to GH; and let LN make the
angle ALN equal to the angle FHI, and be itself to AL as HI to FH.
But AK, KM. AL, LN are to be drawn towards
those sides of the lines AD, AK, AL, that the letters CAKMC, ALKA,
DALND may be carried round in the same order as the letters FGHIF; and
draw MN meeting the right line CE in *i*. Make the angle *i*EP
equal to the angle IGF, and let PE be to E*i* as FG to GI; and
through P draw PQ*f* that may with the right line ADE contain
an angle PQE equal to the angle FIG, and may meet the right line AB in
*f*, and join *fi*. But PE and PQ are to be drawn
towards those sides of the lines CE, PE, that the circular order of
the letters PE*i*P and PEQP may be the same as of the letters
FGHIF; and if upon the line *fi*, in the same order of
letters, and similar to the trapezium FGHI, a trapezium *fghi*
is constructed, and a trajectory given in kind is circumscribed about
it, the Problem will be solved.

So far concerning the finding of the orbits. It remains that we determine the motions of bodies in the orbits so found.

*To find at any assigned time the place of a body moving in, a
given parabolic trajectory.*

Let S be the focus, and A the principal vertex of the parabola; and
suppose 4AS × M equal to the parabolic area to be cut off APS, which
either was described by the radius SP, since the body's departure from
the vertex, or is to be described thereby before its arrival there.
Now the quantity of that area to be cut off is known from the time
which is proportional to it. Bisect AS in G, and erect the
perpendicular GH equal to BM, and a circle described about the centre
H, with the interval HS, will cut the parabola in the place P
required. For letting fall PO perpendicular on the axis, and drawing
PH, there will be

AG^{2} + GH^{2}

(= HP^{2} = (AO
− AG)^{2} + (PO − GH)^{2})

= AO^{2} + PO^{2} − 2GAO + 2 GH +
PO + AG^{2} + GH^{2}.

Whence 2GH × PO (= AO^{2} + PO^{2}
− 2GAO) = AO^{2} + ¾PO^{2}. For AO^{2} write
AO × PO^{2}

4AS; then dividing all the terms by 2PO, and
multiplying them by 2AS, we shall have ^{4}/_{3}GH ×
AS (= ^{1}/_{6}AO × PO + ½AS × PO = AO+3AS

6 × PO = 4AO − 3SO

6 × PO = to the area (APO − SPO))
= to the area APS. But GH was 3M, and therefore ^{4}/_{3}GH
× AS is 4AS × M. Wherefore the
area cut off APS is equal to the area that was to be cut off 4AS
× M. Q.E.D.

Cor. 1. Hence GH is to AS as the time in which the body described the arc AP to the time in which the body described the arc between the vertex A and the perpendicular erected from the focus S upon the axis.

Cor. 2. And supposing a circle ASP perpetually to pass through the moving body P, the velocity of the point H is to the velocity which the body had in the vertex A as 3 to 8; and therefore in the same ratio is the line GH to the right line which the body, in the time of its moving from A to P, would describe with that velocity which it had in the vertex A.

Cor. 3. Hence, also, on the other hand, the time may be found in which the body has described any assigned arc AP. Join AP, and on its middle point erect a perpendicular meeting the right line GH in H.

*There is no oval figure whose area, cut off by right lines at
pleasure, can be universally found by means of equations of any
number of finite terms and dimensions.*

Suppose that within the oval any point is given; about which as a
pole a right line is perpetually revolving with an uniform motion,
while in that right line a moveable point going out from the pole
moves always forward with a velocity proportional to the square of
that right line with in the oval. By this motion that point will
describe a spiral with infinite circumgyrations. Now if a portion of
the area of the oval cut off by that right line could be found by a
finite equation, the distance of the point from the pole, which is
proportional to this area, might be found by the same equation, and
therefore all the points of the spiral might be found by a finite
equation also; and therefore the intersection of a right line given in
position with the spiral might also be found by a finite equation. But
every right line infinitely produced cuts a spiral in an infinite
number of points; and the equation by which any one intersection of
two lines is found at the same time exhibits all their intersections
by as many roots, and therefore rises to as many dimensions as there
are intersections. Be cause two circles mutually cut one another in
two points, one of those intersections is not
to be found but by an equation of two dimensions, by which the other
intersection may be also found. Because there may be four
intersections of two conic sections, any one of them is not to be
found universally, but by an equation of four dimensions, by which
they may be all found together. For if those intersections are
severally sought, be cause the law and condition of all is the same,
the calculus will be the same in every case, and therefore the
conclusion always the same; which must therefore comprehend all those
intersections at once within itself, and exhibit them all
indifferently. Hence it is that the intersections of the conic scions
with the curves of the third order, because they may amount to six,
come out together by equations of six dimensions; and the
intersections of two curves of the third order, because they may
amount to nine, come out together by equations of nine dimensions. If
this did not necessarily happen, we might reduce all solid to plane
Problems, and those higher than solid to solid Problems. But here I
speak of curves irreducible in power. For if the equation by which the
curve is defined may be reduced to a lower power, the curve will not
be one single curve, but composed of two, or more, whose intersections
may be severally found by different calculusses. After the same manner
the two intersections of right lines with the conic sections come out
always by equations of two dimensions; the three intersections of
right lines with the irreducible curves of the third order by
equations of three dimensions; the four intersections of right lines
with the irreducible curves of the fourth order, by equations of four
dimensions; and so on *in infinitum*. Wherefore the
innumerable intersections of a right line with a spiral, since this is
but one simple curve and not reducible to more curves, require
equations infinite in number of dimensions and roots, by which they
may be all exhibited together. For the law and calculus of all is the
same. For if a perpendicular is let fall from the pole upon that
intersecting right line, and that perpendicular together with the
intersecting line revolves about the pole, the intersections of the
spiral will mutually pass the one into the other; and that which was
first or nearest, after one revolution, will be the second; after two,
the third; and so on: nor will the equation in the mean time be
changed but as the magnitudes of those quantities are changed, by
which the position of the intersecting line is determined. Wherefore
since those quantities after every revolution return to their first
magnitudes, the equation will return to its first form; and
consequently one and the same equation will exhibit all the
intersections, and will therefore have an infinite number of roots, by
which they may be all exhibited. And therefore the intersection of a
right line with a spiral cannot be universally found by any finite
equation; and of consequence there is no oval figure whose area, cut
off by right lines at pleasure, can be universally exhibited by any
such equation.

By the same argument, if the interval of the
pole and point by which the spiral is described is taken proportional
to that part of the perimeter of the oval which is cut off; it may be
proved that the length of the perimeter cannot be universally
exhibited by any finite equation. But here I speak of ovals that are
not touched by conjugate figures running out *in infinitum*.

Cor. Hence the area of an ellipsis, described by a radius drawn from the focus to the moving body, is not to be found from the time given by a finite equation; and therefore cannot be determined by the description of curves geometrically rational. Those curves I call geometrically rational, all the points whereof may be determined by lengths that are definable by equations; that is, by the complicated ratios of lengths. Other curves (such as spirals, quadratrixes, and cycloids) I call geometrically irrational. For the lengths which are or are not as number to number (according to the tenth Book of Elements) are arithmetically rational or irrational. And therefore I cut off an area of an ellipsis proportional to the time in which it is described by a curve geometrically irrational, in the following manner.

*To find the place of a body moving in a given elliptic
trajectory at any assigned time.*

Suppose A to be the principal vertex, S the focus, and O the centre of the ellipsis APB; and let P be the place of the body to be found. Produce OA to G so as OG may be to OA as OA to OS. Erect the perpendicular GH; and about the centre O, with the interval OG, describe the circle GEF; and on the ruler GH, as a base, suppose the wheel GEF to move forwards, revolving about its axis, and in the mean time by its point A describing the cycloid ALI. Which done, take GK to the perimeter GEFG of the wheel, in the ratio of the time in which the body proceeding from A described the arc AP, to the time of a whole revolution in the ellipsis. Erect the perpendicular KL meeting the cycloid in L; then LP drawn parallel to KG will meet the ellipsis in P, the required place of the body.

For about the centre O with the interval OA describe the semi-circle AQB, and let LP, produced, if need be, meet the arc AQ in Q, and join SQ, OQ. Let OQ meet the arc EFG in F, and upon OQ let fall the perpendicular SR. The area APS is as the area AQS, that is, as the difference between the sector OQA and the triangle OQS, or as the difference of the rectangles ½OQ × AQ, and ½OQ × SR, that is, because ½OQ is given, as the difference between the arc AQ and the right line SR; and therefore (because of the equality of the given ratios SR to the sine of the arc AQ, OS to OA, OA to OG, AQ to GF; and by division, AQ − SR to GF − sine of the arc AQ) as GK, the difference between the arc GF and the sine of the arc AQ. Q.E.D.

But since the description of this curve is difficult, a solution by
approximation will be preferable. First, then, let there be found a
certain angle B which may be to an angle of 57,29578 degrees, which an
arc equal to the radius subtends, as SH, the distance of the foci, to
AB, the diameter of the ellipsis. Secondly, a certain length L, which
may be to the radius in the same ratio inversely. And these being
found, the Problem may be solved by the following analysis. By any
construction (or even by conjecture), suppose we know P the place of
the body near its true place *p*. Then letting fall on the
axis of the ellipsis the ordinate PR from the proportion of the
diameters of the ellipsis, the ordinate RQ of the circumscribed circle
AQB will be given; which ordinate is the sine of the angle AOQ,
supposing AO to be the radius, and also cuts the ellipsis in P. It
will be sufficient if that angle is found by a rude calculus in
numbers near the truth. Suppose we also know the angle proportional to
the time, that is, which is to four right angles as the time in which
the body described the arc A*p*, to the time of one revolution
in the ellipsis. Let this angle be N. Then take an angle D, which may
be to the angle B as the sine of the angle AOQ to the radius; and an
angle E which may be to the angle N − AOQ + D as the length L to the
same length L diminished by the cosine of the angle AOQ, when that
angle is less than a right angle, or increased thereby when greater.
In the next place, take an angle F that may be to the angle B as the
sine of the angle AOQ + E to the radius, and an angle G, that may be
to the angle N − AOQ − E + F as the length L to the same length L
diminished by the cosine of the angle AOQ + E, when that angle is less
than a right angle, or increased thereby when greater. For the third
time take an angle H, that may be to the angle B as the sine of the
angle AOQ + E + G to the radius; and an angle I to the angle N − AOQ −
E − G + H, as the length L is to the same
length L diminished by the cosine of the angle AOQ + E + G, when that
angle is less than a right angle, or increased thereby when greater.
And so we may proceed *in infinitum*. Lastly, take the angle
AO*q* equal to the angle AOQ + E + G + I +, &c. and from
its cosine O*r* and the ordinate *pr*, which is to its
sine *qr* as the lesser axis of the ellipsis to the greater,
we shall have *p* the correct place of the body. When the
angle N − AOQ + D happens to be negative, the sign + of the angle E
must be every where changed into −, and the sign − into +. And the
same thing is to be understood of the signs of the angles G and I,
when the angles N − AOQ − E + F, and N − AOQ − E − G + H come out
negative. But the infinite series AOQ + E + G + I +, &c. converges
so very fast, that it will be scarcely ever needful to proceed beyond
the second term E. And the calculus is founded upon this Theorem, that
the area APS is as the difference between the arc AQ and the right
line let fall from the focus S perpendicularly upon the radius OQ.

And by a calculus not unlike, the Problem is solved in the hyperbola. Let its centre be O, its vertex A, its focus S, and asymptote OK; and suppose the quantity of the area to be cut off is known, as being proportional to the time. Let that be A, and by conjecture suppose we know the position of a right line SP, that cuts off an area APS near the truth. Join OP, and from A and P to the asymptote draw AI, PK parallel to the other asymptote; and by the table of logarithms the area AIKP will be given, and equal thereto the area OPA, which subducted from the triangle OPS, will leave the area cut off APS. And by applying 2APS − SA, or 2A − SAPS, the double difference of the area A that was to be cut off, and the area APS that is cut off, to the line SN that is let fall from the focus S, perpendicular upon the tangent TP, we shall have the length of the chord PQ. Which chord PQ is to be inscribed between A and P, if the area APS that is cut off be greater than the area A that was to be cut off, but towards the contrary side of the point P, if otherwise: and the point Q will be the place of the body more accurately. And by repeating the computation the place may be found perpetually to greater and greater accuracy.

And by such computations we have a general analytical resolution of the Problem. But the particular calculus that follows is better fitted for astronomical purposes. Supposing AO, OB, OD, to be the semi-axis of the ellipsis, and L its latus rectum, and D the difference betwixt the lesser semi-axis OD, and ½L the half of the latus rectum: let an angle Y be found, whose sine may be to the radius as the rectangle under that difference D, and AO + OD the half sum of the axes to the square of the greater axis AB. Find also an angle Z, whose sine may be to the radius as the double rectangle under the distance of the foci SH and that difference D to triple the square of half the greater semi-axis AO. Those angles being once found, the place of the body may be thus determined. Take the angle T proportional to the time in which the arc BP was described, or equal to what is called the mean motion; and an angle V the first equation of the mean motion to the angle Y, the greatest first equation, as the sine of double the angle T is to the radius; and an angle X, the second equation, to the angle Z, the second greatest equation, as the cube of the sine of the angle T is to the cube of the radius. Then take the angle BHP the mean motion equated equal to T + X + V, the sum of the angles T, V, X, if the angle T is less than a right angle; or equal to T + X − V, the difference of the same, if that angle T is greater than one and less than two right angles; and if HP meets the ellipsis in P, draw SP, and it will cut off the area BSP nearly proportional to the time.

This practice seems to be expeditious enough, because the angles V and X, taken in second minutes, if you please, being very small, it will be sufficient to find two or three of their first figures. But it is likewise sufficiently accurate to answer to the theory of the planet's motions. For even in the orbit of Mars, where the greatest equation of the centre amounts to ten degrees, the error will scarcely exceed one second. But when the angle of the mean motion equated BHP is found, the angle of the true motion BSP, and the distance SP, are readily had by the known methods.

And so far concerning the motion of bodies in curve lines. But it may also come to pass that a moving body shall ascend or descend in a right line; and I shall now go on to explain what belongs to such kind of motions.

*Supposing that the centripetal force is reciprocally
proportional to the square of the distance of the places from the
centre; it is required to define the spaces which a body, falling
directly, describes in given times.*

Case 1. If the body does not fall
perpendicularly, it will (by Cor. 1
Prop. XIII) describe some conic section whose focus is A placed in the
centre of force. Suppose that conic section to be ARPB and its focus
S. And, first, if the figure be an ellipsis, upon the greater axis
thereof AB describe the semi-circle ADB, and let the right line DPC
pass through the falling body, making right angles with the axis; and
drawing DS, PS, the area ASD will be proportional to the area ASP, and
therefore also to the time. The axis AB still remaining the same, let
the breadth of the ellipsis be perpetually diminished, and the area
ASD will always remain proportional to the time. Suppose that breadth
to be diminished *in infinitum*; and the orbit APB in that
case coinciding with the axis AB, and the focus S with the extreme
point of the axis B, the body will descend in the right line AC, and
the area ABD will become proportional to the time. Wherefore the space
AC will be given which the body describes in a given time by its
perpendicular fall from the place A, if the area ABD is taken
proportional to the time, and from the point D the right line DC is
let fall perpendicularly on the right line AB. Q.E.I.

Case 2. If the figure RPB is an hyperbola, on
the same principal diameter AB describe the rectangular hyperbola BED;
and because the areas CSP, CB*f*P, SP*f*B, are severally
to the several areas CSD, CBED, SDEB, in the given ratio of the
heights CP, CD, and the area SP*f*B is proportional to the time
in which the body P will move through the arc P*f*B. the area
SDEB will be also proportional to that time. Let the latus rectum of
the hyperbola RPB be diminished *in infinitum*, the latus
transversum remaining the same; and the arc PB will come to coincide
with the right line CB, and the focus S, with the vertex B, and the
right line SD with the right line BD. And therefore the area BDEB will
be proportional to the time in which the body C, by its perpendicular
descent, describes the line CB. Q.E.I.

Case 3. And by the like argument, if the figure RPB is a parabola, and to the same principal vertex B another parabola BED is described, that may always remain given while the former para bola in whose perimeter the body P moves, by having its latus rectum diminished and reduced to nothing, comes to coincide with the line CB, the parabolic segment BDEB will be proportional to the time in which that body P or C will descend to the centre S or B. Q.E.I

*The things above found being supposed. I say, that the velocity
of a falling body in any place* C *is to the velocity of a
body, describing a circle about the centre* B *at the
distance* BC*, in the subduplicate ratio of* AC*,
the distance of the body from the remoter vertex* A *of
the circle or rectangular hyperbola, to* ½AB*, the
principal semi-diameter of the figure.*

Let AB, the common diameter of both figures RPB, DEB, be bisected in
O; and draw the right line PT that may touch the figure RPB in P, and
likewise cut that common diameter AB (produced, if need be) in T; and
let SY be perpendicular to this line, and BQ to this diameter, and
suppose the latus rectum of the figure RPB to be L. From Cor. 9, Prop.
XVI, it is manifest that the velocity of a body, moving in the line
RPB about the centre S, in any place P, is to the velocity of a body
describing a circle about the same centre, at the distance SP, in the
subduplicate ratio of the rectangle ½L × SP to SY². For by the
properties of the conic sections ACB is to CP² as 2AO to L, and
therefore 2CP^{2} × AO

ACB is equal to L. Therefore those
velocities are to each other in the subduplicate ratio of CP^{2} × AO × SP

ACB to SY². Moreover, by the properties
of the conic sections, CO is to BO as BO to TO, and (by composition or
division) as CB to BT. Whence (by division or composition) BO − or +
CO will be to BO as CT to BT, that is, AC will be to AO as CP to BQ;
and therefore CP^{2} × AO
× SP

ACB is equal to BQ^{2} × AC × SP

AO × BC. Now suppose CP, the breadth of
the figure RPB, to be diminished *in infinitum*, so as the
point P may come to coincide with the point C, and the point S with
the point B, and the line SP with the line BC, and the line SY with
the line BQ; and the velocity of the body now descending
perpendicularly in the line CB will be to the velocity of a
body describing a circle about the centre B, at the distance BC; in
the subduplicate ratio of BQ^{2}
× AC × SP

AO × BC to SY², that is (neglecting the
ratios of equality of SP to BC, and BQ² to SY²), in the subduplicate
ratio of AC to AO, or ½AB. Q.E.D.

Cor. 1. When the points B and S come to coincide, TC will become to TS as AC to AO.

Cor. 2. A body revolving in any circle at a given distance from the Centre, by its motion converted upwards, will ascend to double its distance from the centre.

*If the figure* BED *is a parabola, I say, that the
velocity of a falling body in any place* C *is equal to
the velocity by which a body may uniformly describe a circle about
the centre* B *at half the interval* BC*.*

For (by Cor. 7, Prop. XVI) the velocity of a body describing a
parabola RPB about the centre S, in any place P, is equal to the
velocity of a body uniformly describing a circle about the same centre
S at half the interval SP. Let the breadth CP of the parabola be
diminished *in infinitum*, so as the parabolic arc P*f*B
may come to coincide with the right line CB, the centre S with the
vertex B, and the interval SP with the interval BC, and the
proposition will be manifest. Q.E.D.

*The same things supposed, I say, that the area of the figure*
DES*, described by the indefinite radius* SD*, is equal to
the area which a body with a radius equal to half the latus rectum
of the figure* DES*, by uniformly revolving about the
centre* S*, may describe in the same time.*

For suppose a body C in the smallest moment
of time describes in falling the infinitely little line C*c*,
while another body K, uniformly revolving about the centre S in the
circle OK*k*, describes the arc K*k*. Erect the
perpendiculars CD, *cd*, meeting the figure DES in D, *d*.
Join SD, S*d*, SK, S*k*, and draw D*d* meeting
the axis AS in T, and thereon let fall the perpendicular SY.

Case 1. If the figure DES is a circle, or a
rectangular hyperbola, bisect its transverse diameter AS in O, and SO
will be half the latus rectum. And because TC is to TD as C*c*
to D*d*, and TD to TS as CD to SY; *ex aequo* TC will
be to TS as CD × C*c* to SY × D*d*. But (by Cor. 1,
Prop. XXXIII) TC is to TS as AC to AO; to wit, if in the coalescence
of the points D, *d*, the ultimate ratios of the lines are
taken. Wherefore AC is to AO or SK as CD × C*c* to SY × D*d*.
Farther, the velocity of the descending body in C is to the velocity
of a body describing a circle about the centre S, at the interval SC,
in the subduplicate ratio of AC to AO or SK (by Prop. XXXIII); and
this velocity is to the velocity of a body describing the circle OK*k*
in the subduplicate ratio of SK to SC (by Cor. 6, Prop IV); and, *ex
aequo*, the first velocity to the last, that is, the little line
C*c* to the arc K*k*, in the subduplicate ratio of AC to
SC, that is, in the ratio of AC to CD. Wherefore CD × C*c* is
equal to AC × K*k*, and consequently AC to SK as AC × K*k*
to SY × D*d*, and thence SK × K*k* equal to SY × D*d*,
and ½SK × K*k* equal to ½SY × D*d*, that is, the area KS*k*
equal to the area SD*d*. Therefore in every moment of time two
equal particles, KS*k* and SD*d*, of areas are
generated, which, if their magnitude is diminished, and their number
increased *in infinitum*, obtain the ratio of equality, and
consequently (by Cor. Lem. IV), the whole areas together generated are
always equal. Q.E.D.

Case 2. But if the figure DES is a parabola,
we shall find, as above, CD × C*c* to SY × D*d* as TC to
TS, that is, as 2 to 1; and that therefore ¼CD × C*c* is equal
to ½SY × D*d*. But the velocity of the falling body in C is
equal to the velocity with which a circle may be uniformly described
at the interval ½SC (by Prop. XXXIV). And this velocity to the
velocity with which a circle may be described with the radius SK, that
is, the little line C*c* to the arc K*k*, is (by Cor. 6,
Prop. IV) in the subduplicate ratio of SK to ½SC; that is, in the
ratio of SK to ½CD. Wherefore ½SK × K*k* is equal to ¼CD × C*c*,
and therefore equal to ½SY × D*d*; that is, the area KS*k*
is equal to the area SD*d*, as above. Q.E.D.

*To determine the times of the descent of a body falling from place A.*

Upon the diameter AS, the distance of the body from the centre at the beginning, describe the semi-circle ADS, as likewise the semi-circle OKH equal thereto, about the centre S. From any place C of the body erect the ordinate CD. Join SD, and make the sector OSK equal to the area ASD. It is evident (by Prop. XXXV) that the body in falling will describe the space AC in the same time in which another body, uniformly revolving about the centre S, may describe the arc OK. Q.E.F.

*To define the times of the ascent or descent of a body
projected upwards or downwards from a given place.*

Suppose the body to go off from the given place G, in the direction
of the line GS, with any velocity. In the duplicate ratio of this
velocity to the uniform velocity in a circle, with which the body may
revolve about
the centre S at the given
interval SG, take GA to ½AS. If that ratio is the same as of the
number 2 to 1, the point A is infinitely remote; in which case a
parabola is to be described with any latus rectum to the vertex S, and
axis SG; as appears by Prop. XXXIV. But if that ratio is less or
greater than the ratio of 2 to 1, in the former case a circle, in the
latter a rectangular hyperbola, is to be described on the diameter SA;
as appears by Prop. XXXIII. Then about the centre S, with an interval
equal to half the latus rectum, describe the circle H*k*K; and
at the place G of the ascending or descending body, and at any other
place C, erect the perpendiculars GI, CD, meeting the conic section or
circle in I and D. Then joining SI, SD, let the sectors HSK, HS*k*
be made equal to the segments SEIS, SEDS. and (by Prop. XXXV) the body
G will describe the space GC in the same time
in which the body K may describe the arc K*k*.
Q.E.F.

*Supposing that the centripetal force is proportional to the
altitude or distance of places from the centre. I say, that the
times and velocities of falling bodies, and the spaces which they
describe, are respectively proportional to the arcs, and the right
and versed sines of the arcs.*

Suppose the body to fall from any place A in the right line AS; and about the centre of force S, with the interval AS, describe the quadrant of a circle AE; and let CD be the right sine of any arc AD; and the body A will in the time AD in falling describe the space AC, and in the place C will acquire the velocity CD.

This is demonstrated the same way from Prop. X, as Prop. XXXII was demonstrated from Prop. XI.

Cor. 1. Hence the times are equal in which one body falling from the place A arrives at the centre S, and another body revolving describes the quadrantal arc ADE.

Cor. 2. Wherefore all the times are equal in which bodies falling from whatsoever places arrive at the centre. For all the periodic times of revolving bodies are equal (by Cor. 3, Prop. IV).

*Supposing a centripetal force of any kind, and granting the
quadratures of curvilinear figures; it is required to find the
velocity of a body, ascending or descending in a right line, in
the several places through which it passes; as also the time in
which it will arrive at any place: and* vice versa*.*

Suppose the body E to fall from any place A in the right line ADEC; and from its place E imagine a perpendicular EG always erected proportional to the centripetal force in that place tending to the centre C; and let BFG be a curve line, the locus of the point G. And in the beginning of the motion suppose EG to coincide with the perpendicular AB; and the velocity of the body in any place E will be as a right line whose square is equal to the curvilinear area ABGE. Q.E.I.

In EG take EM reciprocally proportional to a right line whose square is equal to the area ABGE, and let VLM he a curve line wherein the point M is always placed, and to which the right line AB produced is an asymptote; and the time in which the body in falling describes the line AE, will be as the curvilinear area ABTVME. Q.E.I.

For in the right line AE let there be taken the very small line DE of
a given length, and let DLF be the place of the line EMG, when the
body was in D; and if the centripetal force be such, that a right
line, whose square is equal to the area ABGE, is as the velocity of
the descending body, the area itself will be as the square of that
velocity; that is, if for the velocities in D and E we write V and V +
I, the area ABFD will be as VV, and the area ABGE as VV + 2VI + II;
and by division, the area DFGE as 2VI + II, and therefore DFGE

DE will be as 2VI+II

DE; that is, if we take the first
ratios of those quantities when just nascent, the length DF is as the
quantity 2VI

DE, and therefore also as half that
quantity I × V

DE. But the time in which the body in
falling describes the verv small line DE, is as that line directly and
the velocity V inversely; and the force will be as the increment I of
the velocity directly and the time inversely; and therefore if we take
the first ratios when those quantities are just nascent, as
I × V

DE, that is, as the length DF.
Therefore a force proportional to DF or EG will cause the body to
descend with a velocity that is as the right line whose square is
equal to the area ABGE. Q.E.D.

Moreover, since the time in which a very small line DE of a given length may be described is as the velocity inversely, and therefore also inversely as a right line whose square is equal to the area ABFD; and since the line DL, and by consequence the nascent area DLME, will be as the same right line inversely, the time will be as the area DLME, and the sum of all the times will be as the sum of all the areas; that is (by Cor. Lem. IV), the whole time in which the line AE is described will be as the whole area ATVME. Q.E.D.

Cor. 1. Let P be the place from whence a body
ought to fall, so as that, when urged by any known uniform centripetal
force (such as gravity is vulgarly supposed to be), it may acquire in
the place D a velocity equal to the velocity which another body,
falling by any force whatever, hath acquired in that place D. In the
perpendicular DF let there be taken DR, which may be to DF as that
uniform force to the other force in the place D. Complete the
rectangle PDRQ, and cut off the area ABFD equal to that rectangle.
Then A will be the place
from whence the other body fell. For completing the rectangle DRSE,
since the area ABFD is to the area DFGE as VV to 2VI, and therefore as
½V to I, that is, as half the whole velocity to the increment of the
velocity of the body falling by the unequable force; and in like
manner the area PQRD to the area DRSE as half the whole velocity to
the increment of the velocity of the body falling by the uniform
force; and since those increments (by reason of the equality of the
nascent times) are as the generating forces, that is, as the ordinates
DF, DR, and consequently as the nascent areas DFGE, DRSE: therefore, *ex
aequo*, the whole areas ABFD, PQRD will be to one another as the
halves of the whole velocities; and therefore, because the velocities
are equal, they become equal also.

Cor. 2. Whence if any body be projected
either upwards or downwards with a given velocity from any place D,
and there be given the law of centripetal force acting on it, its
velocity will be found in any other place, as *e*, by erecting
the ordinate *eg*, and taking that velocity to the velocity in
the place D as a right line whose square is equal to the rectangle
PQRD, either increased by the curvilinear area DF*ge*, if the
place *e* is below the place D, or diminished by the same area
DF*ge*, if it be higher, is to the right line whose square is
equal to the rectangle PQRD alone.

Cor. 3. The time is also known by erecting
the ordinate *em* reciprocally proportional to the square root
of PQRD + or − DF*ge*, and taking the time in which the body
has described the line D*e* to the time in which another body
has fallen with an uniform force from P, and in falling arrived at D
in the proportion of the curvilinear area DL*me* to the
rectangle 2PD × DL. For the time in which a body falling with an
uniform force hath described the line PD, is to the time in which the
same body has described the line PE in the subduplicate ratio of PD to
PE; that is (the very small line DE being just nascent), in the ratio
of PD to PD + ½DE, or 2PD to 2PD + DE, and, by division, to the time
in which the body hath described the small line DE, as 2PD to DE, and
therefore as the rectangle 2PD × DL to the area DLME; and the time in
which both the bodies described the very small line DE is to the time
in which the body moving unequably hath described the line D*e*
as the area DLME to the area DL*me*; and, *ex aequo*,
the first mentioned of these times is to the last as the rectangle 2PD
× DL to the area DL*me*.

*If a body, acted upon by any centripetal force, is any how
moved, and another body ascends or descends in a right line, and
their velocities be equal in any one case of equal altitudes,
their velocities will be also equal at all equal altitudes.*

Let a body descend from A through D and E, to the centre C; and let
another body move from V in the curve line VIK*k*. From the
centre C, with any distances, describe the concentric circles DI, EK,
meeting the right line AC in D and E, and the curve VIK in I and K.
Draw IC meeting KE in N, and on IK let fall the perpendicular NT; and
let the interval DE or IN between the circumferences of the circles be
very small; and imagine the bodies in D and I to have equal
velocities. Then because the distances CD and CI are equal, the
centripetal forces in D and I will be also equal. Let those forces be
expressed by the equal lineolae DE and IN; and let the force IN (by
Cor. 2 of the Laws of Motion) be resolved into two others, NT and IT.
Then the force NT acting in the direction of the line NT perpendicular
to the path ITK of the body will not at all affect or change the
velocity of the body in that path, but only draw it aside from a
rectilinear course, and make it deflect perpetually from the tangent
of the orbit, and proceed in the curvilinear path ITK*k*. That
whole force, therefore, will be spent in producing this effect; but
the other force IT, acting in the direction of the course of the body,
will be all employed in accelerating it, and in the least given time
will produce an acceleration proportional to itself. Therefore the
accelerations of the bodies in D and I, produced in equal times, are
as the lines DE, IT (if we take the first ratios of the nascent lines
DE, IN, IK, IT, NT); and in unequal times as those lines and the times
conjunctly. But the times in which DE and IK are described, are, by
reason of the equal velocities (in D and I) as the spaces described DE
and IK, and therefore the accelerations in the course of the bodies
through the lines DE and IK are as DE and IT, and DE and IK
conjunctly; that is, as the square of DE to the rectangle IT into IK.
But the rectangle IT × IK is equal to the square of IN, that is, equal
to the square of DE; and therefore the accelerations generated in the
passage of the bodies from D and I to E and K are equal. Therefore the
velocities of the bodies in E and K are also equal, and by the same
reasoning they will always be found equal in any subsequent equal
distances. Q.E.D.

By the same reasoning, bodies of equal velocities and equal distances from the centre will he equally retarded in their ascent to equal distances. Q.E.D.

Cor. 1. Therefore if a body either oscillates by hanging to a string, or by any polished and perfectly smooth impediment is forced to move in a curve line; and another body ascends or descends in a right line, and their velocities be equal at any one equal altitude, their velocities will be also equal at all other equal altitudes. For by the string of the pendulous body, or by the impediment of a vessel perfectly smooth, the same thing will be effected as by the transverse force NT. The body is neither accelerated nor retarded by it, but only is obliged to leave its rectilinear course.

Cor. 2. Suppose the quantity P to be the
greatest distance from the centre to which a body can ascend, whether
it be oscillating, or revolving in a trajectory, and so the same
projected upwards from any point of a trajectory with the velocity it
has in that point. Let the quantity A be the distance of the body from
the centre in any other point of the orbit; and let the centripetal
force be always as the power A^{n−1}, of the quantity A, the
index of which power *n*−1 is any number *n*
diminished by unity. Then the velocity in every altitude A will be as
√(P^{a} − A^{n}) and
therefore will be given. For by Prop. XXXIX, the velocity of a body
ascending and descending in a right line is in that very ratio.

*Supposing a centripetal force of any kind, and granting the
quadratures of curvilinear figures, it is required to find as well
the trajectories in which bodies will move, as the times of their
motions in the trajectories found.*

Let any centripetal force tend to the centre C, and let it be
required to find the trajectory VIK*k*. Let there be given the
circle VR, described from the centre C with any interval CV; and from
the same centre describe any other circles ID, KE cutting the
trajectory in I and K, and the right line CV in D and E. Then draw the
right line CNIX cutting the circles KE, VR in N and X, and the right
line CKY meeting the circle VR in Y. Let the points I and K be
indefinitely near; and let the body go on from V through I and K to *k*;
and let the point A be the place from whence another body is to fall,
so as in the place D to acquire a velocity equal to the velocity of
the first body in I. And things remaining as in Prop. XXXIX, the
lineola IK, described in the least given time will
be as the velocity, and therefore as the right line whose square is
equal to the area ABFD, and the triangle ICK proportional to the time
will be given, and therefore KN will be reciprocally as the altitude
IC; that is (if there be given any quantity Q, and the altitude IC be
called A), as Q

A. This quantity Q

A call Z, and suppose the magnitude of
Q to be such that in some case √(ABFD) may
be to Z as IK to KN, and then in all cases √(ABFD)
will be to Z as IK to KN, and ABFD to ZZ as IK² to KN², and by
division ABFD − ZZ to ZZ as IN² to KN², and therefore √(ABFD
− ZZ) to Z; or Q

A as IN to KN; and therefore A × KN
will be equal to Q × IN

√(ABFD − ZZ). Therefore since YX × XC
is to A × KN as CX², to AA, the rectangle XY × XC will be equal to
Q × IN × CX^{2}

AA√(ABFD − ZZ). Therefore in the
perpendicular DF let there be taken continually D*b*, D*c*
equal to Q

2√(ABFD − ZZ), Q × CX^{2}

2AA√(ABFD − ZZ) respectively, and let
the curve lines *ab, ac,* the foci of the points *b*
and *c*, be described: and from the point V let the
perpendicular V*a* be erected to the line AC, cutting off the
curvilinear areas VD*ba*, VD*ca*, and let the ordinates
E*z*, E*x*, be erected also. Then because the rectangle
D*b* × IN or D*bz*E is equal to half the rectangle A ×
KN, or to the triangle ICK; and the rectangle D*c* × IN or D*cx*E
is equal to half the rectangle YX × XC, or to the triangle XCY; that
is, because the nascent particles D*bz*E, ICK of the areas VD*ba*,
VIC are always equal; and the nascent particles D*cx*E, XCY of
the areas VD*ca*, VCX are always equal: therefore the generated
area VD*ba* will be equal to the generated area VIC, and
therefore proportional to the time; and the generated area VD*ca*
is equal to the generated sector VCX. If, therefore, any time be given
during which the body has been moving from V, there will be also given
the area proportional to it VD*ba*; and thence will be given
the altitude of the body CD or CI; and the area VD*ca*, and the
sector VCX equal thereto, together with its angle VCI. But the angle
VCI, and the altitude CI being given, there is also given the place I,
in which the body will be found at the end of that time.
Q.E.I.

Cor. 1. Hence the greatest and least altitudes of the bodies, that is, the apsides of the trajectories, may be found very readily. For the apsides are those points in which a right line IC drawn through the centre falls perpendicularly upon the trajectory VIK; which comes to pass when the right lines IK and NK become equal; that is, when the area ABFD is equal to ZZ.

Cor. 2. So also the angle KIN, in which the trajectory at any place cuts the line IC, may be readily found by the given altitude IC of the body: to wit, by making the sine of that angle to radius as KN to IK that is, as Z to the square root of the area ABFD.

Cor. 3. If to the centre C, and the principal
vertex V, there be described a conic section VRS; and from any point
thereof, as R, there be drawn the tangent RT meeting the axis CV
indefinitely produced in the point T; and then joining CR there be
drawn the right line CP, equal to the abscissa CT, making an angle VCP
proportional to the sector VCR; and if a centripetal force,
reciprocally proportional to the cubes of the distances of the places
from the centre, tends to the centre C; and from the place V there
sets out a body with a just velocity in the direction of a line
perpendicular to the right line CV; that body will proceed in a
trajectory VPQ, which the point P will always touch; and therefore if
the conic section VRS be an hyberbola, the body will descend to the
centre; but if it be an ellipsis, it will ascend perpetually, and go
farther and farther off *in infinitum*. And, on the contrary,
if a body endued with any velocity goes off from the place V, and
according as it begins either to descend obliquely to the centre, or
ascends obliquely from it, the figure VRS be either an hyperbola or an
ellipsis, the trajectory may be found by increasing or diminishing the
angle VCP in a given ratio. And the centripetal force becoming
centrifugal, the body will ascend obliquely in the trajectory VPQ,
which is found by taking the angle VCP proportional to the elliptic
sector VRC, and the length CP equal to the length CT, as before. All
these things follow from the foregoing Proposition, by the quadrature
of a certain curve, the invention of which, as being easy enough, for
brevity's sake I omit.

*The law of centripetal force being given, it is required to
find the motion of a body setting out from a given place, with a
given velocity, in the direction of a given right line.*

Suppose the same things as in the three preceding propositions; and
let the body go off from the place I in the direction of the little
line, IK, with the same velocity as another body, by falling with an
uniform centripetal force from the place P, may acquire in D; and let
this uniform force be to the force with which the body is
at first urged in I, as DR to DF. Let the body go on towards *k*;
and about the centre C, with the interval C*k*, describe the
circle *ke*, meeting the right line PD in *e*, and let
there be erected the lines *eg, ev, ew*, ordinately applied to
the curves BF*g*, *abv, acw*. From the given rectangle
PDRQ and the given law of centripetal force, by which the first body
is acted on, the curve line BF*g* is also given, by the
construction of Prop. XXVII, and its Cor. 1. Then from the given angle
CIK is given the proportion of the nascent lines IK, KN; and thence,
by the construction of Prob. XXVIII, there is given the quantity Q,
with the curve lines *abv, acw*; and therefore, at the end of
any time D*bve*, there is given both the altitude of the body C*e*
or C*k*, and the area D*cwe*, with the sector equal to
it XC*y*, the angle IC*k*, and the place *k*, in
which the body will then be found. Q.E.I.

We suppose in these Propositions the centripetal force to vary in its recess from the centre according to some law, which any one may imagine at pleasure; but at equal distances from the centre to be everywhere the same.

I have hitherto considered the motions of bodies in immovable orbits. It remains now to add something concerning their motions in orbits which revolve round the centres of force.

*It is required to make a body move in a trajectory that
revolves about the centre of force in the same manner as another
body in the same trajectory at rest.*

In the orbit VPK, given by position, let the body P revolve,
proceeding from V towards K. From the centre C let there be
continually drawn C*p*, equal to CP, making the angle VC*p*
proportional to the angle VCP; and the area which the line C*p*
describes will be to the area VCP, which the line CP describes at the
same time, as the velocity of the describing line C*p* to the
velocity of the describing line CP; that is, as the angle VC*p*
to the angle VCP, therefore in a given ratio, and therefore
proportional to the time. Since, then, the area described by the line
C*p* in an immovable plane is proportional to the time, it is
manifest that a body, being acted upon by a just quantity of
centripetal force may revolve with the point
*p* in the curve line which the same point *p*, by the
method just now explained, may be made to describe an immovable plane.
Make the angle VC*u* equal to the angle PC*p*, and the
line C*u* equal to CV, and the figure *u*C*p*
equal to the figure VCP, and the body being always in the point *p*,
will move in the perimeter of the revolving figure *u*C*p*,
and will describe its (revolving) arc *up* in the same time
that the other body P describes the similar and equal arc VP in the
quiescent figure VPK. Find, then, by Cor. 5, Prop. VI., the
centripetal force by which the body may be made to revolve in the
curve line which the point *p* describes in an immovable
plane, and the Problem will be solved. Q.E.F.

*The difference of the forces, by which two bodies may be made
to move equally, one in a quiescent, the other in the same orbit
revolving, is in a triplicate ratio of their common altitudes inversely.*

Let the parts of the quiescent orbit VP, PK be similar and equal to
the parts of the revolving orbit *up*, pk; and let the
distance of the points P and K be supposed of the utmost smallness.
Let fall a perpendicular *kr* from the point *k* to
the right line *p*C, and produce it to *m*, so that *mr*
may be to *kr* as the angle VC*p* to the angle VCP.
Because the altitudes of the bodies PC and *p*C, KC and *k*C,
are always equal, it is manifest that the increments or decrements of
the lines PC and *p*C are always equal; and therefore if each
of the several motions of the bodies in the places P and *p*
be resolved into two (by Cor. 2 of the Laws of Motion), one of which
is directed towards the centre, or according to the lines PC, *p*C,
and the other, transverse to the former, hath a direction
perpendicular to the lines PC and *p*C; the motions towards
the centre will be equal, and the transverse motion of the body *p*
will be to the transverse motion of the body P as the angular motion
of the line *p*C to the angular motion of the line PC; that
is, as the angle VC*p* to the angle VCP. Therefore, at the same
time that the body P, by both its motions, comes to the point K, the
body *p*, having an equal motion towards the centre, will be
equally moved from *p* towards C; and therefore that time
being expired, it will be found somewhere in the line *mkr*,
which, passing through the point *k*, is perpendicular to the
line *p*C; and by its transverse motion will acquire a
distance from the line *p*C, that will
be to the distance which the other body P acquires from the line PC as
the transverse motion of the body *p* to the transverse motion
of the other body P. Therefore since *kr* is equal to the
distance which the body P acquires from the line PC, and *mr*
is to *kr* as the angle VC*p* to the angle VCP, that
is, as the transverse motion of the body *p* to the transverse
motion of the body P, it is manifest that the body *p*, at the
expiration of that time, will be found in the place *m*. These
things will be so, if the bodies *p* and P are equally moved
in the directions of the lines *p*C and PC, and are therefore
urged with equal forces in those directions, but if we take an angle *p*C*n*
that is to the angle *p*C*k* as the angle VC*p*
to the angle VCP, and *n*C be equal to *k*C, in that
case the body *p* at the expiration of the time will really be
in *n*; and is therefore urged with a greater force than the
body P, if the angle *n*C*p* is greater than the angle
*k*C*p*, that is, if the orbit *upk*, move
either *in consequentia* or *in antecedentia*, with a
celerity greater than the double of that with which the line CP moves
*in consequentia*; and with a less force if the orbit moves
slower *in antecedentia*. And the difference of the forces
will be as the interval *mn* of the places through which the
body would be carried by the action of that difference in that given
space of time. About the centre C with the interval C*n* or C*k*
suppose a circle described cutting the lines *mr, mn* produced
in *s* and *t*, and the rectangle *mn × mt*
will be equal to the rectangle *mk × ms*, and therefore *mn*
will be equal to mk × ms

mt. But since the triangles *p*C*k*,
*p*C*n*, in a given time, are of a given magnitude, *kr*
and *mr*, and their difference *mk*, and their sum *ms*,
are reciprocally as the altitude *p*C, and therefore the
rectangle *mk × ms* is reciprocally as the square of the
altitude *p*C. But, moreover, *mt* is directly as ½*mt*,
that is, as the altitude *p*C. These are the first ratios of
the nascent lines: and hence mk ×
ms

mt, that is, the nascent lineola *mn*,
and the difference of the forces proportional thereto, are
reciprocally as the cube of the altitude *p*C.
Q.E.D.

Cor. 1. Hence the difference of the forces in
the places P and *p*, or K and *k*, is to the force
with which a body may revolve with a circular motion from R to K, in
the same time that the body P in an immovable orb describes the arc
PK, as the nascent line *mn* to the versed sine of the nascent
arc RK, that is, as mk × ms

mt to rk^{2}

2kC, or as *mk × ms* to the
square of *rk*; that is, if we take given quantities F and G
in the same ratio to one another as the angle VCP bears to the angle
VC*p*, as GG − FF to FF. And, therefore, if from the centre C,
with any distance CP or C*p*, there be described a circular
sector equal to the whole area VPC, which the body revolving
in an immovable orbit has by a radius drawn to the centre described in
any certain time, the difference of the forces, with which the body P
revolves in an immovable orbit, and the body *p* in a movable
orbit, will be to the centripetal force, with which another body by a
radius drawn to the centre can uniformly describe that sector in the
same time as the area VPC is described, as GG − FF to FF. For that
sector and the area *p*C*k* are to one another as the
times in which they are described.

Cor. 2. If the orbit VPK be an ellipsis,
having its focus C, and its highest apsis V, and we suppose the the
ellipsis *upk* similar and equal to it, so that *p*C
may be always equal to PC, and the angle VC*p* be to the angle
VCP in the given ratio of G to F; and for the altitude PC or *p*C
we put A, and 2R for the latus rectum of the ellipsis, the force with
which a body may be made to revolve in a movable ellipsis will be as
FF

AA+RGG − RFF

A^{3}, and *vice versa*.
Let the force with which a body may revolve in an immovable ellipsis
be expressed by the quantity FF

AA, and the force in V will be FF

CV^{2}. But the force with
which a body may revolve in a circle at the distance CV, with the same
velocity as a body revolving in an ellipsis has in V, is to the force
with which a body revolving in an ellipsis is acted upon in the apsis
V, as half the latus rectum of the ellipsis to the semi-diameter CV of
the circle, and therefore is as RFF

CV^{3}; and the force which is
to this, as GG − FF to FF, is as RGG
− RFF

CV^{3} : and this force (by
Cor. 1 of this Prop.) is the difference of the forces in V, with which
the body P revolves in the immovable ellipsis VPK, and the body *p*
in the movable ellipsis *upk*. Therefore since by this Prop,
that difference at any other altitude A is to itself at the altitude
CV as 1

A^{3} to 1

CV^{3}, the same difference
in every altitude A will be as RGG
− RFF

A^{3}. Therefore to the force
FF

AA, by which the body may revolve in an
immovable ellipsis VPK add the excess
RGG − RFF

A^{3}, and the sum will be the
whole force FF

AA+RGG − RFF

A^{3} by which a body may
revolve in the same time in the movable ellipsis *upk*.

Cor. 3. In the same manner it will be found,
that, if the immovable orbit VPK be an ellipsis having its centre in
the centre of the forces C, and there be supposed a movable ellipsis *upk*,
similar, equal, and concentrical to it; and 2R be the principal latus
rectum of that ellipsis, and 2T the latus transversum, or greater
axis; and the angle VC*p* be continually to the angle VCP as G
to F; the forces with which bodies may revolve in the immovable and
movable ellipsis, in equal times, will be as FFA

T^{3} and FFA

T^{3}+RGG − RFF

A^{3} respectively.

Cor. 4. And universally, if the greatest
altitude CV of the body be called T, and the radius of the curvature
which the orbit VPK has in V, that is, the radius of a circle equally
curve, be called R, and the centripetal force with which a body may
revolve in any immovable trajectory VPK at the place V be called
VFF

TT, and in other places P be
indefinitely styled X; and the altitude CP be called A, and G be taken
to F in the given ratio of the angle VC*p* to the angle VCP;
the centripetal force with which the same body will perform the same
motions in the same time, in the same trajectory *upk*
revolving with a circular motion, will be as the sum of the forces
X+VRGG − VRFF

A^{3}.

Cor. 5. Therefore the motion of a body in an immovable orbit being given, its angular motion round the centre of the forces may be increased or diminished in a given ratio; and thence new immovable orbits may be found in which bodies may revolve with new centripetal forces.

Cor. 6. Therefore if there be erected the
line VP of an indeterminate length, perpendicular to the line CV given
by position, and CP be drawn, and C*p* equal to it, making the
angle VC*p* having a given ratio to the angle VCP, the force
with which a body may revolve in the curve line V*pk*, which
the point *p* is continually describing, will be reciprocally
as the cube of the altitude C*p*. For the body P, by its *vis
inertiae* alone, no other force impelling it, will proceed
uniformly in the right line VP. Add, then, a force tending to the
centre C reciprocally as the cube of the altitude CP or C*p*,
and (by what was just demonstrated) the body
will deflect from the rectilinear motion into the curve line V*pk*.
But this curve V*pk* is the same with the curve VPQ found in
Cor. 3, Prop XLI, in which, I said, bodies attracted with such forces
would ascend obliquely.

*To find the motion of the apsides in orbits approaching very near to circles.*

This problem is solved arithmetically by reducing the orbit, which a
body revolving in a movable ellipsis (as in Cor. 2 and 3 of the above
Prop.) describes in an immovable plane, to the figure of the orbit
whose apsides are required; and then seeking the apsides of the orbit
which that body describes in an immovable plane. But orbits acquire
the same figure. if the centripetal forces with which they are
described, compared between themselves, are made proportional at equal
altitudes. Let the point V be the highest apsis, and write T for the
greatest altitude CV, A for any other altitude CP or C*p*, and
X for the difference of the altitudes CV − CP; and the force with
which a body moves in an ellipsis revolving about its focus C (as in
Cor. 2), and which in Cor. 2 was as
FF

AA + RGG − RFF

A^{3}, that is as,
FFA + RGG − RFF

A^{3},
by substituting T − X for A, will become as RGG − RFF + TFF − FFX

A^{3}. In like manner any other
centripetal force is to be reduced to a fraction whose denominator is
A³, and the numerators are to be made analogous by collating together
the homologous terms. This will be made plainer by Examples.

Example 1. Let us suppose the centripetal
force to be uniform, and therefore as
A^{3}

A^{3} or, writing T − X for A
in the numerator, as T^{3}
− 3TTX+3TXX − X^{3}

A^{3}. Then collating together
the correspondent terms of the numerators, that is, those that consist
of given quantities, with those of given quantities, and those of
quantities not given with those of quantities not given, it will
become RGG − RFF + TFF to T³ as − FFX to 3TTX + 3TXX − X³, or as −FF
to −3TT + 3TX − XX. Now since the orbit is supposed extremely near to
a circle, let it coincide with a circle; and because in that case R
and T become equal, and X is infinitely diminished, the last ratios
will be, as RGG to T², so −FF to −3TT, or as GG to TT, so FF to 3TT;
and again, as GG to FF, so TT to 3TT, that is, as 1 to 3; and
therefore G is to F, that is, the angle VC*p* to the angle VCP,
as 1 to √3. Therefore since the body, in an immovable ellipsis, in descending from the upper to the lower apsis, describes an angle,
if I may so speak, of 180 deg., the other body in a movable ellipsis,
and therefore in the immovable orbit we are treating of, will in its
descent from the upper to the lower apsis, describe an angle VC*p*
of 180

√3 deg. And this comes to pass by reason of the likeness of
this orbit which a body acted upon by an uniform centripetal force
describes, and of that orbit which a body performing its circuits in a
revolving ellipsis will describe in a quiescent plane. By this
collation of the terms, these orbits are made similar; not
universally, indeed, but then only when they approach very near to a
circular figure. A body, therefore revolving with an uniform
centripetal force in an orbit nearly circular, will always describe an
angle of 180

√3 deg., or 103 deg., 55 m., 23 sec., at the centre; moving
from the upper apsis to the lower apsis when it has once described
that angle, and thence returning to the upper apsis when it has
described that angle again; and so on *in infinitum*.

Exam. 2. Suppose the centripetal force to be
as any power of the altitude A, as, for example, A^{n−3}, or
A^{n}

A^{3}; where *n* − 3 and *n* signify
any indices of powers whatever, whether integers or fractions,
rational or surd, affirmative or negative. That numerator A^{n}
or (T − X)^{n} being reduced to an indeterminate series by my
method of converging series, will become T^{n}
− nXT^{n−1} + nn − n

2XXT^{n−2}, &c. And conferring these
terms with the terms of the other numerator RGG − RFF + TFF − FFX, it
becomes as RGG − RFF + TFF to T^{n}, so − FF to −nT^{n−1}
+ nn − n

2XT^{n−2}, &c. And taking the last
ratios where the orbits approach to circles, it becomes as RGG to T^{n},
so − FF to −*n*T^{n−1}, or as GG to T^{n−1}, so
FF to *n*T^{n−}; and again, GG to FF, so T^{n−1}
to *n*T^{n−1}, that is, as 1 to *n*; and
therefore G is to F, that is the angle VC*p* to the angle VCP,
as 1 to √n. Therefore since the angle VCP, described in the descent of
the body from the upper apsis to the lower apsis in an ellipsis, is of
180 deg., the angle VC*p*, described in the descent of the body
from the upper apsis to the lower apsis in an orbit nearly circular
which a body describes with a centripetal force proportional to the
power A^{n−3}, will be equal to an angle of 180

√n deg., and this angle being repeated, the body will return
from the lower to the upper apsis, and so on *in infinitum*.
As if the centripetal force be as the distance of the body from the
centre, that is, as A, or A^{4}

A^{3}, *n* will be
equal to 4, and √n equal to 2; and therefore the angle between
the upper and the lower apsis will be equal to 180

2 deg., or 90 deg. Therefore the body having performed a
fourth part of one revolution, will arrive at the lower apsis, and
having performed another fourth part, will arrive at the upper apsis,
and so on by turns *in infinitum*. This appears also from
Prop. X. For a body acted on by this centripetal force will revolve in
an immovable ellipsis, whose centre is the centre of force. If the
centripetal force is reciprocally as the distance, that is, directly
as 1

A or A^{2}

A^{3}, *n* will be equal to 2; and therefore
the angle between the upper and lower apsis will be 180

√2 deg., or 127 deg., 16 min., 45 sec.; and therefore a body
revolving with such a force, will by a perpetual repetition of this
angle, move alternately from the upper to the lower and from the lower
to the upper apsis for ever. So, also, if the centripetal force be
reciprocally as the biquadrate root of the eleventh power of the
altitude, that is, reciprocally as A^{11/4} , and, therefore,
directly as 1

A^{11/4} or as A^{1/4}

A^{3}, *n* will be equal to ¼, and 180

√n deg. will be equal to 360 deg.; and therefore the body
parting from the upper apsis, and from thence perpetually descending,
will arrive at the lower apsis when it has completed one entire
revolution; and thence ascending perpetually, when it has completed
another entire revolution, it will arrive again at the upper apsis;
and so alternately for ever.

Exam. 3. Taking *m* and *n*
for any indices of the powers of the altitude, and *b* and *c*
for any given numbers, suppose the centripetal force to be as bA^{m} − ca^{n}

A^{3}, that is, as b
into (T − X)^{m} + c into (T − X)^{n}

A^{3} or (by the method of converging series
above-mentioned) as

bT^{m}+cT^{n}
− mbXT^{m−1}ncXT^{n−1} + mm − m

2bXXT^{m−2} + nn
− n

2cXXT^{n−2}

A^{3} &c.

and comparing the terms of the numerators, there will arise RGG
− RFF + TFF to bT^{m} + cT^{n} as −FF
to −*mb*T^{m−1} − *nc*T^{n}
+ mm − m

2bXT^{m−2} + nn
− n

2cXT^{n−2}, &c. And taking the last
ratios that arise when the orbits come to a circular form, there will
come forth GG to *b*T^{m−1} + *c*T^{n−1}
as FF to *mb*T^{m−1} + *nc*T^{n−1};
and again, GG to FF as *b*T^{m−1} +
*c*T^{n−1} to *mb*T^{n−1}
+ *nc*T^{n−1}. This proportion, by expressing
the greatest altitude CV or T arithmetically by unity, becomes, GG to
FF as *b + c* to *mb + nc*, and therefore as 1
to mb + nc

b + c. Whence G becomes to F, that is, the angle VC*p*
to the angle VCP, as 1 to √
mb + nc

b + c. And therefore since
the angle VCP between the upper and the lower apsis, in an immovable
ellipsis, is of 180 deg., the angle VC*p* between the same
apsides in an orbit which a body describes with a centripetal force,
that is, as bA^{m} + cA^{n}

A^{3}, will be equal to an angle of 180
√b + c

mb + nc deg. And by the same
reasoning, if the centripetal force be as bA^{m}
− cA^{n}

A^{3}, the angle between the apsides will be found
equal to 180√
b − c

mb − nc. After the same
manner the Problem is solved in more difficult cases. The quantity to
which the centripetal force is proportional must always be resolved
into a converging series whose denominator is A³. Then the given part
of the numerator arising from that operation is to be supposed in the
same ratio to that part of it which is not given, as the given part of
this numerator RGG − RFF + TFF − FFX is to
that part of the same numerator which is not given. And taking away
the superfluous quantities, and writing unity for T, the proportion of
G to F is obtained.

Cor. 1 . Hence if the centripetal force be as
any power of the altitude, that power may be found from the motion of
the apsides; and so contrariwise. That is, if the whole angular
motion, with which the body returns to the same apsis, be to the
angular motion of one revolution, or 360 deg., as any number as *m*
to another as *n*, and the altitude called A; the force will
be as the power A^{nnmm−3} of the altitude
A; the index of which power is nn

mm−3. This appears by the
second example. Hence it is plain that the force in its recess from
the centre cannot decrease in a greater than a triplicate ratio of the
altitude. A body revolving with such a force and parting from the
apsis, if it once begins to descend, can never arrive at the lower
apsis or least altitude, but will descend to the centre, describing
the curve line treated of in Cor. 3, Prop. XLI. But if it should, at
its parting from the lower apsis, begin to ascend never so little, it
will ascend *in infinitum*, and never come to the upper apsis;
but will describe the curve line spoken of in the same Cor., and Cor.
6; Prop. XLIV. So that where the force in its recess from the centre
decreases in a greater than a triplicate ratio of the altitude, the
body at its parting from the apsis, will either descend to the centre,
or ascend in infinitum, according as it descends or ascends at the
beginning of its motion. But if the force in its recess from the
centre either decreases in a less than a triplicate ratio of the
altitude, or increases in any ratio of the altitude whatsoever, the
body will never descend to the centre, but will at some time arrive at
the lower apsis; and, on the contrary, if the body alternately
ascending and descending from one apsis to another never comes to the
centre, then either the force increases in the recess from the centre,
or it decreases in a less than a triplicate ratio of the altitude; and
the sooner the body returns from one apsis to another, the farther is
the ratio of the forces from the triplicate ratio. As if the body
should return to and from the upper apsis by an alternate descent and
ascent in 8 revolutions, or in 4, or 2, or 1½; that is, if *m*
should be to *n* as 8, or 4, or 2, or 1½ to 1, and therefore
nn

mm−3, be ^{1}/_{64}
− 3, or ^{1}/_{16} − 3, or
^{1}/_{4} − 3,
or ^{4}/_{9}
− 3; then the force will be as A^{1/64−3};
or A^{1/16−3};
or A^{1/4−3};
or A^{4/9−3};
that is, it will be reciprocally as A^{3−1/64},
or A^{3−1/16},
or A^{3−1/4},
or A^{3−4/9}.
If the body after each revolution returns to the same apsis, and the
apsis remains unmoved, then *m* will be to *n* as 1 to
1, and therefore A^{nn/mm−3}
will be equal to A^{−2}, or 1/AA;
and therefore the decrease of the forces will be in a duplicate ratio
of the altitude; as was demonstrated above. If the body in three
fourth parts, or two thirds, or one third, or one fourth part of an
entire revolution, return to the same apsis; *m* will be to *n*
as ¾ or ⅔ or ⅓ or ¼ to 1, and therefore A^{nn/mm−3}
is equal to A^{16/9−3},
or A^{9/4−3},
or A^{9−3}, or A^{16−3};
and therefore the force is either reciprocally as A^{11/9},
or directly as A^{6} or A^{13}. Lastly if the body in
its progress from the upper apsis to the same upper apsis again, goes
over one entire revolution and three deg. more, and therefore that
apsis in each revolution of the body moves three deg. *in
consequentia*; then *m* will be to *n* as 363
deg. to 360 deg. or as 121 to 120, and therefore A^{nn/mm−3}
will be equal to A^{−29523/14641},
and therefore the centripetal force will be reciprocally as A^{29523/14641},
or reciprocally as A^{24/2 4 3}
very nearly. Therefore the centripetal force decreases in a ratio
something greater than the duplicate; but approaching 59¾ times nearer
to the duplicate than the triplicate.

Cor. 2. Hence also if a body, urged by a
centripetal force which is reciprocally as the square of the altitude,
revolves in an ellipsis whose focus is in the centre of the forces;
and a new and foreign force should be added to or subducted from this
centripetal force, the motion of the apsides arising from that foreign
force may (by the third Example) be known; and so on the contrary. As
if the force with which the body revolves in the ellipsis be
as 1

AA; and the foreign force subducted
as *c*A, and therefore the remaining force as A − cA^{4}

A^{3}; then (by the third
Example) *b* will be equal to 1. *m* equal to 1, and *n*
equal to 4; and therefore the angle of revolution between the apsides
is equal to 180√(1
− c

1 − 4c) deg. Suppose that
foreign force to be 357.45 parts less than the other force with which
the body revolves in the ellipsis; that is, *c* to be 100

35745; A or T being equal to 1; and
then 180√(1 − c

1 − 4c) will be 180√(
35645

35345) or 180.7623, that is,
180 deg., 45 min., 44 sec. Therefore the body, parting from the upper
apsis, will arrive at the lower apsis with an angular motion of 180
deg., 45 min., 44 sec, and this angular motion being repeated, will
return to the upper apsis; and therefore the upper apsis in each
revolution will go forward 1 deg., 31 min., 28 sec. The apsis of the
moon is about twice as swift.

So much for the motion of bodies in orbits whose planes pass through the centre of force. It now remains to determine those motions in eccentrical planes. For those authors who treat of the motion of heavy bodies used to consider the ascent and descent of such bodies, not only in a perpendicular direction, but at all degrees of obliquity upon any given planes; and for the same reason we are to consider in this place the motions of bodies tending to centres by means of any forces whatsoever, when those bodies move in eccentrical planes. These planes are supposed to be perfectly smooth and polished, so as not to retard the motion of the bodies in the least. Moreover, in these demonstrations, instead of the planes upon which those bodies roll or slide, and which are therefore tangent planes to the bodies, I shall use planes parallel to them, in which the centres of the bodies move, and by that motion describe orbits. And by the same method I afterwards determine the motions of bodies performed in curve superficies.

*Any kind of centripetal force being supposed, and the centre of
force, and any plane whatsoever in which the body revolves, being
given, and the quadratures of curvilinear figures being allowed;
it is required to determine the motion of a body going off from a
given place, with a given velocity, in the direction of a given
right line in that plane.*

Let S be the centre of force, SC the least distance of that centre from the given plane, P a body issuing from the place P in the direction of the right line PZ, Q the same body revolving in its trajectory, and PQR the trajectory itself which is required to be found, described in that given plane. Join CQ, QS, and if in QS we take SV proportional to the centripetal force with which the body is attracted towards the centre S, and draw VT parallel to CQ, and meeting SC in T; then will the force SV be resolved into two (by Cor. 2, of the Laws of Motion), the force ST, and the force TV; of which ST attracting the body in the direction of a line perpendicular to that plane, does not at all change its motion in that plane. But the action of the other force TV, coinciding with the position of the plane itself, attracts the body directly towards the given point C in that plane; and therefore causes the body to move in this plane in the same manner as if the force ST were taken away, and the body were to revolve in free space about the centre C by means of the force TV alone. But there being given the centripetal force TV with which the body Q revolves in free space about the given centre C, there is given (by Prop. XLII) the trajectory PQR which the body describes; the place Q, in which the body will be found at any given time; and, lastly, the velocity of the body in that place Q. And so è contra. Q.E.I.

*Supposing the centripetal force to be proportional to the
distance of the body from the centre; all bodies revolving in any
planes whatsoever will describe ellipses, and complete their
revolutions in equal times; and those which move in right lines,
running backwards and forwards alternately, will complete their
several periods of going and returning in the same times.*

For letting all things stand as in the foregoing Proposition, the force SV, with which the body Q revolving in any plane PQR is attracted towards the centre S, is as the distance SQ; and therefore because SV and SQ, TV and CQ are proportional, the force TV with which the body is attracted towards the given point C in the plane of the orbit is as the distance CQ. Therefore the forces with which bodies found in the plane PQR are attracted towards the point C, are in proportion to the distances equal to the forces with which the same bodies are attracted every way towards the centre S; and therefore the bodies will move in the same times, and in the same figures, in any plane PQR about the point C, as they would do in free spaces about the centre S; and therefore (by Cor. 2, Prop. X, and Cor. 2, Prop. XXXVIII.) they will in equal times either describe ellipses in that plane about the centre C, or move to and fro in right lines passing through the centre C in that plane; completing the same periods of time in all cases. Q.E.D.

The ascent and descent of bodies in curve superficies has a near relation to these motions we have been speaking of. Imagine curve lines to be described on any plane, and to revolve about any given axes passing through the centre of force, and by that revolution to describe curve superficies; and that the bodies move in such sort that their centres may be always found in those superficies. If those bodies reciprocate to and fro with an oblique ascent and descent, their motions will be performed in planes passing through the axis, and therefore in the curve lines, by whose revolution those curve superficies were generated. In those cases, therefore, it will be sufficient to consider the motion in those curve lines.

*If a wheel stands upon the outside of a globe at right angles
thereto, and revolving about its own axis goes forward in a great
circle, the length of the curvilinear path which any point, given
in the perimeter of the wheel, hath described since the time that
it touched the globe (which curvilinear path we may call the
cycloid or epicycloid), will be to double the versed sine of half
the arc which since that time has touched the globe in passing
over it, as the sum of the diameters of the globe and the wheel to
the semi-diameter of the globe.*

*If a wheel stand upon the inside of a concave globe at right
angles thereto, and revolving about its own axis go forward in one
of the great circles of the globe, the length of the curvilinear
path which any point, given in the perimeter of the wheel, hath
described since it touched the globe, will be to the double of the
versed sine of half the arc which in all that time has touched the
globe in passing over it, as the difference of the diameters of
the globe and the wheel to the semi-diameter of the globe.*

Let ABL be the globe, C its centre, BPV the wheel insisting thereon,
E the centre of the wheel, B the point of contact, and P the given
point in the perimeter of the wheel. Imagine this wheel to proceed in
the great circle ABL from A through B towards L, and in its progress
to revolve in such a manner that the arcs AB, PB may be always equal
one to the other, and the given point P in the perimeter of the wheel
may describe in the
mean time the curvilinear path AP. Let AP be the whole curvilinear
path described since the wheel touched the globe in A, and the length
of this path AP will be to twice the versed sine of the arc ½PB as 2CE
to CB. For let the right line CE (produced if need be) meet the wheel
in V, and join CP, BP, EP, VP; produce CP, and let fall thereon the
perpendicular VF. Let PH, VH, meeting in H, touch the circle in P and
V, and let PH cut VF in G, and to VP let fall the perpendiculars GI,
HK. From the centre C with any interval let there be described the
circle *nom*, cutting the right line CP in *n*, the
perimeter of the wheel BP in *o*, and the curvilinear path AP
in *m*; and from the centre V with the interval V*o*
let there be described a circle cutting VP produced in *q*.

Because the wheel in its progress always revolves about the point of
contact B, it is manifest that the right line BP is perpendicular to
that curve line AP which the point P of the wheel describes, and
therefore that the right line VP will touch this curve in the point P.
Let the radius of the circle *nom* be gradually increased or
diminished so that at last it become equal to the distance CP; and by
reason of the similitude of the evanescent figure P*nomq*, and
the figure PFGVI, the ultimate ratio of the evanescent lineolae P*m*,
P*n*, P*o*, P*q*, that is, the ratio of the
momentary mutations of the curve AP, the right line CP, the circular
arc BP, and the right line VP, will be the
same as of the lines PV, PF, PG, PI, respectively. But since VF is
perpendicular to CF, and VH to CV, and therefore the angles HVG, VCF
equal; and the angle VHG (because the angles of the quadrilateral
figure HVEP are right in V and P) is equal to the angle CEP, the
triangles VHG, CEP will be similar; and thence it will come to pass
that as EP is to CE so is HG to HV or HP, and so KI to KP, and by
composition or division as CB to CE so is PI to PK, and doubling the
consequents as CB to 2CE so PI to PV, and so is P*q* to P*m*.
Therefore the decrement of the line VP, that is, the increment of the
line BV − VP to the increment of the curve line AP is in a given ratio
of CB to 2CE, and therefore (by Cor. Lem. IV) the lengths BV − VP and
AP, generated by those increments, are in the same ratio. But if BV be
radius, VP is the cosine of the angle BVP or ½BEP, and therefore BV −
VP is the versed sine of the same angle, and therefore in this wheel,
whose radius is ½BV, BV − VP will be double the versed sine of the arc
½BP. Therefore AP is to double the versed sine of the arc ½BP as 2CE
to CB. Q.E.D.

The line AP in the former of these Propositions we shall name the cycloid without the globe, the other in the latter Proposition the cycloid within the globe, for distinction sake.

Cor. 1. Hence if there be described the entire cycloid ASL, and the same be bisected in S, the length of the part PS will be to the length PV (which is the double of the sine of the angle VBP, when EB is radius) as 2CE to CB, and therefore in a given ratio.

Cor. 2. And the length of the semi-perimeter of the cycloid AS will be equal to a right line which is to the diameter of the wheel BV as 2CE to CB.

*To cause a pendulous body to oscillate in a given cycloid.*

Let there be given within the globe QVS described with the centre C, the cycloid QRS, bisected in R, and meeting the superficies of the globe with its extreme points Q and S on either hand. Let there be drawn CR bisecting the arc QS in O, and let it be produced to A in such sort that CA may be to CO as CO to CR. About the centre C, with the interval CA, let there be described an exterior globe DAF; and within this globe, by a wheel whose diameter is AO, let there be described two semi-cycloids AQ, AS, touching the interior globe in Q and S, and meeting the exterior globe in A. From that point A, with a thread APT in length equal to the line AR, let the body T depend, and oscillate in such manner between the two semi-cycloids AQ, AS, that, as often as the pendulum parts from the perpendicular AR, the upper part of the thread AP may be applied to that semi-cycloid APS towards which the motion tends, and fold itself round that curve line, as if it were some solid obstacle, the remaining part of the same thread PT which has not yet touched the semi-cycloid continuing straight. Then will the weight T oscillate in the given cycloid QRS. Q.E.F.

For let the thread PT meet the cycloid QRS in T, and the circle QOS in V, and let CV be drawn; and to the rectilinear part of the thread PT from the extreme points P and T let there be erected the perpendiculars BP, TW, meeting the right line CV in B and W. It is evident, from the construction and generation of the similar figures AS, SR, that those perpendiculars PB, TW, cut off from CV the lengths VB, VW equal the diameters of the wheels OA, OR. Therefore TP is to VP (which is double the sine of the angle VBP when ½BV is radius) as BW to BV, or AO + OR to AO, that is (since CA and CO, CO and CR, and by division AO and OR are proportional), as CA + CO to CA, or, if BV be bisected in E, as 2CE to CB. Therefore (by Cor. 1, Prop. XLIX), the length of the rectilinear part of the thread PT is always equal to the arc of the cycloid PS, and the whole thread APT is always equal to the half of the cycloid APS, that is (by Cor. 2, Prop. XLIX), to the length AR. And therefore contrariwise, if the string remain always equal to the length AR, the point T will always move in the given cycloid QRS. Q.E.D.

Cor. The string AR is equal to the semi-cycloid AS, and therefore has the same ratio to AC the semi-diameter of the exterior globe as the like semi-cycloid SR has to CO the semi-diameter of the interior globe.

*If a centripetal force tending on all sides to the centre*
C *of a globe, be in all places as the distance of the place
from the centre, and by this force alone acting upon it, the body*
T *oscillate (in the manner above described) in the perimeter of
the cycloid* QRS*; I say, that all the oscillations, how
unequal soever in themselves, will be performed in equal times.*

For upon the tangent TW infinitely produced let fall the
perpendicular CX, and join CT. Because the centripetal force with
which the body T is impelled towards C is as the distance CT, let this
(by Cor. 2, of the Laws) be resolved into the parts CX, TX, of which
CX impelling the body directly from P stretches the thread PT, and by
the resistance the thread makes to it is totally employed, producing
no other effect; but the other part TX, impelling the body
transversely or towards X, directly accelerates the motion in the
cycloid. Then it is plain that the acceleration of the body,
proportional to this accelerating force, will be every
moment as the length TX, that is (because CV,
WV, and TX, TW proportional to them are given), as the length TW, that
is (by Cor. 1, Prop. XLIX) as the length of the arc of the cycloid TR.
If therefore two pendulums APT, A*p*t, be unequally drawn aside
from the perpendicular AR, and let fall together, their accelerations
will be always as the arcs to be described TR, *t*R. But the
parts described at the beginning of the motion are as the
accelerations, that is, as the wholes that are to be described at the
beginning, and therefore the parts which remain to be described, and
the subsequent accelerations proportional to those parts, are also as
the wholes, and so on. Therefore the accelerations, and consequently
the velocities generated, and the parts described with those
velocities; and the parts to be described, are always as the wholes;
and therefore the parts to be described preserving a given ratio to
each other will vanish together, that is, the two bodies oscillating
will arrive together at the perpendicular AR. And since on the other
hand the ascent of the pendulums from the lowest place R through the
same cycloidal arcs with a retrograde motion, is retarded in the
several places they pass through by the same forces by which their
descent was accelerated; it is plain that the velocities of their
ascent and descent through the same arcs are equal, and consequently
performed in equal times; and, therefore, since the two parts of the
cycloid RS and RQ lying on either side of the perpendicular are
similar and equal, the two pendulums will perform as well the wholes
as the halves of their oscillations in the same times.
Q.E.D.

Cor. The force with which the body T is accelerated or retarded in any place T of the cycloid, is to the whole weight of the same body in the highest place S or Q as the arc of the cycloid TR is to the arc SR or QR.

*To define the velocities of the pendulums in the several
places, and the times in which both the entire oscillations, and
the several parts of them are performed.*

About any centre G, with the interval GH equal to the arc of the
cycloid RS, describe a semi-circle HKM bisected by the semi-diameter
GK. And if a centripetal force proportional to the distance of the
places from the centre tend to the centre G, and it be in the
perimeter HIK equal to the centripetal force in the perimeter of the
globe QOS tending towards its centre, and at the same time that the
pendulum T is let fall from the highest place S, a body, as L, is let
fall from H to G; then because the
forces which act upon the bodies are equal at the beginning, and
always proportional to the spaces to be described TR, LG, and
therefore if TR and LG are equal, are also equal in the places T and
L, it is plain that those bodies describe at the beginning equal
spaces ST, HL, and therefore are still acted upon equally, and
continue to describe equal spaces. Therefore by Prop. XXXVIII, the
time in which the body describes the arc ST is to the time of one
oscillation, as the arc HI the time in which the body H arrives at L,
to the semi-periphery HKM, the time in which the body H will come to
M. And the velocity of the pendulous body in the place T is to its
velocity in the lowest place R, that is, the velocity of the body H in
the place L to its velocity in the place G; or the momentary increment
of the line HL to the momentary increment of the line HG (the arcs HI,
HK increasing with an equable flux) as the ordinate LI to the radius
GK, or as √(SR^{2} − TR^{2})
to SR. Hence, since in unequal oscillations there are described in
equal time arcs proportional to the entire arcs of the oscillations,
there are obtained from the times given, both the velocities and the
arcs described in all the oscillations universally. Which was first
required.

Let now any pendulous bodies oscillate in different cycloids
described within different globes, whose absolute forces are also
different; and if the absolute force of any globe QOS be called V, the
accelerative force with which the pendulum is acted on in the
circumference of this globe, when it begins to move directly towards
its centre, will be as the distance of the pendulous body from that
centre and the absolute force of the globe conjunctly, that is, as CO
× V. Therefore the lineola HY, which is as this accelerated force CO ×
V, will be described in a given time; and if there be erected the
perpendicular YZ meeting the circumference in Z, the nascent arc HZ
will denote that given time. But that nascent arc HZ is in the
subduplicate ratio of the rectangle GHY, and therefore as √(GH
× CO × V). Whence the time of an entire oscillation in the
cycloid QRS (it being as the semi-periphery HKM, which denotes that
entire oscillation, directly; and as the arc HZ which in like manner
denotes a given time inversely) will be as GH directly and √(GH
× CO × V) inversely; that is, because GH and SR are equal, as
√(SR

CO × V), or (by Cor. Prop.
L,) as √(AR

AC × V). Therefore the
oscillations in all globes and cycloids, performed with what absolute
forces soever, are in a ratio compounded of the subduplicate ratio of
the length of the string directly, and the subduplicate ratio of the
distance between the point of suspension and the centre of the globe
inversely, and the subduplicate ratio of the absolute force of the
globe inversely also. Q.E.I.

Cor. 1. Hence also
the times of oscillating, falling, and revolving bodies may be
compared among themselves. For if the diameter of the wheel with which
the cycloid is described within the globe is supposed equal to the
semi-diameter of the globe, the cycloid will become a right line
passing through the centre of the globe, and the oscillation will be
changed into a descent and subsequent ascent in that right line.
Whence there is given both the time of the descent from any place to
the centre, and the time equal to it in which the body revolving
uniformly about the centre of the globe at any distance describes an
arc of a quadrant. For this time (by Case 2) is to the time of half
the oscillation in any cycloid QRS as 1 to √(
AR

AC).

Cor. 2. Hence also follow what Sir *Christopher
Wren* and M. *Huygens* have discovered concerning the
vulgar cycloid. For if the diameter of the globe be infinitely
increased, its sphaerical superficies will be changed into a plane,
and the centripetal force will act uniformly in the direction of lines
perpendicular to that plane, and this cycloid of our's will become the
same with the common cycloid. But in that case the length of the arc
of the cycloid between that plane and the describing point will become
equal to four times the versed sine of half the arc of the wheel
between the same plane and the describing point, as was discovered by
Sir *Christopher Wren*. And a pendulum between two such
cycloids will oscillate in a similar and equal cycloid in equal times,
as M. *Huygens* demonstrated. The descent of heavy bodies also
in the time of one oscillation will be the same as M. *Huygens*
exhibited.

The propositions here demonstrated are adapted to the true constitution of the Earth, in so far as wheels moving in any of its great circles will describe, by the motions of nails fixed in their perimeters, cycloids without the globe; and pendulums, in mines and deep caverns of the Earth, must oscillate in cycloids within the globe, that those oscillations may be performed in equal times. For gravity (as will be shewn in the third book) decreases in its progress from the superficies of the Earth; upwards in a duplicate ratio of the distances from the centre of the Earth; downwards in a simple ratio of the same.

*Granting the quadratures of curvilinear figures, it is required
to find the forces with which bodies moving in given curve lines
may always perform their oscillations in equal times.*

Let the body T oscillate in any curve line STRQ, whose axis is AR passing through the centre of force C. Draw TX touching that curve in any place of the body T, and in that tangent TX take TY equal to the arc TR. The length of that arc is known from the common methods used for the quadratures of figures. From the point Y draw the right line YZ perpendicular to the tangent. Draw CT meeting that perpendicular in Z, and the centripetal force will be proportional to the right line TZ. Q.E.I.

For if the force with which the body is attracted from T towards C be expressed by the right line TZ taken proportional to it, that force will be resolved into two forces TY, YZ, of which YZ drawing the body in the direction of the length of the thread PT, does not at all change its motion; whereas the other force TY directly accelerates or retards its motion in the curve STRQ. Wherefore since that force is as the space to be described TR, the accelerations or retardations of the body in describing two proportional parts (a greater and a less) of two oscillations, will be always as those parts, and therefore will cause those parts to be described together. But bodies which continually describe together parts proportional to the wholes, will describe the wholes together also. Q.E.D.

Cor. 1. Hence if the body T, hanging by a rectilinear thread AT from the centre A, describe the circular arc STRQ, and in the mean time be acted on by any force tending downwards with parallel directions, which is to the uniform force of gravity as the arc TR to its sine TN, the times of the several oscillations will be equal. For because TZ, AR are parallel, the triangles ATN, ZTY are similar; and therefore TZ will be to AT as TY to TN; that is, if the uniform force of gravity be expressed by the given length AT, the force TZ, by which the oscillations become isochronous, will be to the force of gravity AT, as the arc TR equal to TY is to TN the sine of that arc.

Cor. 2. And therefore in clocks, if forces were impressed by some machine upon the pendulum which preserves the motion, and so compounded with the force of gravity that the whole force tending downwards should be always as a line produced by applying the rectangle under the arc TR and the radius AR to the sine TN, all the oscillations will become isochronous.

*Granting the quadratures of curvilinear figures, it is required
to find the times in which bodies by means of any centripetal
force will descend or ascend in any curve lines described in a
plane passing through the centre of force.*

Let the body descend from any place S, and move in any curve ST*t*R
given in a plane passing through the centre of force C. Join CS, and let
it be divided into innumerable equal parts, and let D*d* be one of
those parts. From the centre C, with the intervals CD, C*d*,
let the circles DT, *dt* be described, meeting the curve line
ST*t*R in T and *t*. And because the law of centripetal
force is given, and also the altitude CS from which the body at first
fell, there will be given the velocity of the body in any other
altitude CT (by Prop. XXXIX). But the time in which the body describes
the lineola T*t* is as the length of that lineola, that is, as
the secant of the angle *t*TC directly, and the velocity
inversely. Let the ordinate DN, proportional to this time, be made
perpendicular to the right line CS at the point D, and because D*d*
is given, the rectangle D*d* × DN, that is, the area DN*nd*,
will be proportional to the same time. Therefore if PN*n* be a
curve line in which the point N is perpetually found, and its
asymptote be the right line SQ standing upon the line CS at right
angles, the area SQPND will be proportional to the time in which the
body in its descent hath described the line ST; and therefore that
area being found, the time is also given. Q.E.I.

*If a body move in any curve superficies, whose axis passes
through the centre of force, and from the body a perpendicular be
let fall upon the axis; and a line parallel and equal thereto be
drawn from any given point of the axis; I say, that this parallel
line will describe an area proportional to the time.*

Let BKL be a curve superficies, T a body revolving in it, STR a trajectory which the body describes in the same, S the beginning of the trajectory, OMK the axis of the curve superficies, TN a right line let fall perpendicularly from the body to the axis; OP a line parallel and equal thereto drawn from the given point O in the axis; AP the orthographic projection of the trajectory described by the point P in the plane AOP in which the revolving line OP is found; A the beginning of that projection, answering to the point S; TC a right line drawn from the body to the centre; TG a part thereof proportional to the centripetal force with which the body tends towards the centre C; TM a right line perpendicular to the curve superficies; TI a part thereof proportional to the force of pressure with which the body urges the superficies, and therefore with which it is again repelled by the superficies towards M; PTF a right line parallel to the axis and passing through the body, and GF, IH right lines let fall perpendicularly from the points G and I upon that parallel PHTF. I say, now. that the area AOP, described by the radius OP from the beginning of the motion, is proportional to the time. For the force TG (by Cor. 2, of the Laws of Motion) is resolved into the forces TF, FG; and the force TI into the forces TH, HI; but the forces TF, TH, acting in the direction of the line PF perpendicular to the plane AOP, introduce no change in the motion of the body but in a direction perpendicular to that plane. Therefore its motion, so far as it has the same direction with the position of the plane, that is, the motion of the point P, by which the projection AP of the trajectory is described in that plane, is the same as if the forces TF, TH were taken away, and the body were acted on by the forces FG, HI alone; that is, the same as if the body were to describe in the plane AOP the curve AP by means of a centripetal force tending to the centre O, and equal to the sum of the forces FG and HI. But with such a force as that (by Prop. 1) the area AOP will be described proportional to the time. Q.E.D.

Cor. By the same reasoning, if a body, acted on by forces tending to two or more centres in any the same right line CO, should describe in a free space any curve line ST, the area AOP would be always proportional to the time.

*Granting the quadratures of curvilinear figures, and supposing
that there are given both the law of centripetal force tending to
a given centre, and the curve superficies whose axis passes
through that centre; it is required to find the trajectory which a
body will describe in that superficies, when going off from a
given place with a given velocity, and in a given direction in that superficies.*

The last construction remaining, let the body T go from the given
place S, in the direction of a line given by position, and turn into
the trajectory sought STR, whose orthographic projection in the plane
BDO is AP. And from the given velocity of the body in the altitude SC,
its velocity in any other altitude TC will be also given. With that
velocity, in a given moment of time, let the body describe the
particle T*t* of its trajectory, and let P*p* be the
projection of that particle described in the plane AOP. Join O*p*,
and a little circle being described upon the curve superficies about
the centre T with the interval T*t*
let the projection of that little circle in the plane AOP be the
ellipsis *p*Q. And because the magnitude of that little circle
T*t*, and TN or PO its distance from the axis CO is also given,
the ellipsis *p*Q will be given both in kind and magnitude, as
also its position to the right line PO. And since the area PO*p*
is proportional to the time, and therefore given because the time is
given, the angle PO*p* will be given. And thence will be given
*p* the common intersection of the ellipsis and the right line
O*p*, together with the angle OP*p*, in which the
projection AP*p* of the trajectory cuts the line OP. But from
thence (by conferring Prop. XLI, with its 2d Cor.) the manner of
determining the curve AP*p* easily appears. Then from the
several points P of that projection erecting to the plane AOP, the
perpendiculars PT meeting the curve superficies in T, there will be
given the several points T of the trajectory. Q.E.I.

I have hitherto been treating of the attractions of bodies towards an immovable centre; though very probably there is no such thing existent in nature. For attractions are made towards bodies, and the actions of the bodies attracted and attracting are always reciprocal and equal, by Law III; so that if there are two bodies, neither the attracted nor the attracting body is truly at rest, but both (by Cor. 4, of the Laws of Motion), being as it were mutually attracted, revolve about a common centre of gravity. And if there be more bodies, which are either attracted by one single one which is attracted by them again, or which all of them, attract each other mutually, these bodies will be so moved among themselves, as that their common centre of gravity will either be at rest, or move uniformly forward in a right line. I shall therefore at present go on to treat of the motion of bodies mutually attracting each other; considering the centripetal forces as attractions; though perhaps in a physical strictness they may more truly be called impulses. But these propositions are to be considered as purely mathematical; and therefore, laying aside all physical considerations, I make use of a familiar way of speaking, to make myself the more easily understood by a mathematical reader.

*Two bodies attracting each other mutually describe similar
figures about their common centre of gravity, and about each other mutually.*

For the distances of the bodies from their common centre of gravity are reciprocally as the bodies; and therefore in a given ratio to each other: and thence, by composition of ratios, in a given ratio to the whole distance between the bodies. Now these distances revolve about their common term with an equable angular motion, because lying in the same right line they never change their inclination to each other mutually. But right lines that are in a given ratio to each other, and revolve about their terms with an equal angular motion, describe upon planes, which either rest with those terms, or move with any motion not angular, figures entirely similar round those terms. Therefore the figures described by the revolution of these distances are similar. Q.E.D.

*If two bodies attract each other mutually with forces of any
kind, and in the mean time revolve about the common centre of
gravity; I say, that, by the same forces, there may be described
round either body unmoved a figure similar and equal to the
figures which the bodies so moving describe round each other mutually.*

Let the bodies S and P revolve about their common centre of gravity
C, proceeding from S to T, and from P to Q. From the given point *s* let
there be continually drawn *sp,
sq*, equal and parallel to SP, TQ; and the curve *pqv*,
which the point *p* describes in its revolution round the
immovable point *s*, will be similar and equal to the curves
which the bodies S and P describe about each other mutually; and
therefore, by Theor. XX, similar to the curves ST and PQV which the
same bodies describe about their common centre of gravity C; and that
because the proportions of the lines SC, CP, and SP or *sp*,
to each other, are given.

Case 1. The common centre of gravity C (by
Cor. 4, of the Laws of Motion) is either at rest, or moves uniformly
in a right line. Let us first suppose it at rest, and in *s*
and *p* let there be placed two bodies, one immovable in *s*,
the other movable in *p*, similar and equal to the bodies S
and P. Then let the right lines PR and *pr* touch the curves
PQ and *pq* in P and *p*, and produce CQ and *sq*
to R and *r*. And because the figures CPRQ, *sprq* are
similar, RQ will be to *rq* as CP to *sp*, and
therefore in a given ratio. Hence if the force with which the body P
is attracted towards the body S, and by consequence towards the
intermediate point the centre C, were to the force with which the body
*p* is attracted towards the centre *s*, in the same
given ratio, these forces would in equal times attract the
bodies from the tangents PR, *pr* to the arcs PQ, *pq*,
through the intervals proportional to them RQ, *rq*; and
therefore this last force (tending to *s*) would make the body
*p* revolve in the curve *pqv*, which would become
similar to the curve PQV, in which the first force obliges the body P
to revolve; and their revolutions would be completed in the same
times. But because those forces are not to each other in the ratio of
CP to *sp*, but (by reason of the similarity and equality of
the bodies S and *s*, P and *p* and the equality of
the distances SP, *sp*) mutually equal, the bodies in equal
times will be equally drawn from the tangents; and therefore that the
body *p* may be attracted through the greater interval *rq*,
there is required a greater time, which will be in the subduplicate
ratio of the intervals; because, by Lemma X, the spaces described at
the very beginning of the motion are in a duplicate ratio of the
times. Suppose, then the velocity of the body *p* to be to the
velocity of the body P in a subduplicate ratio of the distance *sp*
to the distance CP, so that the arcs *pq*, PQ, which are in a
simple proportion to each other, may be described in times that are in
a subduplicate ratio of the distances; and the bodies P, *p*,
always attracted by equal forces, will describe round the quiescent
centres C and *s* similar figures PQV, *pqv*, the
latter of which *pqv* is similar and equal to the figure which
the body P describes round the movable body S. Q.E.D.

Case 2. Suppose now that the common centre of
gravity, together with the space in which the bodies are moved among
themselves, proceeds uniformly in a right line; and (by Cor. 6, of the
Laws of Motion) all the motions in this space will be performed in the
same manner as before; and therefore the bodies will describe mutually
about each other the same figures as before, which will be therefore
similar and equal to the figure *pqv*. Q.E.D.

Cor. 1. Hence two bodies attracting each
other with forces proportional to their distance, describe (by Prop.
X) both round their common centre of gravity, and round each other
mutually concentrical ellipses; and, *vice versa*, if such
figures are described, the forces are proportional to the distances.

Cor. 2. And two bodies, whose forces are
reciprocally proportional to the square of their distance, describe
(by Prop. XI, XII, XIII), both round their common centre of gravity,
and round each other mutually, conic sections having their focus in
the centre about which the figures are described. And, *vice versa*,
if such figures are described, the centripetal forces are reciprocally
proportional to the squares of the distance.

Cor. 3. Any two bodies revolving round their common centre of gravity describe areas proportional to the times, by radii drawn both to that centre and to each other mutually.

*The periodic time of two bodies* S *and* P *revolving
round their common centre of gravity* C*, is to the
periodic time of one of the bodies* P *revolving round the
other* S *remaining unmoved, and describing a figure
similar and equal to those which the bodies describe about each
other mutually, in a subduplicate ratio of the other body* S *to
the sum of the bodies* S + P*.*

For, by the demonstration of the last Proposition, the times in which
any similar arcs PQ, and *pq* are described are in a
subduplicate ratio of the distances CP and SP, or *sp*, that
is, in a subduplicate ratio of the body S to the sum of the bodies S +
P. And by composition of ratios, the sums of the times in which all
the similar arcs PQ and *pq* are described, that is, the whole
times in which the whole similar figures are described are in the same
subduplicate ratio. Q.E.D.

*If two bodies* S *and* P*, attracting each other
with forces reciprocally proportional to the squares of their
distance, revolve about their common centre of gravity; I say,
that the principal axis of the ellipsis which either of the
bodies, as* P*, describes by this motion about the other*
S*, will be to the principal axis of the ellipsis, which the same
body* P *may describe in the same periodical time about
the other body* S *quiescent, as the sum of the two bodies*
S + P *to the first of two mean proportionals between that sum
and the other body* S*.*

For if the ellipses described were equal to each other, their
periodic times by the last Theorem would be in a subduplicate ratio of
the body S to the sum of the bodies *S + P*. Let the periodic
time in the latter ellipsis be diminished in that ratio, and the
periodic times will become equal; but, by Prop. XV, the principal axis
of the ellipsis will be diminished in a ratio sesquiplicate to the
former ratio; that is, in a ratio to which the ratio of S to S + P is
triplicate; and therefore that axis will be to the principal axis of
the other ellipsis as the first of two mean proportionals between S +
P and S to S + P. And inversely the principal axis of the ellipsis
described about the movable body will be to the principal axis of that
described round the immovable as S + P to the first of two mean
proportionals between S + P and S. Q.E.D.

*If two bodies attracting each other with any kind of forces,
and not otherwise agitated or obstructed, are moved in any manner
whatsoever, those motions will be the same as if they did not at
all attract each other mutually, but were both attracted with the
same forces by a third body placed in their common centre of
gravity; and the law of the attracting forces will he the same in
respect of the distance of the bodies from the common centre, as
in respect of the distance between the two bodies.*

For those forces with which the bodies attract each other mutually, by tending to the bodies, tend also to the common centre of gravity lying directly between them; and therefore are the same as if they proceeded from in intermediate body. Q.E.D.

And because there is given the ratio of the distance of either body from that common centre to the distance between the two bodies, there is given, of course, the ratio of any power of one distance to the same power of the other distance; and also the ratio of any quantity derived in any manner from one of the distances compounded any how with given quantities, to another quantity derived in like manner from the other distance, and as many given quantities having that given ratio of the distances to the first. Therefore if the force with which one body is attracted by another be directly or inversely as the distance of the bodies from each other, or as any power of that distance; or, lastly, as any quantity derived after any manner from that distance compounded with given quantities; then will the same force with which the same body is attracted to the common centre of gravity be in like manner directly or inversely as the distance of the attracted body from the common centre, or as any power of that distance; or, lastly, as a quantity derived in like sort from that distance compounded with analogous given quantities. That is, the law of attracting force will be the same with respect to both distances. Q.E.D.

*To determine the motions of two bodies which attract each other
with forces reciprocally proportional to the squares of the
distance between them, and are let fall from given places.*

The bodies, by the last Theorem, will be moved in the same manner as if they were attracted by a third placed in the common centre of their gravity; and by the hypothesis that centre will be quiescent at the beginning of their motion, and therefore (by Cor. 4, of the Laws of Motion) will be always quiescent. The motions of the bodies are therefore to be determined (by Prob. XXV) in the same manner as if they were impelled by forces tending to that centre; and then we shall have the motions of the bodies attracting each other mutually. Q.E.I.

*To determine the motions of two bodies attracting each other
with forces reciprocally proportional to the squares of their
distance, and going off from given places in given directions with
given velocities.*

The motions of the bodies at the beginning being given, there is given also the uniform motion of the common centre of gravity, and the motion of the space which moves along with this centre uniformly in a right line, and also the very first, or beginning motions of the bodies in respect of this space. Then (by Cor. 5. of the Laws, and the last Theorem) the subsequent motions will be performed in the same manner in that space, as if that space together with the common centre of gravity were at rest, and as if the bodies did not attract each other, but were attracted by a third body placed in that centre. The motion therefore in this movable space of each body going off from a given place, in a given direction, with a given velocity, and acted upon by a centripetal force tending to that centre, is to be determined by Prob. IX and XXVI, and at the same time will be obtained the motion of the other round the same centre. With this motion compound the uniform progressive motion of the entire system of the space and the bodies revolving in it, and there will be obtained the absolute motion of the bodies in immovable space. Q.E.I.

*Supposing forces with which bodies mutually attract each other
to increase in a simple ratio of their distances from the centres;
it is required to find the motions of several bodies among
themselves.*

Suppose the first two bodies T and L to have their common centre of gravity in D. These, by Cor. 1, Theor. XXI, will describe ellipses having their centres in D, the magnitudes of which ellipses are known by Prob. V.

Let now a third body S attract the two former T and L with the accelerative forces ST, SL, and let it be attracted again by them. The force ST (by Cor. 2, of the Laws of Motion) is resolved into the forces SD, DT; and the force SL into the forces SD and DL. Now the forces DT, DL, which are as their sum TL, and therefore as the accelerative forces with which the bodies T and L attract each other mutually, added to the forces of the bodies T and L, the first to the first, and the last to the last, compose forces proportional to the distances DT and DL as before, but only greater than those former forces: and therefore (by Cor. 1, Prop. X, and Cor. 1, and 8, Prop. IV) they will cause those bodies to describe ellipses as before, but with a swifter motion. The remaining accelerative forces SD and DL, by the motive forces SD × T and SD × L, which are as the bodies attracting those bodies equally and in the direction of the lines TI, LK parallel to DS, do not at all change their situations with respect to one another, but cause them equally to approach to the line IK; which must be imagined drawn through the middle of the body S, and perpendicular to the line DS. But that approach to the line IK will be hindered by causing the system of the bodies T and L on one side, and the body S on the other, with proper velocities, to revolve round the common centre of gravity C. With such a motion the body S, because the sum of the motive forces SD × T and SD × L is proportional to the distance CS, tends to the centre C, will describe an ellipsis round the same centre C; and the point D, because the lines CS and CD are proportional, will describe a like ellipsis over against it. But the bodies T and L, attracted by the motive forces SD × T and SD × L, the first by the first, and the last by the last, equally and in the direction of the parallel lines TI and LK, as was said before, will (by Cor. 5 and 6, of the Laws of Motion) continue to describe their ellipses round the movable centre D, as before. Q.E.I.

Let there be added a fourth body V, and, by the like reasoning, it will be demonstrated that this body and the point C will describe ellipses about the common centre of gravity B; the motions of the bodies T, L, and S round the centres D and C remaining the same as before; but accelerated. And by the same method one may add yet more bodies at pleasure. Q.E.I

This would be the case, though the bodies T and L attract each other mutually with accelerative forces either greater or less than those with which they attract the other bodies in proportion to their distance. Let all the mutual accelerative attractions be to each other as the distances multiplied into the attracting bodies; and from what has gone before it will easily be concluded that all the bodies will describe different ellipses with equal periodical times about their common centre of gravity B, in an immovable plane. Q.E.I.

*Bodies, whose forces decrease in a duplicate ratio of their
distances from their centres, may move among themselves in
ellipses; and by radii drawn to the foci may describe areas
proportional to the times very nearly.*

In the last Proposition we demonstrated that case in which the motions will be performed exactly in ellipses. The more distant the law of the forces is from the law in that case, the more will the bodies disturb each other's motions; neither is it possible that bodies attracting each other mutually according to the law supposed in this Proposition should move exactly in ellipses, unless by keeping a certain proportion of distances from each other. However, in the following crises the orbits will not much differ from ellipses.

Case I. Imagine several lesser bodies to revolve about some very great one at different distances from it, and suppose absolute forces tending to every one of the bodies proportional to each. And because (by Cor. 4, of the Laws) the common centre of gravity of them all is either at rest, or moves uniformly forward in a right line, suppose the lesser bodies so small that the great body may be never at a sensible distance from that centre; and then the great body will, without any sensible error, be either at rest, or move uniformly forward in a right line; and the lesser will revolve about that great one in ellipses, and by radii drawn thereto will describe areas proportional to the times; if we except the errors that may be introduced by the receding of the great body from the common centre of gravity, or by the mutual actions of the lesser bodies upon each other. But the lesser bodies may be so far diminished, as that this recess and the mutual actions of the bodies on each other may become less than any assignable; and therefore so as that the orbits may become ellipses, and the areas answer to the times, without any error that is not less than any assignable. Q.E.O.

Case 2. Let us imagine a system of lesser bodies revolving about a very great one in the manner just described, or any other system of two bodies revolving about each other to be moving uniformly forward in a right line, and in the mean time to be impelled sideways by the force of another vastly greater body situate at a great distance. And because the equal accelerative forces with which the bodies are impelled in parallel directions do not change the situation of the bodies with respect to each other, but only oblige the whole system to change its place while the parts still retain their motions among themselves, it is manifest that no change in those motions of the attracted bodies can arise from their attractions towards the greater, unless by the inequality of the accelerative attractions, or by the inclinations of the lines towards each other, in whose directions the attractions are made. Suppose, therefore, all the accelerative attractions made towards the great body to be among themselves as the squares of the distances reciprocally; and then, by increasing the distance of the great body till the differences of the right lines drawn from that to the others in respect of their length, and the inclinations of those lines to each other, be less than any given, the motions of the parts of the system will continue without errors that are not less than any given. And because, by the small distance of those parts from each other, the whole system is attracted as if it were but one body, it will therefore be moved by this attraction as if it were one body; that is, its centre of gravity will describe about the great body one of the conic sections (that is, a parabola or hyperbola when the attraction is but languid and an ellipsis when it is more vigorous); and by radii drawn thereto, it will describe areas proportional to the times, without any errors but those which arise from the distances of the parts, which are by the supposition exceedingly small, and may be diminished at pleasure. Q.E.O.

By a like reasoning one may proceed to more compounded cases *in
infinitum*.

Cor. 1. In the second Case, the nearer the very great body approaches to the system of two or more revolving bodies, the greater will the perturbation be of the motions of the parts of the system among themselves; because the inclinations of the lines drawn from that great body to those parts become greater; and the inequality of the proportion is also greater.

Cor. 2. But the perturbation will be greatest of all, if we suppose the accelerative attractions of the parts of the system towards the greatest body of all are not to each other reciprocally as the squares of the distances from that great body; especially if the inequality of this proportion be greater than the inequality of the proportion of the distances from the great body. For if the accelerative force, acting in parallel directions and equally, causes no perturbation in the motions of the parts of the system, it must of course, when it acts unequally, cause a perturbation somewhere, which will be greater or less as the inequality is greater or less. The excess of the greater impulses acting upon some bodies, and not acting upon others, must necessarily change their situation among themselves. And this perturbation, added to the perturbation arising from the inequality and inclination of the lines, makes the whole perturbation greater.

Cor. 3. Hence if the parts of this system move in ellipses or circles without any remarkable perturbation, it is manifest that, if they are at all impelled by accelerative forces tending to any other bodies, the impulse is very weak, or else is impressed very near equally and in parallel directions upon all of them.

*If three bodies whose forces decrease in a duplicate ratio of
the distances attract each other mutually; and the accelerative
attractions of any two towards the third be between themselves
reciprocally as the squares of the distances; and the two least
revolve about the greatest; I say, that the interior of the two
revolving bodies will, by radii drawn to the innermost and
greatest, describe round that body areas more proportional to the
times, and a figure more approaching to that of an ellipsis having
its focus in the point of concourse of the radii, if that great
body be agitated by those attractions, than it would do if that
great body were not attracted at all by the lesser, but remained
at rest; or than, it would if that great body were very much more
or very much less attracted, or very much more or very much less
agitated, by the attractions.*

This appears plainly enough from the demonstration of the second Corollary of the foregoing Proposition; but it maybe made out after this manner by a way of reasoning more distinct and more universally convincing.

Case 1. Let the lesser bodies P and S revolve
in the same plane about the greatest body T, the body P describing the
interior orbit PAB, and S the exterior orbit
ESE. Let SK be the mean distance of the bodies P and S; and let the
accelerative attraction of the body P towards S, at that mean
distance, be expressed by that line SK. Make SL to SK as the
square of SK to the square of SP, and SL will be the accelerative
attraction of the body P towards S at any distance SP. Join PT, and
draw LM parallel to it meeting ST in M; and the attraction SL will be
resolved (by Cor. 2, of the Laws of Motion) into the attractions SM,
LM. And so the body P will be urged with a threefold accelerative
force. One of these forces tends towards T, and arises from the mutual
attraction of the bodies T and P. By this force alone the body P would
describe round the body T, by the radius PT, areas proportional to the
times, and an ellipsis whose focus is in the centre of the body T; and
this it would do whether the body T remained unmoved, or whether it
were agitated by that attraction. This appears from Prop. XI, and Cor.
2 and 3 of Theor. XXI. The other force is that of the attraction LM,
which, because it tends from P to T, will be superadded to and
coincide with the former force; and cause the areas to be still
proportional to the times, by Cor. 3, Theor. XXI. But because it is
not reciprocally proportional to the square of the distance PT, it
will compose, when added to the former, a force varying from that
proportion; which variation will be the greater by how much the
proportion of this force to the former is greater, *caeteris
paribus*. Therefore, since by Prop. XI, and by Cor. 2, Theor.
XXI, the force with which the ellipsis is described about the focus T
ought to be directed to that focus, and to be reciprocally
proportional to the square of the distance PT, that compounded force
varying from that proportion will make the orbit PAB vary from the
figure of an ellipsis that has its focus in the point T; and so much
the more by how much the variation from that proportion is greater;
and by consequence by how much the proportion of the second force LM
to the first force is greater, *caeteris paribus*. But now the
third force SM, attracting the body P in a direction parallel to ST,
composes with the other forces a new force which is no longer directed
from P to T; and which varies so much more from this direction by how
much the proportion of this third force to the other forces is
greater, *caeteris paribus*; and therefore causes the body P
to describe, by the radius TP, areas no longer proportional to the
times; and therefore makes the variation from that proportionality so
much greater by how much the proportion of this force to the others is
greater. But this third force will increase the variation of the orbit
PAB from the elliptical figure
before-mentioned upon two accounts; first because that force is not
directed from P to T; and, secondly, because it is not reciprocally
proportional to the square of the distance PT. These things being
premised, it is manifest that the areas are then most nearly
proportional to the times, when that third force is the least
possible, the rest preserving their former quantity; and that the
orbit PAB does then approach nearest to the elliptical figure
above-mentioned, when both the second and third, but especially the
third force, is the least possible; the first force remaining in its
former quantity.

Let the accelerative attraction of the body T towards S be expressed by the line SN; then if the accelerative attractions SM and SN were equal, these, attracting the bodies T and P equally and in parallel directions would not at all change their situation with respect to each other. The motions of the bodies between themselves would be the same in that case as if those attractions did not act at all, by Cor. 6, of the Laws of Motion. And, by a like reasoning, if the attraction SN is less than the attraction SM, it will take away out of the attraction SM the part SN, so that there will remain only the part (of the attraction) MN to disturb the proportionality of the areas and times, and the elliptical figure of the orbit. And in like manner if the attraction SN be greater than the attraction SM, the perturbation of the orbit and proportion will be produced by the difference MN alone. After this manner the attraction SN reduces always the attraction SM to the attraction MN, the first and second attractions remaining perfectly unchanged; and therefore the areas and times come then nearest to proportionality, and the orbit PAB to the above-mentioned elliptical figure, when the attraction MN is either none, or the least that is possible; that is, when the accelerative attractions of the bodies P and T approach as near as possible to equality; that is, when the attraction SN is neither none at all, nor less than the least of all the attractions SM, but is, as it were; a mean between the greatest and least of all those attractions SM, that is, not much greater nor much less than the attraction SK. Q.E.D.

Case 2. Let now the lesser bodies P, S, revolve about a greater T in different planes; and the force LM, acting in the direction of the line PT situate in the plane of the orbit PAB, will have the same effect as before; neither will it draw the body P from the plane of its orbit. But the other force NM acting in the direction of a line parallel to ST (and which, therefore, when the body S is without the line of the nodes is inclined to the plane of the orbit PAB), besides the perturbation of the motion just now spoken of as to longitude, introduces another perturbation also as to latitude, attracting the body P out of the plane of its orbit. And this perturbation, in any given situation of the bodies P and T to each other, will be as the generating force MN; and therefore becomes least when the force MN is least, that is (as was just now shewn), where the attraction SN is not much greater nor much less than the attraction SK. Q.E.D.

Cor. 1. Hence it may be easily collected, that if several less bodies P, S, R, &c., revolve about a very great body T, the motion of the innermost revolving body P will be least disturbed by the attractions of the others, when the great body is as well attracted and agitated by the rest (according to the ratio of the accelerative forces) as the rest are by each other mutually.

Cor. 2. In a system of three bodies, T, P, S,
if the accelerative attractions of any two of them towards a third be
to each other reciprocally as the squares of the distances, the body
P, by the radius PT, will describe its area about the body T swifter
near the conjunction A and the opposition B than it will near the
quadratures C and D. For every force with which the body P is acted on
and the body T is not, and which does not act in the direction of the
line PT, does either accelerate or retard the description of the area,
according as it is directed, whether *in consequentia* or *in
antecedentia*. Such is the force NM. This force in the passage
of the body P from C to A is directed *in consequentia* to its
motion, and therefore accelerates it; then as far as D *in
antecedentia*, and retards the motion; then *in consequentia*
as far as B; and lastly *in antecedentia* as it moves from B
to C.

Cor. 3. And from the same reasoning it
appears that the body P *caeteris paribus*, moves more swiftly
in the conjunction and opposition than in the quadratures.

Cor. 4. The orbit of the body P, *caeteris
paribus*, is more curve at the quadratures than at the
conjunction and opposition. For the swifter bodies move, the less they
deflect from a rectilinear path. And besides the force KL, or NM, at
the conjunction and opposition, is contrary to the force with which
the body T attracts the body P, and therefore diminishes that force;
but the body P will deflect the less from a rectilinear path the less
it is impelled towards the body T.

Cor. 5. Hence the body P, *caeteris
paribus*, goes farther from the body T at the quadratures than
at the conjunction and opposition. This is said,
however, supposing no regard had to the motion of eccentricity. For if
the orbit of the body P be eccentrical, its eccentricity (as will be
shewn presently by Cor. 9) will be greatest when the apsides are in
the syzygies; and thence it may sometimes come to pass that the body
P, in its near approach to the farther apsis, may go farther from the
body T at the syzygies than at the quadratures.

Cor. 6. Because the centripetal force of the central body T, by which the body P is retained in its orbit, is increased at the quadratures by the addition caused by the force LM, and diminished at the syzygies by the subduction caused by the force KL, and, because the force KL is greater than LM, it is more diminished than increased; and, moreover, since that centripetal force (by Cor. 2, Prop. IV) is in a ratio compounded of the simple ratio of the radius TP directly, and the duplicate ratio of the periodical time inversely; it is plain that this compounded ratio is diminished by the action of the force KL; and therefore that the periodical time, supposing the radius of the orbit PT to remain the same, will be increased, and that in the subduplicate of that ratio in which the centripetal force is diminished; and, therefore, supposing this radius increased or diminished, the periodical time will be increased more or diminished less than in the sesquiplicate ratio of this radius, by Cor. 6, Prop. IV. If that force of the central body should gradually decay, the body P being less and less attracted would go farther and farther from the centre T; and, on the contrary, if it were increased, it would draw nearer to it. Therefore if the action of the distant body S, by which that force is diminished, were to increase and decrease by turns, the radius TP will be also increased and diminished by turns; and the periodical time will be increased and diminished in a ratio compounded of the sesquiplicate ratio of the radius, and of the subduplicate of that ratio in which the centripetal force of the central body T is diminished or increased, by the increase or decrease of the action of the distant body S.

Cor. 7. It also follows, from what was before laid down, that the axis of the ellipsis described by the body P, or the line of the apsides, does as to its angular motion go forwards and backwards by turns, but more forwards than backwards, and by the excess of its direct motion is in the whole carried forwards. For the force with which the body P is urged to the body T at the quadratures, where the force MN vanishes, is compounded of the force LM and the centripetal force with which the body T attracts the body P. The first force LM, if the distance PT be increased, is increased in nearly the same proportion with that distance, and the other force decreases in the duplicate ratio of the distance; and therefore the sum of these two forces decreases in a less than the duplicate ratio of the distance PT; and therefore, by Cor. 1, Prop. XLV, will make the line of the apsides, or, which is the same thing, the upper apsis, to go backward. But at the conjunction and opposition the force with which the body P is urged towards the body T is the difference of the force KL, and of the force with which the body T attracts the body P; and that difference, because the force KL is very nearly increased in the ratio of the distance PT, decreases in more than the duplicate ratio of the distance PT; and therefore, by Cor. 1, Prop. XLV, causes the line of the apsides to go forwards. In the places between the syzygies and the quadratures, the motion of the line of the apsides depends upon both of these causes conjunctly, so that it either goes forwards or backwards in proportion to the excess of one of these causes above the other. Therefore since the force KL in the syzygies is almost twice as great as the force LM in the quadratures, the excess will be on the side of the force KL, and by consequence the line of the apsides will be carried forwards. The truth of this and the foregoing Corollary will be more easily understood by conceiving the system of the two bodies T and P to be surrounded on every side by several bodies S, S, S, &c., disposed about the orbit ESE. For by the actions of these bodies the action of the body T will be diminished on every side, and decrease in more than a duplicate ratio of the distance.

Cor. 8. But since the progress or regress of the apsides depends upon the decrease of the centripetal force, that is, upon its being in a greater or less ratio than the duplicate ratio of the distance TP, in the passage of the body from the lower apsis to the upper; and upon a like increase in its return to the lower apsis again; and therefore becomes greatest where the proportion of the force at the upper apsis to the force at the lower apsis recedes farthest from the duplicate ratio of the distances inversely; it is plain, that, when the apsides are in the syzygies, they will, by reason of the subducting force KL or NM − LM, go forward more swiftly; and in the quadratures by the additional force LM go backward more slowly. Because the velocity of the progress or slowness of the regress is continued for a long time; this inequality becomes exceedingly great.

Cor. 9. If a body is obliged, by a force reciprocally proportional to the square of its distance from any centre, to revolve in an ellipsis round that centre; and afterwards in its descent from the upper apsis to the lower apsis, that force by a perpetual accession of new force is increased in more than a duplicate ratio of the diminished distance; it is manifest that the body, being impelled always towards the centre by the perpetual accession of this new force, will incline more towards that centre than if it were urged by that force alone which decreases in a duplicate ratio of the diminished distance, and therefore will describe an orbit interior to that elliptical orbit, and at the lower apsis approaching nearer to the centre than before. Therefore the orbit by the accession of this new force will become more eccentrical. If now, while the body is returning from the lower to the upper apsis, it should decrease by the same degrees by which it increases before the body would return to its first distance; and therefore if the force decreases in a yet greater ratio, the body, being now less attracted than before, will ascend to a still greater distance, and so the eccentricity of the orbit will be increased still more. Therefore if the ratio of the increase and decrease of the centripetal force be augmented each revolution, the eccentricity will be augmented also; and, on the contrary, if that ratio decrease, it will be diminished.

Now, therefore, in the system of the bodies T, P, S, when the apsides of the orbit PAB are in the quadratures, the ratio of that increase and decrease is least of all, and becomes greatest when the apsides are in the syzygies. If the apsides are placed in the quadratures, the ratio near the apsides is less, and near the syzygies greater, than the duplicate ratio of the distances; and from that greater ratio arises a direct motion of the line of the apsides, as was just now said. But if we consider the ratio of the whole increase or decrease in the progress between the apsides, this is less than the duplicate ratio of the distances. The force in the lower is to the force in the upper apsis in less than a duplicate ratio of the distance of the upper apsis from the focus of the ellipsis to the distance of the lower apsis from the same focus; and, contrariwise, when the apsides are placed in the syzygies, the force in the lower apsis is to the force in the upper apsis in a greater than a duplicate ratio of the distances. For the forces LM in the quadratures added to the forces of the body T compose forces in a less ratio; and the forces KL in the syzygies subducted from the forces of the body T, leave the forces in a greater ratio. Therefore the ratio of the whole increase and decrease in the passage between the apsides is least at the quadratures and greatest at the syzygies; and therefore in the passage of the apsides from the quadratures to the syzygies it is continually augmented, and increases the eccentricity of the ellipsis; and in the passage from the syzygies to the quadratures it is perpetually decreasing, and diminishes the eccentricity.

Cor. 10. That we may give an account of the errors as to latitude, let us suppose the plane of the orbit EST to remain immovable; and from the cause of the errors above explained, it is manifest, that, of the two forces NM, ML, which are the only and entire cause of them, the force ML acting always in the plane of the orbit PAB never disturbs the motions as to latitude; and that the force NM, when the nodes are in the syzygies, acting also in the same plane of the orbit, does not at that time affect those motions. But when the nodes are in the quadratures, it disturbs them very much, and, attracting the body P perpetually out of the plane of its orbit, it diminishes the inclination of the plane in the passage of the body from the quadratures to the syzygies, and again increases the same in the passage from the syzygies to the quadratures. Hence it comes to pass that when the body is in the syzygies, the inclination is then least of all, and returns to the first magnitude nearly, when the body arrives at the next node. But if the nodes are situate at the octants after the quadratures, that is, between C and A, D and B, it will appear, from what was just now shewn, that in the passage of the body P from either node to the ninetieth degree from thence, the inclination of the plane is perpetually diminished; then, in the passage through the next 45 degrees to the next quadrature, the inclination is increased; and afterwards, again, in its passage through another 45 degrees to the next node, it is diminished. Therefore the inclination is more diminished than increased, and is therefore always less in the subsequent node than in the preceding one. And, by a like reasoning, the inclination is more increased than diminished when the nodes are in the other octants between A and D, B and C. The inclination, therefore, is the greatest of all when the nodes are in the syzygies. In their passage from the syzygies to the quadratures the inclination is diminished at each appulse of the body to the nodes: and be comes least of all when the nodes are in the quadratures, and the body in the syzygies; then it increases by the same degrees by which it decreased before; and, when the nodes come to the next syzygies, returns to its former magnitude.

Cor. 11. Because when the nodes are in the
quadratures the body P is perpetually attracted from the plane of its
orbit; and because this attraction is made towards S in its passage
from, the node C through the conjunction A to the node D; and to the
contrary part in its passage from the node D through the opposition B
to the node C; it is manifest that, in its motion from the node C, the
body recedes continually from the former plane CD of its orbit till it
comes to the next node; and therefore at that node, being now at its
greatest distance from the first plane CD, it will pass through the
plane of the orbit EST not in D, the other node of that plane, but in
a point that lies nearer to the body S, which therefore be comes a new
place of the node *in antecedentia* to its former place. And,
by a like reasoning, the nodes will continue to recede in their
passage from this node to the next. The nodes, therefore, when situate
in the quadratures, recede perpetually; and at the syzygies, where no
perturbation can be produced in the motion as to latitude, are
quiescent: in the intermediate places they partake of both conditions,
and recede more slowly; and, therefore, being always either retrograde
or stationary, they will be carried backwards, or *in antecedentia*,
each revolution.

Cor. 12. All the errors described in these corrollaries are a little greater at the conjunction of the bodies P, S, than at their opposition; because the generating forces NM and ML are greater.

Cor. 13. And since the causes and proportions of the errors and variations mentioned in these Corollaries do not depend upon the magnitude of the body S, it follows that all things before demonstrated will happen, if the magnitude of the body S be imagined so great as that the system of the two bodies P and T may revolve about it. And from this increase of the body S, and the consequent increase of its centripetal force, from which the errors of the body P arise, it will follow that all these errors, at equal distances, will be greater in this case, than in the other where the body S revolves about the system of the bodies P and T.

Cor. 14. But since the forces NM, ML, when
the body S is exceedingly distant, are very nearly as the force SK and
the ratio PT to ST conjunctly; that is, if both the distance PT, and
the absolute force of the body S be given, as ST³ reciprocally; and
since those forces NM, ML are the causes of all the errors and effects
treated of in the foregoing Corollaries; it is manifest that all those
effects, if the system of bodies T and P continue as before, and only
the distance ST and the absolute force of the body S be changed, will
be very nearly in a ratio compounded of the direct ratio of the
absolute force of the body S, and the triplicate inverse ratio of the
distance ST. Hence if the system of bodies T and P revolve about a
distant body S, those forces NM, ML, and their effects, will be (by
Cor. 2 and 6, Prop IV) reciprocally in a duplicate ratio of the
periodical time. And thence, also, if the magnitude of the body S be
proportional to its absolute force, those forces NM, ML, and their
effects, will be directly as the cube of the apparent diameter of the
distant body S viewed from T, and so *vice versa*. For these
ratios are the same as the compounded ratio above mentioned.

Cor. 15. And because if the orbits ESE and PAB, retaining their figure, proportions, and inclination to each other, should alter their magnitude; and the forces of the bodies S and T should either remain, or be changed in any given ratio; these forces (that is, the force of the body T, which obliges the body P to deflect from a rectilinear course into the orbit PAB, and the force of the body S, which causes the body P to deviate from that orbit) would act always in the same manner, and in the same proportion; it follows, that all the effects will be similar and proportional, and the times of those effects proportional also; that is, that all the linear errors will be as the diameters of the orbits, the angular errors the same as before; and the times of similar linear errors, or equal angular errors, as the periodical times of the orbits.

Cor. 16. Therefore if the figures of the orbits and their inclination to each other be given, and the magnitudes, forces, and distances of the bodies be any how changed, we may, from the errors and times of those errors in one case, collect very nearly the errors and times of the errors in any other case. But this may be done more expeditiously by the following method. The forces NM, ML, other things remaining unaltered, are as the radius TP; and their periodical effects (by Cor. 2, Lem. X) are as the forces and the square of the periodical time of the body P conjunctly. These are the linear errors of the body P; and hence the angular errors as they appear from the centre T (that is, the motion of the apsides and of the nodes, and all the apparent errors as to longitude and latitude) are in each revolution of the body P as the square of the time of the revolution, very nearly. Let these ratios be compounded with the ratios in Cor. 14, and in any system of bodies T, P, S, where P revolves about T very near to it, and T revolves about S at a great distance, the angular errors of the body P, observed from the centre T, will be in each revolution of the body P as the square of the periodical time of the body P directly, and the square of the periodical time of the body T inversely. And therefore the mean motion of the line of the apsides will be in a given ratio to the mean motion of the nodes; and both those motions will be as the periodical time of the body P directly, and the square of the periodical time of the body T inversely. The increase or diminution of the eccentricity and inclination of the orbit PAB makes no sensible variation in the motions of the apsides and nodes, unless that increase or diminution be very great indeed.

Cor. 17. Since the line LM becomes sometimes
greater and sometimes less than the radius PT, let the mean quantity
of the force LM be expressed
by that radius PT; and then that mean force will be to the mean force SK or SN
(which may be also expressed by ST) as the length PT to the length ST.
But the mean force SN or ST, by which the body T is retained in the
orbit it describes about S, is to the force with which the body P is
retained in its orbit about T in a ratio compounded of the ratio of
the radius ST to the radius PT, and the duplicate ratio of the
periodical time of the body P about T to the periodical time of the
body T about S. And, *ex aequo*, the mean force LM is to the
force by which the body P is retained in its orbit about T (or by
which the same body P might revolve at the distance PT in the same
periodical time about any immovable point T) in the same duplicate
ratio of the periodical times. The periodical times therefore being
given, together with the distance PT, the mean force LM is also given;
and that force being given, there is given also the force MN, very
nearly, by the analogy of the lines PT and MN.

Cor. 18. By the same
laws by which the body P revolves about the body T, let us suppose
many fluid bodies to move round T at equal distances from it; and to
be so numerous, that they may all become contiguous to each other, so
as to form a fluid annulus, or ring, of a round figure, and
concentrical to the body T; and the several parts of this annulus,
performing their motions by the same law as the body P, will draw
nearer to the body T, and move swifter in the conjunction and
opposition of themselves and the body S, than in the quadratures. And
the nodes of this annulus, or its intersections with the plane of the
orbit of the body S or T, will rest at the syzygies; but out of the
syzygies they will be carried backward, or *in antecedentia*;
with the greatest swiftness in the quadratures, and more slowly in
other places. The inclination of this annulus also will vary, and its
axis will oscillate each revolution, and when the revolution is
completed will return to its former situation, except only that it
will be carried round a little by the precession of the nodes.

Cor. 19. Suppose now the sphaerical body T, consisting of some matter not fluid, to be enlarged, and to extend itself on every side as far as that annulus, and that a channel were cut all round its circumference containing water; and that this sphere revolves uniformly about its own axis in the same periodical time. This water being accelerated and retarded by turns (as in the last Corollary), will be swifter at the syzygies, and slower at the quadratures, than the surface of the globe, and so will ebb and flow in its channel after the manner of the sea. If the attraction of the body's were taken away, the water would acquire no motion of flux and reflux by revolving round the quiescent centre of the globe. The case is the same of a globe moving uniformly forwards in a right line, and in the mean time revolving about its centre (by Cor. 5 of the Laws of Motion), and of a globe uniformly attracted from its rectilinear course (by Cor. 6, of the same Laws). But let the body S come to act upon it, and by its unequable attraction the water will receive this new motion; for there will be a stronger attraction upon that part of the water that is nearest to the body, and a weaker upon that part which is more remote. And the force LM will attract the water downwards at the quadratures, and depress it as far as the syzygies; and the force KL will attract it upwards in the syzygies, and withhold its descent, and make it rise as far as the quadratures; except only in so far as the motion of flux and reflux may be directed by the channel of the water, and be a little retarded by friction.

Cor. 20. If, now, the annulus becomes hard,
and the globe is diminished, the motion of flux and reflux will cease;
but the oscillating motion of the inclination and the praecession of
the nodes will remain. Let the globe have the same axis with the
annulus, and perform its revolutions in the same times, and at its
surface touch the annulus within, and adhere to it; then the globe
partaking of the motion of the annulus, this whole compages will
oscillate, and the nodes will go backward, for the globe, as we shall
shew presently, is perfectly indifferent to the receiving of all
impressions. The greatest angle of the inclination of the annulus
single is when the nodes are in the syzygies. Thence in the progress
of the nodes to the quadratures, it endeavours to diminish its
inclination, arid by that endeavour impresses a motion upon the whole
globe. The globe retains this motion impressed, till the annulus by a
contrary endeavour destroys that motion, and impresses a new motion in
a contrary direction. And by this means the greatest motion of the
decreasing inclination happens when the nodes are in the quadratures;
and the least angle of inclination in the octants
after the quadratures; and, again, the greatest motion of reclination
happens when the nodes are in the syzygies; and the greatest angle of
reclination in the octants following. And the case is the same of a
globe without this annulus, if it be a little higher or a little
denser in the equatorial than in the polar regions; for the excess of
that matter in the regions near the equator supplies the place of the
annulus. And though we should suppose the centripetal force of this
globe to be any how increased, so that all its parts were to tend
downwards, as the parts of our earth gravitate to the centre, yet the
phenomena of this and the preceding Corollary would scarce be altered;
except that the places of the greatest and least height of the water
will be different: for the water is now no longer sustained and kept
in its orbit by its centrifugal force, but by the channel in which it
flows. And, besides, the force LM attracts the water downwards most in
the quadratures, and the force KL or NM − LM attracts it upwards most
in the syzygies. And these forces conjoined cease to attract the water
downwards, and begin to attract it upwards in the octants before the
syzygies; and cease to attract the water upwards, and begin to attract
the water downwards in the octants after the syzygies. And thence the
greatest height of the water may happen about the octants after the
syzygies; and the least height about the octants after the
quadratures; excepting only so far as the motion of ascent or descent
impressed by these forces may by the *vis insita* of the water
continue a little longer, or be stopped a little sooner by impediments
in its channel.

Cor. 21. For the same reason that redundant
matter in the equatorial regions of a globe causes the nodes to go
backwards, and therefore by the increase of that matter that
retrogradation is increased, by the diminution is diminished, and by
the removal quite ceases: it follows, that, if more than that
redundant matter be taken away, that is, if the globe be either more
depressed, or of a more rare consistence near the equator than near
the poles, there will arise a motion of the nodes *in consequentia*.

Cor. 22. And thence from the motion of the
nodes is known the constitution of the globe. That is, if the globe
retains unalterably the same poles, and the motion (of the nodes) be *in
antecedentia*, there is a redundance of the matter near the
equator; but if *in consequentia*, a deficiency. Suppose a
uniform and exactly spherical globe to be first at rest in a free
space: then by some impulse made obliquely upon its superficies to be
driven from its place, and to receive a motion partly circular and
partly right forward. Because this globe is perfectly indifferent to
all the axes that pass through its centre, nor has a greater
propensity to one axis or to one situation of the axis than to any
other, it is manifest that by its own force it will never change its
axis, or the inclination of it. Let now this globe be impelled
obliquely by a new impulse in the same part of its superficies as
before, and since the effect of an impulse is not at all changed by
its coming sooner or later, it is manifest that these two impulses,
successively impressed, will produce the same motion as if they were
impressed at the same time: that, is, the same motion as if the globe
had been impelled by a simple force compounded of them both (by Cor.
2, of the Laws), that is, a simple motion about an axis of a given
inclination. And the case is the same if the second impulse were made
upon any other place of the equator of the first motion; and also if
the first impulse were made upon any place in the equator of the
motion which would be generated by the second impulse alone; and
therefore, also, when both impulses are made in any places whatsoever;
for these impulses will generate the same circular motion as if they
were impressed together, and at once, in the place of the
intersections of the equators of those motions, which would be
generated by each of them separately. Therefore, a homogeneous and
perfect globe will not retain several distinct motions, but will unite
all those that are impressed on it, and reduce them into one;
revolving, as far as in it lies, always with a simple and uniform
motion about one single given axis, with an inclination perpetually
invariable. And the inclination of the axis, or the velocity of the
rotation, will not be changed by centripetal force. For if the globe
be supposed to be divided into two hemispheres, by any plane
whatsoever passing through its own centre, and the centre to which the
force is directed, that force will always urge each hemisphere
equally; and therefore will not incline the globe any way as to its
motion round its own axis. But let there be added any where between
the pole and the equator a heap of new matter like a mountain, and
this, by its perpetual endeavour to recede from the centre of its
motion, will disturb the motion of the globe, and cause its poles to
wander about its superficies, describing circles about themselves and
their opposite points. Neither can this enormous evagation of
the poles be corrected, unless by placing that mountain either in one
of the poles; in which case, by Cor. 21, the nodes of the equator will
go forwards; or in the equatorial regions, in which case, by Cor. 20,
the nodes will go backwards; or, lastly, by adding on the other side
of the axis a new quantity of matter, by which the mountain may be
balanced in its motion; and then the nodes will either go forwards or
backwards, as the mountain and this newly added matter happen to be
nearer to the pole or to the equator.

*The same laws of attraction being supposed, I say, that the
exterior body* S *does, by radii drawn to the point* O*,
the common centre of gravity of the interior bodies* P *and*
T*, describe round that centre areas more proportional to the
times, and an orbit more approaching to the form of an ellipsis
having its focus in that centre, than it can describe round the
innermost and greatest body* T *by radii drawn to that body.*

For the attractions of the body S towards T and P compose its absolute attraction, which is more directed towards O, the common centre of gravity of the bodies T and P, than it is to the greatest body T; and which is more in a reciprocal proportion to the square of the distance SO, than it is to the square of the distance ST; as will easily appear by a little consideration.

*The same laws of attraction supposed, I say, that the exterior
body* S *will, by radii drawn to* O*, the common
centre of gravity of the interior bodies* P *and* T*,
describe round that centre areas more proportional to the times,
and an orbit more approaching to the form of an ellipsis having
its focus in that centre, if the innermost and greatest body be
agitated by these attractions as well as the rest, than it would
do if that body were either at rest as not attracted, or were much
more or much less attracted, or much more or much less agitated.*

This may be demonstrated after the same manner as Prop. LXVI, but by a more prolix reasoning, which I therefore pass over. It will be sufficient to consider it after this manner. From the demonstration of the last Proposition it is plain, that the centre, towards which the body S is urged by the two forces conjunctly, is very near to the common centre of gravity of those two other bodies. If this centre were to coincide with that common centre, and moreover the common centre of gravity of all the three bodies were at rest, the body S on one side, and the common centre of gravity of the other two bodies on the other side, would describe true ellipses about that quiescent common centre. This appears from Cor. 2, Prop LVIII, compared with what was demonstrated in Prop. LXIV, and LXV. Now this accurate elliptical motion will be disturbed a little by the distance of the centre of the two bodies from the centre towards which the third body S is attracted. Let there be added, moreover, a motion to the common centre of the three, and the perturbation will be increased yet more. Therefore the perturbation is least when the common centre of the three bodies is at rest; that is, when the innermost and greatest body T is attracted according to the same law as the rest are; and is always greatest when the common centre of the three, by the diminution of the motion of the body T, begins to be moved, and is more and more agitated.

Cor. And hence if more lesser bodies revolve about the great one, it may easily be inferred that the orbits described will approach nearer to ellipses; and the descriptions of areas will be more nearly equable, if all the bodies mutually attract and agitate each other with accelerative forces that are as their absolute forces directly, and the squares of the distances inversely; and if the focus of each orbit be placed in the common centre of gravity of all the interior bodies (that is, if the focus of the first and innermost orbit be placed in the centre of gravity of the greatest and inner most body; the focus of the second orbit in the common centre of gravity of the two innermost bodies; the focus of the third orbit in the common centre of gravity of the three innermost; and so on), than if the innermost body were at rest, and was made the common focus of all the orbits.

*In a system of several bodies* A, B, C, D, *&c., if
any one of those bodies, as* A*, attract all the rest,*
B, C, D, *&c., with accelerative forces that are
reciprocally as the squares of the distances from the attracting
body; and another body, as* B*, attracts also the rest.*
A, C, D, *&c., with forces that are reciprocally as the
squares of the distances from the attracting body; the absolute
forces of the attracting bodies* A *and* B *will
be to each other as those very bodies A and B to which those forces belong.*

For the accelerative attractions of all the bodies B, C, D, towards A, are by the supposition equal to each other at equal distances; and in like manner the accelerative attractions of all the bodies towards B are also equal to each other at equal distances. But the absolute attractive force of the body A is to the absolute attractive force of the body B as the accelerative attraction of all the bodies towards A to the accelerative attraction of all the bodies towards B at equal distances; and so is also the accelerative attraction of the body B towards A to the accelerative attraction of the body A towards B. But the accelerative attraction of the body B towards A is to the accelerative attraction of the body A towards B as the mass of the body A to the mass of the body B; because the motive forces which (by the 2d, 7th, and 8th Definition) are as the accelerative forces and the bodies attracted conjunctly are here equal to one another by the third Law. Therefore the absolute attractive force of the body A is to the absolute attractive force of the body B as the mass of the body A to the mass of the body B. Q.E.D.

Cor. 1. Therefore if each of the bodies of the system A, B, C, D, &c. does singly attract all the rest with accelerative forces that are reciprocally as the squares of the distances from the attracting body, the absolute forces of all those bodies will be to each other as the bodies themselves.

Cor. 2. By a like reasoning, if each of the bodies of the system A, B, C, D, &c., do singly attract all the rest with accelerative forces, which are either reciprocally or directly in the ratio of any power whatever of the distances from the attracting body; or which are defined by the distances from each of the attracting bodies according to any common law; it is plain that the absolute forces of those bodies are as the bodies themselves.

Cor. 3. In a system of bodies whose forces decrease in the duplicate ratio of the distances, if the lesser revolve about one very great one in ellipses, having their common focus in the centre of that great body, and of a figure exceedingly accurate; and moreover by radii drawn to that great body describe areas proportional to the times exactly; the absolute forces of those bodies to each other will be either accurately or very nearly in the ratio of the bodies. And so on the contrary. This appears from Cor. of Prop. XLVIII, compared with the first Corollary of this Prop.

These Propositions naturally lead us to the analogy there is between centripetal forces, and the central bodies to which those forces used to be directed; for it is reasonable to suppose that forces which are directed to bodies should depend upon the nature and quantity of those bodies, as we see they do in magnetical experiments. And when such cases occur, we are to compute the attractions of the bodies by assigning to each of their particles its proper force, and then collecting the sum of them all. I here use the word attraction in general for any endeavour, of what kind soever, made by bodies to approach to each other; whether that endeavour arise from the action of the bodies themselves, as tending mutually to or agitating each other by spirits emitted; or whether it arises from the action of the aether or of the air, or of any medium whatsoever, whether corporeal or incorporeal, any how impelling bodies placed therein towards each other. In the same general sense I use the word impulse, not defining in this treatise the species or physical qualities of forces, but investigating the quantities and mathematical proportions of them; as I observed before in the Definitions. In mathematics we are to investigate the quantities of forces with their proportions consequent upon any conditions supposed; then, when we enter upon physics, we compare those proportions with the phenomena of Nature, that we may know what conditions of those forces answer to the several kinds of attractive bodies. And this preparation being made, we argue more safely concerning the physical species, causes, and proportions of the forces. Let us see, then, with what forces sphaerical bodies consisting of particles endued with attractive powers in the manner above spoken of must act mutually upon one another: and what kind of motions will follow from thence.

*If to every point of a sphaerical surface there tend equal
centripetal forces decreasing in the duplicate ratio of the
distances from those points; I say, that a corpuscle placed within
that superficies will not be attracted by those forces any way.*

Let HIKL, be that sphaerical superficies, and P a corpuscle placed within. Through P let there be drawn to this superficies to two lines HK, IL, intercepting very small arcs HI, KL; and because (by Cor. 3, Lem. VII) the triangles HPI, LPK are alike, those arcs will be proportional to the distances HP, LP; and any particles at HI and KL of the sphaerical superficies, terminated by right lines passing through P, will be in the duplicate ratio of those distances. Therefore the forces of these particles exerted upon the body P are equal between themselves. For the forces are as the particles directly, and the squares of the distances inversely. And these two ratios compose the ratio of equality. The attractions therefore, being made equally towards contrary parts, destroy each other. And by a like reasoning all the attractions through the whole sphaerical superficies are destroyed by contrary attractions. Therefore the body P will not be any way impelled by those attractions. Q.E.D.

*The same things supposed as above, I say, that a corpuscle
placed with out the sphaerical superficies is attracted towards
the centre of the sphere with a force reciprocally proportional to
the square of its distance from that centre.*

Let AHKB, *ahkb*, be two equal sphaerical superficies
described about the centre S, *s*;
their diameters AB, *ab*; and let P and *p* be two
corpuscles situate without the spheres in those diameters produced.
Let there
be drawn from the corpuscles the lines PHK, PIL, *phk, pil*,
cutting off from the great circles AHB,
*ahb*, the equal arcs HK, *hk*, IL, *il*;
and to those lines let fall the perpendiculars SD, *sd*, SE, *se*,
IR, *ir*; of which let SD, *sd*, cut PL, *pl*,
in F and *f*. Let fall also to the diameters the
perpendiculars IQ, *iq*. Let now the angles DPE, *dpe*,
vanish; and because DS and *ds*, ES and *es* are
equal, the lines PE, PF, and *pe, pf*, and the lineolao DF, *df*
may be taken for equal; because their last ratio, when the angles DPE,
*dpe* vanish together, is the ratio of equality. These things
then supposed, it will be, as PI to PF so is RI to DF, and as *pf*
to *pi* so is *df* or DF to *ri*; and, *ex
aequo*, as PI × *pf* to PF × *pi* so is RI to *ri*,
that is (by Cor. 3, Lem VII), so is the arc IH to the arc *ih*.
Again, PI is to PS as IQ to SE, and *ps* to *pi* as *se*
or SE to *iq*; and, *ex aequo*, PI × *ps* to
PS × *pi* as IQ to *iq*. And compounding the ratios
PI² × *pf* × *ps* is to *pi²* × PF × PS, as IH
× IQ to *ih* × *iq*; that is, as the circular
superficies which is described by the arc IH, as the semi-circle AKB
revolves about the diameter AB, is to the circular superficies
described by the arc *ih* as the semi-circle *akb*
revolves about the diameter *ab*. And the forces with which
these superficies attract the corpuscles P and *p* in the
direction of lines tending to those superficies are by the hypothesis
as the superficies themselves directly, and the squares of the
distances of the superficies from those corpuscles inversely; that is,
as *pf* × *ps* to PF × PS. And these forces again are
to the oblique parts of them which (by the resolution of forces as in
Cor. 2, of the Laws) tend to the centres in the directions of the
lines PS, *ps*, as PI to PQ, and *pi* to *pq*;
that is (because of the like triangles PIQ and PSF, *piq* and
*psf*), as PS to PF and *ps* to *pf*. Thence *ex
aequo*, the attraction of the corpuscle P towards S is to the
attraction of the corpuscle *p* towards *s* as
PF × pf × ps

PS is to pf
× PF × ps

ps, that is, as *ps*² to PS² .
And, by a like reasoning, the forces with which the superficies
described by the revolution of the arcs KL, *kl* attract those
corpuscles, will be as *ps²* to PS² . And in the same ratio
will be the forces of all the circular superficies into which each of
the sphaerical superficies may be divided by taking *sd*
always equal to SD, and *se* equal to SE. And therefore, by
composition, the forces of the entire sphaerical superficies exerted
upon those corpuscles will be in the same ratio. Q.E.D.

*If to the several points of a sphere there tend equal
centripetal forces decreasing in a duplicate ratio of the
distances from those points; and there be given both the density
of the sphere and the ratio of the diameter of the sphere to the
distance of the corpuscle from its centre; I say, that the force
with which the corpuscle is attracted is proportional to the
semi-diameter of the sphere.*

For conceive two corpuscles to be severally attracted by two spheres, one by one, the other by the other, and their distances from the centres of the spheres to be proportional to the diameters of the spheres respectively, and the spheres to be resolved into like particles, disposed in a like situation to the corpuscles. Then the attractions of one corpuscle towards the several particles of one sphere will be to the attractions of the other towards as many analogous particles of the other sphere in a ratio compounded of the ratio of the particles directly, and the duplicate ratio of the distances inversely. But the particles are as the spheres, that is, in a triplicate ratio of the diameters, and the distances are as the diameters; and the first ratio directly with the last ratio taken twice inversely, becomes the ratio of diameter to diameter. Q.E.D.

Cor. 1. Hence if corpuscles revolve in circles about spheres composed of matter equally attracting, and the distances from the centres of the spheres be proportional to their diameters, the periodic times will be equal.

Cor. 2. And, *vice versa*, if the
periodic times are equal, the distances will be proportional to the
diameters. These two Corollaries appear from Cor. 3, Prop. IV.

Cor. 3. If to the several points of any two solids whatever, of like figure and equal density, there tend equal centripetal forces decreasing in a duplicate ratio of the distances from those points, the forces, with which corpuscles placed in a like situation to those two solids will be attracted by them, will be to each other as the diameters of the solids.

*If to the several points of a given sphere there tend equal
centripetal forces decreasing in a duplicate ratio of the
distances from the points; I say, that a corpuscle placed within
the sphere is attracted by a force proportional to its distance
from the centre.*

In the sphere ABCD, described about the centre S, let there be placed the corpuscle P; and about the same centre S, with the interval SP, conceive described an interior sphere PEQF. It is plain (by Prop. LXX) that the concentric sphaerical superficies, of which the difference AEBF of the spheres is composed, have no effect at all upon the body P, their attractions being destroyed by contrary attractions. There remains, therefore, only the attraction of the interior sphere PEQF. And (by Prop, LXXII) this is as the distance PS. Q.E.D.

By the superficies of which I here imagine the solids composed, I do not mean superficies purely mathematical, but orbs so extremely thin, that their thickness is as nothing; that is, the evanescent orbs of which the sphere will at last consist when the number of the orbs is increased, and their thickness diminished without end. In like manner, by the points of which lines, surfaces, and solids are said to be composed, are to be understood equal particles, whose magnitude is perfectly inconsiderable.

*The same things supposed, I say, that a corpuscle situate
without the sphere is attracted with a force reciprocally
proportional to the square of its distance from the centre.*

For suppose the sphere to be divided into innumerable concentric sphaerical superficies, and the attractions of the corpuscle arising from the several superficies will be reciprocally proportional to the square of the distance of the corpuscle from the centre of the sphere (by Prop. LXXI). And, by composition, the sum of those attractions, that is, the attraction of the corpuscle towards the entire sphere, will be in the same ratio. Q.E.D.

Cor. 1. Hence the attractions of homogeneous spheres at equal distances from the centres will be as the spheres themselves. For (by Prop. LXXII) if the distances be proportional to the diameters of the spheres, the forces will be as the diameters. Let the greater distance be diminished in that ratio; and the distances now being equal, the attraction will be increased in the duplicate of that ratio; and therefore will be to the other attraction in the triplicate of that ratio; that is, in the ratio of the spheres.

Cor. 2. At any distances whatever the attractions are as the spheres applied to the squares of the distances.

Cor. 3. If a corpuscle placed without an homogeneous sphere is attracted by a force reciprocally proportional to the square of its distance from the centre, and the sphere consists of attractive particles, the force of every particle will decrease in a duplicate ratio of the distance from each particle.

*If to the several points of a given sphere there tend equal
centripetal forces decreasing in a duplicate ratio of the
distances from the points; I say, that another similar sphere will
be attracted by it with a force reciprocally proportional to the
square of the distance of the centres.*

For the attraction of every particle is reciprocally as the square of its distance from the centre of the attracting sphere (by Prop. LXXIV), and is therefore the same as if that whole attracting force issued from one single corpuscle placed in the centre of this sphere. But this attraction is as great as on the other hand the attraction of the same corpuscle would be, if that were itself attracted by the several particles of the attracted sphere with the same force with which they are attracted by it. But that attraction of the corpuscle would be (by Prop. LXXIV) reciprocally proportional to the square of its distance from the centre of the sphere; therefore the attraction of the sphere, equal thereto, is also in the same ratio. Q.E.D.

Cor. 1. The attractions of spheres towards other homogeneous spheres are as the attracting spheres applied to the squares of the distances of their centres from the centres of those which they attract.

Cor. 2. The case is the same when the attracted sphere does also attract. For the several points of the one attract the several points of the other with the same force with which they themselves are attracted by the others again; and therefore since in all attractions (by Law III) the attracted and attracting point are both equally acted on, the force will be doubled by their mutual attractions, the proportions remaining.

Cor. 3. Those several truths demonstrated above concerning the motion of bodies about the focus of the conic sections will take place when an attracting sphere is placed in the focus, and the bodies move without the sphere.

Cor. 4. Those things which were demonstrated before of the motion of bodies about the centre of the conic sections take place when the motions are performed within the sphere.

*If spheres be however dissimilar (as to density of matter and
attractive force) in the same ratio onward from the centre to the
circumference; but every where similar, at every given distance
from the centre, on all sides round about; and the attractive
force of every point decreases in the duplicate ratio of the
distance of the body attracted; I say, that the whole force with
which one of these spheres attracts the other will be reciprocally
proportional to the square of the distance of the centres.*

Imagine several concentric similar spheres, AB, CD, EF, &c., the
innermost of which added to the outermost may compose a matter more
dense towards the centre, or subducted from them may leave the same
more lax and rare. Then, by Prop. LXXV, these spheres will attract
other similar concentric spheres GH, IK, LM,
&c., each the other, with forces reciprocally proportional to the
square of the distance SP. And, by composition or division, the sum of
all those forces, or the excess of any of them above the others; that
is, the entire force with which the whole sphere AB (composed of any
concentric spheres or of their differences) will attract the whole
sphere GH (composed of any concentric spheres or their differences) in
the same ratio. Let the number of the concentric spheres be increased
*in infinitum*, so that the density of the matter together with
the attractive force may, in the progress from the circumference to
the centre, increase or decrease according to any given law; and by
the addition of matter not attractive, let the deficient density be
supplied, that so the spheres may acquire any form desired; and the
force with which one of these attracts the other will be still, by the
former reasoning, in the same ratio of the square of the distance
inversely. Q.E.D.

Cor. 1. Hence if many spheres of this kind, similar in all respects, attract each other mutually, the accelerative attractions of each to each, at any equal distances of the centres, will be as the attracting spheres.

Cor. 2. And at any unequal distances, as the attracting spheres applied to the squares of the distances between the centres.

Cor. 3. The motive attractions, or the weights of the spheres towards one another, will be at equal distances of the centres as the attracting and attracted spheres conjunctly; that is, as the products arising from multiplying the spheres into each other.

Cor. 4. And at unequal distances, as those products directly, and the squares of the distances between the centres inversely.

Cor. 5. These proportions take place also when the attraction arises from the attractive virtue of both spheres mutually exerted upon each other. For the attraction is only doubled by the conjunction of the forces, the proportions remaining as before.

Cor. 6. If spheres of this kind revolve about others at rest, each about each; and the distances between the centres of the quiescent and revolving bodies are proportional to the diameters of the quiescent bodies; the periodic times will be equal.

Cor. 7. And, again, if the periodic times are equal, the distances will be proportional to the diameters.

Cor. 8. All those truths above demonstrated, relating to the motions of bodies about the foci of conic sections, will take place when an attracting sphere, of any form and condition like that above described, is placed in the focus.

Cor. 9. And also when the revolving bodies are also attracting spheres of any condition like that above described.

*If to the several points of spheres there tend centripetal
forces proportional to the distances of the points from the
attracted bodies; I say, that the compounded force with which two
spheres attract each other mutually is as the distance between the
centres of the spheres.*

Case 1. Let AEBF be a sphere; S its centre; P
a corpuscle attracted; PASB the axis of the sphere passing through the
centre of the corpuscle; EF, *ef* two planes cutting the
sphere, and perpendicular to the axis, and equi-distant, one on one
side, the other on the other, from the centre of the sphere; G and *g*
the intersections of the planes and the axis; and H any point in the
plane EF. The centripetal force of the point H upon the corpuscle P,
exerted in the direction of the line PH, is as the distance PH; and
(by Cor. 2, of the Laws) the same exerted in the direction of the line
PG, or towards the centre S, is as the length PG. Therefore the force
of all the points in the plane EF (that is, of that whole plane) by
which the corpuscle P is attracted towards the centre S is as the
distance PG multiplied by the number of those points, that is, as the
solid contained under that plane EF and the distance PG. And in like
manner the force of the plane *ef*, by which the corpuscle P
is attracted towards the centre S, is as that plane drawn into its
distance P*g*, or as the equal plane EF drawn into that
distance P*g*; and the sum of the forces of both planes as the
plane EF drawn into the sum of the distances PG + P*g*, that
is, as that plane drawn into twice the distance PS of the centre and
the corpuscle; that is, as twice the plane EF drawn into the distance
PS, or as the sum of the equal planes EF + *ef* drawn into the
same distance. And, by a like reasoning, the forces of all the planes
in the whole sphere, equi-distant on each side from the centre of the
sphere, are as the sum of those planes drawn into the distance PS,
that is, as the whole sphere and the distance PS conjunctly.
Q.E.D.

Case 2. Let now the corpuscle P attract the sphere AEBF. And, by the same reasoning, it will appear that the force with which the sphere is attracted is as the distance PS. Q.E.D.

Case 3. Imagine another sphere composed of innumerable corpuscles P; and because the force with which every corpuscle is attracted is as the distance of the corpuscle from the centre of the first sphere, and as the same sphere conjunctly, and is therefore the same as if it all proceeded from a single corpuscle situate in the centre of the sphere, the entire force with which all the corpuscles in the second sphere are attracted, that is, with which that whole sphere is attracted, will be the same as if that sphere were attracted by a force issuing from a single corpuscle in the centre of the first sphere; and is therefore proportional to the distance between the centres of the spheres. Q.E.D.

Case 4. Let the spheres attract each other mutually, and the force will be doubled, but the proportion will remain. Q.E.D.

Case 5. Let the corpuscle *p* be
placed within the sphere AEBF; and because the force of the plane *ef*
upon the corpuscle is as the solid contained under that plane and the
distance *pg*; and the contrary force of the plane EP as the
solid contained under that plane and the distance *p*G; the
force compounded of both will be as the difference of the solids, that
is, as the sum of the equal planes drawn into half the difference of
the distances; that is, as that sum drawn into *pS*, the
distance of the corpuscle from the centre of the sphere. And, by a
like reasoning, the attraction of all the planes EF, *ef*,
throughout the whole sphere, that is, the attraction of the whole
sphere, is conjunctly as the sum of all the planes, or as the whole
sphere, and as *p*S, the distance of the corpuscle from the
centre of the sphere. Q.E.D.

Case 6. And if there be composed a new sphere
out of innumerable corpuscles such as *p*, situate within the
first sphere AEBF, it may be proved, as before, that the attraction,
whether single of one sphere towards the other, or mutual of both
towards each other, will be as the distance *p*S of the
centres. Q.E.D.

*If spheres is the progress from the centre to the circumference
be however dissimilar and unequable, but similar on every side
round about at all given distances from the centre; and the
attractive force of every point be as the distance of the
attracted body; I say, that the entire force with which two
spheres of this kind attract each other mutually is proportional
to the distance between the centres of the spheres.*

This is demonstrated from the foregoing Proposition, in the same manner as Proposition LXXVI was demonstrated from Proposition LXXV.

Cor. Those things that were above demonstrated in Prop. X and LXIV, of the motion of bodies round the centres of conic sections, take place when all the attractions are made by the force of sphaerical bodies of the condition above described, and the attracted bodies are spheres of the same kind.

I have now explained the two principal cases of attractions; to wit, when the centripetal forces decrease in a duplicate ratio of the distances, or increase in a simple ratio of the distances, causing the bodies in both cases to revolve in conic sections, and composing sphaerical bodies whose centripetal forces observe the same law of increase or decrease in the recess from the centre as the forces of the particles themselves do; which is very remarkable. It would be tedious to run over the other cases, whose conclusions are less elegant and important, so particularly as I have done these. I choose rather to comprehend and determine them all by one general method as follows.

*If about the centre* S *there be described any circle
as* AEB*, and about the centre* P *there be also
described two circles* EF, ef*, cutting the first in* E
*And* e*, and the line PS in F and f; and there be let
fall to PS the perpendiculars ED, ed; I say, that if the distance
of the arcs* EF, ef *be supposed to be infinitely
diminished, the last ratio of the evanscent line* Dd *to
the evanescent line* Ff *is the same as that of the line*
PE *to the line* PS*.*

For if the line P*e* cut the arc EF in *q*; and the
right line E*e*, which
coincides with the evanescent arc E*e*, be produced, and meet the right
line PS in T; and there be let fall from S to PE the perpendicular SG;
then, because of the like triangles DTE, *d*T*e*, DES,
it will be as D*d* to E*e* so DT to TE, or DE to ES: and
because the triangles, E*eq*, ESG (by Lem. VIII, and Cor. 3,
Lem. VII) are similar, it will be as E*e* to *eq* or F*f*
so ES to SG; and, *ex aequo*, as D*d* to F*f* so
DE to SG; that is (because of the similar triangles PDE, PGS), so is
PE to PS. Q.E.D.

*Suppose a superficies as* EFfe *to have its breadth
infinitely diminished, and to be just vanishing and that the same
superficies by its revolution round the axis* PS *describes
a sphaerical concavo-convex solid, to the several equal particles
of which there tend equal centripetal forces; I say, that the
force with which that solid attracts a corpuscle situate in* P
*Is in a ratio compounded of the ratio of the solid* DE² × Ff
*And the ratio of the force with which the given particle in the
place* Ff *would, attract the same corpuscle.*

For if we consider, first, the force of the sphaerical superficies FE which
is generated by the revolution of the arc FE, and is cut any where, as in
*r*, by the line *de*, the annular part of the
superficies generated by the revolution of the arc *r*E will
be as the lineola D*d*, the radius of the sphere PE remaining
the same; as *Archimedes* has demonstrated in his Book of the
Sphere and Cylinder. And the force of this superficies exerted in the
direction of the lines PE or P*r* situate all round in the
conical superficies, will be as this annular superficies itself; that
is as the lineola D*d*, or, which is the same, as the rectangle
under the given radius PE of the sphere and the lineola D*d*;
but that force, exerted in the direction of the line PS tending to the
centre S, will be less in the ratio PD to PE, and therefore will be as
PD × D*d*. Suppose now the line DF to be divided into
innumerable little equal particles, each of which call D*d*,
and then the superficies FE will be divided into so many equal annuli,
whose forces will be as the sum of all the rectangles PD × D*d*,
that is, as ½PF² − ½PD², and therefore as DE². Let now the superficies
FE be drawn into the altitude F*f*; and the force of the solid
EF*fe* exerted upon the corpuscle P will be as DE² × F*f*;
that is, if the force be given which any given particle as F*f*
exerts upon the corpuscle P at the distance PF. But if that force be
not given, the force of the solid EF*fe* will be as the solid
DE² × F*f* and that force not given, conjunctly.
Q.E.D.

*If to the several equal parts of a sphere* ABE *described
about the centre S there tend equal centripetal forces; and from
the several points* D *in the axis of the sphere* AB *in
which a corpuscle, as* F*, is placed, there be erected the
perpendiculars* DE *meeting the sphere in* E*, and
if in those perpendiculars the lengths* DN *be taken as
the quantity* DE^{2}
× PS

PE, *and as the force which a particle of the sphere
situate in the axis exerts at the distance* PE *upon the
corpuscle* P *conjunctly; I say, that the whole force with
which the corpuscle* P *is attracted towards the sphere is
as the area* ANB*, comprehended under the axis of the
sphere* AB*, and the crrve line* ANB*, the locus of the point N.*

For supposing the construction in the last Lemma and Theorem to
stand, conceive the axis of the sphere AB to be divided into
innumerable equal particles D*d*, and the whole sphere to be
divided into so many sphserical concavo-convex laminae EF*fe*;
and erect the perpendicular *dn*. By the last Theorem, the
force with which the laminae EF*fe* attracts the corpuscle *P*
is as DE² × F*f* and the force of one particle exerted at the
distance PE or PF, conjunctly. But (by the last Lemma) D*d* is to F*f* as PE to PS,
and therefore F*f* is equal to PS
× Dd

PE; and DE² × F*f* is equal to D*d*
× DE^{2} × PS

PE; and therefore the force of the lamina EF*fe*
is as D*d* × DE^{2}
× PS

PE and the force of a particle exerted at the
distance PF conjunctly; that is, by the supposition, as DN × D*d*,
or as the evanescent area DN*nd*. Therefore the forces of all
the laminae exerted upon the corpuscle P are as all the areas DN*nd*,
that is, the whole force of the sphere will be as the whole area ANB.
Q.E.D.

Cor. 1. Hence if the centripetal force
tending to the several particles remain always the same at all
distances, and DN be made as DE^{2}
× PS

PE the whole force with which the corpuscle is attracted by
the sphere is as the area ANB.

Cor. 2. If the centripetal force of the
particles be reciprocally as the distance of the corpuscle attracted
by it, and DN be made as DE^{2}
× PS

PE^{2}, the force with which the corpuscle P is
attracted by the whole sphere will be as the area ANB.

Cor. 3. If the centripetal force of the
particles be reciprocally as the cube of the distance of the corpuscle
attracted by it, and DN be made as DE^{2}
× PS

PE^{4}, the force with which the corpuscle is
attracted by the whole sphere will be as the area ANB.

Cor. 4. And universally if the centripetal
force tending to the several particles of the sphere be supposed to be
reciprocally as the quantity V; and DN be made as DE^{2} × PS

PE × V; the force with which a corpuscle is attracted by the
whole sphere will be as the area ANB.

*The things remaining as above, it is required to measure the area* ANB*.*

From the point P let there be drawn the right line PH touching the
sphere in H; and to the axis PAB, letting fall the perpendicular HI,
bisect PI in L; and (by Prop. XII, Book II, Elem.) PE² is equal to
PS² + SE² + 2PSD. But because the triangles SPH,
SHI are alike, SE² or SH² is equal to the rectangle PSI.
Therefore PE² is equal to the rectangle contained under PS and PS
+ SI + 2SD; that is, under PS and 2LS + 2SD; that is, under PS and
2LD. Moreover DE² is equal to SE² − SD², or SE² − LS² + 2SLD − LD²,
that is, 2SLD − LD² − ALB. For LS² − SE² or LS² − SA² (by Prop. VI,
Book II, Elem.) is equal to the rectangle ALB. Therefore if instead of
DE² we write 2SLD − LD² − ALB, the quantity DE^{2} × PS

PE × V, which (by Cor. 4 of the foregoing Prop.) is as the
length of the ordinate DN, will now resolve itself into three parts
2SLD × PS

PE × V − LD^{2}
× PS

PE × V − ALB
× PS

PE × V; where if instead of V
we write the inverse ratio of the centripetal force, and instead of PE
the mean proportional between PS and 2LD, those three parts will
become ordinates to so many curve lines, whose areas are discovered by
the common methods. Q.E.D.

Example 1. If the centripetal force tending
to the several particles of the sphere be reciprocally as the
distance; instead of V write PE the distance, then 2PS × LD for PE²;
and DN will become as SL − ½LD − ALB

2LD. Suppose DN equal to its double 2SL
− LD − ALB

LD; and 2SL the given part of the
ordinate drawn into the length AB will describe the rectangular area
2SL × AB; and the indefinite part LD, drawn perpendicularly into the
same length with a continued motion, in such sort as in its motion one
way or another it may either by increasing or decreasing remain always
equal to the length LD, will describe the area LB^{2} − LA^{2}

2, that is, the area SL × AB; which
taken from the former area 2SL × AB, leaves the area SL × AB. But the
third part ALB

LD, drawn after the same manner with a
continued motion perpendicularly into the same length,
will describe the area of an hyperbola, which subducted from the area
SL × AB will leave ANB the area sought. Whence arises this
construction of the Problem. At the points, L, A, B, erect the
perpendiculars L*l*, A*a*, B*b*; making A*a*
equal to LB, and B*b* equal to LA. Making L*l* and LB
asymptotes, describe through the points *a, b,* the
hyperbolic curve *ab*. And the chord *ba* being drawn,
will inclose the area *aba* equal to the area sought ANB.

Example 2. If the centripetal force tending
to the several particles of the sphere be reciprocally as the cube of
the distance, or (which is the same thing) as that cube applied to any
given plane; write PE^{3}

2AS^{2} for V, and 2PS × LD for
PE²; and DN will become as SL × AS^{2}

PS × LD − AS^{2}

2PS − ALB
× AS^{2}

2PS × LD^{2} that is
(because PS, AS, SI are continually proportional), as LSI

LD − ^{1}/_{2}SI
− ALB × SI

2LD^{2}. If we draw
then these three parts into the length AB, the first LSI

LD will generate the area of an
hyperbola; the second ½SI the area ½AB × SI; the third ALB × SI

2LD^{2} the area ALB × SI

2LA ALB
× SI

2LB, that is, ½AB × SI. From the first
subduct the sum of the second and third, and there will remain ANB,
the area sought. Whence arises this construction of the problem. At
the points L, A, S, B, erect
the perpendiculars L*l* A*a* S*s*, B*b*, of
which suppose S*s* equal to SI; and through the point *s*,
to the asymptotes L*l*, LB, describe the hyperbola *asb*
meeting the perpendiculars A*a*, B*b*, in *a*
and *b*; and the rectangle 2ASI, subducted from the hyberbolic
area A*asb*B, will leave ANB the area sought.

Example 3. If the centripetal force tending
to the several particles of the spheres decrease in a quadruplicate
ratio of the distance from the particles; write PE^{4}

2AS^{3} for V, then √(2PS+LD)
for PE, and DN will become as SI^{2} × SL

√(2SI) × 1

√LD^{3} − SI^{2}

2√(2SI) × 1

√LD − SI^{2}
× ALB

2√(2SI) × 1

√LD^{5}. These three
parts drawn into the length AB, produce so many areas, viz.
2SI^{2} × SL

√(2SI) into (
1

√(LA) − 1

√(LB) ); SI^{2}

√(2SI) into √LB −
√LA; and SI^{2}
× ALB

3√(2SI) into (
1

√(LA^{3}) − 1

√(LB^{3}) ). And
these after due reduction come forth 2SI^{2}
× SL

LI, SI², and SI² + 2SI^{3}

3LI. And these by subducting the last
from the first, become 4SI^{3}

3LI. Therefore the entire force with
which the corpuscle P is attracted towards the centre of the sphere is
as SI^{3}

PI, that is, reciprocally as PS³ × PI.
Q.E.I.

By the same method one may determine the attraction of a corpuscle situate within the sphere, but more expeditiously by the following Theorem.

*In a sphere described about the centre S with the interval SA,
if there be taken* SI, SA, SP *continually proportional; I
say, that the attraction of a corpuscle within the sphere in any
place* I *is to its attraction without the sphere in the
place* P *in a ratio compounded of the subduplicate ratio
of* IS, PS*, the distances from the centre, and the
subduplicate ratio of the centripetal forces tending to the centre
in those places* P *and* I*.*

As if the centripetal forces of the particles of the sphere be
reciprocally as the distances of the corpuscle attracted by them; the
force with which the corpuscle situate in I is attracted by the entire
sphere will be to the force with which it is attracted in P in a ratio
compounded of the subduplicate ratio of the distance SI to the
distance SP, and the subduplicate ratio of the centripetal force in
the place I arising from any particle in the centre to the centripetal
force in the place P arising from the same particle in the centre;
that is, in the subduplicate ratio of the distances SI, SP to each
other reciprocally. These two subduplicate ratios compose the ratio of
equality, and therefore the attractions in I and P produced by the
whole sphere are equal. By the like calculation, if the forces of the
particles of the sphere are reciprocally in a duplicate ratio of the
distances, it will be found that the attraction in I is to the
attraction in P as the distance SP to the semi-diameter SA of the
sphere. If those forces are reciprocally in a triplicate ratio of the
distances, the attractions in I and P will be to each other as SP² to
SA²; if in a quadruplicate ratio, as SP³ to SA³. Therefore since the
attraction in P was found in this last case to be reciprocally as PS³
× PI, the attraction in I will be reciprocally as SA³ × PI, that is,
because SA³ is given reciprocally as PI. And the progression is the
same *in infinitum*. The demonstration of this Theorem is as
follows:

The things remaining as above constructed, and a corpuscle being in
any place P, the ordinate DN was found to be
as DE^{2} × PS

PE × V. Therefore if IE be drawn, that
ordinate for any other place of the corpuscle, as I, will become (*mutatis
mutandis*) as DE^{2}
× IS

IE × V. Suppose the centripetal forces
flowing from any point of the sphere, as E, to be to each other at the
distances IE and PE as PE^{n} to IE^{n} (where the
number *n* denotes the index of the powers of PE and IE), and
those ordinates will become as DE^{2}
× PS

PE × PE^{n} and DE^{2} × IS

IE × IE^{n} whose ratio to each
other is as PS × IE × IE^{n} to
IS × PE × PE^{n}. Because SI, SE, SP are
in continued proportion, the triangles SPE, SEI are alike; and thence
IE is to PE as IS to SE or SA. For the ratio of IE to PE write the
ratio of IS to SA; and the ratio of the ordinates becomes that of PS ×
IE^{n} to SA × PE^{n}. But the ratio of PS to SA is
subduplicate of that of the distances PS, SI; and the ratio of IE^{n}
to PE^{n} (because IE is to PE as IS to SA) is subduplicate of
that of the forces at the distances PS, IS. Therefore the ordinates,
and consequently the areas which the ordinates describe, and the
attractions proportional to them, are in a ratio compounded of those
subduplicate ratios. Q.E.D.

*To find the force with which a corpuscle placed in the centre
of a sphere is attracted towards any segment of that sphere whatsoever.*

Let P be a body in the centre of that sphere and RBSD a segment
thereof contained under the plane RDS, and thesphaerical superficies
RBS. Let DB be cut in F by a sphaerical superficies EFG described from
the centre P, and let the segment be divided into the parts BREFGS,
FEDG. Let us suppose that segment to be not a purely mathematical but
a physical superficies, having some, but a perfectly inconsiderable
thickness. Let that thickness be called O, and (by what *Archimedes*
has demonstrated) that superficies will be as PF × DF × O. Let us
suppose besides the attractive forces of the particles of the sphere
to be reciprocally as that power of the distances, of which *n*
is index; and the force with which the superficies EFG attracts the
body P will be (by Prop. LXXIX) as DE^{2}
× O

PF^{n}, that is, as 2DF × O

PF^{(n-1)} − DF^{2} × O

PF^{n}. Let the
perpendicular FN drawn into O be proportional
to this quantity; and the curvilinear area BDI, which the ordinate FN,
drawn through the length DB with a continued motion will describe,
will be as the whole force with which the whole segment RBSD attracts
the body P. Q.E.I.

*To find the force with which a corpuscle, placed without the
centre of a sphere in the axis of any segment, is attracted by that segment.*

Let the body P placed in the axis ADB of the segment EBK be attracted by that segment. About the centre P, with the interval PE, let the spherical superficies EFK be described; and let it divide the segment into two parts EBKFE and EFKDE. Find the force of the first of those parts by Prop. LXXXI, and the force of the latter part by Prop. LXXXIII, and the sum of the forces will be the force of the whole segment EBKDE. Q.E.I.

The attractions of sphaerical bodies being now explained, it comes next in order to treat of the laws of attraction in other bodies consisting in like manner of attractive particles; but to treat of them particularly is not necessary to my design. It will be sufficient to subjoin some general propositions relating to the forces of such bodies, and the motions thence arising, because the knowledge of these will be of some little use in philosophical inquiries.

*If a body be attracted by another, and its attraction be vastly
stronger when it is contiguous to the attracting body than when
they are separated from one another by a very small interval; the
forces of the particles of the attracting body decrease, in the
recess of the body attracted, in more than a duplicate ratio of
the distance of the particles.*

For if the forces decrease in a duplicate ratio of the distances from the particles, the attraction towards a sphaerical body being (by Prop. LXXIV) reciprocally as the square of the distance of the attracted body from the centre of the sphere, will not be sensibly increased by the contact, and it will be still less increased by it, if the attraction, in the recess of the body attracted, decreases in a still less proportion. The proposition, therefore, is evident concerning attractive spheres. And the case is the same of concave sphaerical orbs attracting external bodies. And much more does it appear in orbs that attract bodies placed within them, because there the attractions diffused through the cavities of those orbs are (by Prop. LXX) destroyed by contrary attractions, and therefore have no effect even in the place of contact. Now if from these spheres and sphaerical orbs we take away any parts remote from the place of contact, and add new parts any where at pleasure, we may change the figures of the attractive bodies at pleasure; but the parts added or taken away, being remote from the place of contact, will cause no remarkable excess of the attraction arising from the contact of the two bodies. Therefore the proposition holds good in bodies of all figures. Q.E.D.

*If the forces of the particles of which an attractive body is
composed decrease, in the recess of the attractive body, in a
triplicate or more than a triplicate ratio of the distance from
the particles, the attraction will be vastly stronger in the point
of contact than when the attracting and attracted bodies are
separated from each other, though by never so small an interval.*

For that the attraction is infinitely increased when the attracted corpuscle comes to touch an attracting sphere of this kind, appears, by the solution of Problem XLI, exhibited in the second and third Examples. The same will also appear (by comparing those Examples and Theorem XLI together) of attractions of bodies made towards concavo-convex orbs, whether the attracted bodies be placed without the orbs, or in the cavities within them. And by adding to or taking from those spheres and orbs any attractive matter any where without the place of contact, so that the attractive bodies may receive any assigned figure, the Proposition will hold good of all bodies universally. Q.E.D.

*If two bodies similar to each other, and consisting of matter
equally attractive, attract separately two corpuscles proportional
to those bodies, and in a like situation to them, the accelerative
attractions of the corpuscles towards the entire bodies will be as
the accelerative attractions of the corpuscles towards particles
of the bodies proportional to the wholes, and alike situated in them.*

For if the bodies are divided into particles proportional to the wholes, and alike situated in them, it will be, as the attraction towards any particle of one of the bodies to the attraction towards the correspondent particle in the other body, so are the attractions towards the several particles of the first body, to the attractions towards the several correspondent particles of the other body; and, by composition, so is the attraction towards the first whole body to the attraction towards the second whole body. Q.E.D.

Cor. 1 . Therefore if, as the distances of
the corpuscles attracted increase, the attractive forces of the
particles decrease in the ratio of any power of the distances, the
accelerative attractions towards the whole bodies will be as the
bodies directly, and those powers of the distances inversely. As if
the forces of the particles decrease in a duplicate ratio of the
distances from the corpuscles attracted, and the bodies are as A³ and
B³, and therefore both the cubic sides of the bodies, and the distance
of the attracted corpuscles from the bodies, are as A and B; the
accelerative attractions towards the bodies will be as A^{3}

A^{2} and B^{3}

B^{2}, that is, as A and B the
cubic sides of those bodies. If the forces of the particles decrease
in a triplicate ratio of the distances from the attracted corpuscles,
the accelerative attractions towards the whole bodies will be as
A^{3}

A^{3} and B^{3}

B^{3}, that is, equal. If the
forces decrease in a quadruplicate ratio, the attractions towards the
bodies will be as A^{3}

A^{4} and B^{3}

B^{4}, that is, reciprocally as
the cubic sides A and B. And so in other cases.

Cor. 2. Hence, on the other hand, from the forces with which like bodies attract corpuscles similarly situated, may be collected the ratio of the decrease of the attractive forces of the particles as the attracted corpuscle recedes from them; if so be that decrease is directly or inversely in any ratio of the distances.

*If the attractive forces of the equal particles of any body be
as the distance of the places from the particles, the force of the
whole body will tend to its centre of gravity; and will be the
same with the force of a globe, consisting of similar and equal
matter, and having its centre in the centre of gravity.*

Let the particles A, B, of the body RSTV attract any corpuscle Z with forces which, supposing the particles to be equal between themselves, are as the distances AZ, BZ; but, if they are supposed unequal, are as those particles and their distances AZ, BZ, conjunctly, or (if I may so speak) as those particles drawn into their distances AZ, BZ respectively. And let those forces be expressed by the contents under A × AZ, and B × BZ. Join AB, and let it be cut in G, so that AG may be to BG as the particle B to the particle A; and G will be the common centre of gravity of the particles A and B. The force A × AZ will (by Cor. 2, of the Laws) be resolved into the forces A × GZ and A × AG; and the force B × BZ into the forces B × GZ and B × BG. Now the forces A × AG and B × BG, because A is proportional to B, and BG to AG, are equal, and therefore having contrary directions destroy one another. There remain then the forces A × GZ and B × GZ. These tend from Z towards the centre G, and compose the force (A + B) × GZ; that is, the same force as if the attractive particles A and B were placed in their common centre of gravity G, composing there a little globe.

By the same reasoning, if there be added a third particle C, and the
force of it be compounded with the force (A + B) ×
GZ tending to the centre G, the force thence arising will
tend to the common centre of gravity of that globe in G and of the
particle C; that is, to the common centre of gravity of the three
particles A, B, C; and will be the same as if that globe and the
particle C were placed in that common centre composing a greater globe
there; and so we may go on *in infinitum*. Therefore the whole
force of all the particles of any body whatever RSTV is the same as if
that body, without removing its centre of gravity, were to put on the
form of a globe. Q.E.D.

Cor. Hence the motion of the attracted body Z will be the same as if the attracting body RSTV were sphaerical; and therefore if that attracting body be either at rest, or proceed uniformly in a right line, the body attracted will move in an ellipsis having its centre in the centre of gravity of the attracting body.

*If there be several bodies consisting of equal particles whose
forces are as the distances of the places from each, the force
compounded of all the forces by which any corpuscle is attracted
will tend to the common centre of gravity of the attracting
bodies; and will be the same as if those attracting bodies,
preserving their common centre of gravity, should unite there, and
be formed into a globe.*

This is demonstrated after the same manner as the foregoing Proposition.

Cor. Therefore the motion of the attracted body will be the same as if the attracting bodies, preserving their common centre of gravity, should unite there, and be formed into a globe. And, therefore, if the common centre of gravity of the attracting bodies be either at rest, or proceed uniformly in a right line, the attracted body will move in an ellipsis having its centre in the common centre of gravity of the attracting bodies.

*If to the several points of any circle there tend equal
centripetal forces, increasing or decreasing in any ratio of the
distances; it is required to find the force with which a corpuscle
is attracted, that is, situate any where in a right line which
stands at right angles to the plant of the circle at its centre.*

Suppose a circle to be described about the centre A with any interval AD in a plane to which the right line AP is perpendicular; and let it be required to find the force with which a corpuscle P is attracted towards the same. From any point E of the circle, to the attracted corpuscle P, let there be drawn the right line PE. In the right line PA take PF equal to PE, and make a perpendicular FK, erected at F, to be as the force with which the point E attracts the corpuscle P. And let the curve line IKL be the locus of the point K. Let that curve meet the plane of the circle in L. In PA take PH equal to PD, and erect the perpendicular HI meeting that curve in I; and the attraction of the corpuscle P towards the circle will be as the area AHIL drawn into the altitude AP. Q.E.I.

For let there be taken in AE a very small line E*e*. Join P*e*,
and in PE, PA take PC, P*f* equal to P*e*. And because
the force, with which any point E of the annulus described about the
centre A with the interval AE in the aforesaid plane attracts to
itself the body P, is supposed to be as FK; and, therefore, the force
with which that point attracts the body P towards A is as AP × FK

PE; and the force with which the whole
annulus attracts the body P towards A is as the annulus and AP × FK

PE conjunctly; and that annulus also is
as the rectangle under the radius AE and the breadth E*e*, and
this rectangle (because PE and AE, E*e* and CE are
proportional) is equal to the rectangle PE × CE or PE × F*f*;
the force with which that annulus attracts the body P towards A will
be as PE × F*f* and AP ×
FK

PE conjunctly; that is, as the content
under F*f* × FK × AP, or as the area FK*kf* drawn into
AP. And therefore the sum of the forces with which all the annuli, in
the circle described about the centre A with the interval AD, attract
the body P towards A, is as the whole area AHIKL drawn into AP.
Q.E.D.

Cor. 1. Hence if the forces of the points
decrease in the duplicate ratio of the
distances, that is, if FK be as 1

PF^{2} and therefore the
area AHIKL as 1

PA − 1

PH; the attraction of the
corpuscle P towards the circle will be as 1 − PA

PH; that is, as AH

PH.

Cor. 2. And universally if the forces of the
points at the distances D be reciprocally as any power D^{n}
of the distances; that is, if FK be as 1

D^{n} and therefore the area
AHIKL as 1

PA^{n-1} − 1

PH^{n-1}; the
attraction of the corpuscle P towards the circle will be as 1

PA^{n-2} − 1

PH^{n-1}.

Cor. 3. And if the diameter of the circle be
increased *in infinitum*, and the number *n* be
greater than unity; the attraction of the corpuscle P towards the
whole infinite plane will be reciprocally as PA^{n-2}, because
the other term PA

PA^{n-1} vanishes.

*To find the attraction of a corpuscle situate in the axis of a
round solid, to whose several points there tend equal centripetal
forces decreasing in any ratio of the distances whatsoever.*

Let the corpuscle P, situate in the axis AB of the solid DECG, be attracted towards that solid. Let the solid be cut by any circle as RFS, perpendicular to the axis: and in its semi-diameter FS, in any plane PALKB passing through the axis, let there be taken (by Prop. XC) the length FK proportional to the force with which the corpuscle P is attracted towards that circle. Let the locus of the point K be the curve line LKI, meeting the planes of the outermost circles AL and BI in L and I; and the attraction of the corpuscle P towards the solid will be as the area LABI. Q.E.I.

Cor. 1. Hence if the solid be a cylinder
described by the parallelogram ADEB revolved about the axis AB, and
the centripetal forces tending to the several points be reciprocally
as the squares of the distances from the points; the attraction of the
corpuscle P towards this cylinder will be as AB − PE + PD. For the
ordinate FK (by Cor. 1, Prop. XC) will be as 1 − PF

PR. The part 1 of this quantity, drawn
into the length AB, describes
the area 1 × AB; and the other part PF

PR, drawn into the length PB describes the area 1
into (PE − AD) (as may be easily shewn from the quadrature of
the curve LKI); and, in like manner, the same part drawn into the
length PA describes the area 1 into (PD − AD),
and drawn into AB, the difference of PB and PA, describes 1
into (PE − PD), the difference of the areas. From the first
content 1 × AB take away the last content 1 into
(PE − PD), and there will remain the area LABI equal to
1 into (AB − PE + PD). Therefore the force,
being proportional to this area, is as AB − PE +
PD.

Cor. 2. Hence also is known the force by
which a spheroid AGBC attracts any body P situate externally in its
axis AB. Let NKRM be a conic section whose ordinate ER perpendicular
to PE may be always equal to the length of the line PD, continually
drawn to the point D in which that ordinate cuts the spheroid. From
the vertices A, B, of the spheriod, let there be erected to its axis
AB the perpendiculars AK, BM, respectively equal to AP, BP, and
therefore meeting the conic section in K and M; and join KM cutting
off from it the segment KMRK. Let S be the centre of the spheroid, and
SC its greatest semi-diameter; and the force with which the spheroid
attracts the body P will be to the force with which a sphere described
with the diameter AB attracts the same body as AS × CS^{2} − PS × KMRK

PS^{2} + CS^{2} − AS^{2}
is to AS^{3}

3PS^{2}. And by a
calculation founded on the same principles may be found the forces of
the segments of the spheroid.

Cor. 3. If the corpuscle be placed within the spheroid and in its axis, the attraction will be as its distance from the centre. This may be easily collected from the following reasoning, whether the particle be in the axis or in any other given diameter. Let AGOF be an attracting spheroid, S its centre, and P the body attracted. Through the body P let there be drawn the semi-diameter SPA, and two right lines DE, FG meeting the spheroid in D and E, F and G; and let PCM, HLN be the superficies of two interior spheroids similar and concentrical to the exterior, the first of which passes through the body P, and cuts the right lines DE, FG in B and C; and the latter cuts the same right lines in H and I, K and L. Let the spheroids have all one common axis, and the parts of the right lines intercepted on both sides DP and BE, FP and CG, DH and IE, FK and LG, will be mutually equal; because the right lines DE, PB, and HI, are bisected in the same point, as are also the right lines FG, PC, and KL. Conceive now DPF, EPG to represent opposite cones described with the infinitely small vertical angles DPF, EPG, and the lines DH, EI to be infinitely small also. Then the particles of the cones DHKF, GLIE, cut off by the spheroidical superficies, by reason of the equality of the lines DH and EI, will be to one another as the squares of the distances from the body P, and will therefore attract that corpuscle equally. And by a like reasoning if the spaces DPF, EGCB be divided into particles by the superficies of innumerable similar spheroids concentric to the former and having one common axis, all these particles will equally attract on both sides the body P towards contrary parts. Therefore the forces of the cone DPF, and of the conic segment EGCB, are equal, and by their contrariety destroy each other. And the case is the same of the forces of all the matter that lies without the interior spheroid PCBM. Therefore the body P is attracted by the interior spheroid PCBM alone, and therefore (by Cor. 3, Prop. LXXII) its attraction is to the force with which the body A is attracted by the whole spheroid AGOD as the distance PS to the distance AS. Q.E.D.

*An attracting body being given, it is required to find the
ratio of the decrease of the centripetal forces tending to its several points.*

The body given must be formed into a sphere, a cylinder, or some regular figure, whose law of attraction answering to any ratio of decrease may be found by Prop. LXXX, LXXXI, and XCI. Then, by experiments, the force of the attractions must be found at several distances, and the law of attraction towards the whole, made known by that means, will give the ratio of the decrease of the forces of the several parts; which was to be found.

*If a solid be plane on one side, and infinitely extended on all
other sides, and consist of equal particles equally attractive,
whose forces decrease, in the recess from the solid, in the ratio
of any power greater than the square of the distances; and a
corpuscle placed towards either part of the plane is attracted by
the force of the whole solid; I say that the attractive force of
the whole solid, in the recess from its plane superficies, will
decrease in the ratio of a power whose side is the distance of the
corpuscle from the plane, and its index less by* 3 *than
the index of the power of the distances.*

Case 1. Let LG*l* be the plane by
which the solid is terminated. Let the solid lie on that hand of the
plane that is towards I, and let it be resolved into innumerable
planes *m*HM, *n*IN, *o*KO, &c., parallel
to GL. And first let the attracted body C be placed without the solid.
Let there be drawn CGHI perpendicular to those innumerable planes, and
let the attractive forces of the points of the solid decrease in the
ratio of a power of the distances whose index is the number *n*
not less than 3. Therefore (by Cor. 3, Prop. XC) the force with which
any plane *m*HM attracts the point C is reciprocally as CH^{n-2}.
In the plane *m*HM take the length HM reciprocally
proportional to CH^{n-2}, and that force will be as HM. In
like manner in the several planes *l*GL, *n*IN, *o*KO,
&c., take the lengths GL, IN, KO, &c., reciprocally
proportional to CG^{n-2}, CI^{n-2}, CK^{n-2},
&c., and the forces of those planes will be as the lengths so
taken, and therefore the sum of the forces as the sum of the lengths,
that is, the force of the whole solid as the area GLOK produced
infinitely towards OK. But that area (by the known methods of
quadratures) is reciprocally as CG^{n-3}, and therefore the
force of the whole solid is reciprocally as CG^{n-3}.
Q.E.D.

Case 2. Let the corpuscle C be now placed on
that hand of the plane *l*GL that is within the solid, and
take the distance CK equal to the distance CG. And the part of the
solid LG*lo*KO terminated by the parallel planes *l*GL,
*o*KO, will attract the corpuscle C, situate in the middle,
neither one way nor another, the contrary actions of the opposite
points destroying one another by reason of their equality. Therefore
the corpuscle C is attracted by the force only of the solid situate
beyond the plane OK. But this force (by Case 1) is reciprocally as CK^{n-3},
that is, (because CG, CK are equal) reciprocally as CG^{n-3}.
Q.E.D.

Cor. 1. Hence if the solid LGIN be terminated on each side by two infinite parallel places LG, IN, its attractive force is known, subducting from the attractive force of the whole infinite solid LGKO the attractive force of the more distant part NIKO infinitely produced towards KO.

Cor. 2. If the more distant part of this
solid be rejected, because its attraction compared with the attraction
of the nearer part is inconsiderable, the
attraction of that nearer part will, as the distance increases,
decrease nearly in the ratio of the power CG^{n-3}.

Cor. 3. And hence if any finite body, plane on one side, attract a corpuscle situate over against the middle of that plane, and the distance between the corpuscle and the plane compared with the dimensions of the attracting body be extremely small; and the attracting body consist of homogeneous particles, whose attractive forces decrease in the ratio of any power of the distances greater than the quadruplicate; the attractive force of the whole body will decrease very nearly in the ratio of a power whose side is that very small distance, and the index less by 3 than the index of the former power. This assertion does not hold good, however, of a body consisting of particles whose attractive forces decrease in the ratio of the triplicate power of the distances; because, in that case, the attraction of the remoter part of the infinite body in the second Corollary is always infinitely greater than the attraction of the nearer part.

If a body is attracted perpendicularly towards a given plane, and from the law of attraction given, the motion of the body be required; the Problem will be solved by seeking (by Prop. XXXIX) the motion of the body descending in a right line towards that plane, and (by Cor. 2, of the Laws) compounding that motion with an uniform motion performed in the direction of lines parallel to that plane. And, on the contrary, if there be required the law of the attraction tending towards the plane in perpendicular directions, by which the body may be caused to move in any given curve line, the Problem will be solved by working after the manner of the third Problem.

But the operations may be contracted by resolving the ordinates into
converging series. As if to a base A the length B be ordinately
applied in any given angle, and that length be as any power of the
base A m

n; and there be sought the force with
which a body, either attracted towards the base or driven from it in
the direction of that ordinate, may be caused to move in the curve
line which that ordinate always describes with its superior extremity;
I suppose the base to be increased by a very small part O, and I
resolve the ordinate (A+O)^{
mn} into an infinite
series A^{mn} + m

nOA^{m
− nn} + mm − mn

2nnOOA^{m
− 2nn} &c., and I
suppose the force proportional to the term of this series in which O
is of two dimensions, that is, to the term mm − mn

2nn OOA ^{m-2nn}. Therefore the force
sought is as mm − mn

nn A ^{m-2nn}, or, which is the
same thing, as mm
− mn

nn B ^{m-2nn}. As if the ordinate
describe a parabola, *m* being = 2, and *n* = 1, the
force will be as the given quantity 2B°, and therefore is given.
Therefore with a given force the body will move in a parabola, as *Galileo*
has demonstrated. If the ordinate describe an hyperbola, *m*
being = 0 − 1, and *n* = 1, the force will be as 2A^{-3}
or 2B^{3}; and therefore a force which is as the cube of the
ordinate will cause the body to move in an hyperbola. But leaving this
kind of propositions, I shall go on to some others relating to motion
which I have hot yet touched upon.

*If two similar mediums be separated from each other by a space
terminated on both sides by parallel planes, and a body in its
passage through that space be attracted or impelled
perpendicularly towards either of those mediums, and not agitated
or hindered by any other force; and the attraction be every where
the same at equal distances from either plane, taken towards the
same hand of the plane; I say, that the sine of incidence upon
either plane will be to the sine of emergence of the other plane
in a given ratio.*

Case 1. Let A*a* and B*b* be
two parallel planes, and let the body light upon the first plane A*a*
in the direction of the line GH, and in its whole passage through the
intermediate space let it be attracted or impelled towards the medium
of incidence, and by that action let it be made to describe a curve
line HI, and let it emerge in the direction of the line IK. Let there
be erected IM perpendicular to B*b* the plane of emergence, and
meeting the line of incidence GH prolonged in M, and the plane of
incidence A*a* in R; and let the line of emergence KI be
produced and meet HM in L. About the centre L, with the interval LI,
let a circle be described cutting both HM in P and Q, and MI produced
in N; and, first, if the attraction or impulse be supposed uniform,
the curve HI (by what *Galileo* has demonstrated) be a
parabola, whose property is that of a rectangle under
its given latus rectum and the line IM is equal to the square of HM;
and moreover the line HM will be bisected in L. Whence if to MI there
be let fall the perpendicular LO, MO, OR will be equal: and adding the
equal lines ON, OI, the wholes MN, IR will be equal also. Therefore
since IR is given, MN is also given, and the rectangle NMI is to the
rectangle under the latus rectum and IM, that is, to HM² in a given
ratio. But the rectangle NMI is equal to the rectangle PMQ, that is,
to the difference of the squares ML², and PL² or LI²; and HM² hath a
given ratio to its fourth part ML²; therefore the ratio of ML² − LI²
to ML² is given, and by conversion the ratio of LI² to ML², and its
subduplicate, the ratio of LI to ML. But in every triangle, as LMI,
the sines of the angles are proportional to the opposite sides.
Therefore the ratio of the sine of the angle of incidence LMR to the
sine of the angle of emergence LIR is given. Q.E.D.

Case 2. Let now the body pass successively
through several spaces terminated with parallel planes A*ab*B,
E*bc*C, &c., and let it be acted on by a force which is
uniform in each of them separately, but different in the different
spaces; and by what was just demonstrated, the sine of the angle of
incidence on the first plane A*a* is to the sine of emergence
from the second plane B*b* in a given ratio; and this sine of
incidence upon the second plane B*b* will be to the sine of
emergence from the third plane C*c* in a given ratio; and this
sine to the sine of emergence from the fourth plane D*d* in a
given ratio; and so on *in infinitum*; and, by equality, the
sine of incidence on the first plane to the sine of emergence from the
last plane in a given ratio. Let now the intervals of the planes be
diminished, and their number be infinitely increased, so that the
action of attraction or impulse, exerted according to any assigned
law, may become continual, and the ratio of the sine of incidence on
the first plane to the sine of emergence from the last plane being all
along given, will be given then also. Q.E.D.

*The same things being supposed, I say, that the velocity of the
body before its incidence is to its velocity after emergence as
the sine of emergence to the sine of incidence.*

Make AH and I*d* equal, and erect the perpendiculars AG, *d*K
meeting the lines of incidence and emergence GH, IK, in G and K. In GH
take TH equal to IK, and to the plane A*a* let fall a
perpendicular T*v*. And (by Cor. 2 of the Laws of Motion) let
the motion of the body be resolved into two, one perpendicular to the
planes A*a*, B*b*, C*c*,
&c, and another parallel to them. The force of attraction or
impulse, acting in directions perpendicular to those planes, does not
at all alter the motion in parallel directions; and therefore the body
proceeding with this motion will in equal times go through those equal
parallel intervals that lie between the line AG and the point H, and
between the point I and the line *d*K; that is, they will
describe the lines GH, IK in equal times. Therefore the velocity
before incidence is to the velocity after emergence as GH to IK or TH,
that is, as AH or I*d* to *v*H; that is (supposing TH
or IK radius), as the sine of emergence to the sine of incidence.
Q.E.D.

*The same things being supposed, and that the motion before
incidence is swifter than afterwards; I say, that if the line of
incidence be inclined continually, the body will be at last
reflected, and the angle of reflexion will be equal to the angle of incidence.*

For conceive the body passing between the parallel planes A*a*,
B*b*, C*c*, &c., to describe parabolic arcs as
above; and let those arcs be HP, PQ, QR, &c. And let the obliquity
of the line of incidence GH to the first plane A*a* be such
that the sine of incidence may be to the radius of the circle whose
sine it is, in the same ratio which the same sine of incidence hath to
the sine of emergence from the plane D*d* into the space D*de*E;
and because the sine of emergence is now become equal to radius, the
angle of emergence will be a right one, and therefore the line of
emergence will coincide with the plane D*d*. Let the body come
to this plane in the point R; and because the line of emergence
coincides with that plane, it is manifest that the body can proceed no
farther towards the plane E*e*. But neither can it proceed in
the line of emergence R*d*; because it is perpetually attracted
or impelled towards the medium of incidence. It will return,
therefore, between the planes C*c*, D*d*, describing an
arc of a parabola QR*q*, whose principal vertex (by what *Galileo*
has demonstrated) is in R, cutting the plane C*c* in the same
angle at *q*, that it did before at Q; then going on in the
parabolic arcs *qp, ph,* &c., similar and equal to the
former arcs QP, PH, &c., it will cut the rest of the planes in the
same angles at *p, h,* &c., as it did before in P, H,
&c., and will emerge at last with the same obliquity at *h*
with which it first impinged on that plane at H. Conceive now the
intervals of the planes A*a*, B*b*, C*c*, D*d*,
E*e*, &c., to be infinitely diminished, and the number in
finitely increased, so that the action of attraction or impulse,
exerted according to any assigned law, may become continual; and, the
angle of emergence remaining all along equal to the angle of
incidence, will be equal to the same also at last. Q.E.D.

These attractions bear a great resemblance to the reflexions and
refractions of light made in a given ratio of the secants, as was
discovered by *Snellius*; and consequently in a given ratio of
the sines, as was exhibited by *Des Cartes*. For it is now
certain from the phenomena of *Jupiter's* Satellites,
confirmed by the observations of different astronomers, that light is
propagated in succession, and requires about seven or eight minutes to
travel from the sun to the earth. Moreover, the rays of light that are
in our air (as lately was discovered by *Grimaldus*, by the
admission of light into a dark room through a small hole, which I have
also tried) in their passage near the angles of bodies, whether
transparent or opaque (such as the circular and rectangular edges of
gold, silver and brass coins, or of knives, or broken pieces of stone
or glass), are bent or inflected round those bodies as if they were
attracted to them; and those rays which in their passage come nearest
to the bodies are the most inflected, as if they were most attracted:
which tiling I myself have also carefully observed. And those which
pass at greater distances are less inflected; and those at still
greater distances are a little inflected the contrary way, and form
three fringes of colours. In the figure *s* represents the
edge of a knife, or any
kind of wedge A*s*B; and *gowog*, *fnunf*, *emtme*,
*dlsld*, are rays inflected towards the knife in the arcs *owo,
nvn, mtm, lsl*; which inflection is greater or less according to
their distance from the knife. Now since this inflection of the rays
is performed in the air without the knife, it follows that the rays
which fall upon the knife are first inflected in the air before they
touch the knife. And the case is the same of the rays falling upon
glass. The refraction, therefore, is made not in the point of
incidence, but gradually, by a continual inflection of the rays: which
is done partly in the air before they touch the glass, partly (if I
mistake not) within the glass, after they have entered it; as is
represented in the rays *ckzc, biyb, ahxa*, falling upon *r,
q, p,* and inflected between *k* and *z, i* and
*y, h* and *x*. Therefore because of the analogy there
is between the propagation of the rays of light and the motion of
bodies, I thought it not amiss to add the following Propositions for
optical uses: not at all considering the nature of the rays of light,
or inquiring whether they are bodies or not; but only determining the
trajectories of bodies which are extremely like the trajectories of
the rays.

*Supposing the sine of incidence upon any superficies to be in a
given ratio to the sine of emergence; and that the inflection of
the paths of those bodies near that superficies is performed in a
very short space, which may be considered as a point; it is
required to determine such a superficies as may cause all the
corpuscles issuing from any one given place to converge to another given place.*

Let A be the place from whence the corpuscles diverge; B the place to which they should converge; CDE the curve line which by its revolution round the axis AB describes the superficies sought; D, E, any two points of that curve: and EF, EG, perpendiculars let fall on the paths of the bodies AD, DB. Let the point D approach to and coalesce with the point E; and the ultimate ratio of the line DF by which AD is increased, to the line DG by which DB is diminished, will be the same as that of the sine of incidence to the sine of emergence. Therefore the ratio of the increment of the line AD to the decrement of the line DB is given; and therefore if in the axis AB there be taken any where the point C through which the curve CDE must pass, and CM the increment of AC be taken in that given ratio to CN the decrement of BC, and from the centres A, B, with the intervals AM, BN, there be described two circles cutting each other in D; that point D will touch the curve sought CDE, and, by touching it any where at pleasure, will determine that curve. Q.E.I.

Cor. 1. By causing the point A or B to go off
sometimes *in infinitum*, and sometimes to move towards other
parts of the point C, will be obtained all those figures which *Cartesius*
has exhibited in his Optics and Geometry relating to refractions. The
invention of which *Cartesius* having thought fit to conceal,
is here laid open in this Proposition.

Cor. 2. If a body lighting on any superficies
CD in the direction of a right line AD, drawn according to any law,
should emerge in the direction of another right line DK; and from the
point C there be drawn curve lines CP, CQ, always perpendicular to AD,
DK; the increments of the lines PD, QD, and therefore the lines
themselves PD, QD, generated by those increments, will be as the sines
of incidence and emergence to each other, and *è contra*.

*The same things supposed; if round the axis* AB *any
attractive superficies be described as* CD*, regular or
irregular, through which the bodies issuing from the given place*
A *must pass; it is required to find a second attractive
superficies* EF*, which may make those bodies converge to a
given place* B*.*

Let a line joining AB cut the first superficies in C and the second in E, the point D being taken any how at pleasure. And supposing the sine of incidence on the first superficies to the sine of emergence from the same, and the sine of emergence from the second superficies to the sine of incidence on the same, to be as any given quantity M to another given quantity N; then produce AB to G, so that BG may be to CE as M − N to N; and AD to H, so that AH may be equal to AG; and DF to K, so that DK may be to DH as N to M. Join KB, and about the centre D with the interval DH describe a circle meeting KB produced in L, and draw BF parallel to DL; and the point F will touch the line EF, which, being turned round the axis AB, will describe the superficies sought. Q.E.F.

For conceive the lines CP, CQ, to be every where perpendicular to AD, DF, and the lines ER, ES to FB, FD respectively, and therefore QS to be always equal to CE; and (by Cor. 2, Prop. XCVII) PD will be to QD as M to N, and therefore as DL to DK, or FB to FK; and by division as DL − FB or PH − PD − FB to FD or FQ − QD; and by composition as PH − FB to FQ, that is (because PH and CG, QS and CE, are equal), as CE + BG − FR to CE − FS. But (because BG is to CE as M − N to N) it comes to pass also that CE + BG is to CE as M to N; and therefore, by division, FR is to FS as M to N; and therefore (by Cor. 2, Prop XCVII) the superficies EF compels a body, falling upon it in the direction DF, to go on in the line FR to the place B. Q.E.D.

In the same manner one may go on to three or more superficies. But of all figures the spherical is the most proper for optical uses. If the object glasses of telescopes were made of two glasses of a sphaerical figure, containing water between them, it is not unlikely that the errors of the refractions made in the extreme parts of the superficies of the glasses may be accurately enough corrected by the refractions of the water. Such object glasses are to be preferred before elliptic and hyperbolic glasses, not only because they may be formed with more ease and accuracy, but because the pencils of rays situate without the axis of the glass would be more accurately refracted by them. But the different refrangibility of different rays is the real obstacle that hinders optics from being made perfect by sphaerical or any other figures. Unless the errors thence arising can be corrected, all the labour spent in correcting the others is quite thrown away.

*If a body is resisted in the ratio of its velocity, the motion
lost by resistance is as the space gone over in its motion.*

For since the motion lost in each equal particle of time is as the velocity, that is, as the particle of space gone over, then, by composition, the motion lost in the whole time will be as the whole space gone over. Q.E.D.

Cor. Therefore if the body, destitute of all gravity, move by its innate force only in free spaces, and there be given both its whole motion at the beginning, and also the motion remaining after some part of the way is gone over, there will be given also the whole space which the body can describe in an infinite time. For that space will be to the space now described as the whole motion at the beginning is to the part lost of that motion.

*Quantities proportional to their differences are continually proportional.*

Let A be to A − B as B to B − C and C to C − D, &c., and, by conversion, A will be to B as B to C and C to D, &c. Q.E.D.

*If a body is resisted in the ratio of its velocity, and moves,
by its* vis insita *only, through a similar medium, and
the times be taken equal, the velocities in the beginning of each
of the times are in a geometrical progression, and the spaces
described in each of the times are as the velocities.*

Case 1. Let the time be divided into equal
particles; and if at the very beginning of each particle we suppose
the resistance to act with one single impulse which is as the
velocity, the decrement of the velocity in each of the
particles of time will be as the same velocity. Therefore the
velocities are proportional to their differences, and therefore (by
Lem. 1, Book II) continually proportional. Therefore if out of an
equal number of particles there be compounded any equal portions of
time, the velocities at the beginning of those times will be as terms
in a continued progression, which are taken by intervals, omitting
every where an equal number of intermediate terms. But the ratios of
these terms are compounded of the equal ratios of the intermediate
terms equally repeated, and therefore are equal. Therefore the
velocities, being proportional to those terms, are in geometrical
progression. Let those equal particles of time be diminished, and
their number increased *in infinitum*, so that the impulse of
resistance may become continual; and the velocities at the beginnings
of equal times, always continually proportional, will be also in this
case continually proportional. Q.E.D.

Case 2. And, by division, the differences of the velocities, that is, the parts of the velocities lost in each of the times, are as the wholes; but the spaces described in each of the times are as the lost parts of the velocities (by Prop. 1, Book I), and therefore are also as the wholes. Q.E.D.

Corol. Hence if to the rectangular asymptotes AC, CH, the hyperbola BG is described, and AB, DG be drawn perpendicular to the asymptote AC, and both the velocity of the body, and the resistance of the medium, at the very beginning of the motion, be expressed by any given line AC, and, after some time is elapsed, by the indefinite line DC; the time may be expressed by the area ABGD, and the space described in that time by the line AD. For if that area, by the motion of the point D, be uniformly increased in the same manner as the time, the right line DC will decrease in a geometrical ratio in the same manner as the velocity; and the parts of the right line AC, described in equal times, will decrease in the same ratio.

*To define the motion of a body which, in a similar medium,
ascends or descends in a right line, and is resisted in the ratio
of its velocity, and acted upon by an uniform force of gravity.*

The body ascending, let the gravity be expounded by any given
rectangle BACH; and the resistance of the medium, at the beginning of
the ascent, by the rectangle BADE, taken on the contrary side of the
right line AB. Through the point B, with the rectangular asymptotes
AC, CH, describe an hyperbola, cutting the perpendiculars DE, *de*,
in G, *g*; and the body ascending
will in the time DG*gd* describe the space EG*ge*; in
the time DGBA, the space of the whole ascent EGB; in the time ABKI,
the space of descent BFK; and in the time IK*ki* the space of
descent KF*fk*; and the velocities of the bodies (proportional
to the resistance of the medium) in these periods of time will be
ABED, AB*ed*, O, ABFI, AB*fi* respectively; and the
greatest velocity which the body can acquire by descending will be
BACH.

For let the rectangle BACH be resolved into in numerable rectangles A*k*,
K*l*, L*m*, M*n*, &c., which shall be as the
increments of the velocities produced in so many equal times; then
will O, A*k*, A*l*, A*m*, A*n*, &c.,
be as the whole velocities; and therefore (by supposition) as the
resistances of the medium in the beginning of each of the equal times.
Make AC to AK, or ABHC to AB*k*K, as the force of gravity to
the resistance in the beginning of the second time; then from the
force of gravity subduct the resistances, and ABHC, K*k*HC, L*l*HC,
M*m*HC, &c., will be as the absolute forces with which the
body is acted upon in the beginning of each of the times, and
therefore (by Law I) as the increments of the velocities, that is, as
the rectangles A*k*, K*l*, L*m*, M*n*,
&c., and therefore (by Lem. 1, Book II) in a geometrical
progression. Therefore, if the right lines K*k*, L*l*, M*m*,
N*n*, &c., are produced so as to meet the hyperbola in *q,
r, s, t,* &c. the areas AB*q*K, K*qr*L, L*rs*M,
M*st*N, &c., will be equal, and therefore analogous to the
equal times and equal gravitating forces. But the area AB*q*K
(by Corol. 3, Lem. VII and VIII, Book I) is to the area B*kq*
as K*q* to ½*kq*, or AC to ½AK, that is, as the force of
gravity to the resistance in the middle of the first time. And by the
like reasoning, the areas *q*KL*r*, *r*LM*s*,
*s*MN*t*, &c., are to the areas *qklr, rlms,
smnt,* &c., as the gravitating forces to the resistances in
the middle of the second, third, fourth time, and so on. Therefore
since the equal areas BAK*q*, *q*KL*r*, *r*LM*s*,
*s*MN*t*, &c., are analogous to the gravitating
forces, the areas B*kq*, *qklr, rlms, smnt,* &c.,
will be analogous to the resistances in the middle of each of the
times, that is (by supposition), to the velocities, and so to the
spaces described. Take the sums of the analogous quantities, and the
areas B*kq*, B*lr*, B*ms*, B*ut*, &c.,
will be analogous to the whole spaces described; and also the areas AB*q*K,
AB*r*L, AB*s*M, AB*t*N, &c., to the times.
Therefore the body, in descending, will in any time AB*r*L
describe the space B*lr*, and in the time L*rt*N the
space *rlnt*. Q.E.D. And the like
demonstration holds in ascending motion.

Corol. 1. Therefore the greatest velocity that the body can acquire by falling is to the velocity acquired in any given time as the given force of gravity which perpetually acts upon it to the resisting force which opposes it at the end of that time.

Corol. 2. But the time being augmented in an arithmetical progression, the sum of that greatest velocity and the velocity in the ascent, and also their difference in the descent, decreases in a geometrical progression.

Corol. 3. Also the differences of the spaces, which are described in equal differences of the times, decrease in the same geometrical progression.

Corol. 4. The space described by the body is the difference of two spaces, whereof one is as the time taken from the beginning of the descent, and the other as the velocity; which [spaces] also at the beginning of the descent are equal among themselves.

*Supposing the force of gravity in any similar medium to be
uniform, and to tend perpendicularly to the plane of the horizon;
to define the motion of a projectile therein, which suffers
resistance proportional to its velocity.*

Let the projectile go from any place D in the direction of any right
line DP, and let its velocity at the beginning of the motion be
expounded by the length DP. From the point P let fall the
perpendicular PC on the horizontal line DC, and cut DC in A, so that
DA may be to AC as the resistance of the medium arising from the
motion upwards at the beginning to the force of gravity; or (which
comes to the same) so that the rectangle under DA and DP may be to
that under AC and CP as the whole resistance at the beginning of the
motion to the force of gravity. With the asymptotes DC, CP describe
any hyperbola GTBS cutting the perpendiculars DG, AB in G and B;
complete the parallelogram DGKC, and let its side GK cut AB in Q. Take
a line N in the same ratio to QB as DC is in to CP; and from any point
R of the right line DC erect RT perpendicular to it, meeting the
hyperbola in T, and the right lines EH, GK, DP in I, *t*, and
V; in that perpendicular take V*r* equal to tGT

N, or which is the same thing, take R*r*
equal to GTIE

N; and the projectile in the time DRTG
will arrive at the point *r* describing the curve line D*ra*F,
the locus of the point *r*; thence it will come to its
greatest height a in the perpendicular AB; and afterwards ever
approach to the asymptote PC. And its velocity in any point *r*
will be as the tangent *r*L to the curve. Q.E.I.

For N is to QB as DC to CP or DR to RV, and therefore RV is equal to
DR × QB

N, and R*r* (that is, RV − V*r*,
or DR × QB − tGT

N ) is equal to DR × AB − RDGT

N. Now let the time be expounded by the
area RDGT and (by Laws, Cor. 2), distinguish the motion of the body
into two others, one of ascent, the other lateral. And since the
resistance is as the motion, let that also be distinguished into two
parts proportional and contrary to the parts of the motion: and
therefore the length described by the lateral motion will be (by Prop.
II, Book II) as the line DR, and the height (by Prop. III, Book II) as
the area DR × AB − RDGT, that is, as the line R*r*. But in the
very beginning of the motion the area RDGT is equal to the rectangle
DR × AQ, and therefore that line R*r* (or DR × AB − DR × AQ

N ) will then be to DR as AB − AQ or QB
to N, that is, as CP to DC; and therefore as the motion upwards to the
motion lengthwise at the beginning. Since, therefore, R*r* is
always as the height, and DR always as the length, and R*r* is
to DR at the beginning as the height to the length, it follows, that R*r*
is always to DR as the height to the length; and therefore that the
body will move in the line D*ra*F, which is the locus of the
point *r*. Q.E.D.

Cor. 1. Therefore *R*r is equal to
DR × AB

N − RDGT

N , and therefore if RT be
produced to X so that RX may be equal to DR
× AB

N, that is, if the parallelogram ACPY
be completed, and DY cutting CP in Z be drawn, and RT be produced till
it meets DY in X; X*r* will be equal to RDGT

N, and therefore proportional to the
time.

Cor. 2. Whence if innumerable lines CR, or,
which is the same, innumerable lines ZX, be taken in a geometrical
progression, there will be as many lines X*r* in an
arithmetical progression. And hence the curve D*ra*F is easily
delineated by the table of logarithms.

Cor. 3. If a parabola be constructed to the
vertex D, and the diameter DG produced downwards, and its latus rectum
is to 2 DP as the whole resistance at the beginning of the notion to
the gravitating force, the velocity with which the body ought to go
from the place D, in the direction of the right line DP, so as in an
uniform resisting medium to describe the curve D*ra*F, will be
the same as that with which it ought to go from the same place D in
the direction of the same right line DP, so as to describe
a parabola in a non-resisting medium. For the latus rectum of this
parabola, at the very beginning of the motion, is
DV^{2}

Vr; and V*r* is tGT

N DR × Tt

2N. But a right line, which, if drawn,
would touch the hyperbola GTS in G, is parallel to DK, and therefore T*t*
is CK × DR

DC, and N is QB × DC

CP. And therefore V*r* is equal
to DR^{2} × CK × CP

2DC^{2} × QB, that is, (because
DR and DC, DV and DP are proportionals), to DV^{2} × CK × CP

2DP^{2} × QB; and the latus
rectum DV^{2}

Vr comes out 2DP^{2} × QB

CK × CP, that is (because QB and CK,
DA, and AC are proportional), 2DP^{2} × DA

AC × CP, and therefore ist to 2DP as DP
× DA to CP × AC; that is, as the resistance to the gravity.
Q.E.D.

Cor. 4. Hence if a body be projected from any
place D with a given velocity, in the direction of a right line DP
given by position, and the resistance of the medium, at the beginning
of the motion, be given, the curve D*ra*F, which that body will
describe, may be found. For the velocity being given, the latus rectum
of the parabola is given, as is well known. And taking 2DP to that
latus rectum, as the force of gravity to the resisting force, DP is
also given. Then cutting DC in A, so that CP × AC may be to DP × DA in
the same ratio of the gravity to the resistance, the point A will be
given. And hence the curve D*ra*F is also given.

Cor. 5. And, on the contrary, if the curve D*ra*F
be given, there will be given both the velocity of the body and the
resistance of the medium in each of the places *r*. For the
ratio of CP × AC to DP × DA being given, there is given both the
resistance of the medium at the beginning of the motion, and the latus
rectum of the parabola; and thence the velocity at the beginning of
the motion is given also. Then from the length of the tangent L
there is given both the velocity proportional to it, and the
resistance proportional to the velocity in any place *r*.

Cor. 6. But since the length 2DP is to the latus rectum of the parabola as the gravity to the resistance in D; and, from the velocity augmented, the resistance is augmented in the same ratio, but the latus rectum of the parabola is augmented in the duplicate of that ratio, it is plain that the length 2DP is augmented in that simple ratio only; and is therefore always proportional to the velocity; nor will it be augmented or diminished by the change of the angle CDP, unless the velocity be also changed.

Cor. 7. Hence appears the method of
determining the curve D*ra*F nearly from the phenomena, and
thence collecting the resistance and velocity with which the body is
projected. Let two similar and equal bodies be projected with the same
velocity, from the place D, in different angles CDP, CD*p*; and
let the places F, *f*, where they fall upon the horizontal
plane DC, be known. Then taking any length for DP or D*p*
suppose the resistance in D to be to the gravity in any ratio
whatsoever, and let that ratio be expounded by any length SM. Then, by
computation, from that assumed length DP, find the lengths DP, D*f*;
and from the ratio Ff

DF, found by calculation, subduct the
same ratio as found by experiment; and let the difference be expounded
by the perpendicular MN. Repeat the same a second and a third time, by
assuming always a new ratio SM of the resistance to the gravity, and
collecting a new difference MN. Draw the affirmative differences on
one side of the right line SM, and the negative on the other side; and
through the points N, N, N, draw a regular curve NNN. cutting the
right line SMMM in X, and SX will be the true ratio of the resistance
to the gravity, which was to be found. From this ratio the length DF
is to be collected by calculation; and a length, which is to the
assumed length DP as the length DF known by experiment to the length
DF just now found, will be the true length DP. This being known, you
will have both the curve line D*ra*F which the body describes,
and also the velocity and resistance of the body in each place.

But, yet, that the resistance of bodies is in the ratio of the velocity, is more a mathematical hypothesis than a physical one. In mediums void of all tenacity, the resistances made to bodies are in the duplicate ratio of the velocities. For by the action of a swifter body, a greater motion in proportion to a greater velocity is communicated to the same quantity of the medium in a less time; and in an equal time, by reason of a greater quantity of the disturbed medium, a motion is communicated in the duplicate ratio greater; and the resistance (by Law II and III) is as the motion communicated. Let us, therefore, see what motions arise from this law of resistance.

*If a body is resisted in the duplicate ratio of its velocity,
and moves by its innate force only through a similar medium; and
the times be taken in a geometrical progression, proceeding from
less to greater terms: I say, that the velocities at the beginning
of each of the times are in the same geometrical progression
inversely; and that the spaces are equal, which are described in
each of the times.*

For since the resistance of the medium is proportional to the square
of the velocity, and the decrement of the velocity is proportional to
the resistance: if the time be divided into innumerable equal
particles, the squares of the velocities at the beginning of each of
the times will be proportional to the differences of the same
velocities. Let those particles of time be AK, KL, LM, &c., taken
in the right line CD; and erect the perpendiculars AB, K*k*, L*l*,
M*m*, &c., meeting the hyperbola B*klm*G, described
with the centre C, and the rectangular asymptotes CD, CH, in B, *k,
l, m,* &c.; then AB will be to K*k* as CK to CA, and,
by division, AB − K*k* to K*k* as AK to CA, and
alternately, AB − K*k* to AK as K*k* to CA; and
therefore as AB × K*k* to AB × CA. Therefore since AK and AB ×
CA are given, AB − K*k* will be as AB × KA; and, lastly, when
AB and K*k* coincide, as AB². And, by the like reasoning, K*k*
− L*l*, L*l* − M*m*, &c., will be as K*k*²,
L*l*², &c. Therefore the squares of the lines AB, K*k*,
L*l*, M*m*, &c., are as their differences; and,
therefore, since the squares of the velocities were shewn above to be
as their differences, the progression of both will be alike. This
being demonstrated it follows also that the areas described by these
lines are in a like progression with the spaces described by these
velocities. Therefore if the velocity at the beginning of the first
time AK be expounded by the line AB, and the
velocity at the beginning of the second time KL by the line K*k*
and the length described in the first time by the area AK*k*B,
all the following velocities will be expounded by the following lines
L*l*, M*m*, &c. and the lengths described, by the
areas K*l*, L*m*. &c. And, by composition, if the
whole time be expounded by AM, the sum of its parts, the whole length
described will be expounded by AM*m*B the sum of its parts. Now
conceive the time AM to be divided into the parts AK, KL, LM, &c.
so that CA, CK, CL, CM, &c. may be in a geometrical progression;
and those parts will be in the same progression, and the velocities
AB, K*k*, L*l*, M*m*, &c., will be in the
same progression inversely, and the spaces described A*k*, K*l*,
L*m*, &c., will be equal. Q.E.D.

Cor. 1. Hence it appears, that if the time be expounded by any part AD of the asymptote, and the velocity in the beginning of the time by the ordinate AB, the velocity at the end of the time will be expounded by the ordinate DG; and the whole space described by the adjacent hyperbolic area ABGD; and the space which any body can describe in the same time AD, with the first velocity AB, in a non-resisting medium, by the rectangle AB × AD.

Cor 2. Hence the space described in a resisting medium is given, by taking it to the space described with the uniform velocity AB in a nonresisting medium, as the hyperbolic area ABGD to the rectangle AB × AD.

Cor. 3. The resistance of the medium is also given, by making it equal, in the very beginning of the motion, to an uniform centripetal force, which could generate, in a body falling through a non-resisting medium, the velocity AB in the time AC. For if BT be drawn touching the hyperbola in B, and meeting the asymptote in T, the right line AT will be equal to AC, and will express the time in which the first resistance, uniformly continued, may take away the whole velocity AB

Cor. 4. And thence is also given the proportion of this resistance to the force of gravity, or any other given centripetal force.

Cor. 5. And, *vice versa*, if there
is given the proportion of the resistance to any given centripetal
force, the time AC is also given, in which a centripetal force equal
to the resistance may generate any velocity as AB; and thence is given
the point B, through which the hyperbola, having CH, CD for its
asymptotes, is to be described; as also the space ABGD, which a body,
by beginning its motion with that velocity AB, can describe in any
time AD, in a similar resisting medium.

*Homogeneous and equal spherical bodies, opposed by resistances
that are in the duplicate ratio of the velocities, and moving on
by their innate force only, will, in times which are reciprocally
as the velocities at the beginning, describe equal spaces, and
lose parts of their velocities proportional to the wholes.*

To the rectangular asymptotes CD, CH describe any hyperbola B*b*E*e*,
cutting the perpendiculars AB, *ab*, DE, *de* in B, *b*,
E, *e*; let the initial velocities be expounded by the
perpendiculars AB, DE, and the times by the lines A*a*, D*d*.
Therefore as A*a* is to D*d*, so (by the hypothesis) is
DE to AB, and so (from the nature of the hyperbola) is CA to CD; and,
by composition, so is C*a* to C*d*. Therefore the areas
AB*ba*, DE*ed*, that is, the spaces described, are equal
among themselves, and the first velocities AB, DE are proportional to
the last *ab, de*; and therefore, by division, proportional to
the parts of the velocities lost, AB − *ab*, DE − *de*.
Q.E.D.

*If spherical bodies are resisted in the duplicate ratio of
their velocities, in times which are as the first motions
directly, and the first resistances inversely, they will lose
parts of their motions proportional to the wholes, and will
describe spaces proportional to those times and the first
velocities conjunctly.*

For the parts of the motions lost are as the resistances and times conjunctly. Therefore, that those parts may be proportional to the wholes, the resistance and time conjunctly ought to be as the motion. Therefore the time will be as the motion directly and the resistance inversely. Wherefore the particles of the times being taken in that ratio, the bodies will always lose parts of their motions proportional to the wholes, and therefore will retain velocities always proportional to their first velocities. And because of the given ratio of the velocities, they will always describe spaces which are as the first velocities and the times conjunctly. Q.E.D.

Cor. 1. Therefore if bodies equally swift are resisted in a duplicate ratio of their diameters, homogeneous globes moving with any velocities whatsoever, by describing spaces proportional to their diameters, will lose parts of their motions proportional to the wholes. For the motion of each globe will be as its velocity and mass conjunctly, that is, as the velocity and the cube of its diameter; the resistance (by supposition) will be as the square of the diameter and the square of the velocity conjunctly; and the time (by this proposition) is in the former ratio directly, and in the latter inversely, that is, as the diameter directly and the velocity inversely; and therefore the space, which is proportional to the time and velocity is as the diameter.

Cor. 2. If bodies equally swift are resisted in a sesquiplicate ratio of their diameters, homogeneous globes, moving with any velocities whatsoever, by describing spaces that are in a sesquiplicate ratio of the diameters, will lose parts of their motions proportional to the wholes.

Cor. 3. And universally; if equally swift
bodies are resisted in the ratio of any power of the diameters, the
spaces, in which homogeneous globes, moving with any velocity
whatsoever, will lose parts of their motions proportional to the
wholes, will be as the cubes of the diameters applied to that power.
Let those diameters be D and E; and if the resistances, where the
velocities are supposed equal, are as D^{n} and E^{n};
the spaces in which the globes, moving with any velocities whatsoever,
will lose parts of their motions proportional to the wholes, will be
as D^{3−n} and E^{3−n}. And therefore homogeneous
globes, in describing spaces proportional to D^{3−n} and E^{3−n},
will retain their velocities in the same ratio to one another as at
the beginning.

Cor. 4. Now if the globes are not homogeneous, the space described by the denser globe must be augmented in the ratio of the density. For the motion, with an equal velocity, is greater in the ratio of the density, and the time (by this Prop.) is augmented in the ratio of motion directly, and the space described in the ratio of the time.

Cor. 5. And if the globes move in different
mediums, the space, in a medium which, *caeteris paribus*,
resists the most, must be diminished in the ratio of the greater
resistance. For the time (by this Prop.) will be diminished in the
ratio of the augmented resistance, and the space in the ratio of the
time.

*The moment of any genitum is equal to the moments of each of
the generating sides drawn into the indices of the powers of those
sides, and into their co-efficients continually.*

I call any quantity a *genitum* which is not made by addition
or subduction of divers parts, but is generated or produced in
arithmetic by the multiplication, division, or extraction of the root
of any terms whatsoever; in geometry by the invention of contents and
sides, or of the extremes and means of proportionals. Quantities of
this kind are products, quotients, roots, rectangles, squares, cubes,
square and cubic sides, and the like. These quantities I here consider
as variable and indetermined, and increasing or decreasing, as it
were, by a perpetual motion or flux; and I understand their
momentaneous increments or decrements by the name of moments; so that
the increments may be esteemed as added or affirmative moments; and
the decrements as subducted or negative ones. But take care not to
look upon finite particles as such. Finite particles are not moments,
but the very quantities generated by the moments. We are to conceive
them as the just nascent principles of finite magnitudes. Nor do we in
this Lemma regard the magnitude of the moments, but their first
proportion, as nascent. It will be the same thing,
if, instead of moments, we use either the velocities of the increments
and decrements (which may also be called the motions, mutations, and
fluxions of quantities), or any finite quantities proportional to
those velocities. The co-efficient of any generating side is the
quantity which arises by applying the genitum to that side.

Wherefore the sense of the Lemma is, that if the moments of any
quantities A, B, C, &c., increasing or decreasing by a perpetual
flux, or the velocities of the mutations which are proportional to
them, be called *a, b, c,* &c., the moment or mutation of
the generated rectangle AB will be *a*B + *b*A; the
moment of the generated content ABC will be *a*BC + *b*AC
+ *c*AB; and the moments of the generated powers A², A³, A^{4},
A^{½}, A^{3/2}, A^{⅓}, A^{⅔},
A^{−1}, A^{−2}, A^{−½} will be 2*a*A, 3*a*A²,
4*a*A³, ½*a*A^{−½}, ^{3}/_{2}*a*A^{½},
⅓*a*A^{−⅔}, ⅔*a*A^{−⅓}, −*a*A^{−2},
−*2a*A^{−3}, −½*a*A^{−3/2}
respectively; and in general, that the moment of any power A n

m, will be n

m *a*A n−m

m. Also, that the moment of the
generated quantity A²B bill be 2*a*AB + bA²; the moment of the
generated quantity A³ B^{4} C² will be 3*a*A² B^{4}
C² + 4*b*A³B³C² + 2*c*A³B^{4}C; and the moment
of the generated quantity A^{3}

B^{2} or A³B^{−2} will
be 3*a*A²B^{−2}−2*b*A³B^{−3}; and so on.
The Lemma is thus demonstrated.

Case 1. Any rectangle, as AB, augmented by a
perpetual flux, when, as yet, there wanted of the sides A and B half
their moments ½*a* and ½*b*, was A−½*a* into B−½*b*,
or AB − ½*a* B − ½*b* A + ¼*ab*; but as soon as
the sides A and B are augmented by the other half moments, the
rectangle becomes A + ½*a* into B + ½*b*, or AB + ½*a*
B + ½*b* A + ¼*ab*. From this rectangle subduct the
former rectangle, and there will remain the excess *a*B + *b*A.
Therefore with the whole increments *a* and *b* of the
sides, the increment *a*B + *b*A of the rectangle is
generated. Q.E.D.

Case 2. Suppose AB always equal to G, and
then the moment of the content ABC or GC (by Case 1) will be *g*C
+ *c*G, that is (putting AB and *a*B + *b*A
for G and *g*), *a*BC + *b*AC + *c*AB.
And the reasoning is the same for contents under ever so many sides.
Q.E.D.

Case 3. Suppose the sides A, B, and C, to be
always equal among themselves; and the moment *a*B + *b*A,
of A², that is, of the rectangle AB, will be 2*a*A; and the
moment *a*BC + *b*AC + *c*AB of A³, that is,
of the content ABC, will be 3*a*A². And by the same reasoning
the moment of any power A^{n} is *na*A^{n−1}.
Q.E.D

Case 4. Therefore since 1

A into A is 1, the moment of 1

A drawn into A,
together with 1

A drawn into *a*, will be the
moment of 1, that is, nothing. Therefore the moment of 1

A, or of A^{−1}, is −a

A^{2}. And generally since
1

A^{n} into A^{n} is
1, the moment of 1

A^{n} drawn into A^{n}
together with 1

A^{n} into *na*A^{n−1}
will be nothing. And, therefore, the moment of 1

A^{n} or A^{−n} will
be −na

A^{n+1}. Q.E.D.

Case 5. And since A^{½} into A^{½}
is A, the moment of A^{½} drawn into 2A^{½} will be *a*
(by Case 3); and, therefore, the moment of A^{½} will be
a

2A^{1}/_{2} or ½*a*A−½.
And, generally, putting A^{
mn} equal to B, then A^{m}
will be equal to B^{n}, and therefore *ma*A^{m−1}
equal to *nb*B^{n−1}, and *ma*A^{−1}
equal to *nb*B^{−1}, or nbA^{−
mn}; and therefore
m

naA^{m−nn} is equal to *b*,
that is, equal to the moment of A^{
mn}. Q.E.D.

Case 6. Therefore the moment of any generated
quantity A^{m}B^{n} is the moment of A^{m}
drawn into B^{n}, together with the moment of B^{n}
drawn into A^{m}, that is, *ma*A^{m−1} B^{n}
+ *nb*B^{n−1} A^{m}; and that whether the
indices *m* and *n* of the powers be whole numbers or
fractions, affirmative or negative. And the reasoning is the same for
contents under more powers. Q.E.D.

Cor. 1. Hence in quantities continually proportional, if one term is given, the moments of the rest of the terms will be as the same terms multiplied by the number of intervals between them nd the given term. Let A, B, C, D, E, F, be continually proportional; then if the term C is given, the moments of the rest of the terms will be among themselves as −2A, −B, D, 2E, 3F.

Cor. 2. And if in four proportionals the two means are given, the moments of the extremes will be as those extremes. The same is to be understood of the sides of any given rectangle.

Cor. 3. And if the sum or difference of two squares is given, the moments of the sides will be reciprocally as the sides.

In a letter of mine to Mr. *J. Collins*, dated *December*
10, 1672, having described a method of tangents, which I suspected to
be the same with *Slusius's* method, which at that time was
not made public, I subjoined these words: *This is one particular,
or rather a Corollary, of a general method,* *which
extends itself, without any troublesome calculation, not only to the
drawing of tangents to any curve lines, whether geometrical or
mechanical, or any how respecting right lines or other curves, but
also to the resolving other abstruser kinds of problems about the
crookedness, areas, lengths, centres of gravity of curves, &c.;
nor is it (as* Hudden's *method* de Maximis &
Minimis*) limited to equations which are free from surd quantities.
This method I have interwoven with that other of working in
equations, by reducing them to infinite series.* So far that
letter. And these last words relate to a treatise I composed on that
subject in the year 1671. The foundation of that general method is
contained in the preceding Lemma.

*If a body in an uniform medium, being uniformly acted upon by
the force of gravity, ascends or descends in a right line; and the
whole space described be distinguished into equal parts, and in
the beginning of each of the parts (by adding or subducting the
resisting force of the medium to or from the force of gravity,
when the body ascends or descends] you collect the absolute
forces; I say, that those absolute forces are in a geometrical progression.*

For let the force of gravity be expounded by the given line AC; the
force of resistance by the indefinite line AK; the absolute force in
the descent of the body by the difference KC: the velocity of the body
by a line AP, which shall be a mean proportional between AK and AC,
and therefore in a subduplicate ratio of the resistance; the increment
of the resistance made in a given particle of time by the lineola KL,
and the contemporaneous increment of the velocity by the lineola PQ;
and with the centre C, and rectangular asymptotes CA, CH, describe any
hyperbola BNS meeting the erected perpendiculars AB, KN, LO in B, N
and O. Because AK is as AP², the moment KL of the one will be as the
moment 2APQ of the other, that is, as AP × KC; for the increment PQ of
the velocity is (by Law II) proportional to the generating force KC.
Let the ratio of KL be compounded with the ratio KN, and the rectangle
KL × KN will become as AP × KC × KN; that is (because the rectangle KC
× KN is given), as AP. But the ultimate ratio of the hyperbolic area
KNOL to the rectangle KL × KN becomes, when the points K and L
coincide, the ratio of equality. Therefore that hyperbolic evanescent
area is as AP. Therefore the whole hyperbolic area ABOL is composed of
particles KNOL which are always proportional to the velocity AP; and
therefore is itself proportional to the space described with that
velocity. Let that area be now divided into equal parts as
ABMI, IMNK, KNOL, &c., and the absolute forces AC, IC, KC, LC,
&c., will be in a geometrical progression. Q.E.D.
And by a like reasoning, in the ascent of the body,
taking, on the contrary side of the point A, the equal areas AB*mi,
imnk, knol,* &c., it will appear that the absolute forces
AC, *i*C, *k*C, *l*C, &c., are continually
proportional. Therefore if all the spaces in the ascent and descent
are taken equal, all the absolute forces *l*C, *k*C, *i*C,
AC, IC, KC, LC, &c., will be continually proportional.
Q.E.D.

Cor. 1. Hence if the space described be
expounded by the hyperbolic area ABNK, the force of gravity, the
velocity of the body, and the resistance of the medium, may be
expounded by the lines AC, AP, and AK respectively; and *vice
versa*.

Cor. 2. And the greatest velocity which the body can ever acquire in an infinite descent will be expounded by the line AC.

Cor. 3. Therefore if the resistance of the medium answering to any given velocity be known, the greatest velocity will be found, by taking it to that given velocity in a ratio subduplicate of the ratio which the force of gravity bears to that known resistance of the medium.

*Supposing what is above demonstrated, I say, that if the
tangents of the angles of the sector of a circle, and of an
hyperbola, be taken proportional to the velocities, the radius
being of a fit magnitude, all the time of the ascent to the
highest place will be as the sector of the circle, and all the
time of descending from the highest place as the sector of the hyperbola.*

To the right line AC, which expresses the force of gravity, let AD be
drawn perpendicular and equal. From the centre D with the
semi-diameter AD describe as well the quadrant A*t*E of a
circle, as the rectangular hyperbola AVZ, whose axis is AK, principal
vertex A, and asymptote DC. Let D*p*, DP be drawn; and the
circular sector A*t*D will be as all the time of the ascent to
the highest place; and the hyperbolic sector ATD as all the time of
descent from the highest place; if so be that the tangents A*p*,
AP of those sectors be as the velocities.

Case 1. Draw D*vq* cutting off the
moments or least particles *t*D*v* and *q*D*p*,
described in the same time, of the sector AD*t* and of the
triangle AD*p*. Since those particles (because of the common
angle D) are in a duplicate ratio of the sides, the particle *t*D*v*
will be as qDp × tD^{2}

pD^{2}, that is (because
*t*D is given), as qDp

pD^{2}. But *p*D² is
AD² + A*p*², that is, AD² + AD × A*k*, or AD × C*k*;
and *q*D*p* is ½AD × *pq*. Therefore *t*D*v*,
the particle of the sector, is as pq

Ck; that is, as the least decrement *pq*
of the velocity directly, and the force C*k* which diminishes
the velocity, inversely; and therefore as the particle of time
answering to the decrement of the velocity. And, by composition, the
sum of all the particles *t*D*v* in the sector AD*t*
will be as the sum of the particles of time answering to each of the
lost particles *pq* of the decreasing velocity A*p*,
till that velocity, being diminished into nothing, vanishes; that is,
the whole sector AD*t* is as the whole time of ascent to the
highest place. Q.E.D.

Case 2. Draw DQV cutting off the least
particles TDV and PDQ of the sector DAV, and of the triangle DAQ; and
these particles will be to each other as DT² to DP², that is (if TX
and AP are parallel), as DX² to DA² or TX² to AP²; and, by division,
as DX² − TX² to DA² − AP² . But, from the nature of the hyperbola, DX²
− TX² is AD²; and, by the supposition, AP² is AD × AK. Therefore the
particles are to each other as AD² to AD² − AD × AK; that is, as AD to
AD − AK or AC to CK: and therefore the particle TDV of the sector is
PDQ × AC

CK; and therefore (because AC and AD
are given) as PQ

CK; that is, as the increment of the
velocity directly, and as the force generating the increment
inversely; and therefore as the particle of the time answering to the
increment. And, by composition, the sum of the particles of time, in
which all the particles PQ of the velocity AP are generated, will be
as the sum of the particles of the sector ATD; that is, the whole time
will be as the whole sector. Q.E.D.

Cor. 1. Hence if AB be equal to a fourth part
of AC, the space which a body will describe by falling in any time
will be to the space which the body could describe, by moving
uniformly on in the same time with its greatest velocity AC, as the
area ABNK, which expresses the space described in falling to the area
ATD, which expresses the time. For since AC is to AP as AP to AK, then
(by Cor. 1, Lem. II, of this Book) LK is to PQ as 2AK to AP, that is,
as 2AP to AC, and thence LK is to ½PQ as AP to ¼AG or AB; and KN is to
AC or AD as AB to CK; and therefore, *ex
aequo*, LKNO to DPQ as AP to CK. But DPQ was to DTV as CK to AC.
Therefore, *ex aequo*, LKNO is to DTV as AP to AC; that is, as
the velocity of the falling body to the greatest velocity which the
body by falling can acquire. Since, therefore, the moments LKNO and
DTV of the areas ABNK and ATD are as the velocities, all the parts of
those areas generated in the same time will be as the spaces described
in the same time; and therefore the whole areas ABNK and ADT,
generated from the beginning, will be as the whole spaces described
from the beginning of the descent. Q.E.D.

Cor. 2. The same is true also of the space
described in the ascent. That is to say, that all that space is to the
space described in the same time, with the uniform velocity AC, as the
area AB*uk* is to the sector AD*t*.

Cor. 3. The velocity of the body, falling in the time ATD, is to the velocity which it would acquire in the same time in a non-resisting space, as the triangle APD to the hyperbolic sector ATD. For the velocity in a non-resisting medium would be as the time ATD, and in a resisting medium is as AP, that is, as the triangle APD. And those velocities, at the beginning of the descent, are equal among themselves, as well as those areas ATD, APD.

Cor. 4. By the same argument, the velocity in
the ascent is to the velocity with which the body in the same time, in
a non-resisting space, would lose all its motion of ascent, as the
triangle A*p*D to the circular sector A*t*D; or as the
right line A*p* to the arc A*t*.

Cor. 5. Therefore the time in which a body,
by falling in a resisting medium, would acquire the velocity AP, is to
the time in which it would acquire its greatest velocity AC, by
falling in a non-resisting space, as the sector ADT to the triangle
ADC: and the time in which it would lose its velocity A*p*, by
ascending in a resisting medium, is to the time in which it would lose
the same velocity by ascending in a non-resisting space, as the arc A*t*
if to its tangent A*p*.

Cor. 6. Hence from the given time there is
given the space described in the ascent or descent. For the greatest
velocity of a body descending *in infinitum* is given (by
Corol. 2 and 3, Theor. VI, of this Book); and thence the time is given
in which a body would acquire that velocity by falling in a
non-resisting space. And taking the sector ADT or AD*t* to the
triangle ADC in the ratio of the given time to the time just now
found, there will be given both the velocity AP or A*p*, and
the area ABNK or AB*nk*, which is to the sector ADT, or AD*t*,
as the space sought to the space which would, in the given time, be
uniformly described with that greatest velocity found just before.

Cor. 7. And by going backward, from the given
space of ascent or descent AB*nk* or ABNK, there will be given
the time AD*t* or ADT.

*Suppose the uniform force of gravity to tend directly to the
plane of the horizon, and the resistance to be as the density of
the medium and the square of the velocity conjunctly: it is
proposed to find the density of the medium in each place, which
shall make the body move in any given curve line; the velocity of
the body and the resistance of the medium in each place.*

Let PQ, be a plane perpendicular to the plane of the scheme itself;
PFHQ a curve line meeting that plane in the points P and Q; G, H, I, K
four places of the body going on in this curve from F to Q; and GB,
HC, ID, KE four parallel ordinates let fall from these points to the
horizon, and standing on the horizontal line PQ, at the points B, C,
D, E; and let the distances BC, CD, DE, of the ordinates be equal
among themselves. From the points G and H let the right lines GL, HN,
be drawn touching the curve in G and H, and meeting the ordinates CH,
DI, produced upwards, in L and N: and complete the parallelogram HCDM.
And the times in which the body describes the arcs GH, HI, will be in
a subduplicate ratio of the altitudes LH, NI, which the bodies would
describe in those times, by falling from the tangents; and the
velocities will be as the lengths described GH, HI directly, and the
times inversely. Let the times be expounded by T and *t*, and
the velocities by GH

T and HI

t; and the decrement of the velocity
produced in the time *t* will be expounded by GH

T − HI

t . This decrement arises from
the resistance which retards the body, and from the gravity which
accelerates it. Gravity, in a falling body, which in its fall
describes the space NI, produces a velocity with which it would be
able to describe twice that space in the same time, as *Galileo*
has demonstrated; that is, the velocity 2NI

t : but if the body describes the arc
HI, it augments that arc only by the length HI − HN or MI × NI

HI; and therefore generates only the
velocity 2MI × NI

t × HI. Let this velocity be added to
the beforementioned decrement, and we shall have the decrement of the
velocity arising from the resistance alone, that is, GH

T − HI

t + 2MI
× NI

t × HI . Therefore
since, in the same time, the action of gravity generates, in a falling
body, the velocity 2NI

t, the resistance will be to the
gravity as GH

T − HI

t + 2MI
× NI

t × HI or as t × GH

T − HI + 2MI
× NI

HI to 2NI.

Now for the abscissas CB, CD, CE, put −*o, o, 2o*. For the
ordinate CH put P; and for MI put any series Q*o* + R*o*²
+ S*o*³ +, &c. And all the terms of the series after the
first, that is, R*o*² + S*o*³ +, &c., will be NI;
and the ordinates DI, EK, and BG will be P − Q*o* − R*o*²
− S*o*³ −, &c., P − 2Q*o* − 4Ro² − 8S*o*³ −,
&c., and P + Q*o* − R*o*² + S*o*³ −, &c.,
respectively. And by squaring the differences of the ordinates BG − CH
and CH − DI, and to the squares thence produced adding the squares of
BC and CD themselves, you will have *oo* + QQ*oo* − 2QR*o*³
+, &c., and *oo* + QQ*oo* + 2QR*o*³ +,
&c., the squares of the arcs GH, HI; whose roots o√(1+QQ)
− QRoo

√(1+QQ) , and o√(1+QQ)
+ QRoo

√(1+QQ) are the arcs GH and
HI. Moreover, if from the ordinate CH there be subducted half the sum
of the ordinates BG and DI, and from the ordinate DI there be
subducted half the sum of the ordinates CH and EK, there will remain R*oo*
and R*oo* + 3S*o*³, the versed sines of the arcs GI and
HK. And these are proportional to the lineolae LH and NI, and
therefore in the duplicate ratio of the infinitely small times T and *t*:
and thence the ratio t

T is √(
R + 3So

R) or R + ^{3}/_{2}So

R ; and t
× GH

T − HI + 2MI
× NI

HI , by substituting the
values of t

T, GH, HI, MI and NI just found,
becomes 3Soo

2R √(1+QQ). And since 2NI is
2R*oo*, the resistance will be now to the gravity as 3Soo

2R √(1+QQ), that is, as
3S√(1+qq) to 4RR.

And the velocity will be such, that a body going off therewith from
any place H, in the direction of the tangent HN, would describe, in
vacuo, a parabola, whose diameter is HC, and its latus rectum
HN^{2}

NI or 1+QQ

R.

And the resistance is as the density of the medium and the square of
the velocity conjunctly; and therefore the density of the medium is as
the resistance directly, and the square of the velocity inversely;
that is, as 3S√(1+QQ)

4RR directly and 1+QQ

R inversely; that is, as S

R√(1+QQ). Q.E.I.

Cor. 1. If the tangent HN be produced both
ways, so as to meet any ordinate AF in T HT

AC will be equal to √(1+QQ);
and therefore in what has gone before may be put for √(1+QQ).
By this means the resistance will be to the gravity as 3S × HT to 4RR
× AC; the velocity will be as HT

AC√R, and the density of the medium
will be as S × AC

R × HT.

Cor. 2. And hence, if the curve line PFHQ be defined by the relation between the base or abscissa AC and the ordinate CH, as is usual, and the value of the ordinate be resolved into a converging series, the Problem will be expeditiously solved by the first terms of the series; as in the following examples.

Example 1. Let the line PFHQ be a semi-circle described upon the diameter PQ, to find the density of the medium that shall make a projectile move in that line.

Bisect the diameter PQ in A; and call AQ, *n*; AC, *a*;
CH, *e*; and CD, *o*; then DI² or AQ² − AD² = *nn
− aa − 2ao − oo*, or *ee − 2ao − oo*; and the root being
extracted by our method, will give DI = e −
ao

e − oo

2e − aaoo

2e^{3} − ao^{3}

2e^{3} − a^{3}o^{3}

2e^{5} − , &c.
Here put *nn* for *ee + aa*, and DI will become
= e − ao

e − nnoo

2e^{3} − anno^{3}

2e^{5} −, &c

Such series I distinguish into successive terms after this manner: I
call that the first term in which the infinitely small quantity *o*
is not found; the second, in which that quantity is of one dimension
only; the third, in which it arises to two dimensions; the fourth, in
which it is of three; and so *ad infinitum*. And the first
term, which here is *e*, will always denote the length of the
ordinate CH, standing at the beginning of the indefinite quantity *o*.
The second term, which here is ao

e, will denote the difference between
CH and DN; that is, the lineola MN which is cut off by completing the
parallelogram HCDM; and therefore always determines the position of
the tangent HN; as, in this case, by taking MN to HM as ao

e to *o*, or *a* to *e*.
The third term, which here is nnoo

2e^{3}, will represent the
lineola IN, which lies between the tangent and the curve; and
therefore determines the angle of contact IHN, or the curvature which
the curve line
has in H. If that lineola IN is of a finite magnitude, it will be expressed by
the third term, together with those that follow *in infinitum*.
But if that lineola be diminished *in infinitum*, the terms
following become in finitely less than the third term, and therefore
may be neglected. The fourth term determines the variation of the
curvature; the fifth, the variation of the variation; and so on.
Whence, by the way, appears no contemptible use of these series in the
solution of problems that depend upon tangents, and the curvature of
curves.

Now compare the series e − ao

e − nnoo

2e^{3} − anno^{3}

2e^{5} − &c., with
the series P − Q*o* − R*oo* − S*o*³
− &c., and for P, Q, R and S, put *e*, a

e, nn

2e^{3} and ann

2e^{5}, and for √(1
+ QQ) put √(1 + aa

ee ) or n

e : and the density of the medium will
come out as a

ne; that is (because *n* is
given), as a

e or AC

CH, that is, as that length of the
tangent HT, which is terminated at the semi-diameter AF standing
perpendicularly on PQ: and the resistance will be to the gravity as 3*a*
to 2*n*, that is, as 3AC to the diameter PQ of the circle; and
the velocity will be as √(CH). Therefore if
the body goes from the place F, with a due velocity, in the direction
of a line parallel to PQ, and the density of the medium in each of the
places H is as the length of the tangent HT, and the resistance also
in any place H is to the force of gravity as 3AC to PQ, that body will
describe the quadrant FHQ of a circle. Q.E.I.

But if the same body should go from the place P, in the direction of
a line perpendicular to PQ, and should begin to move in an arc of the
semi circle PFQ, we must take AC or *a* on the contrary side
of the centre A; and therefore its sign must be changed, and we must
put −*a* for +*a*. Then the density of the medium would
come out as −a

e. But nature does not admit of a
negative density, that is, a density which accelerates the motion of
bodies; and therefore it cannot naturally come to pass that a body by
ascending from P should describe the quadrant PF of a circle. To
produce such an effect, a body ought to be accelerated by an impelling
medium, and not impeded by a resisting one.

Example 2. Let the line PFQ be a parabola, having its axis AF perpendicular to the horizon PQ, to find the density of the medium, which will make a projectile move in that line.

From the nature of the parabola, the rectangle PDQ is equal to the
rectangle under the ordinate DI and some given right line; that is, if
that right line be called *b*; PC, *a*; PQ, *c*;
CH, *e*; and CD, *o*; the rectangle *a* + *o*
into *c − a − o* or *ac − aa − 2ao + co − oo*, is
equal to the rectangle *b* into DI, and therefore DI is equal
to ac − aa

b + c
− 2a

bo − oo

b . Now the second term
c−2a

bo of this series is to be put
for Q*o*, and the third term oo

b for R*oo*. But since there are
no more terms, the co-efficient S of the fourth term will vanish; and
therefore the quantity S

R√(1+QQ), to which the density of the
medium is proportional, will be nothing. Therefore, where the medium
is of no density, the projectile will move in a parabola; as *Galileo*
hath heretofore demonstrated. Q.E.I.

Example 3. Let the line AGK be an hyperbola, having its asymptote NX perpendicular to the horizontal plane AK, to find the density of the medium that will make a projectile move in that line.

Let MX be the other asymptote, meeting the ordinate DG produced in V;
and from the nature of the hyperbola, the rectangle of XV into VG will
be given. There is also given the ratio of DN to VX, and therefore the
rectangle of DN into VG is given. Let that be *bb*: and,
completing the parallelogram DNXZ, let BN be called *a*; BD, *o*;
NX, *c*; and let the given ratio of VZ to ZX or DN be m

n. Then DN will be equal to *a − o*,
VG equal to bb

a − o, VZ equal to m

n × (a − o), and GD or
NX − VZ − VG equal to c −
m

n a + m

no − bb

a−o . Let the term bb

a−o be resolved into the converging
series bb

a + bb

aao + bb

a^{3}oo + bb

a^{4}o^{3} ,
&c., and GD will become equal to c − m

na − bb

a + m

no − bb

aao − bb

a^{3}o^{2} − bb

a^{4}o^{3} ,
&c. The second term m

no − bb

aao of this series is to be
used for Q*o*; the third bb

a^{3}o^{2} ,
with its sign changed for R*o*²; and the fourth bb

a^{4}o^{3} ,
with its sign changed also for S*o*³, and their coefficients
m

n − bb

aa , bb

a^{3} and bb

a^{4} are to be put for Q, R,
and S in the former rule. Which being done, the density of the medium
will come out as bb

a^{4}

bb

a^{3} √(1 + mm

nn − 2mbb

naa + b^{4}

a^{4}) or
1

√(aa + mm

nnaa − 2mbb

n + b^{4}

aa) , that is, if in
VZ you take VY equal to VG, as 1

XY. For *aa* and
m^{2}

n^{2}a^{2} − 2mbb

n + b^{4}

aa are the squares of XZ and
ZY. But the ratio of the resistance to gravity is found to be that of
3XY to 2YG; and the velocity is that with which the body would
describe a parabola, whose vertex is G, diameter DG, latus rectum
XY^{2}

VG. Suppose, therefore, that the
densities of the medium in each of the places G are reciprocally as
the distances XY, and that the resistance in any place G is to the
gravity as 3XY to 2YG; and a body let go from the place A, with a due
velocity, will describe that hyperbola AGK. Q.E.I.

Example 4. Suppose, indefinitely, the line
AGK to be an hyperbola described with the centre
X, and the asymptotes MX, NX, so that, having constructed the
rectangle XZDN, whose side ZD cuts the hyperbola in G and its
asymptote in V, VG may be reciprocally as any power DN^{n} of
the line ZX or DN, whose index is the number *n*: to find the
density of the medium in which a projected body will describe this
curve.

For BN, BD, NX, put A, O, C, respectively, and let VZ be to XZ or DN
as *d* to *e*, and VG be equal to bb

DN^{n}; then DN will be equal
to A − O, VG = bb

(A − O)^{n} ,
VZ = d

e (A − O), and GD or NX − VZ −
VG equal to

C − d

eA + d

eO − bb

(A − O)^{n} .

eA + d

eO − bb

(A − O)

Let the term bb

(A − O)^{n} be resolved into
an infinite series

bb

A^{n} + nbb

A^{n + 1} × O + nn + n

2A^{n + 2} × bb O^{2}
+ n^{3} + 3nn + 2n

6A^{n + 3} × bb O^{3},&c.,

A

A

2A

6A

And GD will be equal to

C − d

eA + bb

A^{n} + d

e O − nbb

A^{n + 1} O − + nn + n

2A^{n + 2}bb O^{2} −
+ n^{3} + 3nn + 2n

6A^{n + 3} bbO^{3},
&c.

eA + bb

A

e O − nbb

A

2A

6A

The second term d

e O − nbb

A^{n+1} O of this
series is to be used for Q*o*, the third nn+n

2A^{n+2}bb O^{2}
for R*oo*, the fourth n^{3}+3nn+2n

6A^{n+3}bbO^{3}
for So³. And thence the density of the medium S

R√(1+QQ), in any place G, will be

n+2

3√(A^{2} + dd

eeA^{2}− 2dnbb

eA^{n}A+ nnb^{4}

A^{2n} ),

3√(A

eeA

eA

A

and therefore if in VZ you take VY equal to *n* × VG, that
density is reciprocally as XY. For A² and dd

eeA^{2} − 2dnbb

eA^{n}A + nnb^{4}

A^{2n} are the squares
of XZ and ZY. But the resistance in the same place G is to the force
of gravity as 3S × XY

A to 4RR, that is, as XY to
2nn + 2n

n + 2 VG. And the velocity there is the
same wherewith the projected body would move in a parabola, whose
vertex is G, diameter GD, and latus rectum 1
+ QQ

R or 2XY^{2}

(nn + n) × VG. Q.E.I.

In the same manner that the density of the medium comes out to be as
S × AC

R × HT, in Cor. 1, if the resistance is
put as any power V^{n} of the velocity V, the density of the
medium will come out to be as S

R^{4−n/2}
× ( AC

HT)^{n−1}

And therefore if a curve can be found, such that the ratio of
S

R^{4−n/2}
to ( HT

AC )^{n−1}, or of
S^{2}

R^{4−n} to (1+QQ)^{n−1}
may be given; the body, in an uniform medium, whose resistance is as
the power V^{n} of the velocity V, will move in this curve.
But let us return to more simple curves.

Because there can be no motion in a parabola except in a non-resisting medium, but in the hyperbolas here described it is produced by a perpetual resistance; it is evident that the line which a projectile describes in an uniformly resisting medium approaches nearer to these hyperbolas than to a parabola. That line is certainly of the hyperbolic kind, but about the vertex it is more distant from the asymptotes, and in the parts remote from the vertex draws nearer to them than these hyperbolas here described. The difference, however, is not so great between the one and the other but that these latter may be commodiously enough used in practice instead of the former. And perhaps these may prove more useful than an hyperbola that is more accurate, and at the same time more compounded. They may be made use of, then, in this manner.

Complete the parallelogram XYGT, and the right line GT will touch the
hyperbola in G, and therefore the density of the medium in G is
reciprocally as the tangent GT, and the velocity there as √
(GT^{2}

GV); and the resistance is to
the force of gravity as GT to 2nn + 2n

n + 2 × GV.

Therefore if a body projected from the place A, in the direction of
the right line AH, describes the hyperbola AGK and AH produced meets
the asymptote NX in H, and AI drawn parallel to it meets the other
asymptote MX in I; the density of the medium in A will be reciprocally
as AH, and the velocity of the body as √(
AH^{2}

AI), and the resistance there
to the force of gravity as AH to 2nn + 2n

n + 2 × AI. Hence the
following rules are deduced.

Rule 1. If the density of the medium at A, and the velocity with which the body is projected remain the same, and the angle NAH be changed, the lengths AH, AI, HX will remain. Therefore if those lengths, in any one case, are found, the hyperbola may afterwards be easily determined from any given angle NAH.

Rule 2. If the angle NAH, and the density of the medium at A, re main the same, and the velocity with which the body is projected be changed, the length AH will continue the same; and AI will be changed in a duplicate ratio of the velocity reciprocally.

Rule 3. If the angle NAH, the velocity of the
body at A, and the accelerative gravity remain the same, and the
proportion of the resistance at A to the motive gravity be augmented
in any ratio; the proportion of AH to AI will be augmented in the same
ratio, the latus rectum of the abovementioned parabola remaining the
same, and also the length AH^{2}

AI proportional to it; and therefore AH
will be diminished in the same ratio, and AI will be diminished in the
duplicate of that ratio. But the proportion of the resistance to the
weight is augmented, when either the specific gravity is made less,
the magnitude remaining equal, or when the density of the medium is
made greater, or when, by diminishing the magnitude, the resistance
becomes diminished in a less ratio than the weight.

Rule 4. Because the density of the medium is greater near the vertex of the hyperbola than it is in the place A, that a mean density may be preserved, the ratio of the least of the tangents GT to the tangent AH ought to be found, and the density in A augmented in a ratio a little greater than that of half the sum of those tangents to the least of the tangents GT.

Rule 5. If the lengths AH, AI are given, and
the figure AGK is to be described, produce HN to X, so that HX may be
to AI as *n* + 1 to 1; and with the centre X, and the
asymptotes MX, NX, describe an hyperbola through the point A, such
that AI may be to any of the lines VG as XV^{n} to XI^{n}.

Rule 6. By how much the greater the number *n*
is, so much the more accurate are these hyperbolas in the ascent of
the body from A, and less accurate in its descent to K; and the
contrary. The conic hyperbola keeps a mean ratio between these, and is
more simple than the rest. Therefore if the hyperbola be of this kind,
and you are to find the point K, where the projected body falls upon
any right line AN passing through the point A, let AN produced meet
the asymptotes MX, NX in M and N, and take NK equal to AM.

Rule 7. And hence appears an expeditious
method of determining this hyperbola from the phenomena. Let two
similar and equal bodies be projected with the same velocity, in
different angles HAK, *h*A*k*, and let them fall upon
the plane of the horizon in K and *k*; and note the proportion
of AK to A*k*. Let it be as *d* to *e*. Then
erecting a perpendicular AI of any length, assume any how the length
AH or A*h*, and thence graphically, or
by scale and compass, collect the lengths AK, A*k* (by Rule 6).
If the ratio of AK to A*k* be the same with that of *d*
to *e*, the length of AH was
rightly assumed. If not, take on the indefinite right line SM, the
length SM equal to the assumed AH; and erect a perpendicular MN equal
to the difference AK

Ak − d

e of the ratios drawn into any
given right line. By the like method, from several assumed lengths AH,
you may find several points N; and draw through them all a regular
curve NNXN, cutting the right line SMMM in X. Lastly, assume AH equal
to the abscissa SX, and thence find again the length AK; and the
lengths, which are to the assumed length AI, and this last AH, as the
length AK known by experiment, to the length AK last found, will be
the true lengths AI and AH, which were to be found. But these being
given, there will be given also the resisting force of the medium in
the place A, it being to the force of gravity as AH to ^{4}/_{3}AI.
Let the density of the medium be increased by Rule 4, and if the
resisting force just found be increased in the same ratio, it will
become still more accurate.

Rule 8. The lengths AH, HX being found; let
there be now required the position of the line AH, according to which
a projectile thrown with that given velocity shall fall upon any point
K. At the joints A and K, erect the lines AC, KF perpendicular to the
horizon; whereof let AC be drawn downwards, and be equal to AI or ½HX.
With the asymptotes AK, KF, describe an hyperbola, whose conjugate
shall pass through the point C; and from the centre A, with the
interval AH, describe a circle cutting that hyperbola in the point H;
then the projectile thrown in the direction of the right line AH will
fall upon the point K. Q.E.I. For the point H,
because of the given length AH, must be somewhere in the circumference
of the described circle. Draw CH meeting AK and KF in E and F; and
because CH, MX are parallel, and AC, AI equal, AE will be equal to AM,
and therefore also equal to KN. But CE is to AE as FH to KN, and
therefore CE and FH are equal. Therefore the point H falls upon the
hyperbolic curve described with the asymptotes AK, KF whose conjugate
passes through the point C; and is therefore found in the
common intersection of this hyperbolic curve
and the circumference of the described circle. Q.E.D. It
is to be observed that this operation is the same, whether the right
line AKN be parallel to the horizon, or inclined thereto in any angle;
and that from two intersections H, *h*, there arise two angles
NAH, NA*h*; and that in mechanical practice it is sufficient
once to describe a circle, then to apply a ruler CH, of an
indeterminate length, so to the point C, that its part FH, intercepted
between the circle and the right line FK, may be equal to its part CE
placed between the point C and the right line AK

What has been said of hyperbolas may be easily applied to parabolas.
For if a parabola be represented by XAGK, touched by a right line XV
in the vertex X, and the ordinates IA, VG be as any powers XI^{n},
XV^{n}, of the abscissas XI, XV; draw XT, GT, AH, whereof let
XT be parallel to VG, and let GT, AH touch the parabola in G and A:
and a body projected from any place A, in the direction of the right
line AH, with a due velocity, will describe this parabola, if the
density of the medium in each of the places G be reciprocally as the
tangent GT. In that case the velocity in G will be the same as would
cause a body, moving in a nonresisting space, to describe a conic
parabola, having G for its vertex, VG produced downwards for its
diameter, and 2GT^{2}

(nn − n) × VG for its latus
rectum. And the resisting force in G will be to the force of gravity
as GT to 2nn −
2n

n − 2VG. Therefore if NAK
represent an horizontal line, and both the density of the medium at A,
and the velocity with which the body is projected, remaining the same,
the angle NAH be any how altered, the lengths AH, AI, HX will remain;
and thence will be given the vertex X of the parabola, and the
position of the right line XI; and by taking VG to IA as XV^{n}
to XI^{n}, there will be given all the points G of the
parabola, through which the projectile will pass.

*If a body be resisted partly in the ratio and partly in the
duplicate ratio of its velocity, and moves in a similar medium by
its innate force only; and the times be taken in arithmetical
progression; then quantities reciprocally proportional to the
velocities, increased by a certain given quantity, will be in
geometrical progression.*

With the centre C, and the rectangular asymptotes CAD*d* and
CH, describe an hyperbola BE*e*, and let AB, DE, *de,*
be parallel to the asymptote CH. In the asymptote CD let A, G be given
points; and if the time be expounded by the hyperbolic area ABED
uniformly increasing, I say, that the velocity may be expressed by the
length DF, whose reciprocal GD, together with the given line CG,
compose the length CD increasing in a geometrical progression.

For let the areola DE*ed* be the least given increment of the
time, and D*d* will be reciprocally as DE, and therefore
directly as CD. Therefore the decrement of 1

GD, which (by Lem. II. Book II) is
Dd

GD^{2}, will be also as
CD

GD^{2} or CG+GD

GD^{2}, that is, as 1

GD + CG

GD^{2} . Therefore the
time ABED uniformly increasing by the addition of the given particles
ED*de*, it follows that 1

GD decreases in the same ratio with
the velocity. For the decrement of the velocity is as the resistance,
that is (by the supposition), as the sum of two quantities, whereof
one is as the velocity, and the other as the square of the velocity;
and the decrement of 1

GD is as the sum of the quantities
1

GD and CG

GD^{2}, whereof the first is
1

GD itself, and the last CG

GD^{2} is as 1

GD^{2} : therefore 1

GD is as the velocity, the decrements
of both being analogous. And if the quantity GD reciprocally
proportional to 1

GD, be augmented by the given
quantity CG; the sum CD, the time ABED uniformly increasing, will
increase in a geometrical progression. Q.E.D.

Cor. 1. Therefore,
if, having the points A and G given, the time be expounded by the
hyperbolic area ABED, the velocity may be expounded by 1

GD the reciprocal of GD.

Cor. 2. And by taking GA to GD as the reciprocal of the velocity at the beginning to the reciprocal of the velocity at the end of any time ABED, the point G will be found. And that point being found the velocity may be found from any other time given.

*The same things being supposed, I say, that if the spaces
described are taken in arithmetical progression, the velocities
augmented by a certain given quantity will be in geometrical progression.*

In the asymptote CD let there be given the point R, and, erecting the perpendicular RS meeting the hyperbola in S, let the space described be expounded by the hyperbolic area RSED; and the velocity will be as the length GD, which, together with the given line CG, composes a length CD decreasing in a geometrical progression, while the space RSED increases in an arithmetical progression.

For, because the increment ED*de* of the space is given, the
lineola D*d*, which is the decrement of GD, will be
reciprocally as ED, and therefore directly as CD; that is, as the sum
of the same GD and the given length CG. But the decrement of the
velocity, in a time reciprocally proportional thereto, in which the
given particle of space D*de*E is described, is as the
resistance and the time conjunctly, that is, directly as the sum of
two quantities, whereof one is as the velocity, the other as the
square of the velocity, and inversely as the velocity; and therefore
directly as the sum of two quantities, one of which is given, the
other is as the velocity. Therefore the decrement both of the velocity
and the line GD is as a given quantity and a decreasing quantity
conjunctly; and, because the decrements are analogous, the decreasing
quantities will always be analogous; viz., the velocity, and the line
GD. Q.E.D.

Cor. 1. If the velocity be expounded by the length GD, the space described will be as the hyperbolic area DESR.

Cor. 2. And if the point R be assumed any how, the point G will be found, by taking GR to GD as the velocity at the beginning to the velocity after any space RSED is described. The point G being given, the space is given from the given velocity: and the contrary.

Cor. 3. Whence since (by Prop. XI) the velocity is given from the given time, and (by this Prop.) the space is given from the given velocity; the space will be given from the given time: and the contrary.

*Supposing that a body attracted downwards by an uniform gravity
ascends or descends in a right line; and that the same is resisted
partly in the ratio of its velocity, and partly in the duplicate
ratio thereof: I say, that, if right lines parallel to the
diameters of a circle and an hyperbola, be drawn through the ends
of the conjugate diameters, and the velocities be as some segments
of those parallels drawn from a given point, the times will be as
the sectors of the areas cut off by right lines drawn from the
centre to the ends of the segments; and the contrary.*

Case 1. Suppose first that the body is ascending, and from the centre D, with any semi-diameter DB, describe a quadrant BETF of a circle, and through the end B of the semi-diameter DB draw the indefinite line BAP, parallel to the semi-diameter DF. In that line let there be given the point A, and take the segment AP proportional to the velocity. And since one part of the resistance is as the velocity, and another part as the square of the velocity, let the whole resistance be as AP² + 2BAP. Join DA, DP, cutting the circle in E and T, and let the gravity be expounded by DA², so that the gravity shall be to the resistance in P as DA² to AP² + 2BAP; and the time of the whole ascent will be as the sector EDT of the circle.

For draw DVQ, cutting off the moment PQ of the velocity AP, and the moment DTV of the sector DET answering to a given moment of time; and that decrement PQ of the velocity will be as the sum of the forces of gravity DA² and of resistance AP² + 2BAP, that is (by Prop. XII Book II, Elem.), as DP². Then the area DPQ, which is proportional to PQ, is as DP², and the area DTV, which is to the area DPQ as DT² to DP², is as the given quantity DT². Therefore the area EDT decreases uniformly according to the rate of the future time, by subduction of given particles DTV, and is therefore proportional to the time of the whole ascent. Q.E.D.

Case 2. If the velocity in the ascent of the body be expounded by the length AP as before, and the resistance be made as AP² + 2BAP, and if the force of gravity be less than can be expressed by DA²; take BD of such a length, that AB² − BD² maybe proportional to the gravity, and let DF be perpendicular and equal to DB, and through the vertex F describe the hyperbola FTVE, whose conjugate semi-diameters are DB and DF, and which cuts DA in E, and DP, DQ in T and V; and the time of the whole ascent will be as the hyperbolic sector TDE.

For the decrement PQ of the velocity, produced in a given particle of time, is as the sum of the resistance AP² + 2BAP and of the gravity AB² − BD², that is, as BP² − BD². But the area DTV is to the area DPQ as DT² to DP²; and, therefore, if GT be drawn perpendicular to DF, as GT² or GD² − DF² to BD², and as GD² to BP², and, by division, as DF² to BP² − BD². Therefore since the area DPQ is as PQ, that is, as BP² − BD², the area DTV will be as the given quantity DF². Therefore the area EDT decreases uniformly in each of the equal particles of time, by the subduction of so many given particles DTV, and therefore is proportional to the time. Q.E.D.

Case 3. Let AP be the velocity in the descent of the body, and AP² + 2BAP the force of resistance, and BD² − AB² the force of gravity, the angle DBA being a right one. And if with the centre D, and the principal vertex B, there be described a rectangular hyperbola BETV cutting DA, DP, and DQ produced in E, T, and V; the sector DET of this hyperbola will be as the whole time of descent.

For the increment PQ of the velocity, and the area DPQ proportional to it, is as the excess of the gravity above the resistance, that is, as BD² − AB² − 2BA×AP − AP² or BD² − BP². And the area DTV is to the area DPQ as DT² to DP²; and therefore as GT² or GD² − BD² to BP², and as GD² to BD², and, by division, as BD² to BD² − BP². Therefore since the area DPQ is as BD² − BP², the area DTV will be as the given quantity BD². Therefore the area EDT increases uniformly in the several equal particles of time by the addition of as many given particles DTV, and therefore is proportional to the time of the descent. Q.E.D.

Cor. If with the centre D and the
semi-diameter DA there be drawn through the vertex A an arc A*t*
similar to the arc ET, and similarly subtending the angle ADT, the
velocity AP will be to the velocity which the body in the time EDT, in
a non−resisting space, can lose in its ascent, or acquire in its
descent, as the area of the triangle DAP to the area of the sector DA*t*;
and therefore is given from the time given. For the velocity in a
non-resisting medium is proportional to the time, and therefore to
this sector; in a resisting medium, it is as the triangle; and in both
mediums, where it is least, it approaches to the ratio of equality, as
the sector and triangle do.

One may demonstrate also that case in the ascent of the body, where the force of gravity is less than can be expressed by DA² or AB² + BD², and greater than can be expressed by AB² − DB², and must be expressed by AB². But I hasten to other things.

*The same things being supposed, I say, that the space described
in the ascent or descent is as the difference of the area by which
the time is expressed, and of some other area which is augmented
or diminished in an arithmetical progression; if the forces
compounded of the resistance and the gravity be taken, in a
geometrical progression.*

Take AC (in these three figures) proportional to the gravity, and AK
to the resistance; but take them on the same side of the point A, if the
body is descending, otherwise on the
contrary. Erect A*b*, which make to DB as DB² to 4BAC: and to
the rectangular asymptotes CK, CH, describe the hyperbola *b*N;
and, erecting KN perpendicular to CK, the area A*b*NK will be
augmented or diminished in an arithmetical progression, while the
forces CK are taken in a geometrical progression. I say, therefore,
that the distance of the body from its greatest altitude is as the
excess of the area A*b*NK above the area DET.

For since AK is as the resistance, that is, as AP² × 2BAP; assume any
given quantity Z, and put AK equal to AP^{2}+2BAP

Z; then (by
Lem. II of this Book) the moment KL of AK will be equal to 2APQ + 2BA × PQ

Z or 2BPQ

Z, and the moment KLON of the area A*b*NK
will be equal to 2BPQ × LO

Z or BPQ
× BD^{3}

2Z × CK × AB.

Case 1. Now if the body ascends, and the
gravity be as AB² + BD², BET being a circle, the line AC, which is
proportional to the gravity, will be AB^{2}+BD^{2}

Z, and DP² or AP² + 2BAP + AB² + BD²
will be AK × Z + AC × Z or CK × Z; and therefore the area DTV will be
to the area DPQ as DT² or DB² to CK × Z.

Case 2. If the body ascends, and the gravity
be as AB² − BD², the line AC will be AB^{2}+BD^{2}

Z, and DT² will be to DP² as DF² or DB²
to BP² − BD² or AP² + 2BAP + AB² − BD², that is, to AK × Z + AC × Z or CK × Z.
And therefore the area DTV will be to the area DPQ as DB² to CK × Z.

Case 3. And by the same reasoning, if the
body descends, and therefore the gravity is as BD² - AB², and the line
AC becomes equal to BD^{2}-AB^{2}

Z; the area DTV will be to the area
DPQ, as DB² to CK × Z: as above.

Since, therefore, these areas are always in this ratio, if for the
area DTV, by which the moment of the time,
always equal to itself, is expressed, there be put any determinate
rectangle, as BD × *m*, the area DPQ, that is, ½BD × PQ, will
be to BD × *m* as CK × Z to BD². And thence PQ × BD³ becomes
equal to 2BD × *m* × CK × Z, and the moment KLON of the area A*b*NK,
found before, becomes BP × BD × m

AB. From the area DET subduct its
moment DTV or BD × *m*, and there will remain AP × BD × m

AB. Therefore the difference of the
moments, that is, the moment of the difference of the areas, is equal
to AP × BD × m

AB; and therefore (because of the given
quantity BD × m

AB ) as the velocity AP; that is, as
the moment of the space which the body describes in its ascent or
descent. And therefore the difference of the areas, and that space,
increasing or decreasing by proportional moments, and beginning
together or vanishing together, are proportional. Q.E.D.

Cor. If the length, which arises by applying
the area DET to the line BD, be called M; and another length V be
taken in that ratio to the length M, which the line DA has to the line
DE; the space which a body, in a resisting medium, describes in its
whole ascent or descent, will be to the space which a body, in a
non-resisting medium, falling from rest, can describe in the same
time, as the difference of the aforesaid areas to BD × V^{2}

AB; and therefore is given from the
time given. For the space in a non-resisting medium is in a duplicate
ratio of the time, or as V²; and, because BD and AB are given, as
BD × V^{2}

AB. This area is equal to the area
DA^{2} × BD × M^{2}

DE^{2} × AB and the moment of M
is *m*; and therefore the moment ot this area is DA^{2} × BD × 2M × m

DE^{2} × AB. But this moment is
to the moment of the difference of the aforesaid areas DET and A*b*NK,
viz., to AB × BD × m

AB, as DA^{2}
× BD × M

DE^{2} to ½BD × AP, or as
DA^{2}

DE^{2} into DET to DAP; and,
therefore, when the areas DET and DAP are least, in the ratio of
equality. Therefore the area BD ×
V^{2}

AB and the difference of the areas DET
and A*b*NK, when all these areas are least, have equal moments;
and are therefore equal. Therefore since the velocities, and therefore
also the spaces in both mediums described together, in the beginning
of the descent, or the end of the ascent, approach to equality, and
therefore are then one to another as the area
BD × V^{2}

AB, and the difference of the areas DET
and A*b*NK; and moreover since the space, in a non-resisting
medium, is perpetually as BD × V^{2}

AB, and the space, in a resisting
medium, is perpetually as the difference of the areas DET and A*b*NK;
it necessarily follows, that the spaces, in both mediums, described in
any equal times, are one to another as that area BD × V^{2}

AB, and the difference of the areas DET
and A*b*NK. Q.E.D.

The resistance of spherical bodies in fluids arises partly from the
tenacity, partly from the attrition, and partly from the density of
the medium. And that part of the resistance which arises from the
density of the fluid is, as I said, in a duplicate ratio of the
velocity; the other part, which arises from the tenacity of the fluid,
is uniform, or as the moment of the time; and, therefore, we might now
proceed to the motion of bodies, which are resisted partly by an
uniform force, or in the ratio of the moments of the time, and partly
in the duplicate ratio of the velocity. But it is sufficient to have
cleared the way to this speculation in Prop. VIII and IX foregoing,
and their Corollaries. For in those Propositions, instead of the
uniform resistance made to an ascending body arising from its gravity,
one may substitute the uniform resistance which arises from the
tenacity of the medium, when the body moves by its *vis insita*
alone; and when the body ascends in a right line, add this uniform
resistance to the force of gravity, and subduct it when the body
descends in a right line. One might also go on to the motion of bodies
which are resisted in part uniformly, in part in the ratio of the
velocity, and in part in the duplicate ratio of the same velocity. And
I have opened a way to this in Prop. XIII and XIV foregoing, in which
the uniform resistance arising from the tenacity of the medium may be
substituted for the force of gravity, or be compounded with it as
before. But I hasten to other things.

*Let* PQR *be a spiral cutting all the radii* SP,
SQ, SR*, &c., in equal angles. Draw the right line* PT *touching
the spiral in any point* P*, and cutting the radius* SQ
*In* T*; draw* PO, QO *perpendicular to the
spiral, and meeting in* O*, and join* SO*. I say,
that if the points* P *and* Q *approach and
coincide, the angle* PSO *will become a right angle, and
the ultimate ratio of the rectangle* TQ × 2PS *to* PQ²
*Will be the ratio of equality.*

For from the right angles OPQ, OQR, subduct the equal angles SPQ, SQR, and there will remain the equal angles OPS, OQS. Therefore a circle which passes through the points OSP will pass also through the point Q. Let the points P and Q coincide, and this circle will touch the spiral in the place of coincidence PQ, and will therefore cut the right line OP perpendicularly. Therefore OP will become a diameter of this circle, and the angle OSP, being in a semi-circle, becomes a right one. Q.E.D.

Draw QD, SE perpendicular to OP, and the ultimate ratios of the lines
will be as follows: TQ to PD as TS or PS to PE, or 2PO to 2PS; and PD
to PQ as PQ to 2PO; and, *ex aequo perturbatè*, to TQ to PQ as
PQ to 2PS. Whence PQ² becomes equal to TQ × 2PS. Q.E.D.

*If the density of a medium in each place thereof be
reciprocally as the distance of the places from an immovable
centre, and the centripetal force be in the duplicate ratio of the
density; I say, that a body may revolve in a spiral which cuts all
the radii drawn from that centre in a given angle.*

Suppose every thing to be as in the foregoing Lemma, and produce SQ
to V so that SV may be equal to SP. In any time let a body, in a
resisting medium, describe the least arc PQ, and in double the time
the least arc PR; and the decrements of those arcs arising from the
resistance, or their differences from the arcs which would be
described in a non-resisting medium in the same times, will be to each
other as the squares of the times in which they are generated;
therefore the decrement of the arc PQ is the
fourth part of the decrement of the arc PR. Whence also if the area QS*r*
be taken equal to the area PSQ, the decrement of the arc PQ will be
equal to half the lineola R*r*; and therefore the force of
resistance and the centripetal force are to each other as the lineola
½R*r* and TQ which they generate in the same time. Because the
centripetal force with which the body is urged in P is reciprocally as
SP², and (by Lem. X, Book I) the lineola TQ, which is generated by
that force, is in a ratio compounded of the ratio of this force and
the duplicate ratio of the time in which the arc PQ is described (for
in this case I neglect the resistance, as being infinitely less than
the centripetal force), it follows that TQ × SP², that is (by the last
Lemma), ½PQ² × SP, will be in a duplicate ratio of the time, and
therefore the time is as PQ ×√SP; and the
velocity of the body, with which the arc PQ is described in that time,
as PQ

PQ × √SP or 1

√SP, that is, in the subduplicate
ratio of SP reciprocally. And, by a like reasoning, the velocity with
which the arc QR is described, is in the subduplicate ratio of SQ
reciprocally. Now those arcs PQ and QR are as the describing
velocities to each other; that is, in the subduplicate ratio of SQ to
SP, or as SQ to √(SP × SQ); and, because of
the equal angles SPQ, SQ*r*, and the equal areas PSQ, QS*r*,
the arc PQ is to the arc Q*r* as SQ to SP. Take the differences
of the proportional consequents, and the arc PQ will be to the arc R*r*
as SQ to SP − √(SP × SQ), or ½VQ. For the
points P and Q coinciding, the ultimate ratio of SP
− √(SP × SQ) to ½VQ is the ratio of equality. Because the
decrement of the arc PQ arising from the resistance, or its double R*r*,
is as the resistance and the square of the time conjunctly, the
resistance will be as Rr

PQ^{2} × SP. But PQ was to R*r*
as SQ to ½VQ, and thence Rr

PQ^{2} × SP becomes as
½VQ

PQ × SP × SQ, or as ½OS

OP × SP^{2}. For the points P
and Q coinciding, SP and SQ coincide also, and the angle PVQ becomes a
right one; and, because of the similar triangles PVQ, PSO, PQ becomes
to ½VQ as OP to ½OS. Therefore OS

OP × SP^{2} is as the
resistance, that is, in the ratio of the density of the medium in P
and the duplicate ratio of the velocity conjunctly. Subduct the
duplicate ratio of the velocity, namely, the ratio 1

SP, and there will remain the density
of the medium in P, as OS

OP × SP. Let the spiral be given,
and, because of the given ratio of OS to OP, the density of the medium
in P will be as 1

SP. Therefore in a medium whose
density is reciprocally as SP the distance from
the centre, a body will revolve in this spiral. Q.E.D.

Cor. 1. The velocity in any place P, is always the same wherewith a body in a non-resisting medium with the same centripetal force would revolve in a circle, at the same distance SP from the centre.

Cor. 2. The density of the medium, if the
distance SP be given, is as OS

OP, but if that distance is not given,
as OS

OP × SP. And thence a spiral may be
fitted to any density of the medium.

Cor. 3. The force of the resistance in any
place P is to the centripetal force in the same place as ½OS to OP.
For those forces are to each other as ½R*r* and TQ, or as
¼VQ × PQ

SQ and ½PQ^{2}

SP, that is, as ½VQ and PQ, or ½OS and
OP. The spiral therefore being given, there is given the proportion of
the resistance to the centripetal force; and, vice versa, from that
proportion given the spiral is given.

Cor. 4. Therefore the body cannot revolve in this spiral, except where the force of resistance is less than half the centripetal force. Let the resistance be made equal to half the centripetal force, and the spiral will coincide with the right line PS, and in that right line the body will descend to the centre with a velocity that is to the velocity, with which it was proved before, in the case of the parabola (Theor. X, Book I), the descent would be made in a non-resisting medium, in the subduplicate ratio of unity to the number two. And the times of the descent will be here reciprocally as the velocities, and therefore given.

Cor. 5. And because at equal distances from the centre the velocity is the same in the spiral PQR as it is in the right line SP, and the length of the spiral is to the length of the right line PS in a given ratio, namely, in the ratio of OP to OS; the time of the descent in the spiral will be to the time of the descent in the right line SP in the same given ratio, and therefore given.

Cor. 6. If from the centre S, with any two
given intervals, two circles are described; and these circles
remaining, the angle which the spiral makes with the radius PS be any
how changed; the number of revolutions which the body can complete in
the space between the circumferences of those circles, going round in
the spiral from one circumference to another, will be as PS

OS, or as the tangent of the angle
which the spiral makes with the radius PS; and the
time of the same revolutions will be as OP

OS, that is, as the secant of the same
angle, or reciprocally as the density of the medium.

Cor. 7. If a body, in a medium whose density
is reciprocally as the distances of places from the centre, revolves
in any curve AEB about that centre, and cuts the first radius AS in
the same angle in B as it did before in A, and that with a velocity
that shall be to its first velocity in A reciprocally in a
subduplicate ratio of the distances from the centre (that is, as AS to
a mean proportional between AS and BS) that body will continue to
describe innumerable similar revolutions BFC, CGD, &c., and by its
intersections will distinguish the radius AS into parts AS, BS, CS,
DS, &c., that are continually proportional. But the times of the
revolutions will be as the perimeters of the orbits AEB, BFC, CGD,
&c., directly, and the velocities at the beginnings A, B, C of
those orbits inversely; that is as AS^{3/2},
BS^{3/2},
CS^{3/2}. And the
whole time in which the body will arrive at the centre, will be to the
time of the first revolution as the sum of all the continued
proportionals AS^{3/2},
BS^{3/2},
CS^{3/2}, going on
*ad infinitum*, to the first term AS^{3/2};
that is, as the first term AS^{3/2}
to the difference of the two first AS^{3/2}
− BS^{3/2}, or as ⅔AS to AB
very nearly. Whence the whole time may be easily found.

Cor. 8. From hence also may be deduced, near enough, the motions of bodies in mediums whose density is either uniform, or observes any other assigned law. From the centre S, with intervals SA, SB, SC, &c., continually proportional, describe as many circles; and suppose the time of the revolutions between the perimeters of any two of those circles, in the medium whereof we treated, to be to the time of the revolutions between the same in the medium proposed as the mean density of the proposed medium between those circles to the mean density of the medium whereof we treated, between the same circles, nearly: and that the secant of the angle in which the spiral above determined, in the medium whereof we treated, cuts the radius AS, is in the same ratio to the secant of the angle in which the new spiral, in the proposed medium, cuts the same radius: and also that the number of all the revolutions between the same two circles is nearly as the tangents of those angles. If this be done every where between every two circles, the motion will be continued through all the circles. And by this means one may without difficulty conceive at what rate and in what time bodies ought to revolve in any regular medium.

Cor. 9. And although these motions becoming eccentrical should be performed in spirals approaching to an oval figure, yet, conceiving the several revolutions of those spirals to be at the same distances from each other, and to approach to the centre by the same degrees as the spiral above described, we may also understand how the motions of bodies may be performed in spirals of that kind.

*If the density of the medium in each of the places be
reciprocally as the distance of the places from the immoveable
centre, and the centripetal force be reciprocally as any power of
the same distance, I say, that the body may revolve in a spiral
intersecting all the radii drawn from that centre in a given angle.*

This is demonstrated in the same manner as the foregoing Proposition.
For if the centripetal force in P be reciprocally as any power SP^{n+1}
of the distance SP whose index is *n* + 1; it will be
collected, as above, that the time in which the body describes any arc
PQ, will be as PQ × PS^{½n}; and
the resistance in P as Rr

PQ^{2} × SP^{n}, or
as (1 − ½n) × VQ

PQ × SP^{n} × SQ, and therefore
as (1 − ½n) × OS

OP × SP^{n+1}, that is (because
(1 − ½n) × OS

OP is a given quantity), reciprocally
as SP^{n+1}. And therefore, since the velocity is reciprocally
as SP^{½n}, the density in P will be reciprocally as SP.

Cor. 1. The resistance is to the centripetal force as (1 − ½n) × OS to OP.

Cor. 2. If the centripetal force be
reciprocally as SP³, 1 − ½*n* will be = 0; and therefore the
resistance and density of the medium will be nothing, as in Prop. IX,
Book I.

Cor. 3. If the centripetal force be reciprocally as any power of the radius SP, whose index is greater than the number 3, the affirmative resistance will be changed into a negative.

This Proposition and the former, which relate to mediums of unequal
density, are to be understood of the motion of bodies that are so
small, that the greater density of the medium on one side of the body
above that on the other is not to be considered. I suppose also the
resistance, *caeteris paribus*, to be proportional to its
density. Whence, in mediums whose force of
resistance is not as the density, the density must be so much
augmented or diminished, that either the excess of the resistance may
be taken away, or the defect supplied.

*To find the centripetal force and the resisting force of the
medium, by which a body, the law of the velocity being given,
shall revolve in a given spiral.*

Let that spiral be PQR. From the velocity, with which the body goes
over the very small arc PQ, the time will be given; and from the
altitude TQ, which is as the centripetal force, and the square of the
time, that force will be given. Then from the difference RS*r*
of the areas PSQ and QSR described in equal particles of time, the
retardation of the body will be given; and from the retardation will
be found the resisting force and density of the medium.

*The law of centripetal force being given, to find the density
of the medium in each of the places thereof, by which a body may
describe a given spiral.*

From the centripetal force the velocity in each place must be found; then from the retardation of the velocity the density of the medium is found, as in the foregoing Proposition.

But I have explained the method of managing these Problems in the tenth Proposition and second Lemma of this Book; and will no longer detain the reader in these perplexed disquisitions. I shall now add some things relating to the forces of progressive bodies, and to the density and resistance of those mediums in which the motions hitherto treated of, and those akin to them, are performed.

*A fluid is any body whose parts yield to any force impressed on
it, by yielding, are easily moved among themselves.*

*All the parts of a homogeneous and unmoved fluid included in
any unmoved vessel, and compressed on every side (setting aside
the consideration of condensation, gravity, and all centripetal
forces), will be equally pressed on every side, and remain in
their places without any motion arising from that pressure.*

Case 1. Let a fluid be included in the spherical vessel ABC, arid uniformly compressed on every side: I say, that no part of it will be moved by that pressure. For if any part, as D, be moved, all such parts at the same distance from the centre on every side must necessarily be moved at the same time by a like motion; because the pressure of them all is similar and equal; and all other motion is excluded that does not arise from that pressure. But if these parts come all of them nearer to the centre, the fluid must be condensed towards the centre, contrary to the supposition. If they recede from it, the fluid must be condensed towards the circumference; which is also contrary to the supposition. Neither can they move in any one direction retaining their distance from the centre, because for the same reason, they may move in a contrary direction; but the same part cannot be moved contrary ways at the same time. Therefore no part of the fluid will be moved from its place. Q.E.D.

Case 2. I say now, that all the spherical parts of this fluid are equally pressed on every side. For let EF be a spherical part of the fluid; if this be not pressed equally on every side, augment the lesser pressure till it be pressed equally on every side; and its parts (by Case 1) will remain in their places. But before the increase of the pressure, they would remain in their places (by Case 1); and by the addition of a new pressure they will be moved, by the definition of a fluid, from those places. Now these two conclusions contradict each other. Therefore it was false to say that the sphere EF was not pressed equally on every side. Q.E.D.

Case 3. I say besides, that different spherical parts have equal pressures. For the contiguous spherical parts press each other mutually and equally in the point of contact (by Law III). But (by Case 2) they are pressed on every side with the same force. Therefore any two spherical parts not contiguous, since an intermediate spherical part can touch both, will be pressed with the same force. Q.E.D.

Case 4. I say now, that all the parts of the fluid are every where pressed equally. For any two parts may be touched by spherical parts in any points whatever; and there they will equally press those spherical parts (by Case 3), and are reciprocally equally pressed by them (by Law III). Q.E.D.

Case 5. Since, therefore, any part GHI of the fluid is inclosed by the rest of the fluid as in a vessel, and is equally pressed on every side; and also its parts equally press one another, and are at rest among themselves; it is manifest that all the parts of any fluid as GHI, which is pressed equally on every side, do press each other mutually and equally, and are at rest among themselves. Q.E.D.

Case 6. Therefore if that fluid be included in a vessel of a yielding substance, or that is not rigid, and be not equally pressed on every side, the same will give way to a stronger pressure, by the Definition of fluidity.

Case 7. And therefore, in an inflexible or rigid vessel, a fluid will not sustain a stronger pressure on one side than on the other, but will give way to it, and that in a moment of time; because the rigid side of the vessel does not follow the yielding liquor. But the fluid, by thus yielding, will press against the opposite side, and so the pressure will tend on every side to equality. And because the fluid, as soon as it endeavours to recede from the part that is most pressed, is withstood by the resistance of the vessel on the opposite side, the pressure will on every side be reduced to equality, in a moment of time, without any local motion: and from thence the parts of the fluid (by Case 5) will press each other mutually and equally, and be at rest among themselves. Q.E.D.

Cor. Whence neither will a motion of the parts of the fluid among themselves be changed by a pressure communicated to the external superficies, except so far as either the figure of the superficies may be somewhere altered, or that all the parts of the fluid, by pressing one another more in tensely or remissly, may slide with more or less difficulty among them selves.

*If all the parts of a spherical fluid, homogeneous at equal
distances from the centre, lying on a spherical concentric bottom,
gravitate towards the centre of the whole, the bottom will sustain
the weight of a cylinder, whose base is equal to the superficies
of the bottom, and whose altitude is the same with that of the incumbent fluid.*

Let DHM be the superficies of the bottom, and AEI the upper
superficies of the fluid. Let the fluid be distinguished into
concentric orbs of equal thickness, by the innumerable spherical
superficies BFK, CGL: and
conceive the force of gravity to act only in the upper superficies of
every orb, and the actions to be equal on the equal parts of all the
superficies. Therefore the upper superficies AE is pressed by the
single force of its own gravity, by which all the parts of the upper
orb, and the second superficies BFK, will (by Prop. XIX), according to
its measure, be equally pressed. The second superficies BFK is pressed
likewise by the force of its own gravity, which, added to the former
force, makes the pressure double. The third superficies GGL is,
according to its measure, acted on by this pressure and the force of
its own gravity besides, which makes its pressure triple. And in like
manner the fourth superficies receives a quadruple pressure, the fifth
superficies a quintuple, and so on. Therefore the pressure acting on
every superficies is not as the solid quantity of the incumbent fluid,
but as the number of the orbs reaching to the upper surface of the
fluid; and is equal to the gravity of the lowest orb multiplied by the
number of orbs: that is, to the gravity of a solid whose ultimate
ratio to the cylinder above-mentioned (when the number of the orbs is
increased and their thickness diminished, *ad infinitum*, so
that the action of gravity from the lowest superficies to the
uppermost may become continued) is the ratio of equality. Therefore
the lowest superficies sustains the weight of the cylinder above
determined. Q.E.D. And by a like reasoning the
Proposition will be evident, where the gravity of the fluid decreases
in any assigned ratio of the distance from the centre, and also where
the fluid is more rare above and denser below. Q.E.D.

Cor. 1. Therefore the bottom is not pressed by the whole weight of the incumbent fluid, but only sustains that part of it which is described in the Proposition; the rest of the weight being sustained archwise by the spherical figure of the fluid.

Cor. 2. The quantity of the pressure is the same always at equal distances from the centre, whether the superficies pressed be parallel to the horizon, or perpendicular, or oblique; or whether the fluid, continued upwards from the compressed superficies, rises perpendicularly in a rectilinear direction, or creeps obliquely through crooked cavities and canals, whether those passages be regular or irregular, wide or narrow. That the pressure is not altered by any of these circumstances, may be collected by applying the demonstration of this Theorem to the several cases of fluids.

Cor. 3. From the same demonstration it may also be collected (by Prop. XIX), that the parts of a heavy fluid acquire no motion among themselves by the pressure of the incumbent weight, except that motion which arises from condensation.

Cor. 4. And therefore if another body of the same specific gravity, incapable of condensation, be immersed in this fluid, it will acquire no motion by the pressure of the incumbent weight: it will neither descend nor ascend, nor change its figure. If it be spherical, it will remain so, notwithstanding the pressure; if it be square, it will remain square; and that, whether it be soft or fluid; whether it swims freely in the fluid, or lies at the bottom. For any internal part of a fluid is in the same state with the submersed body; and the case of all submersed bodies that have the same magnitude, figure, and specific gravity, is alike. If a submersed body, retaining its weight, should dissolve and put on the form of a fluid, this body, if before it would have ascended, descended, or from any pressure assume a new figure, would now likewise ascend, descend, or put on a new figure; and that, because its gravity and the other causes of its motion remain. But (by Case 5, Prop. XIX) it would now be at rest, and retain its figure. Therefore also in the former case.

Cor. 5. Therefore a body that is specifically
heavier than a fluid contiguous to it will sink; and that which is
specifically lighter will ascend, and attain so much motion and change
of figure as that excess or defect of gravity is able to produce. For
that excess or defect is the same thing as an impulse, by which a
body, otherwise *in equilibrio* with the parts of the fluid,
is acted on; and may be compared with the excess or defect of a weight
in one of the scales of a balance.

Cor. 6. Therefore bodies placed in fluids
have a twofold gravity the one true and absolute, the other apparent,
vulgar, and comparative. Absolute gravity is the whole force with
which the body tends downwards; relative and vulgar gravity is the
excess of gravity with which the body tends downwards more than the
ambient fluid. By the first kind of gravity the parts of all fluids
and bodies gravitate in their proper places; and therefore their
weights taken together compose the weight of the whole. For the whole
taken together is heavy, as may be experienced in vessels full of
liquor; and the weight of the whole is equal to the weights of all the
parts, and is therefore composed of them. By the other kind of gravity
bodies do not gravitate in their places; that is, compared with one
another, they do not preponderate, but, hindering one another's
endeavours to descend, remain in their proper places, as if they were
not heavy. Those things which are in the air, and do not preponderate,
are commonly looked on as not heavy. Those which do preponderate are
commonly reckoned heavy, in as much as they are not sustained by the
weight of the air. The common weights are nothing else but the excess
of the true weights above the weight of the air. Hence also, vulgarly,
those things are called light which are less heavy, and, by yielding
to the preponderating air, mount upwards. But these are only
comparatively light, and not truly so, because they descend
*in vacuo*. Thus, in water, bodies which, by their greater or
less gravity, descend or ascend, are comparatively
and apparently heavy or light; and their comparative and apparent
gravity or levity is the excess or defect by which their true gravity
either exceeds the gravity of the water or is exceeded by it. But
those things which neither by preponderating descend, nor, by yielding
to the preponderating fluid, ascend, although by their true weight
they do increase the weight of the whole, yet comparatively, and in
the sense of the vulgar, they do not gravitate in the water. For these
cases are alike demonstrated.

Cor. 7. These things which have been demonstrated concerning gravity take place in any other centripetal forces.

Cor. 8. Therefore if the medium in which any body moves be acted on either by its own gravity, or by any other centripetal force, and the body be urged more powerfully by the same force; the difference of the forces is that very motive force, which, in the foregoing Propositions, I have considered as a centripetal force. But if the body be more lightly urged by that force, the difference of the forces becomes a centrifugal force, and is to be considered as such.

Cor. 9. But since fluids by pressing the included bodies do not change their external figures, it appears also (by Cor. Prop. XIX) that they will not change the situation of their internal parts in relation to one another; and therefore if animals were immersed therein, and that all sensation did arise from the motion of their parts, the fluid will neither hurt the immersed bodies, nor excite any sensation, unless so far as those bodies may be condensed by the compression. And the case is the same of any system of bodies encompassed with a compressing fluid. All the parts of the system will be agitated with the same motions as if they were placed in a vacuum, and would only retain their comparative gravity; unless so far as the fluid may somewhat resist their motions, or be requisite to conglutinate them by compression.

*Let the density of any fluid be proportional to the
compression, and its parts be attracted downwards by a centripetal
force reciprocally proportional to the distances from the centre:
I say, that, if those distances be taken continually proportional,
the densities of the fluid at the same distances will be also
continually proportional.*

Let ATV denote the spherical bottom of the fluid, S the centre, SA,
SB, SC, SD, SE, SF, &c., distances continually proportional. Erect
the perpendiculars AH, BI, CK, DL, EM, FN, &c., which shall be as
the densities of the medium in the places A, B, C, D, E, F; and the
specific gravities in those places will be AH

AS, BI

BS, CK

CS, &c., or, which is all one, as
AH

AB, BI

BC, CK

CD, &c. Suppose, first, these
gravities to be uniformly continued from A to B, from B to C, from C
to D, &c., the decrements in the points
B, C, D, &c., being taken by steps. And these gravities drawn into
the altitudes AB, BC, CD, &c., will give the pressures AH, BI, CK,
&c., by which the bottom ATV is acted on (by Theor. XV). Therefore
the particle A sustains all the pressures AH, BI, CK, DL, &c.,
proceeding *in infinitum*; and the particle B sustains the
pressures of all but the first AH; and the particle C all but the two
first AH, BI; and so on: and therefore the density AH of the first
particle A is to the density BI of the second particle B as the sum of
all AH + BI + CK + DL, *in infinitum*, to the sum of all BI +
CK + DL, &c. And BI the density of the second particle B is to CK
the density of the third C, as the sum of all BI + CK + DL, &c.,
to the sum of all CK + DL, &c. Therefore these sums are
proportional to their differences AH, BI, CK, &c., and therefore
continually proportional (by Lem. 1 of this Book); and therefore the
differences AH, BI, CK, &c., proportional to the sums, are also
continually proportional. Wherefore since the densities in the places
A, B, C, &c., are as AH, BI, CK, &c., they will also be
continually proportional. Proceed intermissively, and, *ex aequo*,
at the distances SA, SC, SE, continually proportional, the densities
AH, CK, EM will be continually proportional. And by the same
reasoning, at any distances SA, SD, SG, continually proportional, the
densities AH, DL, GO, will be continually proportional. Let now the
points A, B, C, D, E, &c., coincide, so that the progression of
the specific gravities from the bottom A to the top of the fluid may
be made continual; and at any distances SA, SD, SG, continually
proportional, the densities AH, DL, GO, being all along continually
proportional, will still remain continually proportional.
Q.E.D.

Cor. Hence if the density of the fluid in two
places, as A and E, be given, its density in any other place Q may be
collected. With the centre S, and the rectangular asymptotes SQ, SX,
describe an hyperbola cutting the perpendiculars AH, EM, QT in *a,
e,* and *q*, as also the perpendiculars HX, MY, TZ, let
fall upon the asymptote SX, in *h, m*, and *t*. Make
the area Y*mt*Z to the given area Y*mh*X as the given
area E*eq*Q to the given area E*ea*A; and the line Z*t*
produced will cut off the line QT proportional to the density. For if
the lines SA, SE, SQ are continually proportional, the areas E*eq*Q,
E*ea*A will be equal, and thence the
areas Y*mt*Z, X*hm*Y, proportional to them, will be also
equal; and the lines SX, SY, SZ, that is, AH, EM, QT continually
proportional, as they ought to be. And if the lines SA, SE, SQ, obtain
any other order in the series of continued proportionals, the lines
AH, EM, QT, because of the proportional hyperbolic areas, will obtain
the same order in another series of quantities continually
proportional.

*Let the density of any fluid be proportional to the
compression, and its parts be attracted downwards by a gravitation
reciprocally proportional to the squares of the distances from the
centre: I say, that if the distances be taken in harmonic
progression, the densities of the fluid at those distances will be
in a geometrical progression.*

Let S denote the centre, and SA, SB, SC, SD, SE, the distances in
geometrical progression. Erect the perpendiculars AH, BI, CK, &c.,
which shall be as the densities of the fluid in the places A, B, C, D,
E, &c., and the specific gravities thereof in those places will be
as AH

SA^{2}, BI

SB^{2}, CK

SC^{2}, &c. Suppose these
gravities to be uniformly continued, the first from A to B, the second
from B to C, the third from C to D, &c. And these drawn into the
altitudes AB, BC, CD, DE, &c., or, which is the same thing, into
the distances SA, SB, SC, &c., proportional to those altitudes,
will give AH

SA, BI

SB, CK

SC, &c., the exponents of the
pressures. Therefore since the densities are as the sums of those
pressures, the differences AH − BI, BI − CK, &c., of the densities
will be as the differences of those sums AH

SA, BI

SB, CK

SC, &c. With the centre S, and the
asymptotes SA, S*x*, describe any hyperbola, cutting the
perpendiculars AH, BI, CK, &c., in *a, b, c,* &c., and
the perpendiculars H*t*, I*n*, K*w*, let fall
upon the asymptote S*x*, in *h, i, k*; and the
differences of the densities *tu, uw,* &c., will be as
AH

SA, BI

SB, &c. And the rectangles *tu
× th, uw × ui*, &c., or *tp, uq,* &c., as
AH × th

SA, BI
× ui

SB, &c., that is, as A*a*, B*b*,
&c. For, by the nature of the hyperbola, SA is to AH or S*t*
as *th* to A*c*, and therefore AH × th

SA is equal to A*a*. And, by a
like reasoning, BI
× ui

SB is equal to B*b*, &c. But
A*a*, B*b*, C*c*, &c., are continually
proportional, and therefore proportional to their differences A*a*
− B*b*, B*b* − C*c*, &c., therefore the
rectangles *tp, uq,* &c., are proportional to those
differences; as also the sums of the rectangles *tp + uq*, or
*tp + uq + wr* to the sums of the differences A*a* − C*c*
or A*a* − D*d*. Suppose several of these terms, and the
sum of all the differences, as A*a* − F*f*, will be
proportional to the sum of all the rectangles, as *zthn*.
Increase the number of terms, and diminish the distances of the points
A, B, C, &c., *in infinitum*, and those rectangles will
become equal to the hyperbolic area *zthn*, and therefore the
difference A*a* − F*f* is proportional to this area.
Take now any distances, as SA, SD, SF, in harmonic progression, and
the differences A*a* − D*d*, D*d* − F*f*
will be equal; and therefore the areas *thlx, xluz*,
proportional to those differences will be equal among themselves, and
the densities S*t*, S*x*, S*z*, that is, AH, DL,
FN, continually proportional. Q.E.D.

Cor. Hence if any two densities of the fluid,
as AH and BI, be given, the area *thiu*, answering to their
difference *tu*, will be given; and thence the density FN will
be found at any height SF, by taking the area *thnz* to that
given area *thiu* as the difference A*a* − F*f*
to the difference A*a* − B*b*.

By a like reasoning it may be proved, that if the gravity of the
particles of a fluid be diminished in a triplicate ratio of the
distances from the centre; and the reciprocals of the squares of the
distances SA, SB, SC, &c., (namely, SA^{3}

SA^{2}, SA^{3}

SB^{2}, SA^{3}

SC^{2} ) be taken in an
arithmetical progression, the densities AH, BI, CK, &c., will be
in a geometrical progression. And if the gravity be diminished in a
quadruplicate ratio of the distances, and the reciprocals of the cubes
of the distances (as SA^{4}

SA^{3}, SA^{4}

SB^{3}, SA^{4}

SC^{3}, &c.,) be taken in
arithmetical progression, the densities AH, BI, CK, &c., will be
in geometrical progression. And so *in infinitum*. Again; if
the gravity of the particles of the fluid be the same at all
distances, and the distances be in arithmetical progression, the
densities will be in a geometrical progression as Dr. *Halley*
has found. If the gravity be as the distance, and the squares of the
distances be in arithmetical progression, the densities will be in
geometrical progression. And so *in infinitum*. These things
will be so, when the density of the fluid condensed by compression is
as the force of compression; or, which is the same thing, when the
space possessed by the fluid is reciprocally as this force. Other laws
of condensation may be supposed, as that the cube of the compressing
force may be as the biquadrate of the density;
or the triplicate ratio of the force the same with the quadruplicate
ratio of the density: in which case, if the gravity he reciprocally as
the square of the distance from the centre; the density will be
reciprocally as the cube of the distance. Suppose that the cube of the
compressing force be as the quadrato-cube of the density; and if the
gravity be reciprocally as the square of the distance, the density
will be reciprocally in a sesquiplicate ratio of the distance. Suppose
the compressing force to be in a duplicate ratio of the density, and
the gravity reciprocally in a duplicate ratio of the distance, and the
density will be reciprocally as the distance. To run over all the
cases that might be offered would be tedious. But as to our own air,
this is certain from experiment, that its density is either
accurately, or very nearly at least, as the compressing force; and
therefore the density of the air in the atmosphere of the earth is as
the weight of the whole incumbent air, that is, as the height of the
mercury in the barometer.

*If a fluid be composed of particles mutually flying each other,
and the density be as the compression, the centrifugal forces of
the particles will be reciprocally proportional to the distances
of their centres. And,* vice versa*, particles flying each
other, with forces that are reciprocally proportional to the
distances of their centres, compose an elastic fluid, whose
density is as the compression.*

Let the fluid be supposed to be included in a cubic space ACE, and
then to be reduced by compression into a lesser cubic space *ace*;
and the distances of the particles retaining a like situation with
respect to each other in both the spaces, will be as the sides AB, *ab*
of the cubes; and the densities of the mediums will be reciprocally as
the containing spaces AB³, *ab³*. In the plane side of the
greater cube ABCD take the square DP equal to the plane side *db*
of the lesser cube: and, by the supposition, the pressure with which
the square DP urges the inclosed fluid will be to the pressure with
which that square *db* urges the inclosed fluid as the
densities of the mediums are to each other, that is, as *ab³*
to AB³. But the pressure with which the square DB urges the included
fluid is to the pressure with which the square DP urges the same fluid
as the square DB to the square DP, that is, as AB² to *ab²*.
Therefore, *ex aequo*, the pressure with which the square DB
urges the fluid is to the pressure with which the square *db*
urges the fluid as *ab* to AB. Let the planes FGH, *fgh*,
be drawn through the middles of the two cubes, and divide the fluid
into two parts. These parts will press each other mutually with the
same forces with which they are themselves
pressed by the planes AC, *ac*, that is, in the proportion of
*ab* to AB: and therefore the centrifugal forces by which these
pressures are sustained are in the same ratio. The number of the
particles being equal, and the situation alike, in both cubes, the
forces which all the particles exert, according to the planes FGH, *fgh*,
upon all, are as the forces which each exerts on each. Therefore the
forces which each exerts on each, according to the plane FGH in the
greater cube, are to the forces which each exerts on each, according
to the plane *fgh* in the lesser cube, as *ab* to AB,
that is, reciprocally as the distances of the particles from each
other. Q.E.D.

And, *vice versa*, if the forces of the single particles are
reciprocally as the distances, that is, reciprocally as the sides of
the cubes AB, *ab*; the sums of the forces will be in the same
ratio, and the pressures of the sides DB, *db* as the sums of
the forces; and the pressure of the square DP to the pressure of the
side DB as *ab²* to AB² . And, *ex aequo*, the
pressure of the square DP to the pressure of the side *db* as
*ab³* to AB³; that is, the force of compression in the one to
the force of compression in the other as the density in the former to
the density in the latter. Q.E.D.

By a like reasoning, if the centrifugal forces of the particles are
reciprocally in the duplicate ratio of the distances between the
centres, the cubes of the compressing forces will be as the
biquadrates of the densities. If the centrifugal forces be
reciprocally in the triplicate or quadruplicate ratio of the
distances, the cubes of the compressing forces will be as the
quadratocubes, or cubo-cubes of the densities. And universally, if D
be put for the distance, and E for the density of the compressed
fluid, and the centrifugal forces be reciprocally as any power D^{n}
of the distance, whose index is the number *n*, the
compressing forces will be as the cube roots of the power E^{n+2},
whose index is the number *n* + 2; and the contrary. All these
things are to be understood of particles whose centrifugal forces
terminate in those particles that are next them, or are diffused not
much further. We have an example of this in magnetical bodies. Their
attractive virtue is terminated nearly in bodies of their own kind
that are next them. The virtue of the magnet is contracted by the
interposition of an iron plate, and is almost terminated at it: for
bodies further off are not attracted by the magnet so much as by the
iron plate. If in this manner particles repel others of their own kind
that lie next them, but do not exert their virtue on the more remote,
particles of this kind will compose such fluids as are treated of in
this Proposition. If the virtue of any particle diffuse itself every
way *in infinitum*, there will be required a greater force to
produce an equal condensation of a greater quantity of the *fluid*.
But whether elastic fluids do really consist
of particles so repelling each other, is a physical question. We have
here demonstrated mathematically the property of fluids consisting of
particles of this kind, that hence philosophers may take occasion to
discuss that question.

*The quantities of matter in funependulous bodies, whose centres
of oscillation are equally distant from the centre of suspension,
are in a ratio compounded of the ratio of the weights and the
duplicate ratio of the times of the oscillations* in vacuo*.*

For the velocity which a given force can generate in a given matter in a given time is as the force and the time directly, and the matter inversely. The greater the force or the time is, or the less the matter, the greater velocity will be generated. This is manifest from the second Law of Motion. Now if pendulums are of the same length, the motive forces in places equally distant from the perpendicular are as the weights: and therefore if two bodies by oscillating describe equal arcs, and those arcs are divided into equal parts; since the times in which the bodies describe each of the correspondent parts of the arcs are as the times of the whole oscillations, the velocities in the correspondent parts of the oscillations will be to each other as the motive forces and the whole times of the oscillations directly, and the quantities of matter reciprocally: and therefore the quantities of matter are as the forces and the times of the oscillations directly and the velocities reciprocally. But the velocities reciprocally are as the times, and therefore the times directly and the velocities reciprocally are as the squares of the times; and therefore the quantities of matter are as the motive forces and the squares of the times, that is, as the weights and the squares of the times. Q.E.D.

Cor. 1. Therefore if the times are equal, the quantities of matter in each of the bodies are as the weights.

Cor. 2. If the weights are equal, the quantities of matter will be as the squares of the times.

Cor. 3. If the quantities of matter are equal, the weights will be reciprocally as the squares of the times.

Cor. 4. Whence since the squares of the
times, *caeteris paribus*, are as the lengths of the
pendulums, therefore if both the times and quantities of matter are
equal, the weights will be as the lengths of the pendulums.

Cor. 5. And universally, the quantity of matter in the pendulous body is as the weight and the square of the time directly, and the length of the pendulum inversely.

Cor. 6. But in a non-resisting medium, the quantity of matter in the pendulous body is as the comparative weight and the square of the time directly, and the length of the pendulum inversely. For the comparative weight is the motive force of the body in any heavy medium, as was shewn above; and therefore does the same thing in such a non-resisting medium as the absolute weight does in a vacuum.

Cor. 7. And hence appears a method both of comparing bodies one among another, as to the quantity of matter in each; and of comparing the weights of the same body in different places, to know the variation of its gravity. And by experiments made with the greatest accuracy, I have always found the quantity of matter in bodies to be proportional to their weight.

*Funependulous bodies that are, in any medium, resisted in the
ratio of the moments of time, and funependulous bodies that move
in a non-resisting medium of the same specific gravity, perform
their oscillations in a cycloid in the same time, and describe
proportional parts of arcs together.*

Let AB be an arc of a cycloid, which a body D, by vibrating in a
non-resisting medium, shall describe in any time. Bisect that arc in
C, so that C may be the lowest point thereof; and the accelerative
force with which the body is urged in any place D, or *d* or
E, will be as the length of the arc CD, or C*d*, or CE. Let
that force be expressed by that same arc; and since the resistance is
as the moment of the time, and therefore given, let it be expressed by
the given part CO of the cycloidal arc, and take the arc O*d*
in the same ratio to the arc CD that the arc OB has to the arc CB: and
the force with which the body in *d* is urged in a resisting
medium, being the excess of the force C*d* above the resistance
CO, will be expressed by the arc O*d*, and will therefore be to
the force with which the body D is urged in a non-resisting medium in
the place D, as the arc O*d* to the arc CD; and therefore also
in the place B, as the arc OB to the arc CB. Therefore if two bodies
D, *d* go from the place Bc and are urged by these forces;
since the forces at the beginning are as the arc CB and OB, the first
velocities and arcs first described will be in the same ratio. Let
those arcs be BD and B*d*, and the remaining arcs CD,
O*d*, will be in the same ratio. Therefore the forces, being
proportional to those arcs CD, O*d*, will remain in the same
ratio as at the beginning, and therefore the bodies will continue
describing together arcs in the same ratio. Therefore the forces and
velocities and the remaining arcs CD, O*d*, will be always as
the whole arcs CB, OB, and therefore those remaining arcs will be
described together. Therefore the two bodies D and *d* will
arrive together at the places C and O; that which moves in the
non-resisting medium, at the place C, and the other, in the resisting
medium, at the place O. Now since the velocities in C and O are as the
arcs CB, OB, the arcs which the bodies describe when they go farther
will be in the same ratio. Let those arcs be CE and O*e*. The
force with which the body D in a non-resisting medium is retarded in E
is as CE, and the force with which the body *d* in the
resisting medium is retarded in *e*, is as the sum of the
force C*e* and the resistance CO, that is, as O*e*; and
therefore the forces with which the bodies are retarded are as the
arcs CB, OB, proportional to the arcs CE, O*e*; and therefore
the velocities, retarded in that given ratio, remain in the same given
ratio. Therefore the velocities and the arcs described with those
velocities are always to each other in that given ratio of the arcs CB
and OB; and therefore if the entire arcs AB, *a*B are taken in
the same ratio, the bodies D and *d* will describe those arcs
together, and in the places A and *a* will lose all their
motion together. Therefore the whole oscillations are isochronal, or
are performed in equal times; and any parts of the arcs, as BD, B*d*,
or BE, B*e*, that are described together, are proportional to
the whole arcs BA, B*a*. Q.E.D.

Cor. Therefore the swiftest motion in a
resisting medium does not fall upon the lowest point C, but is found
in that point O, in which the whole arc described B*a* is
bisected. And the body, proceeding from thence to *a*, is
retarded at the same rate with which it was accelerated before in its
descent from B to O.

*Funependulous bodies, that are resisted in the ratio of the
velocity, have their oscillations in a cycloid isochronal.*

For if two bodies, equally distant from their centres of suspension, describe, in oscillating, unequal arcs, and the velocities in the correspondent parts of the arcs be to each other as the whole arcs; the resistances, proportional to the velocities, will be also to each other as the same arcs. Therefore if these resistances be subducted from or added to the motive forces arising from gravity which are as the same arcs, the differences or sums will be to each other in the same ratio of the arcs; and since the increments and decrements of the velocities are as these differences or sums, the velocities will be always as the whole arcs; therefore if the velocities are in any one case as the whole arcs, they will remain always in the same ratio. But at the beginning of the motion, when the bodies begin to descend and describe those arcs, the forces, which at that time are proportional to the arcs, will generate velocities proportional to the arcs. Therefore the velocities will be always as the whole arcs to be described, and therefore those arcs will be described in the same time. Q.E.D.

*If funependulous bodies are resisted in the duplicate ratio of
their velocities, the differences between the times of the
oscillations in a resisting medium, and the times of the
oscillations in a non-resisting medium of the same, specific
gravity, will be proportional to the arcs described in oscillating nearly.*

For let equal pendulums in a resisting medium describe the unequal arcs A, B; and the resistance of the body in the arc A will be to the resistance of the body in the correspondent part of the arc B in the duplicate ratio of the velocities, that is, as AA to BB nearly. If the resistance in the arc B were to the resistance in the arc A as AB to AA, the times in the arcs A and B would be equal (by the last Prop.) Therefore the resistance AA in the arc A, or AB in the arc B, causes the excess of the time in the arc A above the time in a non-resisting medium; and the resistance BB causes the excess of the time in the arc B above the time in a non-resisting medium. But those excesses are as the efficient forces AB and BB nearly, that is, as the arcs A and B. Q.E.D.

Cor. 1. Hence from the times of the oscillations in unequal arcs in a resisting medium, may be known the times of the oscillations in a non-resisting medium of the same specific gravity. For the difference of the times will be to the excess of the time in the lesser arc above the time in a non-resisting medium as the difference of the arcs to the lesser arc.

Cor. 2. The shorter oscillations are more isochronal, and very short ones are performed nearly in the same times as in a non-resisting medium. But the times of those which are performed in greater arcs are a little greater, because the resistance in the descent of the body, by which the time is prolonged, is greater, in proportion to the length described in the descent than the resistance in the subsequent ascent, by which the time is contracted. But the time of the oscillations, both short and long, seems to be prolonged in some measure by the motion of the medium. For retarded bodies are resisted somewhat less in proportion to the velocity, and accelerated bodies somewhat more than those that proceed uniformly forwards; because the medium, by the motion it has received from the bodies, going forwards the same way with them, is more agitated in the former case, and less in the latter; and so conspires more or less with the bodies moved. Therefore it resists the pendulums in their descent more, and in their ascent less, than in proportion to the velocity; and these two causes concurring prolong the time.

*If a funependulous body, oscillating in a cycloid, be resisted
in the ratio of the moments of the time, its resistance will be to
the force of gravity as the excess of the arc described in the
whole descent above the arc described in the subsequent ascent to
twice the length of the pendulum.*

Let BC represent the arc described in the descent, C*a* the
arc described in the ascent, and A*a* the difference of the
arcs: and things remaining as they were constructed and demonstrated
in Prop. XXV, the force with which the oscillating body is urged in
any place D will be to the force of resistance as the arc CD to the
arc CO, which is half of that difference A*a*. Therefore the
force with which the oscillating body is urged at the beginning or the
highest point of the cycloid, that is, the force of gravity, will be
to the resistance as the arc of the cycloid, between that highest
point and lowest point C, is to the arc CO; that is (doubling those
arcs), as the whole cycloidal arc, or twice the length of the
pendulum, to the arc A*a*. Q.E.D.

*Supposing that a body oscillating in a cycloid is resisted in a
duplicate ratio of the velocity: to find the resistance in each place.*

Let B*a* be an arc described in one entire oscillation, C the lowest point
of the cycloid, and CZ half the whole cycloidal arc, equal to the length of the pendulum;
and let it be required to find the resistance of the body in any
place D. Cut the indefinite right line OQ in the points O, S, P, Q, so
that (erecting the perpendiculars OK, ST, PI, QE, and with the centre
O, and the aysmptotes OK, OQ, describing the hyperbola TIGE cutting
the perpendiculars ST, PI, QE in T, I, and E, and through the point I
drawing KF, parallel to the asymptote OQ, meeting the asymptote OK in
K, and the perpendiculars ST and QE in L and F) the hyperbolic area
PIEQ may be to the hyperbolic area PITS as the arc BC, described in
the descent of the body, to the arc C*a* described in the
ascent; and that the area IEF may be to the area ILT as OQ to OS. Then
with the perpendicular MN cut off the hyperbolic area PINM, and let
that area be to the hyperbolic area PIEQ as the arc CZ to the arc BC
described in the descent. And if the perpendicular RG cut off the
hyperbolic area PIGR, which shall be to the area PIEQ as any arc CD to
the arc BC described in the whole descent, the resistance in any place
D will be to the force of gravity as the area OR

OQ IEF − IGH to the area PINM.

For since the forces arising from gravity with which the body is
urged in the places Z, B, D, *a*, are as the arcs CZ, CB, CD,
C*a* and those arcs are as the areas PINM, PIEQ, PIGR, PITS;
let those areas be the exponents both of the arcs and of the forces
respectively. Let D*d* be a very small space described by the
body in its descent: and let it be expressed by the very small area RG*gr*
comprehended between the parallels RG, *rg*; and produce *rg*
to *h*, so that GH*hg* and RG*gr* may be the
contemporaneous decrements of the areas IGH, PIGR. And the increment
GH*hg* − Rr

OQ IEF, or R*r* × HG − Rr

OQ IEF, of the area OR

OQ IEF − IGH will be to the decrement
RG*gr*, or R*r* × RG, of the area PIGR, as HG − IEF

OQ to RG; and therefore as OR × HG −
OR

OQ IEF to OR × GR or OP × PI, that is
(because of the equal quantities OR × HG, OR × HR − OR × GR, ORHK −
OPIK, PIHR and PIGR + IGH), as PIGR + IGH − OR

OQ IEF to OPIK. Therefore if the area
OR

OQ IEF − IGH be called Y, and RG*gr*
the decrement of the area PIGR be given, the increment of the area Y
will be as PIGR − Y.

Then if V represent the force arising from the gravity, proportional to the arc CD to be described, by which the body is acted upon in D, and R be put for the resistance, V − R will be the whole force with which the body is urged in D. Therefore the increment of the velocity is as V − R and the particle of time in which it is generated conjunctly. But the velocity itself is as the contemporaneous increment of the space described directly and the same particle of time inversely. Therefore, since the resistance is, by the supposition, as the square of the velocity, the increment of the resistance will (by Lem. II) be as the velocity and the increment of the velocity conjunctly, that is, as the moment of the space and V − R conjunctly; and, therefore, if the moment of the space be given, as V − R; that is, if for the force V we put its exponent PIGR, and the resistance R be expressed by any other area Z, as PIGR − Z.

Therefore the area PIGR uniformly decreasing by the subduction of
given moments, the area Y increases in proportion of PIGR − Y, and the
area Z in proportion of PIGR − Z. And therefore if the areas Y and Z
begin together, and at the beginning are equal, these, by the addition
of equal moments, will continue to be equal and in like manner
decreasing by equal moments, will vanish together. And, *vice
versa*, if they together begin and vanish, they will have equal
moments and be always equal; and that, because if the resistance Z be
augmented, the velocity together with the arc C*a*, described
in the ascent of the body, will be diminished; and the point in which
all the motion together with the resistance ceases coming nearer to
the point C, the resistance vanishes sooner than the area Y. And the
contrary will happen when the resistance is diminished.

Now the area Z begins and ends where the resistance is nothing, that
is, at the beginning of the motion where the arc CD is equal to the arc CB,
and the right line RG falls upon the right line QE;
and at the end of the motion where the arc CD is
equal to the arc C*a*, and RG falls upon the right line ST. And
the area Y or OR

OQ IEF − IGH begins and ends also where
the resistance is nothing, and therefore where OR

OQ IEF and IGH are equal; that is (by
the construction), where the right line RG falls successively upon the
right lines QE and ST. Therefore those areas begin and vanish
together, and are therefore always equal. Therefore the area OR

OQ IEF − IGH is equal to the area Z, by
which the resistance is expressed, and therefore is to the area PINM,
by which the gravity is expressed, as the resistance to the gravity.
Q.E.D.

Cor. 1. Therefore the
resistance in the lowest place C is to the force of gravity as the
area OP

OQ IEF to the area PINM.

Cor. 2. But it becomes greatest where the area PIHR is to the area IEF as OR to OQ. For in that case its moment (that is, PIGR − Y) becomes nothing.

Cor. 3. Hence also may be known the velocity in each place, as being in the subduplicate ratio of the resistance, and at the beginning of the motion equal to the velocity of the body oscillating in the same cycloid without any resistance.

However, by reason of the difficulty of the calculation by which the resistance and the velocity are found by this Proposition, we have thought fit to subjoin the Proposition following.

*If a right line* aB *be equal to the arc of a cycloid
which an oscillating body describes, and at each of its points*
D *the perpendiculars* DK *be erected, which shall be to
the length of the pendulum as the resistance of the body in the
corresponding points of the arc to the force of gravity; I say,
that the difference between the arc described in the whole descent
and the arc described in the whole subsequent ascent drawn into
half the sum of the same arcs will be equal to the area* BKa *which
all those perpendiculars take up.*

Let the arc of the cycloid, described in one entire oscillation, be
expressed by the right line *a*B, equal to it, and the arc
which would have been described *in vacuo* by the length AB.
Bisect AB in C, and the point C will represent B the lowest point of
the cycloid, and CD will be as the force arising from gravity, with
which the body in D is urged in the direction of the tangent of the
cycloid, and will have the same ratio to the length of the pendulum as
the force in D has to the force of gravity. Let that force, therefore,
be expressed by that length CD, and the force of gravity by the length
of the pendulum; and if in DE you take DK in the same ratio to the
length of the pendulum as the resistance has to the gravity, DK will
be the exponent of the resistance. From the centre C with the interval
CA or CB describe a semi-circle BE*e*A. Let the body describe,
in the least time, the space D*d*; and, erecting the
perpendiculars DE, *de*, meeting the circumference in E and *e*,
they will be as the velocities which the body descending *in vacuo*
from the point B would acquire in the places D and *d*. This
appears by Prop LII, Book I. Let therefore,
these velocities be expressed by those perpendiculars DE, *de*;
and let DF be the velocity which it acquires in D by falling from B in
the resisting medium. And if from the centre C with the interval CF we
describe the circle F*f*M meeting the right lines *de*
and AB in *f* and M, then M will be the place to which it
would thenceforward, without farther resistance, ascend, and *df*
the velocity it would acquire in *d*. Whence, also, if F*g*
represent the moment of the velocity which the body D, in describing
the least space D*d*, loses by the resistance of the medium;
and CN be taken equal to C*g*; then will N be the place to
which the body, if it met no farther resistance, would thenceforward
ascend, and MN will be the decrement of the ascent arising from the
loss of that velocity. Draw F*m* perpendicular to *df*,
and the decrement F*g* of the velocity DF generated by the
resistance DK will be to the increment *fm* of the same
velocity, generated by the force CD, as the generating force DK to the
generating force CD. But because of the similar triangles F*mf*,
F*hg*, FDC, *fm* is to F*m* or D*d* as CD
to DF; and, *ex aequo*, F*g* to D*d* as DK to
DF. Also F*h* is to F*g* as DF to CF; and, *ex aequo
perturbatè*, F*h* or MN to D*d* as DK to CF or CM;
and therefore the sum of all the MN × CM will be equal to the sum of
all the D*d* × DK. At the moveable point M suppose always a
rectangular ordinate erected equal to the indeterminate CM, which by a
continual motion is drawn into the whole length A*a*; and the
trapezium described by that motion, or its equal, the rectangle A*a*
× ½*a*B, will be equal to the sum of all the MN × CM, and
therefore to the sum of all the D*d* × DK, that is, to the area
BKVT*a*. Q.E.D.

Cor. Hence from the law of resistance, and
the difference A*a* of the arcs C*a*, CB, may be
collected the proportion of the resistance to the gravity nearly.

For if the resistance DK be uniform, the figure BKT*a* will be
a rectangle under B*a* and DK; and thence the rectangle under
½B*a* and A*a* will be equal to the rectangle under B*a*
and DK, and DK will be equal to ½A*a*. Wherefore since DK is
the exponent of the resistance, and the length of the pendulum the
exponent of the gravity, the resistance will be to the gravity as ½A*a*
to the length of the pendulum; altogether as in Prop. XXVIII is
demonstrated.

If the resistance be as the velocity, the figure BKT*a* will
be nearly an ellipsis. For if a body, in a non-resisting medium, by
one entire oscillation, should describe the length BA, the velocity in
any place D would be as the ordinate DE of the circle described on the
diameter AB. Therefore since B*a* in the resisting medium, and
BA in the non-resisting one, are described nearly in the same times;
and therefore the velocities in each of the points of B*a* are
to the velocities in the correspondent points of the length BA nearly
as B*a* is to BA, the velocity in the point D in the resisting
medium will be as the ordinate of the circle
or ellipsis described upon the diameter B*a*; and therefore the
figure BKVT*a* will be nearly an ellipsis. Since the resistance
is supposed proportional to the velocity, let OV be the exponent of
the resistance in the middle point O; and an ellipsis BRVS*a*
described with the centre O, and the semi-axes OB, OV, will be nearly
equal to the figure BKVT*a*, and to its equal the rectangle A*a*
× BO. Therefore A*a* × BO is to OV × BO as the area of this
ellipsis to OV × BO; that is, A*a* is to OV as the area of the
semi-circle to the square of the radius, or as 11 to 7 nearly; and,
therefore, 7

11 A*a* is to the length of
the pendulum as the resistance of the oscillating body in O to its
gravity.

Now if the resistance DK be in the duplicate ratio of the velocity,
the figure BKVT*a* will be almost a parabola having V for its
vertex and OV for its axis, and therefore will be nearly equal to the
rectangle under B*a* and OV. Therefore the rectangle under ½B*a*
and A*a* is equal to the rectangle ⅔B*a* × OV, and
therefore OV is equal to ¾A*a*; and therefore the resistance in
O made to the oscillating body is to its gravity as ¾Aa to the length
of the pendulum.

And I take these conclusions to be accurate enough for practical
uses. For since an ellipsis or parabola BRVS*a* falls in with
the figure BKVT*a* in the middle point V, that figure, if
greater towards the part BRV or VS*a* than the other, is less
towards the contrary part, and is therefore nearly equal to it.

*If the resistance made to an oscillating body in each of the
proportional parts of the arcs described be augmented or
diminished in a given ratio, the difference between the arc
described in the descent and the arc described in the subsequent
ascent will be augmented or diminished in the same ratio.*

For that difference arises from the retardation of the pendulum by
the resistance of the medium, and therefore is as the whole
retardation and the retarding resistance proportional thereto. In the
foregoing Proposition the rectangle under the right line ½*a*B
and the difference A*a* of the arcs CB, C*a*, was equal
to the area BKT*a*. And that area, if the length *a*B
remains, is augmented or diminished in the ratio of the ordinates DK;
that is, in the ratio of the resistance and is therefore as the length
*a*B and the resistance conjunctly. And therefore the rectangle
under A*a* and ½*a*B is as *a*B and the
resistance conjunctly, and therefore A*a* is as the resistance.
Q.E.D.

Cor. 1. Hence if the resistance be as the velocity, the difference of the arcs in the same medium will be as the whole arc described: and the contrary.

Cor. 2. If the resistance be in the duplicate ratio of the velocity, that difference will be in the duplicate ratio of the whole arc: and the contrary.

Cor. 3. And universally, if the resistance be in the triplicate or any other ratio of the velocity, the difference will be in the same ratio of the whole arc: and the contrary.

Cor. 4. If the resistance be partly in the simple ratio of the velocity, and partly in the duplicate ratio of the same, the difference will be partly in the ratio of the whole arc, and partly in the duplicate ratio of it: and the contrary. So that the law and ratio of the resistance will be the same for the velocity as the law and ratio of that difference for the length of the arc.

Cor. 5. And therefore if a pendulum describe successively unequal arcs, and we can find the ratio of the increment or decrement of this difference for the length of the arc described, there will be had also the ratio of the increment or decrement of the resistance for a greater or less velocity.

From these propositions we may find the resistance of mediums by
pendulums oscillating therein. I found the resistance of the air by
the following experiments. I suspended a wooden globe or ball weighing
57 7

22 ounces troy, its diameter 6
^{7}/_{8} *London*
inches, by a fine thread on a firm hook, so that the distance between
the hook and the centre of oscillation of the globe was 10½ feet. I
marked on the thread a point 10 feet and 1 inch distant from the
centre of suspension; and even with that point I placed a ruler
divided into inches, by the help whereof I observed the lengths of the
arcs described by the pendulum. Then I numbered the oscillations in
which the globe would lose 1

8 part of its motion. If the pendulum
was drawn aside from the perpendicular to the distance of 2 inches,
and thence let go, so that in its whole descent it described an arc of
2 inches, and in the first whole oscillation, compounded of the
descent and subsequent ascent, an arc of almost 4 inches, the same in
164 oscillations lost 1

8 part of its motion, so as in its
last ascent to describe an arc of 1¾ inches. If in the first descent
it described an arc of 4 inches, it lost 1

8 part of its motion in 121
oscillations, so as in its last ascent to describe an arc of 3½
inches. If in the first descent it described an arc of 8, 16, 32, or
64 inches, it lost 1

8 part of its motion in 69, 35½, 18½,
9⅔ oscillations, respectively. Therefore the difference between the
arcs described in the first descent and the last ascent was in the
1st, 2d, 3d, 4th, 5th, 6th cases, ¼, ½, 1, 2, 4, 8 inches
respectively. Divide those differences by the number of oscillations
in each case, and in one mean oscillation, wherein an arc of 3¾, 7½,
15, 30, 60, 120 inches was described, the
difference of the arcs described in the descent and subsequent ascent
will be 1

656, 1

242, 1

69, 4

71, 8

37, 24

29 parts of an inch, respectively.
But these differences in the greater oscillations are in the duplicate
ratio of the arcs described nearly, but in lesser oscillations
something greater than in that ratio; and therefore (by Cor. 2, Prop.
XXXI of this Book) the resistance of the globe, when it moves very
swift, is in the duplicate ratio of the velocity, nearly; and when it
moves slowly, somewhat greater than in that ratio.

Now let V represent the greatest velocity in any oscillation, and let
A, B, and C be given quantities, and let us suppose the difference of
the arcs to be AV + BV^{
32} + CV² . Since the
greatest velocities are in the cycloid as ½ the arcs described in
oscillating, and in the circle as ½ the chords of those arcs; and
therefore in equal arcs are greater in the cycloid than in the circle
in the ratio of ½ the arcs to their chords; but the times in the
circle are greater than in the cycloid, in a reciprocal ratio of the
velocity; it is plain that the differences of the arcs (which are as
the resistance and the square of the time conjunctly) are nearly the
same in both curves: for in the cycloid those differences must be on
the one hand augmented, with the resistance, in about the duplicate
ratio of the arc to the chord, because of the velocity augmented in
the simple ratio of the same; and on the other hand diminished, with
the square of the time, in the same duplicate ratio. Therefore to
reduce these observations to the cycloid, we must take the same
differences of the arcs as were observed in the circle, and suppose
the greatest velocities analogous to the half, or the whole arcs, that
is, to the numbers ½, 1, 2, 4, 8, 16. Therefore in the 2d, 4th, and
6th cases, put 1, 4, and 16 for V; and the difference of the arcs in
the 2d case will become ½

121 = A + B + C; in the 4th case
2

35½ = 4A + 8B + 16C; in the 6th
8

9^{2}/_{3} = 16A +
64B + 256C. These equations reduced give A = 0,0000916, B = 0,0010847,
and C = 0,0029558. Therefore the difference of the arcs is as 0,0000916V
+ 0,0010847V^{32} + 0,0029558V²: and
therefore since (by Cor. Prop. XXX, applied to this case) the
resistance of the globe in the middle of the arc described in
oscillating, where the velocity is V, is to its weight as ^{7}/_{11}AV
+ ^{7}/_{10}BV^{3/2}
+ ¾CV² to the length of the pendulum, if for A, B, and C you
put the numbers found, the resistance of the globe will be to its
weight as 0,0000583V + 0,0007593V^{
32} + 0,0022169V² to
the length of the pendulum between the centre of suspension and the
ruler, that is, to 121 inches. Therefore since V in the second case
represents 1, in the 4th case 4, and in the 6th case 16, the
resistance will be to the weight of the globe, in the 2d case, as
0,0030345 to 121; in the 4th, as 0,041748 to 121; in the 6th, as
0,61705 to 121.

The arc, which the point marked in the thread
described in the 6th case, was of 120 − 8

9^{2}/_{3} ,
or 119^{5}/_{29} inches. And therefore since the
radius was 121 inches, and the length of the pendulum between the
point of suspension and the centre of the globe was 126 inches, the
arc which the centre of the globe described was 124^{3}/_{31}
inches. Because the greatest velocity of the oscillating body, by
reason of the resistance of the air, does not fall on the lowest point
of the arc described, but near the middle place of the whole arc, this
velocity will be nearly the same as if the globe in its whole descent
in a non-resisting medium should describe 62^{3}/_{62}
inches, the half of that arc, and that in a cycloid, to which we have
above reduced the motion of the pendulum; and therefore that velocity
will be equal to that which the globe would acquire by falling
perpendicularly from a height equal to the versed sine of that arc.
But that versed sine in the cycloid is to that arc 62^{3}/_{62}
as the same arc to twice the length of the pendulum 252, and therefore
equal to 15,278 inches. Therefore the velocity of the pendulum is the
same which a body would acquire by falling, and in its fall describing
a space of 15,278 inches. Therefore with such a velocity the globe
meets with a resistance which is to its weight as 0,61705 to 121, or
(if we take that part only of the resistance which is in the duplicate
ratio of the velocity) as 0,56752 to 121.

I found, by an hydrostatical experiment, that the weight of this
wooden globe was to the weight of a globe of water of the same
magnitude as 55 to 97: and therefore since 121 is to 213,4 in the same
ratio, the resistance made to this globe of water, moving forwards
with the above-mentioned velocity, will be to its weight as 0,56752 to
213,4, that is, as 1 to 376^{1}/_{50}. Whence since
the weight of a globe of water, in the time in which the globe with a
velocity uniformly continued describes a length of 30,556 inches, will
generate all that velocity in the falling globe, it is manifest that
the force of resistance uniformly continued in the same time will take
away a velocity, which will be less than the other in the ratio of 1
to 376^{1}/_{50}, that is, the 1

376^{1}/_{50} part of
the whole velocity. And therefore in the time that the globe, with the
same velocity uniformly continued, would describe the length of its
semi-diameter, or 3^{7}/_{16} inches, it would lose
the ^{1}/_{3342} part of its motion.

I also counted the oscillations in which the pendulum lost ¼ part of
its motion. In the following table the upper numbers denote the length
of the arc described in the first descent, expressed in inches and
parts of an inch; the middle numbers denote the length of the arc
described in the last ascent; and in the lowest place are the numbers
of the oscillations. I give an account of this experiment, as being
more accurate than that in which only ^{1}/_{8}
part of the motion was lost. I leave the calculation to such as are
disposed to make it.

First descent | 2 | 4 | 8 | 16 | 32 | 64 |

Last ascent | 1½ | 3 | 6 | 12 | 24 | 48 |

Numb. of oscill. | 374 | 272 | 162½ | 83⅓ | 41⅔ | 22⅔ |

I afterward suspended a leaden globe of 2 inches in diameter,
weighing 26¼ ounces troy by the same thread, so that between the
centre of the globe and the point of suspension there was an interval
of 10½ feet, and I counted the oscillations in which a given part of
the motion was lost. The first of the following tables exhibits the
number of oscillations in which ^{1}/_{8} part of the
whole motion was lost; the second the number of oscillations in which
there was lost part of the same.

First descent | 1 | 2 | 4 | 8 | 16 | 32 | 64 |

Last ascent | ^{7}/_{8} | ^{7}/_{4} | 3½ | 7 | 14 | 28 | 56 |

Numb, of oscill. | 226 | 228 | 193 | 140 | 90½ | 53 | 30 |

First descent | 1 | 2 | 4 | 8 | 16 | 32 | 64 |

Last ascent | ¾ | 1½ | 3 | 6 | 12 | 24 | 48 |

Numb. of oscill. | 510 | 518 | 420 | 318 | 204 | 121 | 70 |

Selecting in the first table the 3d, 5th, and 7th observations, and
expressing the greatest velocities in these observations particularly
by the numbers 1, 4, 16 respectively, and generally by the quantity V
as above, there will come out in the 3d observation ^{1}/_{2}

193 = A + B + C, in the 5th
observation 2

90^{1}/_{2} = 4A +
8B + 16C, in the 7th observation 8

30 = 16A + 64B + 256C. These
equations reduced give A = 0,001414, B = 0,000297, C = 0,000879. And
thence the resistance of the globe moving with the velocity V will be
to its weight 26¼ ounces in the same ratio as 0,0009V
+ 0,000208V^{3/2} + 0,000659V²
to 121 inches, the length of the pendulum. And if we regard that part
only of the resistance which is in the duplicate ratio of the
velocity, it will be to the weight of the globe as 0,000659V² to 121
inches. But this part of the resistance in the first experiment was to
the weight of the wooden globe of 57^{7}/_{22} ounces
as 0,002217V² to 121; and thence the resistance of the wooden globe is
to the resistance of the leaden one (their velocities being equal) as
57^{7}/_{22} into 0,002217 to 26¼ into 0,000659, that
is, as 7⅓ to 1. The diameters of the two globes were 6^{7}/_{8}
and 2 inches, and the squares of these are to each other as 47¼ and 4,
or 11^{13}/_{16} and 1, nearly. Therefore the
resistances of these equally swift globes were in less than a
duplicate ratio of the diameters. But we have not yet considered the
resistance of the thread, which was certainly very considerable, and
ought to be subducted from the resistance of the pendulums here found.
I could not determine this accurately, but I found it greater than a third part of the whole resistance of the lesser pendulum; and
thence I gathered that the resistances of the globes, when the
resistance of the thread is subducted, are nearly in the duplicate
ratio of their diameters.
For the ratio of 7⅓ − ⅓ to 1 − ⅓,
or 10½ to 1 is not very different from the duplicate ratio of the diameters
11^{13}/_{16} to 1.

Since the resistance of the thread is of less moment in greater
globes, I tried the experiment also with a globe whose diameter was
18¾ inches. The length of the pendulum between the point of suspension
and the centre of oscillation was 122½ inches, and between the point
of suspension and the knot in the thread 109½ inches. The arc
described by the knot at the first descent of the pendulum was 32
inches. The arc described by the same knot in the last ascent after
five oscillations was 28 inches. The sum of the arcs, or the whole arc
described in one mean oscillation, was 60 inches. The difference of
the arcs 4 inches. The ^{1}/_{10} part of this, or
the difference between the descent and ascent in one mean oscillation,
is ^{2}/_{5} of an inch. Then as the radius 109½ to
the radius 122½, so is the whole arc of 60 inches described by the
knot in one mean oscillation to the whole arc of 67^{1}/_{8}
inches described by the centre of the globe in one mean oscillation;
and so is the difference ^{3}/_{5} to a new
difference 0,4475. If the length of the arc described were to remain,
and the length of the pendulum should be augmented in the ratio of 126
to 122½, the time of the oscillation would be augmented, and the
velocity of the pendulum would be diminished in the subduplicate of
that ratio; so that the difference 0,4475 of the arcs described in the
descent and subsequent ascent would remain. Then if the arc described
be augmented in the ratio of 124^{3}/_{31} to 67^{1}/_{8},
that difference 0.4475 would be augmented in the duplicate of that
ratio, and so would become 1,5295. These things would be so upon the
supposition that the resistance of the pendulum were in the duplicate
ratio of the velocity. Therefore if the pendulum describe the whole
arc of 124^{3}/_{31} inches, and its length between
the point of suspension and the centre of oscillation be 126 inches,
the difference of the arcs described in the descent and subsequent
ascent would be 1,5295 inches. And this difference multiplied into the
weight of the pendulous globe, which was 208 ounces, produces 318,136.
Again; in the pendulum above-mentioned, made of a wooden globe, when
its centre of oscillation, being 126 inches from the point of
suspension, described the whole arc of 124^{3}/_{31}
inches, the difference of the arcs described in the descent and ascent
was ^{126}/_{121} into 8

9^{2}/_{3}. This
multiplied into the weight of the globe, which was 57^{7}/_{22}
ounces, produces 49,396. But I multiply these differences into the
weights of the globes, in order to find their resistances. For the
differences arise from the resistances, and are as the resistances
directly and the weights inversely. Therefore the resistances are as
the numbers 318,136 and 49,396. But that part of the resistance
of the lesser globe, which is in the duplicate
ratio of the velocity, was to the whole resistance as 0,56752 tor
0,61675, that is, as 45,453 to 49,396; whereas that part of the
resistance of the greater globe is almost equal to its whole
resistance; and so those parts are nearly as 318,136 and 45,453, that
is, as 7 and 1. But the diameters of the globes are 18¾ and 6^{7}/_{8};
and their squares 351^{9}/_{16} and 47^{17}/_{64}
are as 7,438 and 1, that is, as the resistances of the globes 7 and 1,
nearly. The difference of these ratios is scarce greater than may
arise from the resistance of the thread. Therefore those parts of the
resistances which are, when the globes are equal, as the squares of
the velocities, are also, when the velocities are equal, as the
squares of the diameters of the globes.

But the greatest of the globes I used in these experiments was not perfectly spherical, and therefore in this calculation I have, for brevity's sake, neglected some little niceties; being not very solicitous for an accurate calculus in an experiment that was not very accurate. So that I could wish that these experiments were tried again with other globes, of a larger size, more in number, and more accurately formed; since the demonstration of a vacuum depends thereon. If the globes be taken in a geometrical proportion, as suppose whose diameters are 4, 8, 16, 32 inches; one may collect from the progression observed in the experiments what would happen if the globes were still larger.

In order to compare the resistances of different fluids with each
other, I made the following trials. I procured a wooden vessel 4 feet
long, 1 foot broad, and 1 foot high. This vessel, being uncovered, I
filled with spring water, and, having immersed pendulums therein, I
made them oscillate in the water. And I found that a leaden globe
weighing 166^{1}/_{6} ounces, and in diameter 3^{5}/_{8}
inches, moved therein as it is set down in the following table; the
length of the pendulum from the point of suspension to a certain point
marked in the thread being 126 inches, and to the centre of
oscillation 134^{3}/_{8} inches.

The arc described inthe first descent, by a point marked in the thread was inches. | ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ | 64 | . | 32 | . | 16 | . | 8 | . | 4 | . | 2 | . | 1 | . | ½ | . | ¼ |

The arc described inthe last ascent was inches. | ⎫ ⎪ ⎬ ⎪ ⎭ | 48 | . | 24 | . | 12 | . | 6 | . | 3 | . | 1½ | . | ¾ | . | ^{3}/_{8} | . | ^{3}/_{16} |

The difference of thearcs, proportional to the motion lost, was inches. | ⎫ ⎪ ⎬ ⎪ ⎭ | 16 | . | 8 | . | 4 | . | 2 | . | 1 | . | ½ | . | ¼ | . | ^{1}/_{8} | . | ^{1}/_{16} |

The number of theoscillations in water. | ⎫ ⎬ ⎭ | ^{29}/_{60} | . | 1^{1}/_{5} | . | 3 | . | 7 | . | 11¼ | . | 12⅔ | . | 13⅓ | ||||

The number of theoscillations in air. | ⎫ ⎬ ⎭ | 85½ | . | 287 | . | 535 |

In the experiments of the 4th column there
were equal motions lost in 535 oscillations made in the air, and 1^{1}/_{5}
in water. The oscillations in the air were indeed a little swifter
than those in the water. But if the oscillations in the water were
accelerated in such a ratio that the motions of the pendulums might be
equally swift in both mediums, there would be still the same number 1^{1}/_{5}
of oscillations in the water, and by these the same quantity of motion
would be lost as before; because the resistance it increased, and the
square of the time diminished in the same duplicate ratio. The
pendulums, therefore, being of equal velocities, there were equal
motions lost in 535 oscillations in the air, and 1^{1}/_{5}
in the water; and therefore the resistance of the pendulum in the
water is to its resistance in the air as 535 to 1^{1}/_{5}.
This is the proportion of the whole resistances in the case of the 4th column.

Now let AV + CV² represent the difference of the arcs described in
the descent and subsequent ascent by the globe moving in air with the
greatest velocity V; and since the greatest velocity is in the case of
the 4th column to the greatest velocity in the case of the 1st column
as 1 to 8; and that difference of the arcs in the case of the 4th
column to the difference in the case of the 1st column as ^{2}/_{535}
to ^{16}/_{85½}, or as 85½ to 4280; put in these
cases 1 and 8 for the velocities, and 85½ and 4280 for the differences
of the arcs, and A + C will be = 85½, and 8A + 64C
= 4280 or A + 8C = 535; and then
by reducing these equations, there will come out 7C = 449½ and
C = 64^{3}/_{14} and A
= 21^{2}/_{7}; and therefore the resistance,
which is as ^{7}/_{11}AV + ^{3}/_{4}CV²,
will become as 13^{6}/_{11}V + 48^{9}/_{56}V².
Therefore in the case of the 4th column, where the velocity was 1, the
whole resistance is to its part proportional to the square of the
velocity as 13^{6}/_{11} + 48^{9}/_{56}.
or 61^{12}/_{17} to
48^{9}/_{56}; and therefore the
resistance of the pendulum in water is to that part of the resistance
in air, which is proportional to the square of the velocity, and which
in swift motions is the only part that deserves consideration, as
61^{12}/_{17} to 48^{9}/_{56}
and 535 to 1^{1}/_{5} conjunctly, that is, as 571 to
1. If the whole thread of the pendulum oscillating in the water had
been immersed, its resistance would have been still greater; so that
the resistance of the pendulum oscillating in the water, that is, that
part which is proportional to the square of the velocity, and which
only needs to be considered in swift bodies, is to the resistance of
the same whole pendulum, oscillating in air with the same velocity, as
about 850 to 1, that is as, the density of water to the density of air, nearly.

In this calculation we ought also to have taken in that part of the resistance of the pendulum in the water which was as the square of the velocity; but I found (which will perhaps seem strange) that the resistance in the water was augmented in more than a duplicate ratio of the velocity. In searching after the cause, I thought upon this, that the vessel was too narrow for the magnitude of the pendulous globe, and by its narrowness obstructed the motion of the water as it yielded to the oscillating globe. For when I immersed a pendulous globe, whose diameter was one inch only, the resistance was augmented nearly in a duplicate ratio of the velocity, I tried this by making a pendulum of two globes, of which the lesser and lower oscillated in the water, and the greater and higher was fastened to the thread just above the water, and, by oscillating in the air, assisted the motion of the pendulum, and continued it longer. The experiments made by this contrivance proved according to the following table.

Arc descr. in first descent | 16 | . | 8 | . | 4 | . | 2 | . | 1 | . | ½ | . | ¼ |

Arc descr. in last ascent | 12 | . | 6 | . | 3 | . | 1½ | . | ¾ | . | ^{3}/_{8} | . | ^{3}/_{16} |

Diff. of arcs, proport. tomotion lost | 4 | . | 2 | . | 1 | . | ½ | . | ¼ | . | ^{1}/_{8} | . | ^{1}/_{16} |

Number of oscillations | 3^{3}/_{8} | . | 6½ | . | 12^{1}/_{12} | . | 21^{1}/_{5} | . | 34 | . | 53 | . | 62^{1}/_{5} |

In comparing the resistances of the mediums with each other, I also caused iron pendulums to oscillate in quicksilver. The length of the iron wire was about 3 feet, and the diameter of the pendulous globe about ⅓ of an inch. To the wire, just above the quicksilver, there was fixed another leaden globe of a bigness sufficient to continue the motion of the pendulum for some time. Then a vessel, that would hold about 3 pounds of quicksilver, was filled by turns with quicksilver and common water, that, by making the pendulum oscillate successively in these two different fluids, I might find the proportion of their resistances; and the resistance of the quicksilver proved to be to the resistance of water as about 13 or 14 to 1; that is, as the density of quicksilver to the density of water. When I made use of a pendulous globe something bigger, as of one whose diameter was about ½ or ⅔ of an inch, the resistance of the quicksilver proved to be to the resistance of the water as about 12 or 10 to 1. But the former experiment is more to be relied on, because in the latter the vessel was too narrow in proportion to the magnitude of the immersed globe; for the vessel ought to have been enlarged together with the globe. I intended to have repeated these experiments with larger vessels, and in melted metals, and other liquors both cold and hot; but I had not leisure to try all: and besides, from what is already described, it appears sufficiently that the resistance of bodies moving swiftly is nearly proportional to the densities of the fluids in which they move. I do not say accurately; for more tenacious fluids, of equal density, will undoubtedly resist more than those that are more liquid; as cold oil more than warm, warm oil more than rain water, and water more than spirit of wine. But in liquors, which are sensibly fluid enough, as in air, in salt and fresh water, in spirit of wine, of turpentine, and salts, in oil cleared of its faeces by distillation and warmed, in oil of vitriol, and in mercury, and melted metals, and any other such like, that are fluid enough to retail for some time the motion impressed upon them by the agitation of the vessel, and which being poured out are easily resolved into drops, I doubt not but the rule already laid down may be accurate enough, especially if the experiments be made with larger pendulous bodies and more swiftly moved.

Lastly, since it is the opinion of some that there is a certain
aethereal medium extremely rare and subtile, which freely pervades the
pores of all bodies; and from such a medium, so pervading the pores of
bodies, some resistance must needs arise; in order to try whether the
resistance, which we experience in bodies in motion, be made upon
their outward superficies only, or whether their internal parts meet
with any considerable resistance upon their superficies, I thought of
the following experiment. I suspended a round deal box by a thread 11
feet long, on a steel hook, by means of a ring of the same metal, so
as to make a pendulum of the aforesaid length. The hook had a sharp
hollow edge on its upper part, so that the upper arc of the ring
pressing on the edge might move the more freely; and the thread was
fastened to the lower arc of the ring. The pendulum being thus
prepared, I drew it aside from the perpendicular to the distance of
about 6 feet, and that in a plane perpendicular to the edge of the
hook, lest the ring, while the pendulum oscillated, should slide to
and fro on the edge of the hook: for the point of suspension, in which
the ring touches the hook, ought to remain immovable. I therefore
accurately noted the place to which the pendulum was brought, and
letting it go, I marked three other places, to which it returned at
the end of the 1st, 2d, and 3d oscillation. This I often repeated,
that I might find those places as accurately as possible. Then I
filled the box with lead and other heavy metals that were near at
hand. But, first, I weighed the box when empty, and that part of the
thread that went round it, and half the remaining part, extended
between the hook and the suspended box; for the thread so extended
always acts upon the pendulum, when drawn aside from the
perpendicular, with half its weight. To this weight I added the weight
of the air contained in the box. And this whole weight was about
1

78 of the weight of the box when filled
with the metals. Then because the box when full of the metals, by
extending the thread with its weight, increased the length of the
pendulum, I shortened the thread so as to make the length of the
pendulum, when oscillating, the same as before. Then drawing aside the
pendulum to the place first marked, and letting it go, I reckoned
about 77 oscillations before the box returned to the second mark, and
as many afterwards before it came to the third mark, and as many after
that before it came to the fourth mark. From whence I conclude that
the whole resistance of the box, when full, had not a greater
proportion to the resistance of the box, when empty, than 78 to 77.
For if their resistances were equal, the box, when full, by reason of
its *vis insita*, which was 78 times greater than the *vis
insita* of the same when empty, ought to have continued its
oscillating motion so much the longer, and
therefore to have returned to those marks at the end of 78
oscillations. But it returned to them at the end of 77 oscillations.

Let, therefore, A represent the resistance of the box upon its external superficies, and B the resistance of the empty box on its internal superficies; and if the resistances to the internal parts of bodies equally swift be as the matter, or the number of particles that are resisted, then 78B will be the resistance made to the internal parts of the box, when full; and therefore the whole resistance A + B of the empty box will be to the whole resistance A + 78B of the full box as 77 to 78, and, by division, A + B to 77B as 77 to 1; and thence A + B to B as 77 × 77 to 1, and, by division again, A to B as 5928 to 1. Therefore the resistance of the empty box in its internal parts will be above 5000 times less than the resistance on its external superficies. This reasoning depends upon the supposition that the greater resistance of the full box arises not from any other latent cause, but only from the action of some subtile fluid upon the included metal.

This experiment is related by memory, the paper being lost in which I had described it; so that I have been obliged to omit some fractional parts, which are slipt out of my memory; and I have no leisure to try it again. The first time I made it, the hook being weak, the full box was retarded sooner. The cause I found to be, that the hook was not strong enough to bear the weight of the box; so that, as it oscillated to and fro, the hook was bent sometimes this and sometimes that way. I therefore procured a hook of sufficient strength, so that the point of suspension might remain unmoved, and then all things happened as is above described.

*Suppose two similar systems of bodies consisting of an equal
number of particles, and let the correspondent particles be
similar and proportional, each in one system to each in the other,
and have a like situation among themselves, and the same given
ratio of density to each other; and let them begin to move among
themselves in proportional times, and with like motions (that is,
those in one system among one another, and those in the other
among one another). And if the particles that are in the same
system do not touch one another, except it the moments of
reflexion; nor attract, nor repel each other, except with
accelerative forces that are as the diameters of the correspondent
particles inversely, and the squares of the velocities directly; I
say, that the particles of those systems will continue to move
among themselves with like motions and in proportional times.*

Like bodies in like situations are said to be moved among themselves
with like motions and in proportional times, when their situations at
the end of those times are always found alike in respect of each
other; as suppose we compare the particles in one system with the
correspondent particles in the other. Hence the times will be
proportional, in which similar and proportional parts of similar
figures will be described by correspondent particles. Therefore if we
suppose two systems of this kind, the correspondent particles, by
reason of the similitude of the motions at their beginning, will
continue to be moved with like motions, so long as they move without
meeting one another; for if they are acted on by no forces,they will
go on uniformly in right lines, by the 1st Law. But if they do agitate
one another with some certain forces, and those forces are as the
diameters of the correspondent particles inversely and the squares of
the velocities directly, then, because the particles are in like
situations, and their forces are proportional, the whole forces with
which correspondent particles are agitated, and which are compounded
of each of the agitating forces (by Corol. 2 of the Laws), will have
like directions, and have the same effect as if they respected centres
placed alike among the particles; and those whole forces will be to
each other as the several forces which compose them, that is, as the
diameters of the correspondent particles inversely, and the squares of
the velocities directly: and therefore will cause
correspondent particles to continue to describe like figures. These things will be
so (by Cor. 1 and 8, Prop. IV., Book 1), if those centres are at rest
but if they are moved, yet by reason of the similitude of the
translations, their situations among the particles of the system will
remain similar, so that the changes introduced into the figures
described by the particles will still be similar. So that the motions
of correspondent and similar particles will continue similar till
their first meeting with each other; and thence will arise similar
collisions, and similar reflexions; which will again beget similar
motions of the particles among themselves (by what was just now
shown), till they mutually fall upon one another again, and so on *ad infinitum*.

Cor. 1. Hence if any two bodies, which are similar and in like situations to the correspondent particles of the systems, begin to move amongst them in like manner and in proportional times, and their magnitudes and densities be to each other as the magnitudes and densities of the corresponding particles, these bodies will continue to be moved in like manner and in proportional times: for the case of the greater parts of both systems and of the particles is the very same.

Cor. 2. And if all the similar and similarly situated parts of both systems be at rest among themselves; and two of them, which are greater than the rest, and mutually correspondent in both systems, begin to move in lines alike posited, with any similar motion whatsoever, they will excite similar motions in the rest of the parts of the systems, and will continue to move among those parts in like manner and in proportional times; and will therefore describe spaces proportional to their diameters.

*The same things faring supposed, I say, that the greater parts
of the systems are resisted in a ratio compounded of the duplicate
ratio of their velocities, and the duplicate ratio of their
diameters, and the simple ratio of the density of the parts of the systems.*

For the resistance arises partly from the centripetal or centrifugal forces with which the particles of the system mutually act on each other, partly from the collisions and reflexions of the particles and the greater parts. The resistances of the first kind are to each other as the whole motive forces from which they arise, that is, as the whole accelerative forces and the quantities of matter in corresponding parts; that is (by the supposition), as the squares of the velocities directly, and the distances of the corresponding particles inversely, and the quantities of matter in the correspondent parts directly: and therefore since the distances of the particles in one system are to the correspondent distances of the particles of the other as the diameter of one particle or part in the former system to the diameter of the correspondent particle or part in the other, and since the quantities of matter are as the densities of the parts and the cubes of the diameters; the resistances are to each other as the squares of the velocities and the squares of the diameters and the densities of the parts of the systems. Q.E.D. The resistances of the latter sort are as the number of correspondent reflexions and the forces of those reflexions conjunctly; but the number of the reflexions are to each other as the velocities of the corresponding parts directly and the spaces between their reflexions inversely. And the forces of the reflexions are as the velocities and the magnitudes and the densities of the corresponding parts conjunctly; that is, as the velocities and the cubes of the diameters and the densities of the parts. And, joining all these ratios, the resistances of the corresponding parts are to each other as the squares of the velocities and the squares of the diameters and the densities of the parts conjunctly. Q.E.D.

Cor. 1. Therefore if those systems are two elastic fluids, like our air, and their parts are at rest among themselves; and two similar bodies proportional in magnitude and density to the parts of the fluids, and similarly situated among those parts, be any how projected in the direction of lines similarly posited; and the accelerative forces with which the particles of the fluids mutually act upon each other are as the diameters of the bodies projected inversely and the squares of their velocities directly; those bodies will excite similar motions in the fluids in proportional times, and will describe similar spaces and proportional to their diameters.

Cor. 2. Therefore in the same fluid a projected body that moves swiftly meets with a resistance that is, in the duplicate ratio of its velocity, nearly. For if the forces with which distant particles act mutually upon one another should be augmented in the duplicate ratio of the velocity, the projected body would be resisted in the same duplicate ratio accurately; and therefore in a medium, whose parts when at a distance do not act mutually with any force on one another, the resistance is in the duplicate ratio of the velocity accurately. Let there be, therefore, three mediums A, B, C, consisting of similar and equal parts regularly disposed at equal distances. Let the parts of the mediums A and B recede from each other with forces that are among themselves as T and V; and let the parts of the medium C be entirely destitute of any such forces. And if four equal bodies D, E, F, G, move in these mediums, the two first D and E in the two first A and B, and the other two F and G in the third C; and if the velocity of the body D be to the velocity of the body E, and the velocity of the body F to the velocity of the body G, in the subduplicate ratio of the force T to the force V; the resistance of the body D to the resistance of the body E, and the resistance of the body F to the resistance of the body G, will be in the duplicate ratio of the velocities; and therefore the resistance of the body D will be to the resistance of the body F as the resistance of the body E to the resistance of the body G. Let the bodies D and F be equally swift, as also the bodies E and G; and, augmenting the velocities of the bodies D and F in any ratio, and diminishing the forces of the particles of the medium B in the duplicate of the same ratio, the medium B will approach to the form and condition of the medium C at pleasure; and therefore the resistances of the equal and equally swift bodies E and G in these mediums will perpetually approach to equality so that their difference will at last become less than any given. Therefore since the resistances of the bodies D and F are to each other as the resistances of the bodies E and G, those will also in like manner approach to the ratio of equality. Therefore the bodies D and F, when they move with very great swiftness, meet with resistances very nearly equal; and therefore since the resistance of the body F is in a duplicate ratio of the velocity, the resistance of the body D will be nearly in the same ratio.

Cor. 3. The resistance of a body moving very swift in an elastic fluid is almost the same as if the parts of the fluid were destitute of their centrifugal forces, and did not fly from each other; if so be that the elasticity of the fluid arise from the centrifugal forces of the particles, and the velocity be so great as not to allow the particles time enough to act.

Cor. 4. Therefore, since the resistances of similar and equally swift bodies, in a medium whose distant parts do not fly from each other, are as the squares of the diameters, the resistances made to bodies moving with very great and equal velocities in an elastic fluid will be as the squares of the diameters, nearly.

Cor. 5. And since similar, equal, and equally swift bodies, moving through mediums of the same density, whose particles do not fly from each other mutually, will strike against an equal quantity of matter in equal times, whether the particles of which the medium consists be more and smaller, or fewer and greater, and therefore impress on that matter an equal quantity of motion, and in return (by the 3d Law of Motion) suffer an equal re-action from the same, that is, are equally resisted; it is manifest, also, that in elastic fluids of the same density, when the bodies move with extreme swiftness, their resistances are nearly equal, whether the fluids consist of gross parts, or of parts ever so subtile. For the resistance of projectiles moving with exceedingly great celerities is not much diminished by the subtilty of the medium.

Cor. 6. All these things are so in fluids whose elastic force takes its rise from the centrifugal forces of the particles. But if that force arise from some other cause, as from the expansion of the particles after the manner of wool, or the boughs of trees, or any other cause, by which the particles are hindered from moving freely among themselves, the resistance, by reason of the lesser fluidity of the medium, will be greater than in the Corollaries above.

*If in a rare medium, consisting of equal particles freely
disposed at equal distances front each other, a globe and a
cylinder described on equal diameters move with equal velocities
in the direction of the axis of the cylinder, the resistance of
the globe will be but half so great as that of the cylinder.*

For since the action of the medium upon the body is the same (by Cor.
5 of the Laws) whether the body move in a quiescent medium, or whether
the particles of the medium impinge with the same velocity upon the
quiescent body, let us consider the body as if it were quiescent, and
see with what force it would be impelled by the moving medium. Let,
therefore, ABKI represent a spherical body described from the centre O
with the semi-diameter CA, and let the particles of the medium impinge
with a given velocity upon that spherical body in the directions of
right lines parallel to AC; and let FB be one of those right lines. In
FB take LB equal to the semi-diameter CB, and draw BD touching the
sphere in B. Upon KC and BD let fall the perpendiculars BE, LD; and
the force with which a particle of the medium, impinging on the globe
obliquely in the direction FB, would strike the globe in B, will be to
the force with which the same particle, meeting the cylinder ONGQ,
described about the globe with the axis ACI, would strike it
perpendicularly in *b*, as LD to LB, or BE to BC. Again; the
efficacy of this force to move the globe, according to the direction
of its incidence FB or AC, is to the efficacy of the same to move the
globe, according to the direction of its determination, that is, in
the direction of the right line BC in which it impels the globe
directly, as BE to BC. And, joining these ratios, the efficacy of a
particle, falling upon the globe obliquely in the direction of the
right line FB to move the globe in the direction of its incidence, is
to the efficacy of the same particle falling in the same line
perpendicularly on the cylinder, to move it in the same direction, as
BE² to BC². Therefore if in *b*E, which is perpendicular to
the circular base of the cylinder NAO, and equal to the radius AC, we
take *b*H equal to BE^{2}

CB; then *b*H will be to *b*E
as the effect of the particle upon the globe to the effect of the
particle upon the cylinder. And therefore the solid which is formed by
all the right lines *b*H will be to the solid formed by all
the right lines *b*E as the effect of all the particles upon
the globe to the effect of all the particles upon the cylinder. But
the former of these solids is a paraboloid
whose vertex is C, its axis CA, and latus rectum CA, and the latter
solid is a cylinder circumscribing the paraboloid; and it is known
that a paraboloid is half its circumscribed cylinder. Therefore the
whole force of the medium upon the globe is half of the entire force
of the same upon the cylinder. And therefore if the particles of the
medium are at rest, and the cylinder and globe move with equal
velocities, the resistance of the globe will be half the resistance of
the cylinder. Q.E.D.

By the same method other figures may be compared together as to their resistance; and those may be found which are most apt to continue their motions in resisting mediums. As if upon the circular base CEBH from the centre O, with the radius OC, and the altitude OD, one would construct a frustum CBGF of a cone, which should meet with less resistance than any other frustum constructed with the same base and altitude, and going forwards towards D in the direction of its axis: bisect the altitude OD in Q, and produce OQ to S, so that QS may be equal to QC, and S will be the vertex of the cone whose frustum is sought.

Whence, by the bye, since the angle CSB is always acute, it follows, that, if the solid ADBE be generated by the convolution of an elliptical or oval figure ADBE about its axis AB, and the generating figure be touched by three right lines FG, GH, HI, in the points P, B, and I, so that GH shall be perpendicular to the axis in the point of contact B, and FG, HI may be inclined to GH in the angles FGB, BHI of 135 degrees: the solid arising from the convolution of the figure ADFGHIE about the same axis AB will be less resisted than the former solid; if so be that both move forward in the direction of their axis AB, and that the extremity B of each go foremost. Which Proposition I conceive may be of use in the building of ships.

If the figure DNFG be such a curve, that if, from any point thereof, as N, the perpendicular NM be let fall on the axis AB, and from the given point G there be drawn the right line GR parallel to a right line touching the figure in N, and cutting the axis produced in R, MN becomes to GR as GR³ to 4BR × GB²; the solid described by the revolution of tins figure about its axis AB, moving in the before-mentioned rare medium from A towards B, will be less resisted than any other circular solid whatsoever, described of the same length and breadth.

*If a rare medium consist of very small quiescent particles of
equal magnitudes, and freely disposed at equal distances from one
another: to find the resistance of a globe moving uniformly
forward in this medium.*

Case 1. Let a cylinder described with the same diameter and altitude be conceived to go forward with the same velocity in the direction of its axis through the same medium; and let us suppose that the particles of the medium, on which the globe or cylinder falls, fly back with as great a force of reflexion as possible. Then since the resistance of the globe (by the last Proposition) is but half the resistance of the cylinder, and since the globe is to the cylinder as 2 to 3, and since the cylinder by falling perpendicularly on the particles, and reflecting them with the utmost force, communicates to them a velocity double to its own; it follows that the cylinder, in moving forward uniformly half the length of its axis, will communicate a motion to the particles which is to the whole motion of the cylinder as the density of the medium to the density of the cylinder; and that the globe, in the time it describes one length of its diameter in moving uniformly forward, will communicate the same motion to the particles; and in the time that it describes two thirds of its diameter, will communicate a motion to the particles which is to the whole motion of the globe as the density of the medium to the density of the globe. And therefore the globe meets with a resistance, which is to the force by which its whole motion may be either taken away or generated in the time in which it describes two thirds of its diameter moving uniformly forward, as the density of the medium to the density of the globe.

Case 2. Let us suppose that the particles of the medium incident on the globe or cylinder are not reflected; and then the cylinder falling perpendicularly on the particles will communicate its own simple velocity to them, and therefore meets a resistance but half so great as in the former case, and the globe also meets with a resistance but half so great.

Case 3. Let us suppose the particles of the medium to fly back from the globe with a force which is neither the greatest, nor yet none at all, but with a certain mean force; then the resistance of the globe will be in the same mean ratio between the resistance in the first case and the resistance in the second. Q.E.I.

Cor. 1. Hence if the globe and the particles are infinitely hard, and destitute of all elastic force, and therefore of all force of reflexion; the resistance of the globe will be to the force by which its whole motion may be destroyed or generated, in the time that the globe describes four third parts of its diameter, as the density of the medium to the density of the globe.

Cor. 2. The resistance of the globe, *caeteris
paribus*, is in the duplicate ratio of the velocity.

Cor. 3. The resistance of the globe, *caeteris
paribus*, is in the duplicate ratio of the diameter.

Cor. 4. The resistance of the globe is, *caeteris
paribus*, as the density of the medium.

Cor. 5. The resistance of the globe is in a ratio compounded of the duplicate ratio of the velocity, and the duplicate ratio of the diameter, and the ratio of the density of the medium.

Cor. 6. The motion of the globe and its resistance may be thus expounded. Let AB be the time in which the globe may, by its resistance uniformly continued, lose its whole motion. Erect AD, BC perpendicular to AB. Let BC be that whole motion, and through the point C, the asymptotes being AD, AB, describe the hyperbola CF. Produce AB to any point E. Erect the perpendicular EF meeting the hyperbola in F. Complete the parallelogram CBEG, and draw AF meeting BC in H. Then if the globe in any time BE, with its first motion BC uniformly continued, describes in a non-resisting medium the space CBEG expounded by the area of the parallelogram, the same in a resisting medium will describe the space CBEF expounded by the area of the hyperbola; and its motion at the end of that time will be expounded by EF, the ordinate of the hyperbola, there being lost of its motion the part FG. And its resistance at the end of the same time will be expounded by the length BH, there being lost of its resistance the part CH. All these things appear by Cor. 1 and 3, Prop. V., Book II.

Cor. 7. Hence if the globe in the time T by
the resistance R uniformly continued lose its whole motion M, the same
globe in the time *t* in a resisting medium, wherein the
resistance R decreases in a duplicate ratio of the velocity, will lose
out of its motion M the part tM

T+t, the part TM

T+t remaining; and will describe a
space which is to the space described in the same time *t*,
with the uniform motion M, as the logarithm of the number T+t

T multiplied by the number
2,302585092994 is to the number t

T, because the hyperbolic area BCFE is
to the rectangle BCGE in that proportion.

I have exhibited in this Proposition the resistance and retardation of spherical projectiles in mediums that are not continued, and shewn that this resistance is to the force by which the whole motion of the globe may be destroyed or produced in the time in which the globe can describe two thirds of its diameter; with a velocity uniformly continued, as the density of the medium to the density of the globe, if so be the globe and the particles of the medium be perfectly elastic, and are endued with the utmost force of reflexion; and that this force, where the globe and particles of the medium are infinitely hard and void of any reflecting force, is diminished one half. But in continued mediums, as water, hot oil, and quicksilver, the globe as it passes through them does not immediately strike against all the particles of the fluid that generate the resistance made to it, but presses only the particles that lie next to it, which press the particles beyond, which press other particles, and so on; and in these mediums the resistance is diminished one other half. A globe in these extremely fluid mediums meets with a resistance that is to the force by which its whole motion may be destroyed or generated in the time wherein it can describe, with that motion uniformly continued, eight third parts of its diameter, as the density of the medium to the density of the globe. This I shall endeavour to shew in what follows.

*To define the motion of water running out of a cylindrical
vessel through a hole made at the bottom.*

Let ACDB be a cylindrical vessel, AB the mouth of it, CD the bottom
parallel to the horizon, EF a circular hole in the middle of the
bottom, G the centre of the hole, and GH the axis of the cylinder
perpendicular to the horizon. And suppose a cylinder of ice APQB to be
of the same breadth with the cavity of the vessel, and to have the
same axis, and to descend perpetually with an uniform motion, and that
its parts, as soon as they touch the superficies AB, dissolve into
water, and flow down by their weight into the vessel, and in their
fall compose the cataract or column of water ABNFEM, passing through
the hole EF, and filling up the same exactly. Let the uniform velocity
of the descending ice and of the contiguous water in the circle AB be
that which the water would acquire by falling through the space IH;
and let IH and HG lie in the same right line; and through
the point I let there be drawn the right line
KL parallel to the horizon and meeting the ice on both the sides thereof
in K and L. Then the velocity of the water running out at the hole EF will be
the same that it would acquire by falling from I through the space IG.
Therefore, by *Galileo's* Theorems, IG will be to IH in the duplicate ratio
of the velocity of the water that runs out at the hole to the velocity
of the water in the circle AB, that is, in the duplicate ratio of the
circle AB to the circle EF; those circles being reciprocally as the
velocities of the water which in the same time and in equal quantities
passes severally through each of them, and completely fills them both.
We are now considering the velocity with which the water tends to the
plane of the horizon. But the motion parallel to the same, by which
the parts of the falling water approach to each other, is not here
taken notice of; since it is neither produced by gravity, nor at all
changes the motion perpendicular to the horizon which the gravity
produces. We suppose, indeed, that the parts of the water cohere a
little, that by their cohesion they may in falling approach to each
other with motions parallel to the horizon in order to form one single
cataract, and to prevent their being divided into several: but the
motion parallel to the horizon arising from this cohesion does not
come under our present consideration.

Case 1. Conceive now the whole cavity in the vessel, which encompasses the falling water ABNFEM, to be full of ice, so that the water may pass through the ice as through a funnel. Then if the water pass very near to the ice only, without touching it; or, which is the same thing, if by reason of the perfect smoothness of the surface of the ice, the water, though touching it, glides over it with the utmost freedom, and without the least resistance; the water will run through the hole EF with the same velocity as before, and the whole weight of the column of water ABNFEM will be all taken up as before in forcing out the water, and the bottom of the vessel will sustain the weight of the ice encompassing that column.

Let now the ice in the vessel dissolve into water; yet will the efflux of the water remain, as to its velocity, the same as before. It will not be less, because the ice now dissolved will endeavour to descend; it will not be greater, because the ice, now become water, cannot descend without hindering the descent of other water equal to its own descent. The same force ought always to generate the same velocity in the effluent water.

But the hole at the bottom of the vessel, by reason of the oblique
motions of the particles of the effluent water, must be a little
greater than before. For now the particles of the water do not all of
them pass through the hole perpendicularly, but, flowing down on all
parts from the sides of the vessel, and converging towards the hole,
pass through it with oblique motions; and in tending downwards meet in
a stream whose diameter is a little smaller below the hole than at the
hole itself; its diameter being to the diameter
of the hole as 5 to 6, or as 5½ to 6½, very nearly, if I took the
measures of those diameters right. I procured a very thin flat plate,
having a hole pierced in the middle, the diameter of the circular hole
being 5

8 parts of an inch. And that the stream
of running waters might not be accelerated in falling, and by that
acceleration become narrower, I fixed this plate not to the bottom,
but to the side of the vessel, so as to make the water go out in the
direction of a line parallel to the horizon. Then, when the vessel was
full of water, I opened the hole to let it run out; and the diameter
of the stream, measured with great accuracy at the distance of about
half an inch from the hole, was 21

40 of an inch. Therefore the diameter
of this circular hole was to the diameter of the stream very nearly as
25 to 21. So that the water in passing through the hole converges on
all sides, and, after it has run out of the vessel, becomes smaller by
converging in that manner, and by becoming smaller is accelerated till
it comes to the distance of half an inch from the hole, and at that
distance flows in a smaller stream and with greater celerity than in
the hole itself, and this in the ratio of 25 × 25 to 21 × 21, or 17 to
12, very nearly; that is, in about the subduplicate ratio of 2 to 1.
Now it is certain from experiments, that the quantity of water running
out in a given time through a circular hole made in the bottom of a
vessel is equal to the quantity, which, flowing with the aforesaid
velocity, would run out in the same time through another circular
hole, whose diameter is to the diameter of the former as 21 to 25. And
therefore that running water in passing through the hole itself has a
velocity downwards equal to that which a heavy body would acquire in
falling through half the height of the stagnant water in the vessel,
nearly. But, then, after it has run out, it is still accelerated by
converging, till it arrives at a distance from the hole that is nearly
equal to its diameter, and acquires a velocity greater than the other
in about the subduplicate ratio of 2 to 1; which velocity a heavy body
would nearly acquire by falling through the whole height of the
stagnant water in the vessel.

Therefore in what follows let the diameter of the stream be represented by that lesser hole which we called EF. And imagine another plane VW above the hole EF, and parallel to the plane there of, to be placed at a distance equal to the diameter of the same hole, and to be pierced through with a greater hole ST, of such a magnitude that a stream which will exactly fill the lower hole EF may pass through it; the diameter of which hole will therefore be to the diameter of the lower hole as 25 to 21, nearly. By this means the water will run perpendicularly out at the lower hole; and the quantity of the water running out will be, according to the magnitude of this last hole, the same, very nearly, which the solution of the Problem requires. The space included between the two planes and the falling stream may be considered as the bottom of the vessel. But, to make the solution more simple and mathematical, it is better to take the lower plane alone for the bottom of the vessel, and to suppose that the water which flowed through the ice as through a funnel, and ran out of the vessel through the hole EF made in the lower plane, preserves its motion continually, and that the ice continues at rest. Therefore in what follows let ST be the diamter of a circular hole described from the centre Z, and let the stream run out of the vessel through that hole, when the water in the vessel is all fluid. And let EF be the diameter of the hole, which the stream, in falling through, exactly fills up, whether the water runs out of the vessel by that upper hole ST, or flows through the middle of the ice in the vessel, as through a funnel. And let the diameter of the upper hole ST be to the diameter of the lower EF as about 25 to 21, and let the perpendicular distance between the planes of the holes be equal to the diameter of the lesser hole EF. Then the velocity of the water downwards, in running out of the vessel through the hole ST, will be in that hole the same that a body may acquire by falling from half the height IZ; and the velocity of both the falling streams will be in the hole EF, the same which a body would acquire by falling from the whole height IG.

Case 2. If the hole EF be not in the middle
of the bottom of the vessel, but in some other part thereof, the water
will still run out with the same velocity as before, if the magnitude
of the hole be the same. For though an heavy body takes a longer time
in descending to the same depth, by an oblique line, than by a
perpendicular line, yet in both cases it acquires in its descent the
same velocity; as *Galileo* has demonstrated.

Case 3. The velocity of the water is the same when it runs out through a hole in the side of the vessel. For if the hole be small, so that the interval between the superficies AB and KL may vanish as to sense, and the stream of water horizontally issuing out may form a parabolic figure: from the latus rectum of this parabola may be collected, that the velocity of the effluent water is that which a body may acquire by falling the height IG or HG of the stagnant water in the vessel. For, by making an experiment, I found that if the height of the stagnant water above the hole were 20 inches, and the height of the hole above a plane parallel to the horizon were also 20 inches, a stream of water springing out from thence would fall upon the plane, at the distance of 37 inches, very nearly, from a perpendicular let fall upon that plane from the hole. For without resistance the stream would have fallen upon the plane at the distance of 40 inches, the latus rectum of the parabolic stream being 80 inches.

Case 4. If the effluent water tend upward, it will still issue forth with the same velocity. For the small stream of water springing upward; ascends with a perpendicular motion to GH or GI, the height of the stagnant water in the vessel; excepting in so far as its ascent is hindered a little by the resistance of the air; and therefore it springs out with the same velocity that it would acquire in falling from that height. Every particle of the stagnant water is equally pressed on all sides (by Prop. XIX., Book II), and, yielding to the pressure, tends always with an equal force, whether it descends through the hole in the bottom of the vessel, or gushes out in an horizontal direction through a hole in the side, or passes into a canal, and springs up from thence through a little hole made in the upper part of the canal. And it may not only be collected from reasoning, but is manifest also from the well-known experiments just mentioned, that the velocity with which the water runs out is the very same that is assigned in this Proposition.

Case 5. The velocity of the effluent water is the same, whether the figure of the hole be circular, or square, or triangular, or any other figure equal to the circular; for the velocity of the effluent water does not depend upon the figure of the hole, but arises from its depth below the plane KL.

Case 6. If the lower part of the vessel ABDC be immersed into stagnant water, and the height of the stagnant water above the bottom of the vessel be GR, the velocity with which the water that is in the vessel will run out at the hole EF into the stagnant water will be the same which the water would acquire by falling from the height IR; for the weight of all the water in the vessel that is below the superficies of the stagnant water will be sustained in equilibrio by the weight of the stagnant water, and therefore does riot at all accelerate the motion of the descending water in the vessel. This case will also appear by experiments, measuring the times in which the water will run out.

Cor. 1. Hence if CA the depth of the water be produced to K, so that AK may be to CK in the duplicate ratio of the area of a hole made in any part of the bottom to the area of the circle AB, the velocity of the effluent water will be equal to the velocity which the water would acquire by falling from the height KC.

Cor. 2. And the force with which the whole motion of the effluent water may be generated is equal to the weight of a cylindric column of water, whose base is the hole EF, and its altitude 2GI or 2CK. For the effluent water, in the time it becomes equal to this column, may acquire, by falling by its own weight from the height GI, a velocity equal to that with which it runs out.

Cor. 3. The weight of all the water in the vessel ABDC is to that part of the weight which is employed in forcing out the water as the sum of the circles AB and EF to twice the circle EF. For let IO be a mean proportional between IH and IG, and the water running out at the hole EF will, in the time that a drop falling from I would describe the altitude IG, become equal to a cylinder whose base is the circle EF and its altitude 2IG, that is, to a cylinder whose base is the circle AB, and whose altitude is 2IO. For the circle EF is to the circle AB in the subduplicate ratio of the altitude IH to the altitude IG; that is, in the simple ratio of the mean proportional IO to the altitude IG. Moreover, in the time that a drop falling from I can describe the altitude IH, the water that runs out will hare become equal to a cylinder whose base is the circle AB, and its altitude 2IH; and in the time that a drop falling from I through H to G describes HG, the difference of the altitudes, the effluent water, that is, the water contained within the solid ABNFEM, will be equal to the difference of the cylinders, that is, to a cylinder whose base is AB, and its altitude 2HO. And therefore all the water contained in the vessel ABDC is to the whole falling water contained in the said solid ABNFEM as HG to 2HO, that is, as HO + OG to 2HO, or IH + IO to 2IH. But the weight of all the water in the solid ABNFEM is employed in forcing out the water: and therefore the weight of all the water in the vessel is to that part of the weight that is employed in forcing out the water as IH + IO to 2IH, and therefore as the sum of the circles EF and AB to twice the circle EF.

Cor. 4. And hence the weight of all the water in the vessel ABDC is to the other part of the weight which is sustained by the bottom of the vessel as the sum of the circles AB and EF to the difference of the same circles.

Cor. 5. And that part of the weight which the bottom of the vessel sustains is to the other part of the weight employed in forcing out the water as the difference of the circles AB and EF to twice the lesser circle EF, or as the area of the bottom to twice the hole.

Cor. 6. That part of the weight which presses
upon the bottom is to the whole weight of the water perpendicularly
incumbent thereon as the circle AB to the sum of the circles AB and
EF, or as the circle AB to the excess of twice the circle AB above the
area of the bottom. For that part of the weight which presses upon the
bottom is to the weight of the whole water in the vessel as the
difference of the circles AB and EF to the sum of the same circles (by
Cor. 4); and the weight of the whole water in the vessel is to the
weight of the whole water perpendicularly incumbent on the bottom as
the circle AB to the difference of the circles AB and EF. Therefore, *ex
aequo perturbatè*, that part of the weight which presses upon
the bottom is to the weight of the whole water perpendicularly
incumbent thereon as the circle AB to the sum
of the circles AB and EF, or the excess of twice the circle AB above
the bottom.

Cor. 7. If in the middle of the hole EF there be placed the little circle PQ described about the centre G, and parallel to the horizon, the weight of water which that little circle sustains is greater than the weight of a third part of a cylinder of water whose base is that little circle and its height GH. For let ABNFEM be the cataract or column of falling water whose axis is GH, as above, and let all the water, whose fluidity is not requisite for the ready and quick descent of the water, be supposed to A be congealed, as well round about the cataract, as above the little circle. And let PHQ be the column of water congealed above the little circle, whose vertex is H, and its altitude GH. And suppose this cataract to fall with its whole weight downwards, and not in the least to lie against or to press PHQ, but to glide freely by it without any friction, unless, perhaps, just at the very vertex of the ice, where the cataract at the beginning of its fall may tend to a concave figure. And as the congealed water AMEC, BNFD, lying round the cataract, is convex in its internal superficies AME, BNF, towards the falling cataract, so this column PHQ will be convex towards the cataract also, and will therefore be greater than a cone whose base is that little circle PQ and its altitude GH; that is, greater than a third part of a cylinder described with the same base and altitude. Now that little circle sustains the weight of this column, that is, a weight greater than the weight of the cone, or a third part of the cylinder.

Cor. 8. The weight of water which the circle
PQ, when very small, sustains, seems to be less than the weight of two
thirds of a cylinder of water whose base is that little circle, and
its altitude HG. For, things standing as above supposed, imagine the
half of a spheroid described whose base is that little circle, and its
semi-axis or altitude HG. This figure will be equal to two thirds of
that cylinder, and will comprehend within it the column of congealed
water PHQ, the weight of which is sustained by that little circle. For
though the motion of the water tends directly downwards, the external
superficies of that column must yet meet the base PQ in an angle
somewhat acute, because the water in its fall is perpetually
accelerated, and by reason of that acceleration become narrower.
Therefore, since that angle is less than a right one, this column in
the lower parts thereof will lie within the hemi-spheroid. In the
upper parts also it will be acute or pointed; because to make it
otherwise, the horizontal motion of the water must be at the vertex
infinitely more swift than its motion towards the horizon. And the
less this circle PQ is, the more acute will the
vertex of this column be; and the circle being diminished *in
infinitum* the angle PHQ will be diminished *in infinitum*,
and therefore the column will lie within the hemi-spheroid. Therefore
that column is less than that hemi-spheroid, or than two-third parts
of the cylinder whose base is that little circle, and its altitude GH.
Now the little circle sustains a force of water equal to the weight of
this column, the weight of the ambient water being employed in causing
its efflux out at the hole.

Cor. 9. The weight of water which the little circle PQ sustains, when it is very small, is very nearly equal to the weight of a cylinder of water whose base is that little circle, and its altitude ½GH; for this weight is an arithmetical mean between the weights of the cone and the hemi-spheroid above mentioned. But if that little circle be not very small, but on the contrary increased till it be equal to the hole EF, it will sustain the weight of all the water lying perpendicularly above it, that is, the weight of a cylinder of water whose base is that little circle, and its altitude GH.

Cor. 10. And (as far as I can judge) the weight which this little circle sustains is always to the weight of a cylinder of water whose base is that little circle, and its altitude ½GH, as EF² to EF² − ½PQ², or as the circle EF to the excess of this circle above half the little circle PQ, very nearly.

*If a cylinder move uniformly forward in the direction of its
length, the resistance made thereto is not at all changed by
augmenting or diminishing that length; and is therefore the same
with the resistance of a circle, described with the same diameter,
and moving forward with the same velocity in the direction, of a
right line perpendicular to its plane.*

For the sides are not at all opposed to the motion; and a cylinder
becomes a circle when its length is diminished *in infinitum*.

*If a cylinder move uninformly forward in a compressed,
infinite, and non-elastic fluid, in the direction of its length,
the resistance arising from the magnitude of its transverse
section is to the force by which its whole motion may be destroyed
or generated, in the time that it moves four times its length, as
the density of the medium to the density of the cylinder, nearly.*

For let the vessel ABDC touch the surface of stagnant water with its bottom CD, and let the water run out of this vessel into the stagnant water through the cylindric canal EFTS perpendicular co the horizon; and let the little circle PQ be placed parallel to the horizon any where in the middle of the canal; and produce CA to K, so that AK may be to CK in the duplicate of the ratio, which the excess of the orifice of the canal EF above the little circle PQ bears to the circle AB. Then it is manifest (by Case 5, Case 6, and Cor. 1, Prop. XXXVI) that the velocity of the water passing through the annular space between the little circle and the sides of the vessel will be the very same which the water would acquire by falling, and in its fall describing the altitude KC or IG.

And (by Cor. 10, Prop. XXXVI) if the breadth of the vessel be infinite, so that the lineola HI may vanish, and the altitudes IG, HG become equal; the force of the water that flows down and presses upon the circle will be to the weight of a cylinder whose base is that little circle, and the altitude ½IG, as EF² to EF² − ½PQ², very nearly. For the force of the water flowing downward uniformly through the whole canal will be the same upon the little circle PQ in whatsoever part of the canal it be placed.

Let now the orifices of the canal EF, ST be closed, and let the little circle ascend in the fluid compressed on every side, and by its ascent let it oblige the water that lies above it to descend through the annular space between the little circle and the sides of the canal. Then will the velocity of the ascending little circle be to the velocity of the descending water as the difference of the circles EF and PQ, is to the circle PQ; and the velocity of the ascending little circle will be to the sum of the velocities, that is, to the relative velocity of the descending water with which it passes by the little circle in its ascent, as the difference of the circles EF and PQ to the circle EF, or as EF² − PQ² to EF². Let that relative velocity be equal to the velocity with which it was shewn above that the water would pass through the annular space, if the circle were to remain unmoved, that is, to the velocity which the water would acquire by falling, and in its fall describing the altitude IG; and the force of the water upon the ascending circle will be the same as before (by Cor. 5, of the Laws of Motion); that is, the resistance of the ascending little circle will be to the weight of a cylinder of water whose base is that little circle, and its altitude ½IG, as EF² to EF² − ½PQ², nearly. But the velocity of the little circle will be to the velocity which the water acquires by falling, and in its fall describing the altitude IG, as EF² − PQ² to EF² .

Let the breadth of the canal be increased *in infinitum*; and
the ratios between EF² − PQ² and EF², and between EF² and EF² − ½PQ²,
will become at last ratios of equality. And therefore the velocity of
the little circle will now be the same which the water would acquire
in falling, and in its fall describing the altitude IG: and the
resistance will become equal to the weight of
a cylinder whose base is that little circle, and its altitude half the
altitude IG, from which the cylinder must fall to acquire the velocity
of the ascending circle; and with this velocity the cylinder in the
time of its fall will describe four times its length. But the
resistance of the cylinder moving forward with this velocity in the
direction of its length is the same with the resistance of the little
circle (by Lem. IV), and is therefore nearly equal to the force by
which its motion may be generated while it describes four times its
length.

If the length of the cylinder be augmented or diminished, its motion, and the time in which it describes four times its length, will be augmented or diminished in the same ratio, and therefore the force by which the motion so increased or diminished, may be destroyed or generated, will continue the same; because the time is increased or diminished in the same proportion; and therefore that force remains still equal to the resistance of the cylinder, because (by Lem. IV) that resistance will also remain the same.

If the density of the cylinder be augmented or diminished, its motion, and the force by which its motion may be generated or destroyed in the same time, will be augmented or diminished in the same ratio. Therefore the resistance of any cylinder whatsoever will be to the force by which its whole motion may be generated or destroyed, in the time during which it moves four times its length, as the density of the medium to the density of the cylinder, nearly. Q.E.D.

A fluid must be compressed to become continued; it must be continued and non-elastic, that all the pressure arising from its compression may be propagated in an instant; and so, acting equally upon all parts of the body moved, may produce no change of the resistance. The pressure arising from the motion of the body is spent in generating a motion in the parts of the fluid, and this creates the resistance. But the pressure arising from the compression of the fluid, be it ever so forcible, if it be propagated in an instant, generates no motion in the parts of a continued fluid, produces no change at all of motion therein; and therefore neither augments nor lessens the resistance. This is certain, that the action of the fluid arising from the compression cannot be stronger on the hinder parts of the body moved than on its fore parts, and therefore cannot lessen the resistance described in this proposition. And if its propagation be infinitely swifter than the motion of the body pressed, it will not be stronger on the fore parts than on the hinder parts. But that action will be infinitely swifter, and propagated in an instant, if the fluid be continued and non-elastic.

Cor. 1. The resistances, made to cylinders going uniformly forward in the direction of their lengths through continued infinite mediums are in a ratio compounded of the duplicate ratio of the velocities and the duplicate ratio of the diameters, and the ratio of the density of the mediums.

Cor. 2. If the breadth of the canal be not infinitely increased but the cylinder go forward in the direction of its length through an included quiescent medium, its axis all the while coinciding with the axis of the canal, its resistance will be to the force by which its whole motion, in the time in which it describes four times its length, may be generated or destroyed, in a ratio compounded of the ratio of EF² to EF² − ½PQ² once, and the ratio of EF² to EF² − PQ² twice, and the ratio of the density of the medium to the density of the cylinder.

Cor. 3. The same thing supposed, and that a length L is to the quadruple of the length of the cylinder in a ratio compounded of the ratio EF² − ½PQ² to EF² once, and the ratio of EF² − PQ² to EF² twice; the resistance of the cylinder will be to the force by which its whole motion, in the time during which it describes the length L, may be destroyed or generated, as the density of the medium to the density of the cylinder.

In this proposition we have investigated that resistance alone which arises from the magnitude of the transverse section of the cylinder, neglecting that part of the same which may arise from the obliquity of the motions. For as, in Case 1, of Prop. XXXVI., the obliquity of the motions with which the parts of the water in the vessel converged on every side to the hole EF hindered the efflux of the water through the hole, so, in this Proposition, the obliquity of the motions, with which the parts of the water, pressed by the antecedent extremity of the cylinder, yield to the pressure, and diverge on all sides, retards their passage through the places that lie round that antecedent extremity, toward the hinder parts of the cylinder, and causes the fluid to be moved to a greater distance; which increases the resistance, and that in the same ratio almost in which it diminished the efflux of the water out of the vessel, that is, in the duplicate ratio of 25 to 21, nearly. And as, in Case 1, of that Proposition, we made the parts of the water pass through the hole EF perpendicularly and in the greatest plenty, by supposing all the water in the vessel lying round the cataract to be frozen, and that part of the water whose motion was oblique, and useless to remain without motion, so in this Proposition, that the obliquity of the motions may be taken away, and the parts of the water may give the freest passage to the cylinder, by yielding to it with the most direct and quick motion possible, so that only so much resistance may remain as arises from the magnitude of the transverse section, and which is incapable of diminution, unless by diminishing the diameter of the cylinder; we must conceive those parts of the fluid whose motions are oblique and useless, and produce resistance, to be at rest among themselves at both extremities of the cylinder, and there to cohere, and be joined to the cylinder. Let ABCD be a rectangle, and let AE and BE be two parabolic arcs, described with the axis AB, and with a latus rectum that is to the space HG, which must be described by the cylinder in falling, in order to acquire the velocity with which it moves, as HG to ½AB. Let CF and DF be two other parabolic arcs described with the axis CD, and a latus rectum quadruple of the former; and by the convolution of the figure about the axis EF let there be generated a solid, whose middle part ABDC is the cylinder we are here speaking of, and whose extreme parts ABE and CDF contain the parts of the fluid at rest among themselves, and concreted into two hard bodies, adhering to the cylinder at each end like a head and tail. Then if this solid EACFDB move in the direction of the length of its axis FE toward the parts beyond E, the resistance will be the same which we have here determined in this Proposition, nearly; that is, it will have the same ratio to the force with which the whole motion of the cylinder may be destroyed or generated, in the time that it is describing the length 4AC with that motion uniformly continued, as the density of the fluid has to the density of the cylinder, nearly. And (by Cor. 7, Prop. XXXVI) the resistance must be to this force in the ratio of 2 to 3, at the least.

*If a cylinder, a sphere, and a spheroid, of equal breadths be
placed successively in the middle of a cylindric canal, so that
their axes may coincide with the axis of the canal, these bodies
will equally hinder the passage of the water through the canal.*

For the spaces lying between the sides of the canal, and the cylinder, sphere, and spheroid, through which the water passes, are equal; and the water will pass equally through equal spaces.

This is true, upon the supposition that all the water above the cylinder, sphere, or spheroid, whose fluidity is not necessary to make the passage of the water the quickest possible, is congealed, as was explained above in Cor. 7, Prop. XXXVI.

*The same supposition remaining, the fore-mentioned bodies are
equally acted on by the water flowing through the canal.*

This appears by Lem. V and the third Law. For the water and the bodies act upon each other mutually and equally.

*If the water be at rest in the canal, and these bodies move
with equal velocity and the contrary way through the canal, their
resistances will be equal among themselves.*

This appears from the last Lemma, for the relative motions remain the same among themselves.

The case is the same of all convex and round bodies, whose axes
coincide with the axis of the canal. Some difference may arise from a
greater or less friction; but in these *Lemmata* we suppose
the bodies to be perfectly smooth, and the medium to be void of all
tenacity and friction; and that those parts of the fluid which by
their oblique and superfluous motions may disturb, hinder, and retard
the flux of the water through the canal, are at rest among themselves;
being fixed like water by frost, and adhering to the fore and hinder
parts of the bodies in the manner explained in the Scholium of the
last Proposition; for in what follows we consider the very least
resistance that round bodies described with the greatest given
transverse sections can possibly meet with.

Bodies swimming upon fluids, when they move straight forward, cause the fluid to ascend at their fore parts and subside at their hinder parts, especially if they are of an obtuse figure; and thence they meet with a little more resistance than if they were acute at the head and tail. And bodies moving in elastic fluids, if they are obtuse behind and before, condense the fluid a little more at their fore parts, and relax the same at their hinder parts; and therefore meet also with a little more resistance than if they were acute at the head and tail. But in these Lemmas and Propositions we are not treating of elastic but non-elastic fluids; not of bodies floating on the surface of the fluid, but deeply immersed therein. And when the resistance of bodies in non-elastic fluids is once known, we may then augment this resistance a little in elastic fluids, as our air; and in the surfaces of stagnating fluids, as lakes and seas.

*If a globe move uniformly forward in a compressed, infinite,
and non-elastic fluid, its resistance is to the force by which its
whole motion may be destroyed or generated, in the time that it
describes eight third parts of its diameter, as the density of the
fluid to the density of the globe, very nearly.* For
the globe is to its circumscribed cylinder as two to three; and
therefore the force which can destroy all the motion of the
cylinder, while the same cylinder is describing the length of four
of its diameters, will destroy all the motion of the globe, while
the globe is describing two thirds of this length, that is, eight
third parts of its own diameter. Now the resistance of the cylinder
is to this force very nearly as the density of the fluid to the
density of the cylinder or globe (by Prop. XXXVII), and the
resistance of the globe is equal to the resistance of the cylinder
(by Lem. V, VI, and VII). Q.E.D.

Cor. 1. The resistances of globes in infinite compressed mediums are in a ratio compounded of the duplicate ratio of the velocity, and the duplicate ratio of the diameter, and the ratio of the density of the mediums.

Cor. 2. The greatest velocity, with which a globe can descend by its comparative weight through a resisting fluid, is the same which it may acquire by falling with the same weight, and without any resistance, and in its fall describing a space that is, to four third parts of its diameter as the density of the globe to the density of the fluid. For the globe in the time of its fall, moving with the velocity acquired in falling, will describe a space that will be to eight third parts of its diameter as the density of the globe to the density of the fluid; and the force of its weight which generates this motion will be to the force that can generate the same motion, in the time that the globe describes eight third parts of its diameter, with the same velocity as the density of the fluid to the density of the globe; and therefore (by this Proposition) the force of weight will be equal to the force of resistance, and therefore cannot accelerate the globe.

Cor. 3. If there be given both the density of the globe and its velocity at the beginning of the motion, and the density of the compressed quiescent fluid in which the globe moves, there is given at any time both the velocity of the globe and its resistance, and the space described by it (by Cor. 7, Prop. XXXV).

Cor. 4. A globe moving in a compressed quiescent fluid of the same density with itself will lose half its motion before it can describe the length of two of its diameters (by the same Cor. 7).

*If a globe move uniformly forward through a fluid inclosed and
compressed in a cylindric canal, its resistance is to the force by
which its whole motion may be generated or destroyed, in the time
in which it describes eight third parts of its diameter, in a
ratio compounded of the ratio of the orifice of the canal to the
excess of that orifice above half the greatest circle of the
globe; and the duplicate ratio of the orifice of the canal, to the
excess of that orifice above the greatest circle of the globe; and
the ratio of the density of the fluid to the density of the globe,
nearly.*
This appears by Cor. 2,
Prop. XXXVII, and the demonstration proceeds in the same manner as
in the foregoing Proposition.

In the last two Propositions we suppose (as was done before in Lem. V) that all the water which precedes the globe, and whose fluidity increases the resistance of the same, is congealed. Now if that water becomes fluid, it will somewhat increase the resistance. But in these Propositions that increase is so small, that it may be neglected, because the convex superficies of the globe produces the very same effect almost as the congelation of the water.

*To find by phenomena the resistance of a globe moving through a
perfectly fluid compressed medium.*

Let A be the weight of the globe *in vacuo*, B its weight in
the resisting medium, D the diameter of the globe. F a space which is
to ^{4}/_{3}D as the
density of the globe to the density of the medium, that is, as A to A
− B, G the time in which the globe falling with the weight B without
resistance describes the space F, and H the velocity which the body
acquires by that fall. Then H will be the greatest velocity with which
the globe can possibly descend with the weight B in the resisting
medium, by Cor. 2, Prop XXXVIII; and the resistance which the globe
meets with, when descending with that velocity, will be equal to its
weight B; and the resistance it meets with in any other velocity will
be to the weight B in the duplicate ratio of that velocity to the
greatest velocity H, by Cor. 1, Prop. XXXVIII.

This is the resistance that arises from the inactivity of the matter of the fluid. That resistance which arises from the elasticity, tenacity, and friction of its parts, may be thus investigated.

Let the globe be let fall so that it may descend in the fluid by the
weight B; and let P be the time of falling, and let that time be
expressed in seconds, if the time G be given in seconds. Find the
absolute number N agreeing to the logarithm 0,4342944819
2P

G, and let L be the logarithm of the number N + 1

N; and the velocity acquired in falling will be N − 1

N + 1H, and the height described will be 2PF

G − 1,3862943611F + 4,605170186LF. If the fluid be
of a sufficient depth, we may neglect the term 4,605170186LF; and
2PF

G − 1,3862943611F will be the altitude described,
nearly. These things appear by Prop. IX, Book II, and its Corollaries,
and are true upon this supposition, that the globe meets with no other
resistance but that which arises from the inactivity of matter. Now if
it really meet with any resistance of another kind, the descent will
be slower, and from the quantity of that retardation will be known the
quantity of this new resistance.

That the velocity and descent of a body falling in a fluid might more
easily be known, I have composed the following table; the first column
of which denotes the times of descent; the second shews the velocities
acquired in falling, the greatest velocity being 100000000: the third
exhibits the spaces described by falling in those times, 2F being the
space which the body describes in the time G with the greatest
velocity; and the fourth gives the spaces described with the greatest
velocity in the same times. The numbers in the fourth column are
2P

G, and by subducting the number 1,3862944 − 4,6051702L, are
found the numbers in the third column; and these numbers must be
multiplied by the space F to obtain the spaces described in falling. A
fifth column is added to all these, containing the spaces described in
the same times by a body falling *in vacuo* with the force of
B its comparative weight.

The Times P. | Velocities of the body falling in the fluid. | The spaces described in falling in the fluid. | The spaces described with the greatest motion. | The spaces described by falling In vacuo. |

0,001G 0,01G 0,1G 0,2G 0,3G 0,4G 0,5G 0,6G 0,7G 0,8G 0,9G 1G 2G 3G 4G 5G 6G 7G 8G 9G 10G | 99999^{29}/_{30}999967 9966799 19737532 29131261 37994896 46211716 53704957 60436778 66403677 71629787 76159416 96402758 99505475 99932930 99990920 99998771 99999834 99999980 99999997 99999999 ^{3}/_{5} | 0,000001F 0,0001F 0,0099834F 0,0397361F 0,0886815F 0,1559070F 0,2402290F 0,3402706F 0,4545405F 0,5815071F 0,7196609F 0,8675617F 2,6500055F 4,6186570F 6,6143765F 8,6137964F 10,6137179F 12,6137073F 14,6137059F 16,6137057F 18,6137056F | 0,002F 0,02F 0,2F 0,4F 0,6F 0,8F 1,0F 1,2F 1,4F 1,6F 1,8F 2F 4F 6F 8F 10F 12F 14F 16F 18F 20F | 0,000001F 0,0001F 0,01F 0,04F 0,09F 0,16F 0,25F 0,36F 0,49F 0,64F 0,81F 1F 4F 9F 16F 25F 36F 49F 64F 81F 100F |

In order to investigate the resistances of fluids from experiments, I
procured a square wooden vessel, whose length and breadth on the
inside was 9 inches *English* measure, and its depth 9 feet ½;
this I filled with rainwater: and having provided globes made up of
wax, and lead included therein, I noted the times of the descents of
these globes, the height through which they descended being 112
inches. A solid cubic foot of *English* measure contains 76
pounds troy weight of rainwater; and a solid inch contains 19

36 ounces troy weight, or 253⅓ grains;
and a globe of water of one inch in diameter contains 132,645 grains
in air, or 132,8 grains *in vacuo*; and any other globe will
be as the excess of its weight *in vacuo* above its weight in
water.

Exper. 1. A globe whose weight was 156¼ grains in air, and 77 grains in water, described the whole height of 112 inches in 4 seconds. And, upon repeating the experiment, the globe spent again the very same time of 4 seconds in falling.

The weight of this globe *in vacuo* is 156 13

38 grains; and the excess of this
weight above the weight of the globe in water is 79 13

38 grains. Hence the diameter of the
globe appears to be 0,84224 parts of an inch. Then it will be, as that
excess to the weight of the globe *in vacuo*, so is the
density of the water to the density of the globe; and so is ^{8}/_{3}
parts of the diameter of the globe (viz. 2,24597 inches) to the space
2F, which will be therefore 4,4256 inches. Now a globe falling *in
vacuo* with its whole weight of 156 13

38 grains in one second of time will
describe 193⅓ inches; and falling in water in the same time with the
weight of 77 grains without resistance, will describe 95,219 inches;
and in the time G, which is to one second of time in the subduplicate
ratio of the space F, or of 2,2128 inches to 95,219 inches, will
describe 2,2128 inches, and will acquire the greatest velocity H with
which it is capable of descending in water. Therefore the time G is
0″.15244. And in this time G, with that greatest velocity H, the globe
will describe the space 2F, which is 4,4256 inches; and therefore in 4
seconds will describe a space of 116,1245 inches. Subduct the space
1,3862944F, or 3,0676 inches, and there will remain a space of
113,0569 inches, which the globe falling through water in a very wide
vessel will describe in 4 seconds. But this space, by reason of the
narrowness of the wooden vessel before mentioned, ought to be
diminished in a ratio compounded of the subduplicate ratio of the
orifice of the vessel to the excess of this orifice above half a great
circle of the globe, and of the simple ratio of the same orifice to
its excess above a great circle of the globe, that is, in a ratio of 1
to 0,9914. This done, we have a space of 112,08 inches, which a globe
falling through the water in this wooden vessel in 4 seconds of time
ought nearly to describe by this theory; but it described 112 inches
by the experiment.

Exper. 2. Three equal
globes, whose weights were severally 76⅓ grains in air, and 5 ^{1}/_{16}
grains in water, were let fall successively; and every one fell
through the water in 15 seconds of time, describing in its fall a
height of 112 inches.

By computation, the weight of each globe *in vacuo* is 76
5

12 grains; the excess of this weight
above the weight in water is 71 grains 17

48; the diameter of the globe 0,81296
of an inch; ^{8}/_{3} parts
of this diameter 2,16789 inches; the space 2F is 2,3217 inches; the
space which a globe of 5 ^{1}/_{16}
grains in weight would describe in one second without resistance,
12,808 inches, and the time G0″,301056. Therefore the globe, with the
greatest velocity it is capable of receiving from a weight of 5
^{1}/_{16} grains in its descent
through water, will describe in the time 0″,301056 the space of 2,3217
inches; and in 15 seconds the space 115,678 inches. Subduct the space
1,3862944F, or 1,609 indies, and there remains the space 114.069
inches, which therefore the falling globe ought to describe in the
same time, if the vessel were very wide. But because our vessel was
narrow, the space ought to be diminished by about 0,895 of an inch.
And so the space will remain 113,174 inches, which a globe falling in
this vessel ought nearly to de scribe in 15 seconds, by the theory.
But by the experiment it described 112 inches. The difference is not
sensible.

Exper. 3. Three equal globes, whose weights were severally 121 grains in air, and 1 grain in water, were successively let fall; and they fell through the water in the times 46″, 47″, and 50″, describing a height of 112 inches.

By the theory, these globes ought to have fallen in about 40″. Now whether their falling more slowly were occasioned from hence, that in slow motions the resistance arising from the force of inactivity does really bear a less proportion to the resistance arising from other causes; or whether it is to be attributed to little bubbles that might chance to stick to the globes, or to the rarefaction of the wax by the warmth of the weather, or of the hand that let them fall; or, lastly, whether it proceeded from some insensible errors in weighing the globes in the water, I am not certain. Therefore the weight of the globe in water should be of several grains, that the experiment may be certain, and to be depended on.

Exper. 4. I began the foregoing experiments
to investigate the resistances of fluids, before I was acquainted with
the theory laid down in the Propositions immediately preceding.
Afterward, in order to examine the theory after it was discovered, I
procured a wooden vessel, whose breadth on the inside was 8⅔ inches,
and its depth 15 feet and ⅓. Then I made four globes of wax, with lead
included, each of which weighed 139¼ grains in air, and 7 1

8 grains in water. These I let fall,
measuring the times of their falling in the water with a pendulum
oscillating to half seconds. The globes were cold, and had remained so
some time, both when they were weighed and
when they were let fall; because warmth rarefies the wax, and by
rarefying it diminishes the weight of the globe in the water; and wax,
when rarefied, is not instantly reduced by cold to its former density.
Before they were let fall, they were totally immersed under water,
lest, by the weight of any part of them that might chance to be above
the water, their descent should be accelerated in its beginning. Then,
when after their immersion they were perfectly at rest, they were let
go with the greatest care, that they might not receive any impulse
from the hand that let them down. And they fell successively in the
times of 47½, 48½, 50, and 51 oscillations, describing a height of 15
feet and 2 inches. But the weather was now a little colder than when
the globes were weighed, and therefore I repeated the experiment
another day; and then the globes fell in the times of 49; 49½, 50. and
53; and at a third trial in the times of 49½, 50, 51, and 53
oscillations. And by making the experiment several times over, I found
that the globes fell mostly in the times of 49½ and 50 oscillations.
When they fell slower, I suspect them to have been retarded by
striking against the sides of the vessel.

Now, computing from the theory, the weight of the globe *in vacuo*
is 139 2

5 grains; the excess of this weight
above the weight of the globe in water 132 11

40 grains; the diameter of the globe
0,99868 of an inch; ^{8}/_{3}
parts of the diameter 2,66315 inches; the space 2F 2,8066 inches; the
space which a globe weighing 7 1

8 grains falling without resistance
describes in a second of time 9,88164 inches; and the time GO″,376843.
Therefore the globe with the greatest velocity with which it is
capable of descending through the water by the force of a weight of 7
1

8 grains, will in the time 0″,376843
describe a space of 2,8066 inches, and in one second of time a space
of 7,44766 inches, and in the time 25″, or in 50 oscillations, the
space 186,1915 inches. Subduct the space 1,386294F, or 1,9454 inches,
and there will remain the space 184,2461 inches which the globe will
describe in that time in a very wide vessel. Because our vessel was
narrow, let this space be diminished in a ratio compounded of the
subduplicate ratio of the orifice of the vessel to the excess of this
orifice above half a great circle of the globe, and of the simple
ratio of the same orifice to its excess above a great circle of the
globe; and we shall have the space of 181,86 inches, which the globe
ought by the theory to describe in this vessel in the time of 50
oscillations, nearly. But it described the space of 182 inches, by
experiment, in 49½ or 50 oscillations.

Exper. 5. Four globes weighing 154^{3}/_{8}
grains in air, and 21½ grains in water, being let fall several times,
fell in the times of 28½, 29, 29½, and 30, and sometimes of 31, 32,
and 33 oscillations, describing a height of 15 feet and 2 inches.

They ought by the theory to have fallen in the time of 29 oscillations, nearly.

Exper. 6. Five
globes, weighing 212 3

8 grains in air, and 79½ in water,
being several times let fall, fell in the times of 15, 15½, 16, 17,
and 18 oscillations, describing a height of 15 feet and 2 inches.

By the theory they ought to have fallen in the time of 15 oscillations, nearly.

Exper. 7. Four globes, weighing 293^{3}/_{8}
grains in air, and 35 ^{7}/_{8}
grains in water, being let fall several times, fell in the times of
29½, 30, 30½, 31, 32, and 33 oscillations, describing a height of 15
feet and 1 inch and ½.

By the theory they ought to have fallen in the time of 28 oscillations, nearly.

In searching for the cause that occasioned these globes of the same weight and magnitude to fall, some swifter and some slower, I hit upon this; that the globes, when they were first let go and began to fall, oscillated about their centres; that side which chanced to be the heavier descending first, and producing an oscillating motion. Now by oscillating thus, the globe communicates a greater motion to the water than if it descended without any oscillations; and by this communication loses part of its own motion with which it should descend; and therefore as this oscillation is greater or less, it will be more or less retarded. Besides, the globe always recedes from that side of itself which is descending in the oscillation, and by so receding comes nearer to the sides of the vessel, so as even to strike against them sometimes. And the heavier the globes are, the stronger this oscillation is; and the greater they are, the more is the water agitated by it. Therefore to diminish this oscillation of the globes, I made new ones of lead and wax, sticking the lead in one side of the globe very near its surface; and I let fall the globe in such a manner, that, as near as possible, the heavier side might be lowest at the beginning of the descent. By this means the oscillations became much less than before, and the times in which the globes fell were not so unequal: as in the following experiments.

Exper. 8. Four globes weighing 139 grains in air, and 6½ in water, were let fall several times, and fell mostly in the time of 51 oscillations, never in more than 52, or in fewer than 50, describing a height of 182 inches.

By the theory they ought to fall in about the time of 52 oscillations

Exper. 9. Four globes weighing 273¼ grains in air, and 140¾ in water, being several times let fall, fell in never fewer than 12, and never more than 13 oscillations, describing a height of 182 inches.

These globes by the theory ought to have fallen in the time of 11⅓ oscillations, nearly.

Exper. 10. Four globes, weighing 384 grains in air, and 119½ in water, being let fall several times, fell in the times of 17¾ 18, 18½, and 19 oscillations, describing a height of 181½ inches. And when they fell in the time of 19 oscillations, I sometimes heard them hit against the sides of the vessel before they reached the bottom.

By the theory they ought to have fallen in the time of 15^{5}/_{9}
oscillations, nearly.

Exper. 11. Three equal globes, weighing 48
grains in the air, and 3 29

32 in water, being several times let
fall, fell in the times of 43½, 44, 44½, 45, and 46 oscillations, and
mostly in 44 and 45, describing a height of 182½ inches, nearly.

By the theory they ought to have fallen in the time of 46
oscillations and^{5}/_{9}, nearly.

Exper. 12. Three equal globes, weighing 141
grains in air, and 4^{3}/_{8} in water, being let fall
several times, fell in the times of 61, 62, 63, 64, and 65
oscillations, describing a space of 182 inches.

And by the theory they ought to have fallen in 64½ oscillations nearly.

From these experiments it is manifest, that when the globes fell slowly, as in the second, fourth, fifth, eighth, eleventh, and twelfth experiments, the times of falling are rightly exhibited by the theory; but when the globes fell more swiftly, as in the sixth, ninth, and tenth experiments, the resistance was somewhat greater than in the duplicate ratio of the velocity. For the globes in falling oscillate a little; and this oscillation, in those globes that are light and fall slowly, soon ceases by the weakness of the motion; but in greater and heavier globes, the motion being strong, it continues longer, and is not to be checked by the ambient water till after several oscillations. Besides, the more swiftly the globes move, the less are they pressed by the fluid at their hinder parts; and if the velocity be perpetually increased, they will at last leave an empty space behind them, unless the compression of the fluid be increased at the same time. For the compression of the fluid ought to be increased (by Prop. XXXII and XXXIII) in the duplicate ratio of the velocity, in order to preserve the resistance in the same duplicate ratio. But because this is not done, the globes that move swiftly are not so much pressed at their hinder parts as the others; and by the defect of this pressure it comes to pass that their resistance is a little greater than in a duplicate ratio of their velocity.

So that the theory agrees with the phaenomena of bodies falling in water. It remains that we examine the phaenomena of bodies falling in air.

Exper. 13. From the top of St. *Paul's*
Church in *London*, in *June* 1710, there were let
fall together two glass globes, one full of quicksilver, the other of
air; and in their fall they described a height of 220 *English*
feet. A wooden table was suspended upon iron hinges on one side, and
the other side of the same was supported by a wooden pin. The two
globes lying upon this table were let fall together by pulling out the
pin by means of an iron wire reaching from thence quite down to the
ground; so that, the pin being removed, the
table, which had then no support but the iron hinges, fell downward,
and turning round upon the hinges, gave leave to the globes to drop
off from it. At the same instant, with the same pull of the iron wire
that took out the pin, a pendulum oscillating to seconds was let go,
and began to oscillate. The diameters and weights of the globes, and
their times of falling, are exhibited in the following table.

The globes filled with mercury. | The globes full of air. | ||||

Weights. | Diameters. | Times in falling. | Weights. | Diameters. | Times in falling. |

908 grains 983 grains 866 grains 747 grains 808 grains 784 grains | 0,8 of an inch 0,8 of an inch 0,8 of an inch 0,75 of an inch 0,75 of an inch 0,75 of an inch | 4″ 4- 4 4+ 4 4+ | 510 grains 642 grains 599 grains 515 grains 483 grains 641 grains | 5,1 inches 5,2 inches 5,1 inches 5,0 inches 5,0 inches 5,2 inches | 8″½ 8 8 8¼ 8½ 8 |

But the times observed must be corrected; for the globes of mercury
(by *Galileo's* theory), in 4 seconds of time, will describe
257 *English* feet, and 220 feet in only 3″ 42‴. So that the
wooden table, when the pin was taken out, did not turn upon its hinges
so quickly as it ought to have done; and the slowness of that
revolution hindered the descent of the globes at the beginning. For
the globes lay about the middle of the table, and indeed were rather
nearer to the axis upon which it turned than to the pin. And hence the
times of falling were prolonged about 18‴; and therefore ought to be
corrected by subducting that excess, especially in the larger globes,
which, by reason of the largeness of their diameters, lay longer upon
the revolving table than the others. This being done, the times in
which the six larger globes fell will come forth 8″ 12‴, 7″ 42‴, 7″
42‴, 7″ 57‴, 8″ 12‴ and 7″ 42‴.

Therefore the fifth in order among the globes that were full of air
being 5 inches in diameter, and 483 grains in weight, fell in 8″ 12‴,
describing a space of 220 feet. The weight of a bulk of water equal to
this globe is 16600 grains; and the weight of an equal bulk of air is
16600

860 grains, or 19^{3}/_{10}
grains; and therefore the weight of the globe *in vacua* is
502^{3}/_{10} grains; and this weight is to the weight
of a bulk of air equal to the globe as 502^{3}/_{10}
to 19^{3}/_{10}; and so is 2F to ^{8}/_{3}
of the diameter of the globe, that is, to 13⅓ inches. Whence 2F
becomes 28 feet 11 inches. A globe, falling *in vacua* with
its whole weight of 502^{3}/_{10} grains, will in one
second of time describe 193⅓ inches as above; and with the weight of
483 grains will describe 185,905 inches; and with that weight 483
grains *in vacua* will describe the space F, or 14 feet 5½
inches, in the time of 57‴ 58″″, and acquire the greatest velocity it
is capable of descending with in the air. With this velocity the globe
in 8″ 12‴ of time will describe 245 feet and 5⅓ inches. Subduct
1,3863F, or 20 feet and ½ an inch, and there remain 225 feet 5 inches.
This space, therefore, the falling globe ought by the theory
to describe in 8″ 12‴. But by the experiment it described a space of
220 feet. The difference is insensible.

By like calculations applied to the other globes full of air, I composed the following table.

The weights of the globe. | The diameters. | The times falling from a height of 220 feet. | The spaces which they would describe by the theory. | The excesses. |

510 grains 642 grains 599 grains 515 grains 483 grains 641 grains | 5,1 inches 5,2 inches 5,1 inches 5 inches 5 inches 5,2 inches | 8″ 12‴ 7″ 42‴ 7″ 42‴ 7″ 57‴ 8″ 12‴ 7″ 42‴ | 226 feet 11 inch. 230 feet 9 inch. 227 feet 10 inch. 224 feet 5 inch. 225 feet 5 inch 230 feet 7 inch. | 6 feet 11 inch 10 feet 9 inch 7 feet 0 inch 4 feet 5 inch 5 feet 5 inch 10 feet 7 inch |

Exper. 14. *Anno* 1719, in the month
of *July*, Dr. *Desaguliers* made some experiments of
this kind again, by forming hogs' bladders into spherical orbs; which
was done by means of a concave wooden sphere, which the bladders,
being wetted well first, were put into. After that being blown full of
air, they were obliged to fill up the spherical cavity that contained
them; and then, when dry, were taken out. These were let fall from the
lantern on the top of the cupola of the same church, namely, from a
height of 272 feet; and at the same moment of time there was let fall
a leaden globe, whose weight was about 2 pounds *troy* weight.
And in the mean time some persons standing in the upper part of the
church where the globes were let fall observed the whole times of
falling; and others standing on the ground observed the differences of
the times between the fall of the leaden weight and the fall of the
bladder. The times were measured by pendulums oscillating to half
seconds. And one of those that stood upon the ground had a machine
vibrating four times in one second; and another had another machine
accurately made with a pendulum vibrating four times in a second also.
One of those also who stood at the top of the church had a like
machine; and these instruments were so contrived, that their motions
could be stopped or renewed at pleasure. Now the leaden globe fell in
about four seconds and ¼ of time; and from the addition of this time
to the difference of time above spoken of, was collected the whole
time in which the bladder was falling. The times which the five
bladders spent in falling, after the leaden globe had reached the
ground, were, the first time, 14¾″, 12¾″, 14^{5}/_{8}″,
17¾″, and 16^{7}/_{8}″; and the second time, 14½″,
14¼″, 14″, 19″, and 16¾″. Add to these 4¼″, the time in which the
leaden globe was falling, and the whole times in which the five
bladders fell were, the first time, 19″, 17″, 18^{7}/_{8}″,
22″, and 21^{1}/_{8}″; and the second time, 18¾″,
18½″, 18¼″, 23¼″, and 21″. The times observed at the top of the church
were, the first time, 19^{3}/_{8}″, 17¼″, 18¾″, 22^{1}/_{8}″,
and 21^{5}/_{8}″; and the second time, 19″, 18^{5}/_{8}″,
18^{3}/_{8}″, 24″, and 21¼″. But the bladders did not
always fall directly down, but sometimes fluttered a little in the
air, and waved to and fro, as they were
descending. And by these motions the times of their falling were
prolonged, and increased by half a second sometimes, and sometimes by
a whole second. The second and fourth bladder fell most directly the
first time, and the first and third the second time. The fifth bladder
was wrinkled, and by its wrinkles was a little retarded. I found their
diameters by their circumferences measured with a very fine thread
wound about them twice. In the following table I have compared the
experiments with the theory; making the density of air to be to the
density of rain-water as 1 to 860, and computing the spaces which by
the theory the globes ought to describe in falling.

The weight of the bladders. | The diameters. | The times of falling from a height of 272 feet. | The spaces which by the theory ought to have been described in those times. | The difference between the theory and the experiments. |

128 grains 156 grains 137½ grains 97½ grains 99 ^{1}/_{8} grains | 5,28 inches 5,19 inches 5,3 inches 5,26 inches 5 inches | 19″ 17″ 18″ 22″ 21 ^{1}/_{8}″ | 271 feet 11 in. 272 feet 0½ in. 272 feet 7 in. 277 feet 4 in. 282 feet 0 in. | - 0 ft 1 in. + 0 ft 0½ in. + 0 ft 7 in. + 5 ft 4 in. + 10 ft 0 in. |

Our theory, therefore, exhibits rightly, within a very little, all the resistance that globes moving either in air or in water meet with; which appears to be proportional to the densities of the fluids in globes of equal velocities and magnitudes.

In the Scholium subjoined to the sixth Section, we shewed, by
experiments of pendulums, that the resistances of equal and equally
swift globes moving in air, water, and quicksilver, are as the
densities of the fluids. We here prove the same more accurately by
experiments of bodies falling in air and water. For pendulums at each
oscillation excite a motion in the fluid always contrary to the motion
of the pendulum in its return; and the resistance arising from this
motion, as also the resistance of the thread by which the pendulum is
suspended, makes the whole resistance of a pendulum greater than the
resistance deduced from the experiments of falling bodies. For by the
experiments of pendulums described in that Scholium, a globe of the
same density as water in describing the length of its semidiameter in
air would lose the 1

3342 part of its motion. But by the
theory delivered in this seventh Section, and confirmed by experiments
of falling bodies, the same globe in describing the same length would
lose only a part of its motion equal to 1

4586, supposing the density of water to
be to the density of air as 860 to 1. Therefore the resistances were
found greater by the experiments of pendulums (for the reasons just
mentioned) than by the experiments of falling globes; and that in the
ratio of about 4 to 3. Bat yet since the resistances of pendulums
oscillating in air, water, and quicksilver, are alike increased by
like causes, the proportion of the resistances in these mediums will
be rightly enough exhibited by the experiments
of pendulums, as well as by the experiments of falling bodies. And
from all this it may be concluded, that the resistances of bodies,
moving in any fluids whatsoever, though of the most extreme fluidity,
are, *caeteris paribus*, as the densities of the fluids.

These things being thus established, we may now determine what part
of its motion any globe projected in any fluid whatsoever would nearly
lose in a given time. Let D be the diameter of the globe, and V its
velocity at the beginning of its motion, and T the time in which a
globe with the velocity V can describe *in vacuo* a space that
is, to the space ^{8}/_{3}D
as the density of the globe to the density of the fluid; and the globe
projected in that fluid will, in any other time *t* lose the
part tV

T+t, the part TV

T+t remaining; and will describe a
space, which will be to that described in the same time *in vacuo*
with the uniform velocity V, as the logarithm of the number T+t

T multiplied by the number 2,302585093
is to the number t

T, by Cor. 7, Prop. XXXV. In slow
motions the resistance may be a little less, because the figure of a
globe is more adapted to motion than the figure of a cylinder
described with the same diameter. In swift motions the resistance may
be a little greater, because the elasticity and compression of the
fluid do not increase in the duplicate ratio of the velocity. But
these little niceties I take no notice of.

And though air, water, quicksilver, and the like fluids, by the
division of their parts *in infinitum*, should be subtilized,
and become mediums infinitely fluid, nevertheless, the resistance they
would make to projected globes would be the same. For the resistance
considered in the preceding Propositions arises from the inactivity of
the matter; and the inactivity of matter is essential to bodies, and
always proportional to the quantity of matter. By the division of the
parts of the fluid the resistance arising from the tenacity and
friction of the parts may be indeed diminished; but the quantity of
matter will not be at all diminished by this division; and if the
quantity of matter be the same, its force of inactivity will be the
same; and therefore the resistance here spoken of will be the same, as
being always proportional to that force. To diminish this resistance,
the quantity of matter in the spaces through which the bodies move
must be diminished; and therefore the celestial spaces, through which
the globes of the planets and comets are perpetually passing towards
all parts, with the utmost freedom, and without the least sensible
diminution of their motion, must be utterly void of any corporeal
fluid, excepting, perhaps, some extremely rare vapours and the rays of
light.

Projectiles excite a motion in fluids as they pass through them, and this motion arises from the excess of the pressure of the fluid at the fore parts of the projectile above the pressure of the same at the hinder parts; and cannot be less in mediums infinitely fluid than it is in air, water, and quicksilver, in proportion to the density of matter in each. Now this excess of pressure does, in proportion to its quantity, not only excite a motion in the fluid, but also acts upon the projectile so as to retard its motion; and therefore the resistance in every fluid is as the motion excited by the projectile in the fluid; and cannot be less in the most subtile aether in proportion to the density of that aether, than it is in air, water, and quicksilver, in proportion to the densities of those fluids.

*A pressure is not propagated through a fluid in rectilinear
directions unless where the particles of the fluid lie in a right line.*

If the particles *a, b, c, d, e,* lie in a right line, the
pressure may be indeed directly propagated from *a* to *e*;
but then the particle *e* will urge the obliquely posited
particles *f* and *g* obliquely, and those particles *f*
and *g* will not sustain this pressure, unless they be
supported by the particles *h* and *k* lying beyond
them; but the particles that support them are also pressed by them;
and those particles cannot sustain that pressure, without being
supported by, and pressing upon, those particles that lie still
farther, as *l* and *m*, and so on *in infinitum*.
Therefore the pressure, as soon as it is propagated to particles that
lie out of right lines, begins to deflect towards one hand and the
other, and will be propagated obliquely *in infinitum*; and
after it has begun to be propagated obliquely, if it reaches more
distant particles lying out of the right line, it will deflect again
on each hand and this it will do as often as it lights on particles
that do not lie exactly in a right line. Q.E.D.

Cor. If any part of a pressure, propagated
through a fluid from a given point, be intercepted by any obstacle,
the remaining part, which is not intercepted, will deflect into the
spaces behind the obstacle. This may be demonstrated also after the
following manner. Let a pressure be propagated from the point A
towards any part, and, if it be possible, in rectilinear
directions; and the obstacle NBCK being perforated in BC, let all the
pressure be intercepted but the coniform part APQ passing through the
circular hole BC. Let the cone APQ be divided into frustums by the
transverse plants, *de, fg, hi*. Then while the cone ABC,
propagating the pressure, urges the conic frustum *degf*
beyond it on the superficies *de*, and this frustum urges the
next frustum *fgih* on the superficies *fg*, and that
frustum urges a third frustum, and so *in infinitum*; it is
manifest (by the third Law) that the first frustum *defg* is,
by the re-action of the second frustum *fghi*, as much urged
and pressed on the superficies *fg*, as it urges and presses
that second frustum. Therefore the frustum *degf* is
compressed on both sides, that is, between the cone A*de* and
the frustum *fhig*; and therefore (by Case 6, Prop. XIX)
cannot preserve its figure, unless it be compressed with the same
force on all sides. Therefore with the same force with which it is
pressed on the superficies *de, fg*, it will endeavour to
break forth at the sides *df, eg*; and there (being not in the
least tenacious or hard, but perfectly fluid) it will run out,
expanding itself, unless there be an ambient fluid opposing that
endeavour. Therefore, by the effort it makes to run out, it will press
the ambient fluid, at its sides *df, eg*, with the same force
that it does the frustum *fghi*; and therefore, the pressure
will be propagated as much from the sides *df, eg*, into the
spaces NO, KL this way and that way, as it is propagated from the
superficies *fg* towards PQ. Q.E.D.

*All motion propagated through a fluid diverges from a
rectilinear progress into the unmoved spaces.*

Case 1. Let a motion be propagated from the
point A through the hole BC, and, if it be possible, let it proceed in
the conic space BCQP according to right lines diverging from the point
A. And let us first suppose this motion to be that of waves in the
surface of standing water; and let *de, fg, hi, kl,* &c.,
be the tops of the several waves, divided from each other by as many
intermediate valleys or hollows. Then, because the water in the
ridges of the waves is higher than in the unmoved
parts of the fluid KL, NO, it will run down from off the tops of those
ridges, *e, g, i, l,* &c., *d, f, h, k,* &c.,
this way and that way towards KL and NO; and because the water is more
depressed in the hollows of the waves than in the unmoved parts of the
fluid KL, NO, it will run down into those hollows out of those unmoved
parts. By the first deflux the ridges of the waves will dilate
themselves this way and that way, and be propagated towards KL and NO.
And because the motion of the waves from A towards PQ is carried on by
a continual deflux from the ridges of the waves into the hollows next
to them, and therefore cannot be swifter than in proportion to the
celerity of the descent; and the descent of the water on each side
towards KL and NO must be performed with the same velocity; it follows
that the dilatation of the waves on each side towards KL and NO will
be propagated with the same velocity as the waves themselves go
forward with directly from A to PQ. And therefore the whole space this
way and that way towards KL and NO will be filled by the dilated waves
*rfgr, shis, tklt, vmnv,* &c. Q.E.D.
That these things are so, any one may find by making the
experiment in still water.

Case 2. Let us suppose that *de, fg, hi,
kl, mn,* represent pulses successively propagated from the point
A through an elastic medium. Conceive the pulses to be propagated by
successive condensations and rarefactions of the medium, so that the
densest part of every pulse may occupy a spherical superficies
described about the centre A, and that equal intervals intervene
between the successive pulses. Let the lines *de, fg, hi, kl,*
&c., represent the densest parts of the pulses, propagated through
the hole BC; and because the medium is denser there than in the spaces
on either side towards KL and NO, it will dilate itself as well
towards those spaces KL, NO, on each hand, as towards the rare
intervals between the pulses; and thence the medium, becoming always
more rare next the intervals, and more dense next the pulses, will
partake of their motion. And because the progressive motion of the
pulses arises from the perpetual relaxation of the denser parts
towards the antecedent rare intervals; and since the pulses will relax
themselves on each hand towards the quiescent parts of the medium KL,
NO, with very near the same celerity; therefore the pulses will dilate
themselves on all sides into the unmoved parts KL, NO, with almost the
same celerity with which they are propagated directly from the centre
A; and therefore will fill up the whole space KLON. Q.E.D.
And we find the same by experience also in sounds which
are heard through a mountain interposed; and, if they come into a
chamber through the window, dilate themselves into all the parts of
the room, and are heard in every corner; and not as reflected from the
opposite walls, but directly propagated from the window, as far as our
sense can judge.

Case 3 Let us suppose, lastly, that a motion of any kind is propagated from A through the hole BC. Then since the cause of this propagation is that the parts of the medium that are near the centre A disturb and agitate those which lie farther from it; and since the parts which are urged are fluid, and therefore recede every way towards those spaces where they are less pressed, they will by consequence recede towards all the parts of the quiescent medium; as well to the parts on each hand, as KL and NO, as to those right before, as PQ; and by this means all the motion, as soon as it has passed through the hole BC, will begin to dilate itself, and from thence, as from its principle and centre, will be propagated directly every way. Q.E.D.

*Every tremulous body in an elastic medium propagates the motion
of the pulses on every side right forward; but in a non-elastic
medium excites a circular motion.*

Case. 1. The parts of the tremulous body,
alternately going and returning, do in going urge and drive before
them those parts of the medium that lie nearest, and by that impulse
compress and condense them; and in returning suffer those compressed
parts to recede again, and expand themselves. Therefore the parts of
the medium that lie nearest to the tremulous body move to and fro by
turns, in like manner as the parts of the tremulous body itself do;
and for the same cause that the parts of this body agitate these parts
of the medium, these parts, being agitated by like tremors, will in
their turn agitate others next to themselves; and these others,
agitated in like manner, will agitate those that lie beyond them, and
so on *in infinitum*. And in the same manner as the first
parts of the medium were condensed in going, and relaxed in returning,
so will the other parts be condensed every time they go, and expand
themselves every time they re turn. And therefore they will not be all
going and all returning at the same instant (for in that case they
would always preserve determined distances from each other, and there
could be no alternate condensation and rarefaction); but since, in the
places where they are condensed, they approach to, and, in the places
where they are rarefied, recede from each other, therefore some of
them will be going while others are returning; and so on *in
infinitum*. The parts so going, and in their going condensed,
are pulses, by reason of the progressive motion with which they strike
obstacles in their way; and therefore the successive pulses produced
by a tremulous body will be propagated in rectilinear directions; and
that at nearly equal distances from each other, because of the equal
intervals of time in which the body, by its several tremors produces
the several pulses. And though the parts of the tremulous body go and
return in some certain and determinate direction, yet the pulses
propagated from thence through the medium will dilate themselves
towards the sides, by the foregoing Proposition; and will
be propagated on all sides from that tremulous body, as from a common
centre, in superficies nearly spherical and concentrical. An example
of this we have in waves excited by shaking a finger in water, which
proceed not only forward and backward agreeably to the motion of the
finger, but spread themselves in the manner of concentrical circles
all round the finger, and are propagated on every side. For the
gravity of the water supplies the place of elastic force.

Case 2. If the medium be not elastic, then, because its parts cannot
be condensed by the pressure arising from the vibrating parts of the
tremulous body, the motion will be propagated in an instant towards
the parts where the medium yields most easily, that is, to the parts
which the tremulous body would otherwise leave vacuous behind it. The
case is the same with that of a body projected in any medium whatever.
A medium yielding to projectiles does not recede *in infinitum*,
but with a circular motion comes round to the spaces which the body
leaves behind it. Therefore as often as a tremulous body tends to any
part, the medium yielding to it comes round in a circle to the parts
which the body leaves; and as often as the body returns to the first
place, the medium will be driven from the place it came round to, and
return to its original place. And though the tremulous body be not
firm and hard, but every way flexible, yet if it continue of a given
magnitude, since it cannot impel the medium by its tremors any where
without yielding to it somewhere else, the medium receding from the
parts of the body where it is pressed will always come round in a
circle to the parts that yield to it. Q.E.D.

Cor. It is a mistake, therefore, to think, as some have done, that the agitation of the parts of flame conduces to the propagation of a pressure in rectilinear directions through an ambient medium. A pressure of that kind must be derived not from the agitation only of the parts of flame, but from the dilatation of the whole.

*If water ascend and descend alternately in the erected legs*
KL, MN, *of a canal or pipe; and a pendulum be constructed whose
length between the point of suspension and the centre of
oscillation is equal to half the length of the water in the canal;
I say, that the water will ascend and descend in the same times in
which the pendulum oscillates.*

I measure the length of the water along the axes of the canal and its legs, and make it equal to the sum of those axes; and take no notice of the resistance of the water arising from its attrition by the sides of the canal. Let, therefore, AB, CD, represent the mean height of the water in both legs; and when the water in the leg KL ascends to the height EF, the water will descend in the leg MN to the height GH. Let P be a pendulous body, VP the thread, V the point of suspension, RPQS the cycloid which the pendulum describes, P its lowest point, PQ an arc equal to the height AE. The force with which the motion of the water is accelerated and retarded alternately is the excess of the weight of the water in one leg above the weight in the other; and, therefore, when the water in the leg KL ascends to EF, and in the other leg descends to GH, that force is double the weight of the water EABF, and therefore is to the weight of the whole water as AE or PQ to VP or PR. The force also with which the body P is accelerated or retarded in any place, as Q, of a cycloid, is (by Cor. Prop. LI) to its whole weight as its distance PQ from the lowest place P to the length PR of the cycloid. Therefore the motive forces of the water and pendulum, describing the equal spaces AE, PQ, are as the weights to be moved; and therefore if the water and pendulum are quiescent at first, those forces will move them in equal times, and will cause them to go and return together with a reciprocal motion. Q.E.D.

Cor. 1. Therefore the reciprocations of the water in ascending and descending are all performed in equal times, whether the motion be more or less intense or remiss.

Cor. 2. If the length of the whole water in
the canal be of 6 1

9 feet of *French* measure, the
water will descend in one second of time, and will ascend in another
second, and so on by turns *in infinitum*; for a pendulum of 3
1

18 such feet in length will oscillate
in one second of time.

Cor. 3. But if the length of the water be increased or diminished, the time of the reciprocation will be increased or diminished in the subduplicate ratio of the length.

*The velocity of waves is in the subduplicate ratio of the breadths.*

This follows from the construction of the following Proposition.

*To find the velocity of waves.*

Let a pendulum be constructed, whose length between the point of suspension and the centre of oscillation is equal to the breadth of the waves and in the time that the pendulum will perform one single oscillation the waves will advance forward nearly a space equal to their breadth.

That which I call the breadth of the waves is the transverse measure lying between the deepest part of the hollows, or the tops of the ridges. Let ABCDEF represent the surface of stagnant water ascending and descending in successive waves; and let A, C, E, &c., be the tops of the waves; and let B, D, F, &c., be the intermediate hollows. Because the motion of the waves is carried on by the successive ascent and descent of the water, so that the parts thereof, as A, C, E, &c., which are highest at one time become lowest immediately after; and because the motive force, by which the highest parts descend and the lowest ascend, is the weight of the elevated water, that alternate ascent and descent will be analogous to the reciprocal motion of the water in the canal, and observe the same laws as to the times of its ascent and descent; and therefore (by Prop. XLIV) if the distances between the highest places of the waves A, C, E, and the lowest B, D, F, be equal to twice the length of any pendulum, the highest parts A, C, E, will become the lowest in the time of one oscillation, and in the time of another oscillation will ascend again. Therefore between the passage of each wave, the time of two oscillations will intervene; that is, the wave will describe its breadth in the time that pendulum will oscillate twice; but a pendulum of four times that length, and which therefore is equal to the breadth of the waves, will just oscillate once in that time. Q.E.I.

Cor. 1. Therefore waves, whose breadth is
equal to 3 1

18 *French* feet, will advance
through a space equal to their breadth in one second of time; and
therefore in one minute will go over a space of 183⅓ feet; and in an
hour a space of 11000 feet, nearly.

Cor. 2. And the velocity of greater or less waves will be augmented or diminished in the subduplicate ratio of their breadth.

These things are true upon the supposition that the parts of water ascend or descend in a right line; but, in truth, that ascent and descent is rather performed in a circle; and therefore I propose the time defined by this Proposition as only near the truth.

*If pulses are propagated through a fluid, the several particles
of the fluid, going and returning with the shortest reciprocal
motion, are always accelerated or retarded according to the law of
the oscillating pendulum.*

Let AB, BC, CD, &c., represent equal distances of successive
pulses, ABC the line of direction of the motion of the successive
pulses propagated from A to B; E, F, G three
physical points of the quiescent medium situate in the right line AC
at equal distances from each other; E*e*, F*f*, G*g*,
equal spaces of extreme shortness,
through which those points go and return with a reciprocal motion in each
vibration; *ε, Φ, γ,* any intermediate places of the same
points; EF, FG physical lineolae, or linear parts of the medium lying
between those points, and successively transferred into the places *εΦ,
Φγ,* and *ef, fg*. Let there be drawn the right line PS
equal to the right line E*e*. Bisect the same in O, and from
the centre O, with the interval OP, describe the circle SIP*i*.
Let the whole time of one vibration; with its proportional parts, be
expounded by the whole circumference of this circle and its parts, in
such sort, that, when any time PH or PHS*h* is completed, if
there be let fall to
PS the perpendicular HL or *hl*,
and there be taken E*ε* equal to PL or P*l*, the
physical point E may be found in *ε*. A point, as E, moving
according to this law with a reciprocal motion, in its going from E
through *ε* to *e*, and returning again through *ε*
to E, will perform its several vibrations with the same degrees of
acceleration and retardation with those of an oscillating pendulum. We
are now to prove that the several physical points of the medium will
be agitated with such a kind of motion. Let us suppose, then, that a
medium hath such a motion excited in it from any cause whatsoever, and
consider what will follow from thence.

In the circumference PHS*h* let there be taken the equal arcs,
HI, IK, or *hi, ik*, having the same ratio to the whole
circumference as the equal right lines EF, FG have to BC, the whole
interval of the pulses. Let fall the perpendiculars IM, KN, or *im,
kn*; then because the points E, F, G are successively agitated
with like motions, and perform their entire vibrations composed of
their going and return, while the pulse is transferred from B to C; if
PH or PHS*h* be the time elapsed since the beginning of the
motion of the point E, then will PI or PHS*i* be the time
elapsed since the beginning of the motion of the point F, and PK or
PHS*k* the time elapsed since the beginning of the motion of
the point G; and therefore E*ε*, F*Φ*, G*γ*, will
be respectively equal to PL, PM, PN, while the points are going, and
to P*l*, P*m*, P*n*, when the points are
returning. Therefore *εγ* or EG + G*γ* − E*ε*
will, when the points are going, be equal to EG − LN and
in their return equal to EG + *ln*. But *εγ* is the
breadth or expansion of the part EG of the medium in the place *εγ*;
and therefore the expansion of that part in its going is to its mean
expansion as EG − LN to EG; and in its return, as EG + *ln* or
EG + LN to EG. Therefore since LN is to KH as IM to the radius OP, and
KH to EG as the circumference PHS*h*P to BC; that is, if we put
V for the radius of a circle whose circumference is equal to BC the
interval of the pulses, as OP to V; and, *ex aequo*, LN to EG
as IM to V; the expansion of the part EG, or of the physical point F
in the place *εγ*, to the mean expansion of the same part in
its first place EG, will be as V − IM to V in going, and as V + *im*
to V in its return. Hence the elastic force of the point P in the
place *εγ* to its mean elastic force in the place EG is as
1

V − IM to 1

V in its going, and 1

V + im to 1

V in its return. And by the same
reasoning the elastic forces of the physical points E and G in going
are as 1

V − HL and 1

V − KN to 1

V; and the difference of the forces
to the mean elastic force of the medium as HL
− KN

VV − V × HL − V × KN + HL × KN to
1

V; that is, as HL − KN

VV to 1

V, or as HL − KN to V; if we suppose
(by reason of the very short extent of the vibrations) HL and KN to be
indefinitely less than the quantity V. Therefore since the quantity V
is given, the difference of the forces is as HL − KN; that is (because
HL − KN is proportional to HK, and OM to OI or OP; and because HK and
OP are given) as OM; that is, if F*f* be bisected in Ω, as Ω*Φ*.
And for the same reason the difference of the elastic forces of the
physical points *ε* and *γ*, in the return of the
physical lineola *εγ*, is as Ω*Φ*. But that difference
(that is, the excess of the elastic force of the point *ε*
above the elastic force of the point γ) is the very force by which the
intervening physical lineola *εγ* of the medium is accelerated
in going, and retarded in returning; and therefore the accelerative
force of the physical lineola *εγ* is as its distance from Ω,
the middle place of the vibration. Therefore (by Prop. XXXVIII, Book
I) the time is rightly expounded by the arc PI; and the linear part of
the medium *εγ* is moved according to the law abovementioned,
that is, according to the law of a pendulum oscillating; and the case
is the same of all the linear parts of which the whole medium is
compounded. Q.E.D.

Cor. Hence it appears that the number of the
pulses propagated is the same with the number of the vibrations of the
tremulous body, and is not multiplied in their progress. For the
physical lineola *εγ* as soon as it returns to its first place
is at rest; neither will it move again, unless it receives
a new motion either from the impulse of the tremulous body, or of the
pulses propagated from that body. As soon, therefore, as the pulses
cease to be propagated from the tremulous body, it will return to a
state of rest, and move no more.

*The velocities of pulses propagated in an elastic fluid are in
a ratiο compounded of the subduplicate ratio of the elastic force
directly, and the subduplicate ratio of the density inversely;
supposing the elastic force of the fluid to be proportional to its condensation.*

Case 1. If the mediums be homogeneous, and the distances of the pulses in those mediums be equal amongst themselves, but the motion in one medium is more intense than in the other, the contractions and dilatations of the correspondent parts will be as those motions; not that this proportion is perfectly accurate. However, if the contractions and dilatations are not exceedingly intense, the error will not be sensible; and therefore this proportion may be considered as physically exact. Now the motive elastic forces are as the contractions and dilatations; and the velocities generated in the same time in equal parts are as the forces. Therefore equal and corresponding parts of corresponding pulses will go and return together, through spaces proportional to their contractions and dilatations, with velocities that are as those spaces; and therefore the pulses, which in the time of one going and returning advance forward a space equal to their breadth, and are always succeeding into the places of the pulses that immediately go before them, will, by reason of the equality of the distances, go forward in both mediums with equal velocity.

Case 2. If the distances of the pulses or their lengths are greater in one medium than in another, let us suppose that the correspondent parts describe spaces, in going and returning, each time proportional to the breadths of the pulses; then will their contractions and dilatations be equal: and therefore if the mediums are homogeneous, the motive elastic forces, which agitate them with a reciprocal motion, will be equal also. Now the matter to be moved by these forces is as the breadth of the pulses; and the space through which they move every time they go and return is in the same ratio. And, moreover, the time of one going and returning is in a ratio compounded of the subduplicate ratio of the matter, and the subduplicate ratio of the space; and therefore is as the space. But the pulses advance a space equal to their breadths in the times of going once and returning once; that is, they go over spaces proportional to the times, and therefore are equally swift.

Case 3. And therefore in mediums of equal density and elastic force, all the pulses are equally swift. Now if the density or the elastic force of the medium were augmented, then, because the motive force is increased in the ratio of the elastic force, and the matter to be moved is increased in the ratio of the density, the time which is necessary for producing the same motion as before will be increased in the subduplicate ratio of the density, and will be diminished in the subduplicate ratio of the elastic force. And therefore the velocity of the pulses will be in a ratio compounded of the subduplicate ratio of the density of the medium inversely, and the subduplicate ratio of the elastic force directly. Q.E.D.

This Proposition will be made more clear from the construction of the following Problem.

*The density and elastic force of a medium being given, to find
the velocity of the pulses.*

Suppose the medium to be pressed by an incumbent weight after the manner of our air; and let A be the height of a homogeneous medium, whose weight is equal to the incumbent weight, and whose density is the same with the density of the compressed medium in which the pulses are propagated. Suppose a pendulum to be constructed whose length between the point of suspension and the centre of oscillation is A: and in the time in which that pendulum will perform one entire oscillation composed of its going and returning, the pulse will be propagated right onwards through a space equal to the circumference of a circle described with the radius A.

For, letting those things stand which were constructed in Prop.
XLVII, if any physical line, as EF, describing the space PS in each
vibration, be acted on in the extremities P and S of every going and
return that it makes by an elastic force that is equal to its weight,
it will perform its several vibrations in the time in which the same
might oscillate in a cycloid whose whole perimeter is equal to the
length PS; and that because equal forces will impel equal corpuscles
through equal spaces in the same or equal times. Therefore since the
times of the oscillations are in the subduplicate ratio of the lengths
of the pendulums, and the length of the pendulum is equal to half the
arc of the whole cycloid, the time of one vibration would be to the
time of the oscillation of a pendulum whose length is A in the
subduplicate ratio of the length ½PS or PO to the length A. But the
elastic force with which the physical lineola EG is urged, when it is
found in its extreme places P, S, was (in the demonstration of Prop.
XLVII) to its whole elastic force as HL − KN to V, that is (since the
point K now falls upon P), as HK to V: and all that force, or which is
the same thing, the incumbent weight by which the lineola EG is
compressed, is to the weight of the lineola as the altitude A of the
incumbent weight to EG the length of the lineola; and therefore, *ex
aequo*, the force
with which the lineola EG is urged in the places P and S is to the
weight of that lineola as HK × A to V × EG; or as PO × A to VV;
because HK was to EG as PO to V. Therefore since the times in which
equal bodies are impelled through equal spaces are reciprocally in the
subduplicate ratio of the forces, the time of one vibration, produced
by the action of that elastic force, will be to the time of a
vibration, produced by the impulse of the weight in a subduplicate
ratio of VV to PO × A, and therefore to the time of the oscillation of
a pendulum whose length is A in the subduplicate ratio of VV to PO ×
A, and the subduplicate ratio of PO to A conjunctly; that is, in the
entire ratio of V to A.
But in the time of one vibration composed of the going and returning of the pendulum,
the pulse will be propagated right onward through a space equal to its breadth BC.
Therefore the time in which a pulse runs over the space BC is to the
time of one oscillation composed of the going and returning of the
pendulum as V to A, that is, as BC to the circumference of a circle
whose radius is A. But the time in which the pulse will run over the
space BC is to the time in which it will run over a length equal to
that circumference in the same ratio; and therefore in the time of
such an oscillation the pulse will run over a length equal to that
circumference. Q.E.D.

Cor. 1. The velocity of the pulses is equal to that which heavy bodies acquire by falling with an equally accelerated motion, and in their fall describing half the altitude A. For the pulse will, in the time of this fall, supposing it to move with the velocity acquired by that fall, run over a space that will be equal to the whole altitude A; and therefore in the time of one oscillation composed of one going and return, will go over a space equal to the circumference of a circle described with the radius A; for the time of the fall is to the time of oscillation as the radius of a circle to its circumference.

Cor. 2. Therefore since that altitude A is as the elastic force of the fluid directly, and the density of the same inversely, the velocity of the pulses will be in a ratio compounded of the subduplicate ratio of the density inversely, and the subduplicate ratio of the elastic force directly.

*To find the distances of the pulses.*

Let the number of the vibrations of the body, by whose tremor the pulses are produced, be found to any given time. By that number divide the space which a pulse can go over in the same time, and the part found will be the breadth of one pulse. Q.E.I.

The last Propositions respect the motions of light and sounds; for
since light is propagated in right lines, it is certain that it cannot
consist in action alone (by Prop. XLI and XLII). As to sounds, since
they arise from tremulous bodies, they can be nothing else but pulses
of the air propagated through it (by Prop. XLIII); and this is
confirmed by the tremors which sounds, if they be loud and deep,
excite in the bodies near them, as we experience in the sound of
drums; for quick and short tremors are less easily excited. But it is
well known that any sounds, falling upon strings in unison with the
sonorous bodies, excite tremors in those strings. This is also
confirmed from the velocity of sounds; for since the specific
gravities of rain-water and quicksilver are to one another as about 1
to 13⅔, and when the mercury in the barometer is at the height of 30
inches of our measure, the specific gravities of the air and of
rain-water are to one another as about 1 to 870, therefore the
specific gravity of air and quicksilver are to each other as 1 to
11890. Therefore when the height of the quicksilver is at 30 inches, a
height of uniform air, whose weight would be sufficient to compress
our air to the density we find it to be of, must be equal to 356700
inches, or 29725 feet of our measure; and this is that very height of
the medium, which I have called A in the construction of the foregoing
Proposition. A circle whose radius is 29725 feet is 186768 feet in
circumference. And since a pendulum 39 1

5 inches in length completes one
oscillation, composed of its going and return, in two seconds of time,
as is commonly known, it follows that a pendulum 29725 feet, or 356700
inches in length will perform a like oscillation in 190¾ seconds.
Therefore in that time a sound will go right onwards 186768 feet, and
therefore in one second 979 feet.

But in this computation we have made no allowance for the crassitude
of the solid particles of the air, by which the sound is propagated
instantaneously. Because the weight of air is to the weight of water
as 1 to 870, and because salts are almost twice as dense as water; if
the particles of air are supposed to be of near the same density as
those of water or salt, and the rarity of the air arises from the
intervals of the particles; the diameter of one particle of air will
be to the interval between the centres of the
particles as 1 to about 9 or 10, and to the interval between the
particles themselves as 1 to 8 or 9. Therefore to 979 feet, which,
according to the above calculation, a sound will advance forward in
one second of time, we may add 979

9, or about 109 feet, to compensate for
the crassitude of the particles of the air: and then a sound will go
forward about 1088 feet in one second of time.

Moreover, the vapours floating in the air being of another spring, and a different tone, will hardly, if at all, partake of the motion of the true air in which the sounds are propagated. Now if these vapours remain unmoved, that motion will be propagated the swifter through the true air alone, and that in the subduplicate ratio of the defect of the matter. So if the atmosphere consist of ten parts of true air and one part of vapours, the motion of sounds will be swifter in the subduplicate ratio of 11 to 10, or very nearly in the entire ratio of 21 to 20, than if it were propagated through eleven parts of true air: and therefore the motion of sounds above discovered must be increased in that ratio. By this means the sound will pass through 1142 feet in one second of time.

These things will be found true in spring and autumn, when the air is rarefied by the gentle warmth of those seasons, and by that means its elastic force becomes somewhat more intense. But in winter, when the air is condensed by the cold, and its elastic force is somewhat remitted, the motion of sounds will be slower in a subduplicate ratio of the density; and, on the other hand, swifter in the summer.

Now by experiments it actually appears that sounds do really advance
in one second of time about 1142 feet of *English* measure, or
1070 feet of *French* measure.

The velocity of sounds being known, the intervals of the pulses are
known also. For M. *Sauveur*, by some experiments that he
made, found that an open pipe about five *Paris* feet in
length gives a sound of the same tone with a viol-string that vibrates
a hundred times in one second. Therefore there are near 100 pulses in
a space of 1070 *Paris* feet, which a sound runs over in a
second of time; and therefore one pulse fills up a space of about 10
7

10 *Paris* feet, that is, about
twice the length of the pipe. From whence it is probable that the
breadths of the pulses, in all sounds made in open pipes, are equal to
twice the length of the pipes.

Moreover, from the Corollary of Prop. XLVII appears the reason why the sounds immediately cease with the motion of the sonorous body, and why they are heard no longer when we are at a great distance from the sonorous bodies than when we are very near them. And besides, from the foregoing principles, it plainly appears how it comes to pass that sounds are so mightily increased in speaking-trumpets; for all reciprocal motion uses to be increased by the generating cause at each return. And in tubes hindering the dilatation of the sounds, the motion decays more slowly, and recurs more forcibly; and therefore is the more increased by the new motion impressed at each return. And these are the principal phaenomena of sounds.

*The resistance arising from the want of lubricity in the parts
of a fluid, is,* caeteris paribus*, proportional to the
velocity with which the parts of the fluid are separated from each other.*

*If a solid cylinder infinitely long, in an uniform and infinite
fluid, revolve with an uniform motion about an axis given in
position, and the fluid be forced round by only this impulse of
the cylinder, and every part of the fluid persevere uniformly in
its motion; I say, that the periodic times of the parts of the
fluid are as their distances from the axis of the cylinder.*

Let AFL be a cylinder turning uniformly about the axis S, and let the
concentric circles BGM, CHN, DIO, EKP, &c., divide the fluid into
innumerable concentric cylindric solid orbs of the same thickness.
Then, because the fluid is homogeneous, the impressions which the
contiguous orbs make upon each other mutually will be (by the
Hypothesis) as their translations from each other, and as the
contiguous superficies upon which the impressions are made. If the
impression made upon any orb be greater or less on its concave than on
its convex side, the stronger impression will prevail, and will either
accelerate or retard the motion of the orb, according as it agrees
with, or is contrary to, the motion of the same. Therefore, that every
orb may persevere uniformly in its motion, the impressions made on
both sides must be equal and their directions contrary. Therefore
since the impressions are as the contiguous superficies, and as their
translations from one another, the translations will be inversely as
the superficies, that is, inversely as the distances of the
superficies from the axis. But the differences of the
angular motions about the axis are as those translations applied to
the distances, or as the translations directly and the distances
inversely; that is, joining these ratios together, as the squares of
the distances inversely. Therefore if there be erected the lines A*a*,
B*b*, C*c*, D*d*, E*e*, &c.,
perpendicular to the several parts of he infinite right line SABCDEQ,
and reciprocally proportional to the squares of SA, SB, SC, SD, SE,
&c., and through the extremities of those perpendiculars there be
supposed to pass an hyperbolic curve, the sums of the differences,
that is, the whole angular motions, will be as the correspondent sums
of the lines A*a*, B*b*, C*c*, D*d*, E*e*,
that is (if to constitute a medium uniformly fluid the number of the
orbs be increased and their breadth diminished *in infinitum*),
as the hyperbolic areas A*a*Q, B*b*Q, C*c*Q, D*d*Q,
E*e*Q, &c., analogous to the sums; and the times,
reciprocally proportional to the angular motions, will be also
reciprocally proportional to those areas. Therefore the periodic time
of any particle as D, is reciprocally as the area D*d*Q, that
is (as appears from the known methods of quadratures of curves),
directly as the distance SD. Q.E.D.

Cor. 1. Hence the angular motions of the particles of the fluid are reciprocally as their distances from the axis of the cylinder, and the absolute velocities are equal.

Cor. 2. If a fluid be contained in a cylindric vessel of an infinite length, and contain another cylinder within, and both the cylinders revolve about one common axis, and the times of their revolutions be as their semi-diameters, and every part of the fluid perseveres in its motion, the periodic times of the several parts will be as the distances from the axis of the cylinders.

Cor. 3. If there be added or taken away any common quantity of angular motion from the cylinder and fluid moving in this manner; yet because this new motion will not alter the mutual attrition of the parts of the fluid, the motion of the parts among themselves will not be changed; for the translations of the parts from one another depend upon the attrition. Any part will persevere in that motion, which, by the attrition made on both sides with contrary directions, is no more accelerated than it is retarded.

Cor. 4. Therefore if there be taken away from this whole system of the cylinders and the fluid all the angular motion of the outward cylinder, we shall have the motion of the fluid in a quiescent cylinder.

Cor. 5. Therefore if the fluid and outward cylinder are at rest, and the inward cylinder revolve uniformly, there will be communicated a circular motion to the fluid, which will be propagated by degrees through the whole fluid; and will go on continually increasing, till such time as the several parts of the fluid acquire the motion determined in Cor. 4.

Cor. 6. And because the fluid endeavours to propagate its motion still farther, its impulse will carry the outmost cylinder also about with it, unless the cylinder be violently detained; and accelerate its motion till the periodic times of both cylinders become equal among themselves. But if the outward cylinder be violently detained, it will make an effort to retard the motion of the fluid; and unless the inward cylinder preserve that motion by means of some external force impressed thereon, it will make it cease by degrees.

All these things will be found true by making the experiment in deep standing water.

*If a solid sphere, in an uniform and infinite fluid, revolves
about an axis given in position, with an uniform motion, and the
fluid be forced round by only this impulse of the sphere; and
every part of the fluid perseveres uniformly in its motion; I say,
that the periodic times of the parts of the fluid are as the
squares of their distances from the centre of the sphere.*

Case 1. Let AFL be a sphere turning uniformly
about the axis S, and let the concentric circles BGM, CHN, DIO, EKP,
&c., divide the fluid into innumerable concentric orbs of the same
thickness. Suppose those orbs to be solid; and, because the fluid is
homogeneous, the impressions which the contiguous orbs make one upon
another will be (by the supposition) as their translations from one
another, and the contiguous superficies upon which the impressions are
made. If the impression upon any orb be greater or less upon its
concave than upon its convex side, the more forcible impression will
prevail, and will either accelerate or retard the velocity of the orb,
according as it is directed with a conspiring or contrary motion to
that of the orb. Therefore that every orb may persevere uniformly in
its motion, it is necessary that the impressions made upon both sides
of the orb should be equal, and have contrary directions. Therefore
since the impressions are as the contiguous superficies, and as their
translations from one another, the translations will be inversely as
the superficies, that is, inversely as the squares of the distances of
the superficies from the centre. But the differences of the angular
motions about the axis are as those translations applied to the
distances, or as the translations directly and the distances
inversely; that is, by compounding those ratios, as the cubes of the
distances inversely. Therefore if upon the several parts of the
infinite right line SABCDEQ there be erected
the perpendiculars A*a*, B*b*, C*c*, D*d*,
E*e*, &c., reciprocally proportional to the cubes of SA,
SB, SC, SD, SE, &c., the sums of the differences, that is, the
whole angular motions will be as the corresponding sums of the lines A*a*,
B*b*, C*c*, D*d*, E*e*, &c., that is
(if to constitute an uniformly fluid medium the number of the orbs be
increased and their thickness diminished *in infinitum*), as
the hyperbolic areas A*a*Q, B*b*Q, C*c*Q, D*d*Q,
E*e*Q, &c., analogous to the sums; and the periodic times
being reciprocally proportional to the angular motions, will be also
reciprocally proportional to those areas. Therefore the periodic time
of any orb DIO is reciprocally as the area D*d*Q, that is (by
the known methods of quadratures), directly as the square of the
distance SD. Which was first to be demonstrated.

Case 2. From the centre of the sphere let
there be drawn a great number of indefinite right lines, making given
angles with the axis, exceeding one another by equal differences; and,
by these lines revolving about the axis, conceive the orbs to be cut
into innumerable annuli; then will every annulus have four annuli
contiguous to it, that is, one on its inside, one on its outside, and
two on each hand. Now each of these annuli cannot be impelled equally
and with contrary directions by the attrition of the interior and
exterior annuli, unless the motion be communicated according to the
law which we demonstrated in Case 1. This appears from that
demonstration. And therefore any series of annuli, taken in any right
line extending itself *in infinitum* from the globe, will move
according to the law of Case 1, except we should imagine it hindered
by the attrition of the annuli on each side of it. But now in a
motion, according to this law, no such is, and therefore cannot be,
any obstacle to the motions persevering according to that law. If
annuli at equal distances from the centre revolve either more swiftly
or more slowly near the poles than near the ecliptic, they will be
accelerated if slow, and retarded if swift, by their mutual attrition;
and so the periodic times will continually approach to equality,
according to the law of Case 1. Therefore this attrition will not at
all hinder the motion from going on according to the law of Case 1,
and therefore that law will take place; that is, the periodic times of
the several annuli will be as the squares of their distances from the
centre of the globe. Which was to be demonstrated in the second place.

Case 3. Let now every annulus be divided by transverse sections into innumerable particles constituting a substance absolutely and uniformly fluid; and because these sections do not at all respect the law of circular motion, but only serve to produce a fluid substance, the law of circular motion will continue the same as before. All the very small annuli will either not at all change their asperity and force of mutual attrition upon account of these sections, or else they will change the same equally. Therefore the proportion of the causes remaining the same, the proportion of the effects will remain the same also; that is, the proportion of the motions and the periodic times. Q.E.D. But now as the circular motion, and the centrifugal force thence arising, is greater at the ecliptic than at the poles, there must be some cause operating to retain the several particles in their circles; otherwise the matter that is at the ecliptic will always recede from the centre, and come round about to the poles by the outside of the vortex, and from thence return by the axis to the ecliptic with a perpetual circulation.

Cor. 1. Hence the angular motions of the parts of the fluid about the axis of the globe are reciprocally as the squares of the distances from the centre of the globe, and the absolute velocities are reciprocally as the same squares applied to the distances from the axis.

Cor. 2. If a globe revolve with a uniform
motion about an axis of a given position in a similar and infinite
quiescent fluid with an uniform motion, it will communicate a whirling
motion to the fluid like that of a vortex, and that motion will by
degrees be propagated onward *in infinitum*; and this motion
will be increased, continually in every part of the fluid, till the
periodical times of the several parts become as the squares of the
distances from the centre of the globe.

Cor. 3. Because the inward parts of the vortex are by reason of their greater velocity continually pressing upon and driving forward the external parts, and by that action are perpetually communicating motion to them, and at the same time those exterior parts communicate the same quantity of motion to those that lie still beyond them, and by this action preserve the quantity of their motion continually unchanged, it is plain that the motion is perpetually transferred from the centre to the circumference of the vortex, till it is quite swallowed up and lost in the boundless extent of that circumference. The matter between any two spherical superficies concentrical to the vortex will never be accelerated; because that matter will be always transferring the motion it receives from the matter nearer the centre to that matter which lies nearer the circumference.

Cor. 4. Therefore, in order to continue a vortex in the same state of motion, some active principle is required from which the globe may receive continually the same quantity of motion which it is always communicating to the matter of the vortex. Without such a principle it will undoubtedly come to pass that the globe and the inward parts of the vortex, being always propagating their motion to the outward parts, and not receiving any new motion, will gradually move slower and slower, and at last be carried round no longer.

Cor. 5. If another globe should be swimming
in the same vortex at a certain distance from its centre, and in the
mean time by some force revolve constantly about an axis of a given
inclination, the motion of this globe will drive the fluid round after
the manner of a vortex; and at first this new
and small vortex will revolve with its globe about the centre of the
other; and in the mean time its motion will creep on farther and
farther, and by degrees be propagated *in infinitum*, after
the manner of the first vortex. And for the same reason that the globe
of the new vortex was carried about before by the motion of the other
vortex, the globe of this other will be carried about by the motion of
this new vortex, so that the two globes will revolve about some
intermediate point, and by reason of that circular motion mutually fly
from each other, unless some force restrains them. Afterward, if the
constantly impressed forces, by which the globes persevere in their
motions, should cease, and every thing be left to act according to the
laws of mechanics, the motion of the globes will languish by degrees
(for the reason assigned in Cor. 3 and 4), and the vortices at last
will quite stand still.

Cor. 6. If several globes in given places
should constantly revolve with determined velocities about axes given
in position, there would arise from them as many vortices going on *in
infinitum*. For upon the same account that any one globe
propagates its motion *in infinitum*, each globe apart will
propagate its own motion *in infinitum* also; so that every
part of the infinite fluid will be agitated with a motion resulting
from the actions of all the globes. Therefore the vortices will not be
confined by any certain limits, but by degrees run mutually into each
other; and by the mutual actions of the vortices on each other, the
globes will be perpetually moved from their places, as was shewn in
the last Corollary; neither can they possibly keep any certain
position among themselves, unless some force restrains them. But if
those forces, which are constantly impressed upon the globes to
continue these motions, should cease, the matter (for the reason
assigned in Cor. 3 and 4) will gradually stop, and cease to move in
vortices.

Cor. 7. If a similar fluid be inclosed in a spherical vessel, and, by the uniform rotation of a globe in its centre, is driven round in a vortex; and the globe and vessel revolve the same way about the same axis, and their periodical times be as the squares of the semi-diameters; the parts of the fluid will not go on in their motions without acceleration or retardation, till their periodical times are as the squares of their distances from the centre of the vortex. No constitution of a vortex can be permanent but this.

Cor. 8. If the vessel, the inclosed fluid, and the globe, retain this motion, and revolve besides with a common angular motion about any given axis, because the mutual attrition of the parts of the fluid is not changed by this motion, the motions of the parts among each other will not be changed; for the translations of the parts among themselves depend upon this attrition. Any part will persevere in that motion in which its attrition on one side retards it just as much as its attrition on the other side accelerates it.

Cor. 9. Therefore if the vessel be quiescent, and the motion of the globe be given, the motion of the fluid will be given. For conceive a plane to pass through the axis of the globe, and to revolve with a contrary motion; and suppose the sum of the time of this revolution and of the revolution of the globe to be to the time of the revolution of the globe as the square of the semi-diameter of the vessel to the square of the semi-diameter of the globe; and the periodic times of the parts of the fluid in respect of this plane will be as the squares of their distances from the centre of the globe.

Cor. 10. Therefore if the vessel move about the same axis with the globe, or with a given velocity about a different one, the motion of the fluid will be given. For if from the whole system we take away the angular motion of the vessel, all the motions will remain the same among themselves as before, by Cor. 8, and those motions will be given by Cor. 9.

Cor. 11. If the vessel and the fluid are quiescent, and the globe revolves with an uniform motion, that motion will be propagated by degrees through the whole fluid to the vessel, and the vessel will be carried round by it, unless violently detained; and the fluid and the vessel will be continually accelerated till their periodic times become equal to the periodic times of the globe. If the vessel be either withheld by some force, or revolve with any constant and uniform motion, the medium will come by little and little to the state of motion defined in Cor. 8, 9, 10, nor will it ever persevere in any other state. But if then the forces, by which the globe and vessel revolve with certain motions, should cease, and the whole system be left to act according to the mechanical laws, the vessel and globe, by means of the intervening fluid, will act upon each other, and will continue to propagate their motions through the fluid to each other, till their periodic times become equal among themselves, and the whole system revolves together like one solid body.

In all these reasonings I suppose the fluid to consist of matter of uniform density and fluidity; I mean, that the fluid is such, that a globe placed any where therein may propagate with the same motion of its own, at distances from itself continually equal, similar and equal motions in the fluid in the same interval of time. The matter by its circular motion endeavours to recede from the axis of the vortex, and therefore presses all the matter that lies beyond. This pressure makes the attrition greater, and the separation of the parts more difficult; and by consequence diminishes the fluidity of the matter. Again; if the parts of the fluid are in any one place denser or larger than in the others, the fluidity will be less in that place, because there are fewer superficies where the parts can be separated from each other. In these cases I suppose the defect of the fluidity to be supplied by the smoothness or softness of the parts, or some other condition; otherwise the matter where it is less fluid will cohere more, and be more sluggish, and therefore will receive the motion more slowly, and propagate it farther than agrees with the ratio above assigned. If the vessel be not spherical, the particles will move in lines not circular, but answering to the figure of the vessel; and the periodic times will be nearly as the squares of the mean distances from the centre. In the parts between the centre and the circumference the motions will be slower where the spaces are wide, and swifter where narrow; but yet the particles will not tend to the circumference at all the more for their greater swiftness; for they then describe arcs of less curvity, and the conatus of receding from the centre is as much diminished by the diminution of this curvature as it is augmented by the increase of the velocity. As they go out of narrow into wide spaces, they recede a little farther from the centre, but in doing so are retarded; and when they come out of wide into narrow spaces, they are again accelerated; and so each particle is retarded and accelerated by turns for ever. These things will come to pass in a rigid vessel; for the state of vortices in an infinite fluid is known by Cor. 6 of this Proposition.

I have endeavoured in this Proposition to investigate the properties of vortices, that I might find whether the celestial phenomena can be explained by them; for the phenomenon is this, that the periodic times of the planets revolving about Jupiter are in the sesquiplicate ratio of their distances from Jupiter's centre; and the same rule obtains also among the planets that revolve about the sun. And these rules obtain also with the greatest accuracy, as far as has been yet discovered by astronomical observation. Therefore if those planets are carried round in vortices revolving about Jupiter and the sun, the vortices must revolve according to that law. But here we found the periodic times of the parts of the vortex to be in the duplicate ratio of the distances from the centre of motion; and this ratio cannot be diminished and reduced to the sesquiplicate, unless either the matter of the vortex be more fluid the farther it is from the centre, or the resistance arising from the want of lubricity in the parts of the fluid should, as the velocity with which the parts of the fluid are separated goes on increasing, be augmented with it in a greater ratio than that in which the velocity increases. But neither of these suppositions seem reasonable. The more gross and less fluid parts will tend to the circumference, unless they are heavy towards the centre. And though, for the sake of demonstration, I proposed, at the beginning of this Section, an Hypothesis that the resistance is proportional to the velocity, nevertheless, it is in truth probable that the resistance is in a less ratio than that of the velocity; which granted, the periodic times of the parts of the vortex will be in a greater than the duplicate ratio of the distances from its centre. If, as some think, the vortices move more swiftly near the centre, then slower to a certain limit, then again swifter near the circumference, certainty neither the sesquiplicate, nor any other certain and determinate ratio, can obtain in them. Let philosophers then see how that phenomenon of the sesquiplicate ratio can be accounted for by vortices.

*Bodies carried about in a vortex, and returning in the same
orb, are of the same density with the vortex, and are moved
according to the same law with the parts of the vortex, as to
velocity and direction of motion.*

For if any small part of the vortex, whose particles or physical
points preserve a given situation among each other, be supposed to be
congealed, this particle will move according to the same law as
before, since no change is made either in its density, *vis insita*,
or figure. And again; if a congealed or solid part of the vortex be of
the same density with the rest of the vortex, and be resolved into a
fluid, this will move according to the same law as before, except in
so far as its particles, now become fluid, may be moved among
themselves. Neglect, therefore, the motion of the particles among
themselves as not at all concerning the progressive motion of the
whole, and the motion of the whole will be the same as before. But
this motion will be the same with the motion of other parts of the
vortex at equal distances from the centre; because the solid, now
resolved into a fluid, is become perfectly like to the other parts of
the vortex. Therefore a solid, if it be of the same density with the
matter of the vortex, will move with the same motion as the parts
thereof, being relatively at rest in the matter that surrounds it. If
it be more dense, it will endeavour more than before to recede from
the centre; and therefore overcoming that force of the vortex, by
which, being, as it were, kept in equilibrio, it was retained in its
orbit, it will recede from the centre, and in its revolution describe
a spiral, returning no longer into the same orbit. And, by the same
argument, if it be more rare, it will approach to the centre.
Therefore it can never continually go round in the same orbit, unless
it be of the same density with the fluid. But we have shewn in that
case that it would revolve according to the same law with those parts
of the fluid that are at the same or equal distances from the centre
of the vortex.

Cor. 1. Therefore a solid revolving in a vortex, and continually going round in the same orbit, is relatively quiescent in the fluid that carries it.

Cor. 2. And if the vortex be of an uniform density, the same body may revolve at any distance from the centre of the vortex.

Hence it is manifest that the planets are not carried round in
corporeal vortices; for, according to the *Copernican*
hypothesis, the planets going
round the sun revolve in ellipses, having the sun in their common
focus; and by radii drawn to the sun describe areas proportional to
the times. But now the parts of a vortex can never revolve with such a
motion. Let AD, BE, CF, represent three orbits described about the sun
S, of which let the utmost circle CF be concentric to the sun; and let
the aphelia of the two innermost be A, B; and their perihelia D, E.
Therefore a body revolving in the orb CF, describing, by a radius
drawn to the sun, areas proportional to the times, will move with an
uniform motion. And, according to the laws of astronomy, the body
revolving in the orb BE will move slower in its aphelion B, and
swifter in its perihelion E; whereas, according to the laws of
mechanics, the matter of the vortex ought to move more swiftly in the
narrow space between A and C than in the wide space between D and F;
that is, more swiftly in the aphelion than in the perihelion. Now
these two conclusions contradict each other. So at the beginning of
the sign of Virgo, where the aphelion of Mars is at present, the
distance between the orbits of Mars and Venus is to the distance
between the same orbits, at the beginning of the sign of Pisces, as
about 3 to 2; and therefore the matter of the vortex between those
orbits ought to be swifter at the beginning of Pisces than at the
beginning of Virgo in the ratio of 3 to 2; for the narrower the space
is through which the same quantity of matter passes in the same time
of one revolution, the greater will be the velocity with which it
passes through it. Therefore if the earth being relatively at rest in
this celestial matter should be carried round by it, and revolve
together with it about the sun, the velocity of the earth at the
beginning of Pisces would be to its velocity at the beginning of Virgo
in a sesquialteral ratio. Therefore the sun's apparent diurnal motion
at the beginning of Virgo ought to be above 70 minutes, and at the
beginning of Pisces less than 48 minutes; whereas, on the contrary,
that apparent motion of the sun is really greater at the beginning of
Pisces than at the beginning of Virgo, as experience testifies; and
therefore the earth is swifter at the beginning of Virgo than at the
beginning of Pisces; so that the hypothesis of vortices is utterly
irreconcileable with astronomical phaenomena, and rather serves to
perplex than explain the heavenly motions. How these motions are
performed in free spaces without vortices, may be understood by the
first Book; and I shall now more fully treat of it in the following
Book.

In the preceding Books I have laid down the principles of philosophy, principles not philosophical, but mathematical: such, to wit, as we may build our reasonings upon in philosophical inquiries. These principles are the laws and conditions of certain motions, and powers or forces, which chiefly have respect to philosophy: but, lest they should have appeared of themselves dry and barren, I have illustrated them here and there with some philosophical scholiums, giving an account of such things as are of more general nature, and which philosophy seems chiefly to be founded on; such as the density and the resistance of bodies, spaces void of all bodies, and the motion of light and sounds. It remains that, from the same principles, I now demonstrate the frame of the System of the World. Upon this subject I had, indeed, composed the third Book in a popular method, that it might be read by many; but afterward, considering that such as had not sufficiently entered into the principles could not easily discern the strength of the consequences, nor lay aside the prejudices to which they had been many years accustomed, therefore, to prevent the disputes which might be raised upon such accounts, I chose to reduce the substance of this Book into the form of Propositions (in the mathematical way), which should be read by those only who had first made themselves masters of the principles established in the preceding Books: not that I would advise any one to the previous study of every Proposition of those Books; for they abound with such as might cost too much time, even to readers of good mathematical learning. It is enough if one carefully reads the Definitions, the Laws of Motion, and the first three Sections of the first Book. He may then pass on to this Book, and consult such of the remaining Propositions of the first two Books, as the references in this, and his occasions, shall require.

*We are to admit no more causes of natural things than such as
are both time and sufficient to explain their appearances.*

To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes.

*Therefore to the same natural effects we must, as far as
possible, assign the same causes.*

As to respiration in a man and in a beast; the descent of stones in *Europe*
and in *America*; the light of our culinary fire and of the
sun; the reflection of light in the earth, and in the planets.

*The qualities of bodies, which admit neither intension nor
remission of degrees, and which are found to belong to all bodies
within the reach of our experiments, are to be esteemed the
universal qualities of all bodies whatsoever.*

For since the qualities of bodies are only known to us by
experiments, we are to hold for universal all such as universally
agree with experiments; and such as are not liable to diminution can
never be quite taken away. We are certainly not to relinquish the
evidence of experiments for the sake of dreams and vain fictions of
our own devising; nor are we to recede from the analogy of Nature,
which uses to be simple, and always consonant to itself. We no other
way know the extension of bodies than by our senses, nor do these
reach it in all bodies; but because we perceive extension in all that
are sensible, therefore we ascribe it universally to all others also.
That abundance of bodies are hard, we learn by experience; and because
the hardness of the whole arises from the hardness of the parts, we
therefore justly infer the hardness of the undivided particles not
only of the bodies we feel but of all others. That all bodies are
impenetrable, we gather not from reason, but from sensation. The
bodies which we handle we find impenetrable, and thence conclude
impenetrability to be an universal property of all bodies whatsoever.
That all bodies are moveable, and endowed with certain powers (which
we call the *vires inertiae*) of persevering in their motion,
or in their rest, we only infer from the like properties observed in
the bodies which we have seen. The extension,
hardness, impenetrability, mobility, and *vis inertiae* of the
whole, result from the extension, hardness, impenetrability, mobility,
and *vires inertiae* of the parts; and thence we conclude the
least particles of all bodies to be also all extended, and hard and
impenetrable, and moveable, and endowed with their proper *vires
inertia*. And this is the foundation of all philosophy.
Moreover, that the divided but contiguous particles of bodies may be
separated from one another, is matter of observation; and, in the
particles that remain undivided, our minds are able to distinguish yet
lesser parts, as is mathematically demonstrated. But whether the parts
so distinguished, and not yet divided, may, by the powers of Nature,
be actually divided and separated from one an other, we cannot
certainly determine. Yet, had we the proof of but one experiment that
any undivided particle, in breaking a hard and solid body, suffered a
division, we might by virtue of this rule conclude that the undivided
as well as the divided particles may be divided and actually separated
to infinity.

Lastly, if it universally appears, by experiments and astronomical
observations, that all bodies about the earth gravitate towards the
earth, and that in proportion to the quantity of matter which they
severally contain; that the moon likewise, according to the quantity
of its matter, gravitates towards the earth; that, on the other hand,
our sea gravitates towards the moon; and all the planets mutually one
towards another; and the comets in like manner towards the sun; we
must, in consequence of this rule, universally allow that all bodies
whatsoever are endowed with a principle of mutual gravitation. For the
argument from the appearances concludes with more force for the
universal gravitation of all bodies than for their impenetrability; of
which, among those in the celestial regions, we have no experiments,
nor any manner of observation. Not that I affirm gravity to be
essential to bodies: by their *vis insita* I mean nothing but
their *vis inertiae*. This is immutable. Their gravity is
diminished as they recede from the earth.

*In experimental philosophy we are to look upon propositions
collected by general induction from phaenomena as accurately or
very nearly true, notwithstanding any contrary hypotheses that may
be imagined, till such time as other phaenomena occur, by which
they may either be made more accurate, or liable to exceptions.*

This rule we must follow, that the argument of induction may not be evaded by hypotheses.

*That the circumjovial planets, by radii drawn to Jupiter's
centre, describe areas proportional to the times of description;
and that their periodic times, the fixed stars being at rest, are
in the sesquiplicate proportion of their distances from, its centre.*

This we know from astronomical observations. For the orbits of these planets differ but insensibly from circles concentric to Jupiter; and their motions in those circles are found to be uniform. And all astronomers agree that their periodic times are in the sesquiplicate proportion of the semi-diameters of their orbits; and so it manifestly appears from the following table.

*The periodic times of the satellites of Jupiter.*

1^{d}.18^{h}.27′.34″. 3^{d}.13^{h}.13′42″.
7^{d}.3^{h}.42′36″. 16^{d}.16^{h}.32′9″.

*The distances of the satellites from Jupiter's centre.*

From the observations of | 1 | 2 | 3 | 4 | |

Borelli Townly by the Microm.Cassini by the TelescopeCassini by the eclip. of the satel. | 5⅔ 5,52 5 5⅔ | 8⅔ 8,78 8 9 | 14 13,47 13 14 ^{23}/_{60} | 24⅔ 24,72 23 25 ^{3}/_{10} | semi-diameter of Jupiter. |

From the periodic times | 5,667 | 9,017 | 14,384 | 25,299 |

Mr. *Pound* has determined, by the help of excellent
micrometers, the diameters of Jupiter and the elongation of its
satellites after the following manner. The greatest heliocentric
elongation of the fourth satellite from Jupiter's centre was taken
with a micrometer in a 15 feet telescope, and at the mean distance of
Jupiter from the earth was found about 8′ 16″. The elongation of the
third satellite was taken with a micrometer in a telescope of 123
feet, and at the same distance of Jupiter from the earth was found 4′
42″. The greatest elongations of the other satellites, at the same
distance of Jupiter from the earth, are found from the periodic times
to be 2′ 56″ 47‴, and 1′ 51″ 6‴.

The diameter of Jupiter taken with the micrometer in a 123 feet
telescope several times, and reduced to Jupiter's mean distance from
the earth, proved always less than 40″, never less than 38″, generally
39″. This diameter in shorter telescopes is 40″, or 41″; for Jupiter's
light is a little dilated by the unequal refrangibility of the rays,
and this dilatation bears less ratio to the diameter of Jupiter in the
longer and more perfect telescopes than in those which are shorter and
less perfect. The times in which two
satellites, the first and the third, passed over Jupiter's body, were
observed, from the beginning of the ingress to the beginning of the
egress, and from the complete ingress to the complete egress, with the
long telescope. And from the transit of the first satellite, the
diameter of Jupiter at its mean distance from the earth came forth 37
1

8 “. and from the transit of the third
37 3

8 “. There was observed also the time
in which the shadow of the first satellite passed over Jupiter's body,
and thence the diameter of Jupiter at its mean distance from the earth
came out about 37″. Let us suppose its diameter to be 37¼″ very
nearly, and then the greatest elongations of the first, second, third,
and fourth satellite will be respectively equal to 5,965, 9,494,
15,141, and 26,63 semi-diameters of Jupiter.

*That the circumsaturnal planets, by radii drawn to Saturn's
centre, describe areas proportional to the times of description;
and that their periodic times, the fixed stars being at rest, are
in the sesquiplicate proportion of their distances from its centre.*

For, as *Cassini* from his own observations has determined,
their distances from Saturn's centre and their periodic times are as
follow.

*The periodic times of the satellites of Saturn.*

1^{d}.21^{h}.18′27″. 2^{d}.17^{h}.41′22″.
4^{d}.12h.25′12″. 15^{d}.22^{h}.41′14″. 79^{d}.7^{h}.48′00″.

*The distances of the satellites from Saturn's centre, in semi-diameters of its ring.*

From observations | 1 19 20. | 2½. | 3½. | 8. | 24. |

From the periodic times | 1,93. | 2,47. | 3,45. | 8. | 23,35. |

The greatest elongation of the fourth satellite from Saturn's centre
is commonly determined from the observations to be eight of those
semi-diameters very nearly. But the greatest elongation of this
satellite from Saturn's centre, when taken with an excellent
micrometer in Mr. *Huygens'* telescope of 123 feet, appeared
to be eight semi-diameters and 7

10 of a semi-diameter. And from this
observation and the periodic times the distances of the satellites
from Saturn's centre in semi-diameters of the ring are 2.1. 2,69.
3,75. 8,7. and 25,35. The diameter of Saturn observed in the same
telescope was found to be to the diameter of the ring as 3 to 7; and
the diameter of the ring, *May* 28-29, 1719, was found to be
43″; and thence the diameter of the ring when Saturn is at its mean
distance from the earth is 42″, and the diameter of Saturn 18″. These
things appear so in very long and excellent telescopes, because in
such telescopes the apparent magnitudes of the heavenly bodies bear a
greater proportion to the dilatation of light in the extremities of
those bodies than in shorter telescopes. If
we, then, reject all the spurious light, the diameter of Saturn will
not amount to more than 16″.

*That the five primary planets, Mercury, Venus, Mars, Jupiter,
and Saturn, with their several orbits, encompass the sun.*

That Mercury and Venus revolve about the sun, is evident from their
moon-like appearances. When they shine out with a full face, they are,
in respect of us, beyond or above the sun; when they appear half full,
they are about the same height on one side or other of the sun; when
horned, they are below or between us and the sun; and they are
sometimes, *when directly under*, seen like spots traversing
the sun's disk. That Mars surrounds the sun, is as plain from its full
face when near its conjunction with the sun, and from the gibbous
figure which it shews in its quadratures. And the same thing is
demonstrable of Jupiter and Saturn, from their appearing full in all
situations; for the shadows of their satellites that appear sometimes
upon their disks make it plain that the light they shine with is not
their own, but borrowed from the sun.

*That the fixed stars being at rest, the periodic times of the
five primary planets, and (whether of the sun, about the earth,
or) of the earth about the sun, are in the sesquiplicate
proportion of their mean distances from the sun.*

This proportion, first observed by *Kepler*, is now received
by all astronomers; for the periodic times are the same, and the
dimensions of the orbits are the same, whether the sun revolves about
the earth, or the earth about the sun. And as to the measures of the
periodic times, all astronomers are agreed about them. But for the
dimensions of the orbits, *Kepler* and *Bullialdus*,
above all others, have determined them from observations with the
greatest accuracy; and the mean distances corresponding to the
periodic times differ but insensibly from those which they have
assigned, and for the most part fall in between them; as we may see
from the following table.

*The periodic times with respect to the fixed stars, of the
planets and earth* revolving *about the sun, in days and
decimal parts of a day.*

♄ | ♃ | ♂ | ♁ | ♀ | ☿ |

10759,275. | 4332,514. | 686,9785. | 365,2565. | 224,6176. | 87,9692. |

*The mean distances of the planets and of the earth from the sun.*

♄ | ♃ | ♂ | |

According to Kepler | 951000. | 519650. | 152350. |

According to Bullialdus | 954198. | 522520. | 152350. |

According to the periodic times | 954006. | 520096. | 152369 |

♁ | ♀ | ☿ | |

According to Kepler | 100000. | 72400. | 38806. |

According to Bullialdus | 100000. | 72398. | 38585. |

According to the periodic times | 100000. | 72333. | 38710 |

As to Mercury and Venus, there can be no doubt about their distances from the sun; for they are determined by the elongations of those planets from the sun; and for the distances of the superior planets, all dispute is cut off by the eclipses of the satellites of Jupiter. For by those eclipses the position of the shadow which Jupiter projects is determined; whence we have the heliocentric longitude of Jupiter. And from its heliocentric and geocentric longitudes compared together, we determine its distance.

*Then the primary planets, by radii drawn to the earth, describe
areas no wise proportional to the times; but that the areas which
they describe by radii drawn to the sun are proportional to the
times of description.*

For to the earth they appear sometimes direct, sometimes stationary, nay, and sometimes retrograde. But from the sun they are always seen direct, and to proceed with a motion nearly uniform, that is to say, a little swifter in the perihelion and a little slower in the aphelion distances, so as to maintain an equality in the description of the areas. This a noted proposition among astronomers, and particularly demonstrable in Jupiter, from the eclipses of his satellites; by the help of which eclipses, as we have said, the heliocentric longitudes of that planet, and its distances from the sun, are determined.

*That the moon, by a radius drawn to the earth's centre,
describes an area proportional to the time of description.*

This we gather from the apparent motion of the moon, compared with its apparent diameter. It is true that the motion of the moon is a little disturbed by the action of the sun: but in laying down these Phenomena I neglect those small and inconsiderable errors.

*That the forces by which the circumjovial planets are continually
drawn off from rectilinear motions, and retained in their proper
orbits, tend to Jupiter's centre; and are reciprocally as the
squares of the distances of the places of those planets from that centre.*

The former part of this Proposition appears from Phaen. I, and Prop. II or III, Book I; the latter from Phaen. I, and Cor. 6, Prop. IV, of the same Book.

The same thing we are to understand of the planets which encompass Saturn, by Phaen. II.

*That the forces by which the primary planets are continually
drawn off from rectilinear motions, and retained in their proper
orbits, tend to the sun; and are reciprocally as the squares of the
distances of the places of those planets from the suits centre.*

The former part of the Proposition is manifest from Phaen. V, and Prop.
II, Book I; the latter from Phaen. IV, and *Cor*. 6, Prop. IV,
of the same Book. But this part of the Proposition is, with great
accuracy, demonstrable from the quiescence of the aphelion points; for a
very small aberration from the *reciprocal* duplicate proportion
would (by Cor. 1, Prop. XLV, Book I) produce a motion of the apsides
sensible enough in every single revolution, and in many of them
enormously great.

*That the force by which the moon is retained in its orbit tends
to the earth; and is reciprocally as the square of the distance of
its place from the earth's centre.*

The former part of the Proposition is evident from Phaen. VI, and Prop.
II or III, Book I; the latter from the very slow motion of the moon's
apogee; which in every single revolution amounting but to 3° 3′ *in
consequentia*, may be neglected. For (by Cor. 1. Prop. XLV, Book
I) it appears, that, if the distance of the moon from the earth's centre
is to the semi-diameter of the earth as D to 1, the force, from which
such a motion will result, is reciprocally as D² ^{4}/_{243},
i. e., reciprocally as the power of D, whose exponent is 2^{4}/_{243};
that is to say, in the proportion of the distance something greater than
reciprocally duplicate, but which comes 59¾ times nearer to the
duplicate than to the triplicate proportion. But in regard that this
motion is owing to the action of the sun (as we shall afterwards
shew), it is here to be neglected. The action of the
sun, attracting the moon from the earth, is nearly as the moon's
distance from the earth; and therefore (by what we have shewed in Cor.
2, Prop. XLV, Book I) is to the centripetal force of the moon as 2 to
357,45, or nearly so; that is, as 1 to 178^{29}/_{40}.
And if we neglect so inconsiderable a force of the sun, the remaining
force, by which the moon is retained in its orb, will be reciprocally as
D². This will yet more fully appear from comparing this force with the
force of gravity, as is done in the next Proposition.

Cor. If we augment the mean centripetal force
by which the moon is retained in its orb, first in the proportion of 177
^{29}/_{40} to 178^{29}/_{40},
and then in the duplicate proportion of the semi-diameter of the earth
to the mean distance of the centres of the moon and earth, we shall have
the centripetal force of the moon at the surface of the earth; supposing
this force, in descending to the earth's surface, continually to
increase in the reciprocal duplicate proportion of the height.

*That the moon gravitates towards the earth, and by the force of
gravity is continually drawn off from a rectilinear motion, and
retained in its orbit.*

The mean distance of the moon from the earth in the syzygies in
semi-diameters of the earth, is, according to *Ptolemy* and most
astronomers, 59; according to *Vendelin* and *Huygens*,
60; to *Copernicus*, 60⅓; to *Street*, 60^{2}/_{5};
and to *Tycho*, 56½. But *Tycho*, and all that follow
his tables of refraction, making the refractions of the sun and moon
(altogether against the nature of light) to exceed the refractions of
the fixed stars, and that by four or five minutes *near the horizon*,
did thereby increase the moon's *horizontal* parallax by a like
number of minutes, that is, by a twelfth or fifteenth part of the whole
parallax. Correct this error, and the distance will become about 60½
semi-diameters of the earth, near to what others have assigned. Let us
assume the mean distance of 60 diameters in the syzygies; and suppose
one revolution of the moon, in respect of the fixed stars, to be
completed in 27^{d}.7^{h}.43′, as astronomers have
determined; and the circumference of the earth to amount to 123249600 *Paris*
feet, as the French have found by mensuration. And now if we imagine the
moon, deprived of all motion, to be let go, so as to descend towards the
earth with the impulse of all that force by which (by Cor. Prop. III) it
is retained in its orb, it will in the space of one minute of time,
describe in its fall 15^{1}/_{12}
*Paris* feet. This we gather by a calculus, founded either upon
Prop. XXXVI, Book I, or (which comes to the same thing) upon Cor. 9,
Prop. IV, of the same Book. For the versed sine of that arc, which the
moon, in the space of one minute of time, would by its mean motion
describe at the distance of 60 semi-diameters of the earth, is nearly 15
^{1}/_{12} *Paris* feet,
or more accurately 15 feet, 1 inch, and 1 line ^{4}/_{9}.
Where fore, since that force, in approaching to the earth, increases in
the reciprocal duplicate proportion of the distance, and, upon that
account, at the surface of the earth, is 60 × 60 times greater than at
the moon, a body in our regions, falling with that force, ought in the
space of one minute of time, to describe 60 × 60 × 15^{1}/_{12}
*Paris* feet; and, in the space of one second of time, to
describe 15^{1}/_{12} of those
feet; or more accurately 15 feet, 1 inch, and 1 line ^{4}/_{9}.
And with this very force we actually find that bodies here upon earth do
really descend; for a pendulum oscillating seconds in the latitude of
Paris will be 3 Paris feet, and 8 lines ½ in length, as Mr. *Huygens*
has observed. And the space which a heavy body describes by falling in
one second of time is to half the length of this pendulum in the
duplicate ratio of the circumference of a circle to its diameter (as Mr.
*Huygens* has also shewn), and is therefore 15 *Paris*
feet, 1 inch, 1 line ^{7}/_{9}.
And therefore the force by which the moon is retained in its orbit
becomes, at the very surface of the earth, equal to the force of gravity
which we observe in heavy bodies there. And therefore (by Rule I and II)
the force by which the moon is retained in its orbit is that very same
force which we commonly call gravity; for, were gravity another force
different from that, then bodies descending to the earth with the joint
impulse of both forces would fall with a double velocity, and in the
space of one second of time would describe 30^{1}/_{6}
*Paris* feet; altogether against experience.

This calculus is founded on the hypothesis of the earth's standing still; for if both earth and moon move about the sun, and at the same time about their common centre of gravity, the distance of the centres of the moon and earth from one another will be 60½ semi-diameters of the earth; as may be found by a computation from Prop. LX, Book I.

The demonstration of this Proposition may be more diffusely explained
after the following manner. Suppose several moons to revolve about the
earth, as in the system of Jupiter or Saturn: the periodic times of
these moons (by the argument of induction) would observe the same law
which *Kepler* found to obtain among the planets; and therefore
their centripetal forces would be reciprocally as the squares of the
distances from the centre of the earth, by Prop. I, of this Book. Now if
the lowest of these were very small, and were so near the earth as
almost to touch the tops of the highest mountains, the centripetal force
thereof, retaining it in its orb, would be very nearly equal to the
weights of any *terrestrial* bodies that should be found upon
the tops of those mountains, as may be known by the foregoing
computation. Therefore if the same little moon should be deserted by its
centrifugal force that carries it through its orb; and so be disabled
from going onward therein, it would descend to the earth; and that with
the same velocity as heavy bodies do actually fall with upon the tops of
those very mountains; because of the equality of the forces that oblige
them both to descend. And if the force by which that lowest moon would
descend were different from gravity, and if that moon were to gravitate
towards the earth, as we find terrestrial bodies do upon the tops of
mountains, it would then descend with twice the velocity, as being impel
led by both these forces conspiring together. Therefore since both these
forces, that is, the gravity of heavy bodies, and the centripetal forces
of the moons, respect the centre of the earth, and are similar and equal
between themselves, they will (by Rule I and II) have one and the same
cause. And therefore the force which retains the moon in its orbit is
that very force which we commonly call gravity; because otherwise this
little moon at the top of a mountain must either be without gravity, or
fall twice as swiftly as heavy bodies are wont to do.

*That the circumjovial planets gravitate towards Jupiter; the
circumsaturnal towards Saturn; the circumsolar towards the sun; and
by the forces of their gravity are drawn off from rectilinear
motions, and retained in curvilinear orbits.*

For the revolutions of the circumjovial planets about Jupiter, of the circumsaturnal about Saturn, and of Mercury and Venus, and the other circumsolar planets, about the sun, are appearances of the same sort with the revolution of the moon about the earth; and therefore, by Rule II, must be owing to the same sort of causes; especially since it has been demonstrated, that the forces upon which those revolutions depend tend to the centres of Jupiter, of Saturn, and of the sun; and that those forces, in receding from Jupiter, from Saturn, and from the sun, decrease in the same proportion, and according to the same law, as the force of gravity does in receding from the earth.

Cor. 1. There is, therefore, a power of gravity tending to all the planets; for, doubtless, Venus, Mercury, and the rest, are bodies of the same sort with Jupiter and Saturn. And since all attraction (by Law III) is mutual, Jupiter will therefore gravitate towards all his own satellites, Saturn towards his, the earth towards the moon, and the sun towards all the primary planets.

Cor. 2. The force of gravity which tends to any one planet is reciprocally as the square of the distance of places from that planet's centre.

Cor. 3. All the planets do mutually gravitate towards one another, by Cor. 1 and 2. And hence it is that Jupiter and Saturn, when near their conjunction; by their mutual attractions sensibly disturb each other's motions. So the sun disturbs the motions of the moon; and both sun and moon disturb our sea, as we shall hereafter explain.

The force which retains the celestial bodies in their orbits has been hitherto called centripetal force; but it being now made plain that it can be no other than a gravitating force, we shall hereafter call it gravity. For the cause of that centripetal force which retains the moon in its orbit will extend itself to all the planets, by Rule I, II, and IV.

*That all bodies gravitate towards every planet; and that the
weights of bodies towards any the same planet, at equal distances
from the centre of the planet, are proportional to the quantities of
matter which they severally contain.*

It has been, now of a long time, observed by others, that all sorts of
heavy bodies (allowance being made for the inequality of retardation
which they suffer from a small power of resistance in the air) descend
to the earth *from equal heights* in equal times; and that
equality of times we may distinguish to a great accuracy, by the help of
pendulums. I tried the thing in gold, silver, lead, glass, sand, common
salt, wood, water, and wheat. I provided two wooden boxes, round and
equal: I filled the one with wood, and suspended an equal weight of gold
(as exactly as I could) in the centre of oscillation of the other. The
boxes hanging by equal threads of 11 feet made a couple of pendulums
perfectly equal in weight and figure, and equally receiving the
resistance of the air. And, placing the one by the other, I observed
them to play together forward and backward, for a long time, with equal
vibrations. And therefore the quantity of matter in the gold (by Cor. 1
and 6, Prop. XXIV, Book II) was to the quantity of matter in the wood as
the action of the motive force (or *vis motrix*) upon all the
gold to the action of the same upon all the wood: that is, as the weight
of the one to the weight of the other: and the like happened in the
other bodies. By these experiments, in bodies of the same weight, I
could manifestly have discovered a difference of matter less than the
thousandth part of the whole, had any such been. But, without all doubt,
the nature of gravity towards the planets is the same as towards the
earth. For, should we imagine our terrestrial bodies removed to the orb
of the moon, and there, together with the moon, deprived of all motion,
to be let go, so as to fall together towards the earth, it is certain,
from what we have demonstrated before, that, in equal times, they would
describe equal spaces with the moon, and of consequence are to the moon,
in quantity of matter, as their weights to its weight. Moreover, since
the satellites of Jupiter perform their
revolutions in times which observe the sesquiplicate proportion of their
distances from Jupiter's centre, their accelerative gravities towards
Jupiter will be reciprocally as the squares of their distances from
Jupiter's centre; that is, equal, at equal distances. And, therefore,
these satellites, if supposed to fall *towards Jupiter* from
equal heights, would describe equal spaces in equal times, in like
manner as heavy bodies do on our earth. And, by the same argument, if
the circumsolar planets were supposed to be let fall at equal distances
from the sun, they would, in their descent towards the sun, describe
equal spaces in equal times. But forces which equally accelerate unequal
bodies must be as those bodies: that is to say, the weights of the
planets *towards the sun*, must be as their quantities of
matter. Further, that the weights of Jupiter and of his satellites
towards the sun are proportional to the several quantities of their
matter, appears from the exceedingly regular motions of the satellites
(by Cor. 3, Prop. LXV, Book 1). For if some of those bodies were more
strongly attracted to the sun in proportion to their quantity of matter
than others, the motions of the satellites would be disturbed by that
inequality of attraction (by Cor. 2, Prop. LXV, Book I). If, at equal
distances from the sun, any satellite, in proportion to the quantity of
its matter, did gravitate towards the sun with a force greater than
Jupiter in proportion to his, according to any given proportion, suppose
of *d* to *e*; then the distance between the centres of
the sun and of the satellite's orbit would be always greater than the
distance between the centres of the sun and of Jupiter nearly in the
subduplicate of that proportion: as by some computations I have found.
And if the satellite did gravitate towards the sun with a force, lesser
in the proportion of *e* to *d*, the distance of the
centre of the satellite's orb from the sun would be less than the
distance of the centre of Jupiter from the sun in the subduplicate of
the same proportion. Therefore if, at equal distances from the sun, the
accelerative gravity of any satellite towards the sun were greater or
less than the accelerative gravity of Jupiter towards the sun but by one
^{1}/_{1000} part of the whole
gravity, the distance of the centre of the satellite's orbit from the
sun would be greater or less than the distance of Jupiter from the sun
by one ^{1}/_{2000} part of
the whole distance; that is, by a fifth part of the distance of the
utmost satellite from the centre of Jupiter; an eccentricity of the
orbit which would be very sensible. But the orbits of the satellites are
concentric to Jupiter, and therefore the accelerative gravities of
Jupiter, and of all its satellites towards the sun, are equal among
themselves. And by the same argument, the weights of Saturn and of his
satellites towards the sun, at equal distances from the sun, are as
their several quantities of matter; and the weights of the moon and of
the earth towards the sun are either none, or accurately proportional to
the masses of matter which they contain. But some they are, by Cor. 1
and 3, Prop. V.

But further; the weights of all the parts of every planet towards any other planet are one to another as the matter in the several parts; for if some parts did gravitate more, others less, than for the quantity of their matter, then the whole planet, according to the sort of parts with which it most abounds, would gravitate more or less than in proportion to the quantity of matter in the whole. Nor is it of any moment whether these parts are external or internal; for if, for example, we should imagine the terrestrial bodies with us to be raised up to the orb of the moon, to be there compared with its body: if the weights of such bodies were to the weights of the external parts of the moon as the quantities of matter in the one and in the other respectively; but to the weights of the internal parts in a greater or less proportion, then likewise the weights of those bodies would be to the weight of the whole moon in a greater or less proportion; against what we have shewed above.

Cor. 1. Hence the weights of bodies do not depend upon their forms and textures; for if the weights could be altered with the forms, they would be greater or less, according to the variety of forms, in equal matter; altogether against experience.

Cor. 2. Universally, all bodies about the earth
gravitate towards the earth; and the weights of all, at equal distances
from the earth's centre, are as the quantities of matter which they
severally contain. This is the quality of all bodies within the reach of
our experiments; and therefore (by Rule III) to be affirmed of all
bodies whatsoever. If the *aether*, or any other body, were
either altogether void of gravity, or were to gravitate less in
proportion to its quantity of matter, then, because (according to *Aristotle*,
*Des Cartes*, and others) there is no diiference betwixt that and
other bodies but in *mere* form of matter, by a successive
change from form to form, it might be changed at last into a body of the
same condition with those which gravitate most in proportion to their
quantity of matter; and, on the other hand, the heaviest bodies,
acquiring the first form of that body, might by degrees quite lose their
gravity. And therefore the weights would depend upon the forms of
bodies, and with those forms might be changed: contrary to what was
proved in the preceding Corollary.

Cor. 3. All spaces are not equally full; for if all spaces were equally full, then the specific gravity of the fluid which fills the region of the air, on account of the extreme density of the matter, would fall nothing short of the specific gravity of quicksilver, or gold, or any other the most dense body; and, therefore, neither gold, nor any other body, could descend in air; for bodies do not descend in fluids, unless they are specifically heavier than the fluids. And if the quantity of matter in a given space can, by any rarefaction, be diminished, what should hinder a diminution to infinity?

Cor. 4. If all the solid particles of all
bodies are of the same density, nor can be rarefied without pores, a
void, space, or vacuum must be granted. By
bodies of the same density, I mean those whose *vires inertiae*,
are in the proportion of their bulks.

Cor. 5. The power of gravity is of a different nature from the power of magnetism; for the magnetic attraction is not as the matter attracted. Some bodies are attracted more by the magnet; others less; most bodies not at all. The power of magnetism in one and the same body may be increased and diminished; and is sometimes far stronger, for the quantity of matter, than the power of gravity; and in receding from the magnet decreases not in the duplicate but almost in the triplicate proportion of the distance, as nearly as I could judge from some rude observations.

*That there is a power of gravity tending to all bodies,
proportional to the several quantities of matter which they contain.*

That all the planets mutually gravitate one towards another, we have proved before; as well as that the force of gravity towards every one of them, considered apart, is reciprocally as the square of the distance of places from the centre of the planet. And thence (by Prop. LXIX, Book I, and its Corollaries) it follows, that the gravity tending towards all the planets is proportional to the matter which they contain.

Moreover, since all the parts of any planet A gravitate towards any other planet B; and the gravity of every part is to the gravity of the whole as the matter of the part to the matter of the whole; and (by Law III) to every action corresponds an equal re-action; therefore the planet B will, on the other hand, gravitate towards all the parts of the planet A; and its gravity towards any one part will be to the gravity towards the whole as the matter of the part to the matter of the whole. Q.E.D.

Cor. 1. Therefore the force of gravity towards
any whole planet arises from, and is compounded of, the forces of
gravity towards all its parts. Magnetic and electric attractions afford
us examples of this; for all attraction towards the whole arises from
the attractions towards the several parts. The thing may be easily
understood in gravity, if we consider a greater planet, as formed of a
number of lesser planets, meeting together in one globe; for *hence
it would appear that* the force of the whole must arise from the
forces of the component parts. If it is objected, that, according to
this law, all bodies with us must mutually gravitate one towards
another, whereas no such gravitation any where appears, I answer, that
since the gravitation towards these bodies is to the gravitation towards
the whole earth as these bodies are to the whole earth, the gravitation
towards them must be far less than to fall under the observation of our
senses.

Cor. 2. The force of gravity towards the several equal particles of any body is reciprocally as the square of the distance of places from the particles; as appears from Cor. 3, Prop. LXXIV, Book I.

*In two spheres mutually gravitating each towards the other, if
the matter in places on all sides round about and equi-distant from
the centres is similar, the weight of either sphere towards the
other will be reciprocally as the square of the distance between their centres.*

After I had found that the force of gravity towards a whole planet did arise from and was compounded of the forces of gravity towards all its parts, and towards every one part was in the reciprocal proportion of the squares of the distances from the part, I was yet in doubt whether that reciprocal duplicate proportion did accurately hold, or but nearly so, in the total force compounded of so many partial ones; for it might be that the proportion which accurately enough took place in greater distances should be wide of the truth near the surface of the planet, where the distances of the particles are unequal, and their situation dissimilar. But by the help of Prop. LXXV and LXXVI, Book I, and their Corollaries, I was at last satisfied of the truth of the Proposition, as it now lies before us.

Cor. 1. Hence we may find and compare together
the weights of bodies towards different planets; for the weights of
bodies revolving in circles about planets are (by Cor. 2, Prop. IV, Book
I) as the diameters of the circles directly, and the squares of their
periodic times reciprocally; and their weights at the surfaces of the
planets, or at any other distances from their centres, are (by this
Prop.) greater or less in the reciprocal duplicate proportion of the
distances. Thus from the periodic times of Venus, revolving about the
sun, in 224^{d}.16¾^{h}, of the utmost circumjovial
satellite revolving about Jupiter, in 16^{d}.16^{8}/_{15}^{h}.;
of the Huygenian satellite about Saturn in 15^{d}.22⅔^{h}.;
and of the moon about the earth in 27^{d}.7^{h}.43′;
compared with the mean distance of Venus from the sun, and with the
greatest heliocentric elongations of the outmost circumjovial satellite
from Jupiter's centre, 8′ 16″; of the Huygenian satellite from the
centre of Saturn, 3′4″; and of the moon from the earth, 10′33″: by
computation I found that the weight of equal bodies, at equal distances
from the centres of the sun, of Jupiter, of Saturn, and of the earth,
towards the sun, Jupiter, Saturn, and the earth, were one to another, as
1, ^{1}/_{1067}, ^{1}/_{3021},
and ^{1}/_{169282}
respectively. Then because as the distances are increased or diminished,
the weights are diminished or increased in a duplicate ratio, the
weights of equal bodies towards the sun, Jupiter, Saturn, and the earth,
at the distances 10000, 997, 791, and 109 from their centres, that is,
at their very superficies, will be as 10000, 943, 529, and 435
respectively. How much the weights of bodies are at the superficies of
the moon, will be shewn hereafter.

Cor. 2. Hence likewise we discover the quantity
of matter in the several planets; for their
quantities of matter are as the forces of gravity at equal distances
from their centres; that is, in the sun, Jupiter, Saturn, and the earth,
as 1, ^{1}/_{1067}, ^{1}/_{3021}
and ^{1}/_{169282}
respectively. If the parallax of the sun be taken greater or less than
10″ 30‴, the quantity of matter in the earth must be augmented or
diminished in the triplicate of that proportion.

Cor. 3. Hence also we find the densities of the planets; for (by Prop. LXXII, Book I) the weights of equal and similar bodies towards similar spheres are, at the surfaces of those spheres, as the diameters of the spheres and therefore the densities of dissimilar spheres are as those weights applied to the diameters of the spheres. But the true diameters of the Sun, Jupiter, Saturn, and the earth, were one to another as 10000, 997, 791, and 109; and the weights towards the same as 10000, 943, 529, and 435 respectively; and therefore their densities are as 100, 94½, 67, and 400. The density of the earth, which comes out by this computation, does not depend upon the parallax of the sun, but is determined by the parallax of the moon, and therefore is here truly defined. The sun, therefore, is a little denser than Jupiter, and Jupiter than Saturn, and the earth four times denser than the sun; for the sun, by its great heat, is kept in a sort of a rarefied state. The moon is denser than the earth, as shall appear afterward.

Cor. 4. The smaller the planets are, they are,
*caeteris paribus*, of so much the greater density; for so the
powers of gravity on their several surfaces come nearer to equality.
They are likewise, *caeteris paribus*, of the greater density,
as they are nearer to the sun. So Jupiter is more dense than Saturn, and
the earth than Jupiter; for the planets were to be placed at different
distances from the sun, that, according to their degrees of density,
they might enjoy a greater or less proportion to the sun's heat. Our
water, if it were removed as far as the orb of Saturn, would be
converted into ice, and in the orb of Mercury would quickly fly away in
vapour; for the light of the sun, to which its heat is proportional, is
seven times denser in the orb of Mercury than with us: and by the
thermometer I have found that a sevenfold heat of our summer sun will
make water boil. Nor are we to doubt that the matter of Mercury is
adapted to its heat, and is therefore more dense than the matter of our
earth; since, in a denser matter, the operations of Nature require a
stronger heat.

*That the force of gravity, considered downward from the surface
of the planets, decreases nearly in the proportion of the distances from their centres.*

If the matter of the planet were of an uniform density, this Proposition would be accurately true (by Prop. LXXIII. Book I). The error, therefore, can be no greater than what may arise from the inequality of the density.

*That the motions of the planets in the heavens may subsist an exceedingly long time.*

In the Scholium of Prop. XL, Book II, I have shewed that a globe of
water frozen into ice, and moving freely in our air, in the time that it
would describe the length of its semi-diameter, would lose by the
resistance of the air ^{1}/_{4586}
part of its motion; and the same proportion holds nearly in all globes,
how great soever, and moved with whatever velocity. But that our globe
of earth is of greater density than it would be if the whole consisted
of water only, I thus make out. If the whole consisted of water only,
whatever was of less density than water, because of its less specific
gravity, would emerge and float above. And upon this account, if a globe
of terrestrial matter, covered on all sides with water, was less dense
than water, it would emerge somewhere; and, the subsiding water falling
back, would be gathered to the opposite side. And such is the condition
of our earth, which in a great measure is covered with seas. The earth,
if it was not for its greater density, would emerge from the seas, and,
according to its degree of levity, would be raised more or less above
their surface, the water of the seas flowing backward to the opposite
side. By the same argument, the spots of the sun, which float upon the
lucid matter thereof, are lighter than that matter; and, however the
planets have been formed while they were yet in fluid masses, all the
heavier matter subsided to the centre. Since, therefore, the common
matter of our earth on the surface thereof is about twice as heavy as
water, and a little lower, in mines, is found about three, or four, or
even five times more heavy, it is probable that the quantity of the
whole matter of the earth may be five or six times greater than if it
consisted all of water; especially since I have before shewed that the
earth is about four times more dense than Jupiter. If, therefore,
Jupiter is a little more dense than water, in the space of thirty days,
in which that planet describes the length of 459 of its semi-diameters,
it would, in a medium of the same density with our air, lose almost a
tenth part of its motion. But since the resistance of mediums decreases
in proportion to their weight or density, so that water, which is 13
^{3}/_{5} times lighter than
quicksilver, resists less in that proportion; and air, which is 860
times lighter than water, resists less in the same proportion; therefore
in the heavens, where the weight of the medium in which the planets move
is immensely diminished, the resistance will almost vanish.

It is shewn in the Scholium of Prop. XXII, Book II, that at the height of 200 miles above the earth the air is more rare than it is at the superficies of the earth in the ratio of 30 to 0,0000000000003998, or as 75000000000000 to 1 nearly. And hence the planet Jupiter, revolving in a medium of the same density with that superior air, would not lose by the resistance of the medium the 1000000th part of its motion in 1000000 years. In the spaces near the earth the resistance is produced only by the air, exhalations, and vapours. When these are carefully exhausted by the air-pump from under the receiver, heavy bodies fall within the receiver with perfect freedom, and without the least sensible resistance: gold itself, and the lightest down, let fall together, will descend with equal velocity; and though they fall through a space of four, six, and eight feet, they will come to the bottom at the same time; as appears from experiments. And therefore the celestial regions being perfectly void of air and exhalations, the planets and comets meeting no sensible resistance in those spaces will continue their motions through them for an immense tract of time.

*That the centre of the system of the world is immovable.*

This is acknowledged by all, while some contend that the earth, others that the sun, is fixed in that centre. Let us see what may from hence follow.

*That the common centre of gravity of the earth, the sun, and all the planets, is immovable.*

For (by Cor. 4 of the Laws) that centre either is at rest, or moves uniformly forward in a right line; but if that centre moved, the centre of the world would move also, against the Hypothesis.

*That the sun is agitated by a perpetual motion, but never recedes
far from the common centre of gravity of all the planets.*

For since (by Cor. 2, Prop. VIII) the quantity of matter in the sun is to the quantity of matter in Jupiter as 1067 to 1; and the distance of Jupiter from the sun is to the semi-diameter of the sun in a proportion but a small matter greater, the common centre of gravity of Jupiter and the sun will fall upon a point a little without the surface of the sun. By the same argument, since the quantity of matter in the sun is to the quantity of matter in Saturn as 3021 to 1, and the distance of Saturn from the sun is to the semi-diameter of the sun in a proportion but a small matter less, the common centre of gravity of Saturn and the sun will fall upon a point a little within the surface of the sun. And, pursuing the principles of this computation, we should find that though the earth and all the planets were placed on one side of the sun, the distance of the common centre of gravity of all from the centre of the sun would scarcely amount to one diameter of the sun. In other cases, the distances of those centres are always less; and therefore, since that centre of gravity is in perpetual rest, the sun, according to the various positions of the planets, must perpetually be moved every way, but will never recede far from that centre.

Cor. Hence the common centre of gravity of the earth, the sun, and all the planets, is to be esteemed the centre of the world; for since the earth, the sun, and all the planets, mutually gravitate one towards another, and are therefore, according to their powers of gravity, in perpetual agitation, as the Laws of Motion require, it is plain that their moveable centres can not be taken for the immovable centre of the world. If that body were to be placed in the centre, towards which other bodies gravitate most (according to common opinion), that privilege ought to be allowed to the sun; but since the sun itself is moved, a fixed point is to be chosen from which the centre of the sun recedes least, and from which it would recede yet less if the body of the sun were denser and greater, and therefore less apt to be moved.

*The planets move in ellipses which have their common focus in the
centre of the sun; and, by radii drawn to that centre, they describe
areas proportional to the times of description.*

We have discoursed above of these motions from the Phaenomena. Now that
we know the principles on which they depend, from those principles we
deduce the motions of the heavens *à priori*. Because the
weights of the planets towards the sun are reciprocally as the squares
of their distances from the sun's centre, if the sun was at rest, and
the other planets did not mutually act one upon another, their orbits
would be ellipses, having the sun in their common focus; and they would
describe areas proportional to the times of *description*, by
Prop, I and XI, and Cor. 1, Prop. XIII, Book I. But the mutual actions
of the planets one upon another are so very small, that they may be
neglected; and by Prop. LXVI, Book I, they less disturb the motions of
the planets around the sun in motion than if those motions were
performed about the sun at rest.

It is true, that the action of Jupiter upon Saturn is not to be
neglected; for the force of gravity towards Jupiter is to the force of
gravity towards the sun (at equal distances, Cor. 2, Prop. VIII) as 1 to
1067; and therefore in the conjunction of Jupiter and Saturn, because
the distance of Saturn from Jupiter is to the distance of Saturn from
the sun almost as 4 to 9, the gravity of Saturn towards Jupiter will be
to the gravity of Saturn towards the sun as 81 to 16 × 1067; or, as 1 to
about 211. And hence arises a perturbation of the orb of Saturn in every
conjunction of this planet with Jupiter, so sensible, that astronomers
are puzzled with it. As the planet is
differently situated in these conjunctions, its eccentricity is
sometimes augmented, sometimes diminished; its aphelion is sometimes
carried forward, sometimes backward, and its mean motion is by turns
accelerated and retarded; yet the whole error in its motion about the
sun, though arising from so great a force, may be almost avoided (except
in the mean motion) by placing the lower focus of its orbit in the
common centre of gravity of Jupiter and the sun (according to Prop.
LXVII, Book I), and therefore that error, when it is greatest, scarcely
exceeds two minutes; and the greatest error in the mean motion scarcely
exceeds two minutes yearly. But in the conjunction of Jupiter and
Saturn, the accelerative forces of gravity of the sun towards Saturn, of
Jupiter towards Saturn, and of Jupiter towards the sun, are almost as
16,81, and 16 × 81 × 3021

25; or 156609: and therefore the
difference of the forces of gravity of the sun towards Saturn, and of
Jupiter towards Saturn, is to the force of gravity of Jupiter towards
the sun as 65 to 156609, or as 1 to 2409. But the greatest power of
Saturn to disturb the motion of Jupiter is proportional to this
difference; and therefore the perturbation of the orbit of Jupiter is
much less than that of Saturn's. The perturbations of the other orbits
are yet far less, except that the orbit of the earth is sensibly
disturbed by the moon. The common centre of gravity of the earth and
moon moves in an ellipsis about the sun in the focus thereof, and, by a
radius drawn to the sun, describes areas proportional to the times of
description. But the earth in the mean time by a menstrual motion is
revolved about this common centre.

*The aphelions and nodes of the orbits of the planets are fixed.*

The aphelions are immovable by Prop. XI, Book I; and so are the planes of the orbits, by Prop. I of the same Book. And if the planes are fixed, the nodes must be so too. It is true, that some inequalities may arise from the mutual actions of the planets and comets in their revolutions; but these will be so small, that they may be here passed by.

Cor. 1. The fixed stars are immovable, seeing they keep the same position to the aphelions and nodes of the planets.

Cor. 2. And since these stars are liable to no sensible parallax from the annual motion of the earth, they can have no force, because of their immense distance, to produce any sensible effect in our system. Not to mention that the fixed stars, every where promiscuously dispersed in the heavens, by their contrary attractions destroy their mutual actions, by Prop. LXX, Book I.

Since the planets near the sun (viz. Mercury, Venus, the Earth, and
Mars) are so small that they can act with but little
force upon each other, therefore their aphelions and nodes must be
fixed, excepting in so far as they are disturbed by the actions of
Jupiter and Saturn, and other higher bodies. And hence we may find, by
the theory of gravity, that their aphelions move a little *in
consequentia*, in respect of the fixed stars, and that in the
sesquiplicate proportion of their several distances from the sun. So
that if the aphelion of Mars, in the space of a hundred years, is
carried 33′ 20″ *in consequentia*, in respect of the fixed
stars; the aphelions of the Earth, of Venus, and of Mercury, will in a
hundred years be carried forwards 17′ 40″, 10′ 53″, and 4′ 16″,
respectively. But these motions are so inconsiderable, that we have
neglected them in this Proposition,

*To find the principal diameters of the orbits of the planets.*

They are to be taken in the sub-sesquiplicate proportion of the periodic times, by Prop. XV, Book I, and then to be severally augmented in the proportion of the sum of the masses of matter in the sun and each planet to the first of two mean proportionals betwixt that sum and the quantity of matter in the sun, by Prop. LX, Book I.

*To find the eccentricities and aphelions of the planets.*

This Problem is resolved by Prop. XVIII, Book I.

*That the diurnal motions of the planets are uniform, and that the
libration of the moon arises from its diurnal motion.*

The Proposition is proved from the first Law of Motion, and Cor. 22,
Prop. LXVI, Book I. Jupiter, with respect to the fixed stars, revolves
in 9^{h}.56′; Mars in 24^{h}.39′; Venus in about 23^{h}.;
the Earth in 23^{h}.56′; the Sun in 25½ days, and the moon in 27
days, 7 hours, 43′. These things appear by the Phaenomena. The spots in
the sun's body return to the same situation on the sun's disk, with
respect to the earth, in 27½ days; and therefore with respect to the
fixed stars the sun revolves in about 25½ days. But because the lunar
day, arising from its uniform revolution about its axis, is menstrual, *that
is, equal to the time of its periodic revolution in its orb*,
therefore the same face of the moon will be always nearly turned to the
upper focus of its orb; but, as the situation of that focus requires,
will deviate a little to one side and to the other from the earth in the
lower focus; and this is the libration in longitude; for the libration
in latitude arises from the moon's latitude, and the inclination of its
axis to the plane of the ecliptic. This theory of the libration of the
moon, Mr. *N. Mercator* in his
Astronomy, published at the beginning of the year 1676, explained more
fully out of the letters I sent him. The utmost satellite of Saturn
seems to revolve about its axis with a motion like this of the moon,
respecting Saturn continually with the same face; for in its revolution
round Saturn, as often as it comes to the eastern part of its orbit, it
is scarcely visible, and generally quite disappears; which is like to be
occasioned by some spots in that part of its body, which is then turned
towards the earth, as M. *Cassini* has observed. So also the
utmost satellite of Jupiter seems to revolve about its axis with a like
motion, because in that part of its body which is turned from Jupiter it
has a spot, which always appears as if it were in Jupiter's own body,
whenever the satellite passes between Jupiter and our eye.

*That the axes of the planets are less than the diameters drawn
perpendicular to the axes.*

The equal gravitation of the parts on all sides would give a spherical figure to the planets, if it was not for their diurnal revolution in a circle. By that circular motion it comes to pass that the parts receding from the axis endeavour to ascend about the equator; and therefore if the matter is in a fluid state, by its ascent towards the equator it will enlarge the diameters there, and by its descent towards the poles it will shorten the axis. So the diameter of Jupiter (by the concurring observations of astronomers) is found shorter betwixt pole and pole than from east to west. And, by the same argument, if our earth was not higher about the equator than at the poles, the seas would subside about the poles, and, rising towards the equator, would lay all things there under water.

*To find the proportion of the axis of a planet to the diameter,
perpendicular thereto.*

Our countryman, Mr. *Norwood*, measuring a distance of 905751
feet of *London* measure between *London* and *York*,
in 1635, and observing the difference of latitudes to be 2° 28′,
determined the measure of one degree to be 367196 feet of *London*
measure, that is 57300 *Paris* toises. M. *Picart*,
measuring an arc of one degree, and 22′ 55″ of the meridian between *Amiens*
and *Malvoisine*, found an arc of one degree to be 57060 *Paris*
toises. M. *Cassini*, the father, measured the distance upon the
meridian from the town of *Collioure* in *Roussillon* to
the Observatory of *Paris*; and his son added the distance from
the Observatory to the Citadel of *Dunkirk*. The whole distance
was 486156½ toises and the difference of the latitudes of *Collioure*
and *Dunkirk* was 8 degrees, and 31′ 11
^{5}/_{6}″. Hence an arc of one
degree appears to be 57061 *Paris* toises. And from these
measures we conclude that the circumference of the earth is 123249600,
and its semi-diameter 19615800 *Paris* feet, upon the
supposition that the earth is of a spherical figure.

In the latitude of *Paris* a heavy body falling in a second of
time describes 15 *Paris* feet, 1 inch, 1^{7}/_{9}
line, as above, that is, 2173 lines ^{7}/_{9}.
The weight of the body is diminished by the weight of the ambient air.
Let us suppose the weight lost thereby to be ^{1}/_{11000}
part of the whole weight; then that heavy body falling *in vacua*
will describe a height of 2174 lines in one second of time.

A body in every sidereal day of 23^{h}.56′4″ uniformly
revolving in a circle at the distance of 19615800 feet from the centre,
in one second of time describes an arc of 1433,46 feet; the versed sine
of which is 0,05236561 feet, or 7,54064 lines. And therefore the force
with which bodies descend in the latitude of *Paris* is to the
centrifugal force of bodies in the equator arising from the diurnal
motion of the earth as 2174 to 7,54064.

The centrifugal force of bodies in the equator is to the centrifugal
force with which bodies recede directly from the earth in the latitude
of *Paris* 48° 50′ 10″ in the duplicate proportion of the radius
to the cosine of the latitude, that is, as 7,54064 to 3,267. Add this
force to the force with which bodies descend by their weight in the
latitude of *Paris*, and a body, in the latitude of *Paris*,
falling by its whole undiminished force of gravity, in the time of one
second, will describe 2177,267 lines, or 15 *Paris* feet, 1
inch, and 5,267 lines. And the total force of gravity in that latitude
will be to the centrifugal force of bodies in the equator of the earth
as 2177,267 to 7,54064, or as 289 to 1.

Wherefore if APBQ represent the figure of the earth, now no longer
spherical, but generated by the rotation of an ellipsis about its lesser
axis PQ; and ACQ*qca* a canal full of water, reaching from the
pole Q*q* to the centre C*c*, and thence rising to the
equator A*a*; the weight of the water in the leg of the canal AC*ca*
will be to the weight of water in the other leg QC*cq* as 289 to
288, because the centrifugal force arising from the circular motion
sustains and takes off one of the 289 parts of the weight (in the one
leg), and the weight of 288 in the other sustains the rest. But by
computation (from Cor. 2, Prop. XCI, Book I) I find, that, if the matter
of the earth was all uniform, and without any motion, and its axis PQ
were to the diameter AB as 100 to 101, the force of gravity in the place
Q towards the earth would be to the force of gravity in the same place Q
towards a sphere described about the centre C with the radius PC, or QC,
as 126 to 125. And, by the same argument, the force of gravity in the
place A towards the spheroid generated by the rotation of the
ellipsis APBQ about the axis AB is to the force of gravity in the same
place A, towards the sphere described about the centre C with the radius
AC, as 125 to 126. But the force of gravity in the place A towards the
earth is a mean proportional betwixt the forces of gravity towards the
spheroid and this sphere; because the sphere, by having its diameter PQ
diminished in the proportion of 101 to 100, is transformed into the
figure of the earth; and this figure, by having a third diameter
perpendicular to the two diameters AB and PQ diminished in the same
proportion, is converted into the said spheroid; and the force of
gravity in A, in either case, is diminished nearly in the same
proportion. Therefore the force of gravity in A towards the sphere
described about the centre C with the radius AC, is to the force of
gravity in A towards the earth as 126 to 125½. And the force of gravity
in the place Q towards the sphere described about the centre C with the
radius QC, is to the force of gravity in the place A towards the sphere
described about the centre C, with the radius AC, in the proportion of
the diameters (by Prop. LXXII, Book I), that is, as 100 to 101. If,
therefore, we compound those three proportions 126 to 125, 126 to 125½,
and 100 to 101, into one, the force of gravity in the place Q towards
the earth will be to the force of gravity in the place A towards the
earth as 126 × 126 × 100 to 125 × 125½ × 101; or as 501 to 500.

Now since (by Cor. 3, Prop. XCI, Book I) the force of gravity in either
leg of the canal AC*ca*, or QC*cq*, is as the distance of
the places from the centre of the earth, if those legs are conceived to
be divided by transverse, parallel, and equidistant surfaces, into parts
proportional to the wholes, the weights of any number of parts in the
one leg AC*ca* will be to the weights of the same number of parts
in the other leg as their magnitudes and the accelerative forces of
their gravity conjunctly, that is, as 101 to 100, and 500 to 501, or as
505 to 501. And therefore if the centrifugal force of every part in the
leg AC*c*a, arising from the diurnal motion, was to the weight of
the same part as 4 to 505, so that from the weight of every part,
conceived to be divided into 505 parts, the centrifugal force might take
off four of those parts, the weights would remain equal in each leg, and
therefore the fluid would rest in an equilibrium. But the centrifugal
force of every part is to the weight of the same part as 1 to 289; that
is, the centrifugal force, which should be ^{4}/_{505}
parts of the weight, is only ^{1}/_{289}
part thereof. And, therefore, I say, by the rule of proportion, that if
the centrifugal force ^{4}/_{505}
make the height of the water in the leg AC*ca* to exceed the
height of the water in the leg QC*cq* by one ^{1}/_{100}
part of its whole height, the centrifugal force ^{1}/_{289}
will make the excess of the height in the leg AC*ca* only
^{1}/_{289} part of the height of
the water in the other leg QC*cq*; and therefore the diameter of
the earth at the equator, is to its diameter from pole to pole as 230 to
229. And since the mean semi-diameter of the
earth, according to *Picart's* mensuration, is 19615800 *Paris*
feet, or 3923,16 miles (reckoning 5000 feet to a mile), the earth will
be higher at the equator than at the poles by 85472 feet, or 17^{1}/_{10}
miles. And its height at the equator will be about 19658600 feet, and at
the poles 19573000 feet.

If, the density and periodic time of the diurnal revolution remaining
the same, the planet was greater or less than the earth, the proportion
of the centrifugal force to that of gravity, and therefore also of the
diameter betwixt the poles to the diameter at the equator, would
likewise remain the same. But if the diurnal motion was accelerated or
retarded in any proportion, the centrifugal force would be augmented or
diminished nearly in the same duplicate proportion; and therefore the
difference of the diameters will be increased or diminished in the same
duplicate ratio very nearly. And if the density of the planet was
augmented or diminished in any proportion, the force of gravity tending
towards it would also be augmented or diminished in the same proportion:
and the difference of the diameters contrariwise would be diminished in
proportion as the force of gravity is augmented, and augmented in
proportion as the force of gravity is diminished. Wherefore, since the
earth, in respect of the fixed stars, revolves in 23^{h}.56′,
but Jupiter in 9^{h}.56′, and the squares of their periodic
times are as 29 to 5, and their densities as 400 to 94½, the difference
of the diameters of Jupiter will be to its lesser diameter as 29

5 × 400

94^{1}/_{2} × 1

229 to 1, or as 1 to 9⅓, nearly.
Therefore the diameter of Jupiter from east to west is to its diameter
from pole to pole nearly as 10⅓ to 9⅓. Therefore since its greatest
diameter is 37″, its lesser diameter lying between the poles will be 33″
25‴. Add thereto about 3″ for the irregular refraction of light, and the
apparent diameters of this planet will become 40″ and 36″ 25‴; which are
to each other as 11^{1}/_{6}
to 10^{1}/_{6}, very nearly.
These things are so upon the supposition that the body of Jupiter is
uniformly dense. But now if its body be denser towards the plane of the
equator than towards the poles, its diameters may be to each other as 12
to 11, or 13 to 12, or perhaps as 14 to 13.

And *Cassini* observed in the year 1691, that the diameter of
Jupiter reaching from east to west is greater by about a fifteenth part
than the other diameter. Mr. *Pound* with his 123 feet
telescope, and an excellent micrometer, measured the diameters of
Jupiter in the year 1719, and found them as follow.

The Times. | Greatest diam. | Lesser diam. | The diam. to each other. | |||||

January March March April | Day. 28 6 9 9 | Hours 6 7 7 9 | Parts 13,40 13,12 13,12 12,32 | Parts 12,28 12,20 12,08 11,48 | As As As As | 12 13¾ 12⅔ 14½ | to to to to | 11 12¾ 11⅔ 13½ |

So that the theory agrees with the phaenomena; for the planets are more heated by the sun's rays towards their equators, and therefore are a little more condensed by that heat than towards their poles.

Moreover, that there is a diminution of gravity occasioned by the diurnal rotation of the earth, and therefore the earth rises higher there than it does at the poles (supposing that its matter is uniformly dense), will appear by the experiments of pendulums related under the following Proposition.

*To find and compare together the weights of bodies in the
different regions of our earth.*

Because the weights of the unequal legs of the canal of water ACQ*qca*
are equal; and the weights of the parts proportional to the whole legs,
and alike situated in them, are one to another as the weights of the
wholes, and therefore equal betwixt themselves; the weights of equal
parts, and alike situated in the legs, will be reciprocally as the legs,
that is, reciprocally as 230 to 229. And the case is the same in all
homogeneous equal bodies alike situated in the legs of the canal. Their
weights are reciprocally as the legs, that is, reciprocally as the
distances of the bodies from the centre of the earth. Therefore if the
bodies are situated in the uppermost parts of the canals, or on the
surface of the earth, their weights will be one to another reciprocally
as their distances from the centre. And, by the same argument, the
weights in all other places round the whole surface of the earth are
reciprocally as the distances of the places from the centre; and,
therefore, in the hypothesis of the earth's being a spheroid are given
in proportion.

Whence arises this Theorem, that the increase of weight in passing from
the equator to the poles is nearly as the versed sine of double the
latitude; or, which comes to the same thing, as the square of the right
sine of the latitude; and the arcs of the degrees of latitude in the
meridian increase nearly in the same proportion. And, therefore, since
the latitude of *Paris* is 48° 50′, that of places under the
equator 00° 00′, and that of places under the poles 90°; and the versed
sines of double those arcs are 11334,00000 and 20000, the radius being
10000; and the force of gravity at the pole is to the force of gravity
at the equator as 230 to 229; and the excess of the force of gravity at
the pole to the force of gravity at the equator as 1 to 229; the excess
of the force of gravity in the latitude of Paris will be to the force of
gravity at the equator as 1 × ^{11334}/_{20000}
to 229, or as 5667 to 2290000. And therefore the whole forces of gravity
in those places will be one to the other as 2295667 to 2290000.
Wherefore since the lengths of pendulums vibrating in equal times are as
the forces of gravity, and in the latitude of *Paris*,
the length of a pendulum vibrating seconds is 3 *Paris* feet,
and 8½ lines, or rather because of the weight of the air, 8^{5}/_{9}
lines, the length of a pendulum vibrating in the same time under the
equator will be shorter by 1,087 lines. And by a like calculus the
following table is made.

Latitude of the place. | Length of the pendulum | Measure of one degree in the meridian. |

Deg. 0 5 10 15 20 25 30 35 40 1 2 3 4 45 6 7 8 9 50 55 60 65 70 75 80 85 90 | Feet Lines 3 . 7,468 3 . 7,482 3 . 7,526 3 . 7,596 3 . 7,692 3 . 7,812 3 . 7,948 3 . 8,099 3 . 8,261 3 . 8,294 3 . 8,327 3 . 8,361 3 . 8,394 3 . 8,428 3 . 8,461 3 . 8,494 3 . 8,528 3 . 8,561 3 . 8,594 3 . 8,756 3 . 8,907 3 . 9,044 3 . 9,162 3 . 9,258 3 . 9,329 3 . 9,372 3 . 9,387 | Toises. 56637 56642 56659 56687 56724 56769 56823 56882 56945 56958 56971 56984 56997 57010 57022 57035 57048 57061 57074 57137 57196 57250 57295 57332 57360 57377 57382 |

By this table, therefore, it appears that the inequality of degrees is so small, that the figure of the earth, in