The Mathematical Principles of Natural Philosophy

by Isaac Newton

*
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
x C x 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 x BD, and Ab^{2}
= Ag x 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.