The Mathematical Principles of Natural Philosophy

by Isaac Newton

*
Of the motions of bodies which are resisted partly in the ratio
of the velocities, and partly in the duplicate of the same ratio.
*

*
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 − 2BAxAP − 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 x 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 x PQ

Z or 2BPQ

Z, and the moment KLON of the area A*b*NK
will be equal to 2BPQ x LO

Z or BPQ
x BD^{3}

2Z x CK x 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 x Z + AC x Z or CK x Z; and therefore the area DTV will be
to the area DPQ as DT² or DB² to CK x 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 x Z + AC x Z or CK x Z.
And therefore the area DTV will be to the area DPQ as DB² to CK x 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 x 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 x *m*, the area DPQ, that is, ½BD x PQ, will
be to BD x *m* as CK x Z to BD². And thence PQ x BD³ becomes
equal to 2BD x *m* x CK x Z, and the moment KLON of the area A*b*NK,
found before, becomes BP x BD x m

AB. From the area DET subduct its
moment DTV or BD x *m*, and there will remain
AP x BD x m

AB. Therefore the difference of the
moments, that is, the moment of the difference of the areas, is equal
to AP x BD x m

AB; and therefore (because of the given
quantity BD x 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 x 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 x V^{2}

AB. This area is equal to the area
DA^{2} x BD x M^{2}

DE^{2} x AB and the moment of M
is *m*; and therefore the moment ot this area is
DA^{2} x BD x 2M x m

DE^{2} x AB. But this moment is
to the moment of the difference of the aforesaid areas DET and A*b*NK,
viz., to AB x BD x m

AB, as DA^{2}
x BD x M

DE^{2} to ½BD x 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 x
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 x 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 x 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 x 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.