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, supjx sing the body
A to be let fall from the point S, the velocity thereof in the place of re
flexion A, without sensible error, will be the same as if it had descended
m vacit.o 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 loAvest
point is as the chord of the arc which it has described in its descent. Aftci
OF NATUltAL PHILOSOPHY. 9 I
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 is
equal to tv, and let the chord of the arc tA 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 e way we are to
correct the place k to which the body B ascends, by finding the place I 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 /A, that we may have its mo
tion 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/, 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 re
flexion. 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 back
wards 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 J4,
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 tht
92 THE MATHEMATICAL PRINCIPLES
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 ky to which the bodies ascended after congress. Nay, and some errors,
too, might have happened from the unequal density of the parts of the pen
dulous bodies themselves, and from the irregularity of the texture pro
ceeding 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), 1 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 di
minish the reflexion in such a certain proportion as the quantity of the
elastic force requires. By the theory of Wren and Huygens, bodies abso
lutely hard return one from another with the same velocity with which
they meet. But this may be affirmed with more certainty of bodies per
fectly 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 per
ceive) 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 computa
tion 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 some-^
thing 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. Sup
pose 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
OF NATURAL PHILOSOPHY. 93
towards the parts on which B lies
;
arid in free spaces, to go forward in
infmitiim with a motion perpetually accelerated ; which is absurd and
contrary to the first Law. For, by the first Law, the system ought to per
severe 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 partFJ
EGI is cut into two parts EGKH and HKI.
whereof HKI is equal to the part EFG, first cut
oft,
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 in
deed if those weights were not equal, the whole earth floating in the nonresisting
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
upw ards 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 oi the balance : but if
they are turned aside by the interposition of oblique planes, or other ob
stacles, and made to ascend or descend obliquely, those bodies will be
equipollent, wThich are reciprocally as the heights of their ascent and de
scent taken according to the perpendicular ; and that on account of the
determination of gravity downwards.
94 THE MATHEMATICAL PRINCIPLES
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, whethel
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 propress
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 ; 7
move a given weight with a given power, or with a given force to over
come 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 ve
locity 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 re
sistances are overcome, will produce an acceleration of motion proportional
thereto, as well in the parts of {he 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 ot
motion. For if we estimate the action of the agent from its force and
velocity conjunctly, and likewise the reaction of the impediment conjuncth
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 YL the use of all sorts of machines will b" found
always equal to one another. And so far as the action is propagated by
the intervening instruments, and at last impressed upon tic resisting
body, the ultimate determination of the action will be always contrary to
the determination of the reaction.
OF NATURAL PHILOSOPHY 95
BOOK I.
OF THE MOTION OF BODIES.
SECTION I.
Of the method offirst and last ratios of quantities, by the help wJicreoj
we demonstrate the propositions that follow.
LEMMA I.
Quantities, and the ratios of quantities, which in anyfinite 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,
LEMMA II.
If in any figure AacE, terminated by the right (f
lines A a. AE, and the curve acE, there be in
scribed any number of parallelograms Ab, Be,
Cd, fyc., 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 com
pleted. Then if the breadth of those parallelo- \
grams be supposed to be diminisJied, and their X BF C D |;
number to be augmented in infinitum : / say, that :he ultimate ratios
which the inscribed fignre AKbLcMdD, the tin nmscribed figure
AalbmcndoE, and enrvilijiear 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 K7, Lw, M//. Do. that is (from the equality of all
their bases), the rectangle under one of their bases K6 and the sum of their
altitudes Aa, that is, the rectangle ABla. But this rectangle, because
M
a
96 THE MATHEMATICAL PRINCIPLES [BOOK 1
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.
LEMMA III.
The same ultimate ratios are also ratios of equality, when the breadth^
AB, BC, DC, fyc., of the parallelograms are unequal, and are all di
minished in infinitum.
For suppose AF equal to the greatest breadth, and
complete the parallelogram FAaf. This parallelo
gram will be greater than the difference of the in
scribed 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 evanes
cent parallelograms will in all parts coincide with
the curvilinear figure. A BF C D E
COR. 2. Much more will the rectilinear figure^comprehendcd under tne
chords of the evanescent arcs ab, be, cd, (fee., ultimately coincide with tl.c
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 acE)
are not rectilinear, but curvilinear limi s of rectilinear figures.
LEMMA IV.
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 respec
tively, 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 p
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. Ill) 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 A
SEC. I.] OF NATURAL PHILOSOPHY. 97
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 aug
mented, and their magnitudes diminished in infinitum, those sums will be
in the ultimate ratio of the parallelogram in the one figure to the corres
pondent 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.
LEMMA V.
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.
LEMMA VI.
If any arc ACB, given in position, is snb- _j
tended by its chord AB, and in any point
A, in the middle of the contiinied curva
ture, is touched by a right line AD, pro
duced both ways ; then if the points A R
and B approach one another and meet,
I say, the angle RAT), contained between,
the chord and the tangent, will be dimin- ?
ished in infinitum, a/id 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 cur
vature at the point A will not be continued, which is against the supposi
tion.
LEMMA VII.
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 Acb be always similar to the arc
ACB. Then, supposing the points A and B to coincide, the angle dAb
will vanish, by the preceding Lemma; and therefore the right lines Ab,
Arf (which are always finite), and the intermediate arc Acb, will coincide,
and become equal among themselves. Wheref ,re, the right lines AB, AD,
98 THE MATHEMATICAL PRINCIPLES [SEC. I.
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 A
BP parallel to the tangent, always cutting
any right line AF passing through A in F/ i-
P, this line BP will be ultimately in the
ratio of equality with the evanescent arc ACB ; because, completing the
parallelogram APBD, 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,
I5D, AF, AG, cutting the tangent AD and its parallel BP : 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.
LEMMA VIII.
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 oj
these evanescent triangles is that of similitude, and their ultimate
ratio that of equality.
For while the point B approaches towards A
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 Acb be always similar to the arc
ACB. Then supposing the points A and B
to coincide, the angle bAd will vanish ; and
therefore the three triangles rAb, rAcb,rAd
^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 indif
ferently use any one of those triangles for any other.
LEMMA IX.
If a ngnt line AE. and a curve tine 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 : / say, that the areas of the triangles ABD, ACE, wilt
ultimately be one to the other in the duplicate ratio of the sides.
BOOK LI OF NATURAL PHILOSOPHY.
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 Ad, Ae 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 D
in b and c. Let the curve Abe be similar
to the curve A BC, and draw the right line
Ag- so as to touch both curves in A, and
cut the ordinates DB, EC, db ec, in F, G,
J] g. Then, supposing the length Ae to remain the same, let the points B
and C meet in the point A ; and the angle cAg vanishing, the curvilinear
areas AW, Ace will coincide with the rectilinear areas A/rf, Age ; and
therefore (by Lem. V) will be one to the other in the duplicate ratio of
the sides Ad, Ae. 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.
LEMMA X.
The spaces which a bodij describes by anyfinite 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 des
cribing similar parts of similar figures in proportional times, are nearly
