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 sim
ilar 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, sim
ilarly applied to the bodies at similar parts of the similar figures, are as
the forces and the squares of the times conjuiu tly.
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 beg
nning of the motion, are as the forces and the squares of the times conjunctly.
100 THE MATHEMATICAL PRINCIPLES I SEC. 1
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 direct
ly, und the forces inversely.
SCHOLIUM.
If in comparing indetermined quantities of different sorts one with
another, any one is said to be as any other directly or inversely, the mean
ing 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 aug
mented 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 mean
ing is, that A is augmented or diminished in the same ratio with B X C
X -jj-,
that is to say, that A and - arc one to the other in a given ratio.
LEMMA XL
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 dupli
cate rati 1) 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 tan
gent 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,) AE2= AG X BD, and
A62=Ag X bd ; and therefore the ratio of AB2 to Ab2
is compounded oi
the ratios of AG to Ag, and of Ed to bd. But because GJ may be as
sumed of less length than any assignable, the ratio of AG to Ag may be
such as to differ from the ratio of equality by less than any assignable
difference ; and therefore the ratio of AB2 to Ab2 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 Ab2
is the same with tho ul
timate ratio of BD to bd. Q.E.D.
CASE 2. Now let BD be inclined to AD in any given an*r1 r
, and the
ultimate ratio of BD to bd will always be the same as before, and there
fore the same with the ratio of AB2 to Ab2
. Q.E-P
BOOK I.] OF NATURAL PHILOSOPHY. 101
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, arid approach nearer to
each other than by any assigned difference, and therefore, by Lem. I, will at
lust be* equal ; and therefore the lines BD; bd arc in the same ratio to each
other as before. Q.E.D.
COR. 1. Therefore since the tangents AD, Ad, the arcs AB, Ab, and
their sines, BC, be, become ultimately equal to the chords AB, Ab} 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, Adb are
ultimately in the triplicate ratio of the sides AD, Ad, c
and in a sesquiplicate ratio of the sides DB, db ; as
being in the ratio compounded of the sides AD to DB,
and of Ad to db. So also the triangles ABC, Abe
are ultimately in the triplicate ratio of the sides BC, be.
What I call the sesquiplicate ratio is the subduplicate
of the triplicate, as being compounded of the simple
and subduplicate ratio. j
COR. 5. And because DB, db are ultimately paral- g
lei and in the duplicate ratio of the lines AD, Ad, the
ultimate curvilinear areas ADB, Adb will be (by the nature of the para
bola) two thirds of the rectilinear triangles ADB, Adb and the segments
AB, Ab will be one third of the same triangles. And thence those areas
and those segments will be in the triplicite ratio as well of the tangents
AD, Ad, as of the chords and arcs AB, AB.
SCHOLIUM.
But we have all along supposed the angle of contact to be neither infi
nitely greater nor infinitely less than the angles of contact made by cir
cles 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 mag
nitude. 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 AD4
, AD5
, AD8
,
AD7
, etc., we shall have a series of angles of contact, proceeding in itifinitum,
wherein every succeeding term is infinitely less than the pre
102 THE MATHEMATICAL PRINCIPLES [BOOK 1
ceding. And if DB be made successively as AD2
, AD|, AD^, AD], AD|
AD7
, &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 AD2 and AD3 there were interposed
the series AD f, ADy, AD4
9
, AD|, AD?, AD|, AD^1
, AD^, AD^7
, &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
euperfices 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 absurdnm, 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 follow
ing propositions to the first and last sums and ratios of nascent and evane
scent 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 un-
(lerstood 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 cer
tain place, and there stopping has no ultimate velocity : because the velo
city, before the body comes to the place, is not its ultimate velocity ; when
it has arrived, is none i ut the answer is easy; for by the ultimate ve
locity 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 ra
tio of evanescent quantities is to Le understood the ratio of the ijuantitiea
SEC. II.] OF NATURAL PHILOSOPHY. 103
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 ve
locity 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 pro
portions 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 de
monstrating any other thing that is likewise geometrical.
It may also be objected, that if the ultimate ratios of evanescent quan
tities 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 wfinitum. This thing will appear
more evident in quantities infinitely great. If two quantities, whose dif
ference is given, be augmented in infin&um, 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 al
ways diminished without end.
SECTION II.
Of the Invention of Centripetal Forces.
PROPOSITION I. THEOREM 1.
The areas, which revolving bodies describe by radii drawn to an ^mmovable
centra offorce do lie in tJ:e 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,
proceel directly to c, alo ILJ; the line Be equal to AB ; so that by the radii
AS, BS, cS, draw. i to the centre, the equal areas ASB, BSc, would be de
104 THE MATHEMATICAL PRINCIPLES [BOOK I
scribed. But when the body
is arrived at B, suppose
that a centripetal force acts
at once with a great im
pulse, and, turning aside the
body from the right line Be,
compels it afterwards to con
tinue its motion along the
right line BC. Draw cC
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
A SB. Join SC, and, because s
SB and Cc are parallel, the triangle SBC will be equal to the triangle SBc,
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, EF7 &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 wjinitum ; 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 de
scription, 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 veloci
ties 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 irifinitum, is produced both ways, it will pass through the centre of force.
COR. 3. If the chords AB, BC, and DE, EF, cf arcs described in equal
SEC. II.] OF NATURAL PHILOSOPHY. 105
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 ulti
mate 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 Be, BV, and E/", EZ : but BV and
EZ, which are equal to Cc and F/, in the demonstration of this Proposi
tion, 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 para
bolic 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 uni
formly forward in right lines.
PROPOSITION II. THEOREM II.
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 propor
tional to the times, is urged by a centripetal force directed to thatpoint
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
offfrom 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 par
1U6 THE MATHEMATICAL PRINCIPLES [BOOK f.
allel K cC. that is, in the direction of the line BS. and in the place C,
accordii g to the direction of a line parallel to dD, that is, in the direction
of the line CS, (fee.; and therefore acts always in the direction of lines
tending to the immovable point S. Q.E.I).
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 propor
tional to the times, the forces are not directed to the point in which the
radii meet ; but deviate therefrom in. consequently 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.
SCHOLIUM.
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 per
petually 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.
PROPOSITION III. THEOREM III.
Every body, that by a radius drawn to the centre of another body, how
soever moved, describes areas about that centre proportional to iJie times,
is urged by a force compounded out of the centripetal force Bending fo
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. of the Laws)
if both bodies are urged in the direction of parallel lines, by a neT force
equal and contrary to that by which the second body T is tinned, 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 ;f the forces is directed to the other body T as its centre. Q.E.D
SEC. IL] OF NATURAL PHILOSOPHY. 107
Co.*. 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 firr.t 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 who_e remaining force by which the first body is urged will tend to the
( ther body T, as its centre.
COR. 2. And, if these areas are proportional to the times nearly, the re
maining 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 moveaJble) centre. The same thing ob
tains, when the other body is moved by any motion whatsoever ; provided
that centripetal force is taken, wrhich remains after subducting that whole
force acting upon that other body T.
SCHOLIUM.
Because the equable description of areas indicates that a centre is re
spected 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 de
scription of areas as an indication of a centre, about which all circular
motion is performed in free spaces ?
PROPOSITION IV. THEOREM IV.
The centripetal forces of bodies, which by equable motions describe differ
ent circles, tend to the centres of the same circles ; and are one to tJie
other as the squares of t/ie 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. L), and are one to another as the versed sines of the least arcs de
scribed 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 there
fore since those arcs are as arcs described in any equal times, and the diame
ers ace as the radii, the forces will be as the squares of any arcs descr
bed in the same time applied to the radii of the circles. Q.E.D.
^OR. 1. Therefore, since those arcs are as the velocities of the bodies.
I OS THE MATHEMATICAL PRINCIPLES [BOOK .
the centripetal forces are in a ratio compounded of the duplicate ra jio 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 cen
tripetal 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
tke 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 them
selves ; and the contrary.
COR. 5. If the periodic times are as the radii, and therefore the veloci
ties equal, the centripetal forces will be reciprocally as the radii
; and the
contrary.
