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 s.ime 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, w T hich 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 &lt; 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 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 be- g 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 itifini- tum, 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 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 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 ne T 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, w r hich 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 dia- me 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.
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 in versely :
and the contrary. COR. 7. And universally, if the periodic time is as any power
Rn of the radius R, and therefore the velocity reciprocally as the power Rn ]
of the radius, the centripetal force will be reciprocally as the power R2n 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 centri petal 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. SCHOLIUM. 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 SEC. II.] OF NATURAL PHILOSOPHY.
109 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 re volves
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 ve locity 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 in creased 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 inftnitum, coin cides
with the circle, as the square of the arc described in a given time ap plied 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. PROPOSITION V. PROBLEM I. There
being given, in any places, the velocity with which a body de scribes 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 tan gents 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 re quired. 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 110 THE
MATHEMATICAL PRINCIPLES [BOOK 1 (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; YE meet. Q.E.D. PROPOSITION VL THEOREM
V. In a space void of resistance, if a body revolves in any orbit about an im
movable centre, and in the least time describes any arc just then, na scent ;
and the versed sine of that arc is supposed to be drawn bisect ing the chord,
and produced passing through the centre offorce: the centripetal force in the
middle of the arc will be as the versed sine di rectly 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. XL), 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, arid the square of the time
inversely. Q.E.D. And the same thing may also be easily demonstrated by Corol.
4 ? T,em. 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, QJl is drawn parallel to the distance SP, meeting the
tangent in R ; and QT is drawn perpen- (licular to the distance SP ; the
centripetal force will be reciprocally as the sp2 x Q/r2 solid- : , if the
solid be taken of that magnitude which it ulti- mately acquires when the points
P and Q, coincide. For Q,R is equal to the versed sine of double the arc QP,
whose middle is P : and double the triangle SQP, or SP X Q,T 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 SY2 X QJP2 solid -7^5-; if SY is a perpendicular from the
centre of force on PR the tangent of the orbit. For the rectangles SY X QP and
SP X Q,T are equal. SEC. II.] OF NATURAL PHILOSOPHY. Ill COR. 3. If the orbit
is cither a circle, or touches or cuts a circle c&lt; ncentrically,
that is, contains with a circle the least angle of contact or sec tion, having
the same curvature rnd 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 QP2 SY2 X PV. For PV is
-. 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 bcj continually drawn back from a
rectilinear course, and. being detained in the perimeter of that figure, will
describe the same by a perpetual revolu- SP2 x QT2 tion. That is, we are to
find, by computation, either the solid ----- or the solid SY2 X PV,
reciprocally proportional to this force. Example: of this we shall give in the
following Problems. PROPOSITION VII. PROBLEM II. Tf a body revolves in the
circumference of a circle; it is proposed to finii 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 mov ing 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 v
chord PV, and the diameter VA of the circle : join AP, and draw Q,T perpen
dicular 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 QRlj x PV2 SI3 -
therefore - TS--is equal to QT2 . Multiply those equals by - . and the points P
and Q, coinciding, for RL write PV ; then we shall have SP- X PV5 SP2 x QT2 And
therefore flr Cor 1 and 5. Prop. VI.) 112 THE MATHEMATICAL PRINCIPLES [BOOK I,
SP2 X PV3 the centripetal force is reciprocally as - ry^~ J that is (because
AV2 ia given), reciprocally as the square of the distance or altitude SP, and
the 3ube of the chord PV conjunctly. Q.E.L The same otherwise. On the tangent
PR produced let fall the perpendicular SY ; and (be cause of the similar
triangles SYP, VPA), we shall have AV to PV as SP SP X PV SP2
&gt;&lt; PV3 to SY, and therefore --^~ - = SY, and - ^- = SY2 X
PV. A V A V And therefore (by Corol. 3 and 5, Prop. VI), the centripetal force
is recipSP2 X PV3 rocally as - ~~ry~~~ I *na* *s (because AV is given),
reciprocally as SP" X PV3 . Q.E.I. Con. 1. Hence if the given point S,
to which the centripetal force al ways tends, is placed in the circumference of
the circle, as at V, the cen tripetal 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 re volve in the same circle, and in the same
periodic time, about any other centre of force R, as RP2 X 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 RP2 X PT3 to SP2 X PV3 ; that is, as SP3 X PV3
SP X RP2 to -- p ; or (because of the similar triangles PSG, TPV) to SGS . 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
X RP2 , contained under the distance of the body from the first centre of force
S, and the square of its distance from the sec ond 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 fch*3 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. SJSG. IL] OF NATURAL PHILOSOPHY. 113
PROPOSITION VIII. PROBLEM III. If a body mi ues in the semi-circuwferencePQA:
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 na ture of the circle, PR2 is equal to the rect angle QR X RN +
QN, or, the points P, Q coinciding, to the rectangle QR x 2PM. Therefore CP2 is
to PM2 as QR X 2PM to QT2 ; and QT2 2PM3 QT2 X SP2 2PM3 X SP2 therefore (by QR
Corol. 8PM3 X SP2 , and QR And 1 and 5, Prop. VI.), the centripetal force is
reciprocally as 2SP2 . that is (neglecting the given ratio -ppr)&gt;
reciprocally as PM3 . Q.E.L And the same thing is likewise easily inferred from
the preceding Pro position. SCHOLIUM. And by a like reasoning, a body will be
moved in an ellipsis, or even ia 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. PROPOSITION IX. PROBLEM IV. If a body
revolves in a spiral PQS, cutting all the radii SP, SQ, fyc., in a given angle;
it is proposed to find thelaio of the centripetal force tending to tJie centre
of that spiral. Suppose the inde finitely small angle AY PSQ to be given ; be
cause, then, all the angles are given, the figure be given SPRQT in specie.
will QT v ,_ Q,T2 Therefore the ratio -7^- is also given, and is as QT, that is
(be lot IX QK cause 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 QPU tU
THE MATHEMATICAL PRINCIPLES [BOOK J (by Lemma XI) will be changed in the
duplicate ratio of PR or QT QT2 Therefore the ratio ~TVD~remains the same as
before, that is, as SP. And QT2 x SP2 -^ is as SP3 , and therefore (by Corol. 1
and 5, Prop. YI) the centripetal force is reciprocally as the cube of the
distance SP. Q.E.I. The same otherwise. The perpendicular SY let fall upon the
tangent, and the chord PY of the circle concentrically cutting the spiral, are
in given ratios to the height SP ; and therefore SP3 is as SY2 X PY, that is
(by Corol. 3 and 5, Prop. YI) reciprocally as the centripetal force. LEMMA XII.
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. PROPOSITION X. PROBLEM Y. If a body revolves in
an ellipsis ; it is proposed to find the law of thi centripetal force tending
to the centre of the ellipsis. Suppose CA, CB to be semi-axes of the ellipsis;
GP, DK, con jugate diameters ; PF, Q,T perpendiculars to those diameters; Qvan
^rdinate to the diame ter GP ; and if the parallelogram QvPR be completed, then
(by the properties of the jonic sections) the rec- langle PvG will be to Qv2 as
PC2 to CD2 ; and (because of the similar triangles Q^T, PCF), Qi&gt; 2
to QT2 as PC2 to PF2 ; and, by com position, the ratio of PvG to QT2 is
compounded of the ratio of PC2 1&lt; QT2 CD2 , and of the ratio of PC2
to PF2 , that is, vG to -p as PC; to _92L^_ P_ ] ^_. Put QR for Pr, and (by
Lem. XII) BC X CA for CD K PF ; also (the points P and Q coinciding) 2PC for
rG; and multiply- SEC. II.] OF NATURAL PHILOSOPHY. 115 QT2 x PC2 ing the
extremes and means together, we shall have rfo~ equal to 2BC2 X CA2 pp .
Therefore (by Cor. 5, Prop. VI), the centripetal force is 2BC2 X CA2
reciprocally as ry~ ; that is (because 2I3C2 X CA2 is given), re ciprocally
as-r^v; that is, directly as the distance PC. QEI. I O TJie same otherwise. [n
the right line PG on the other side of the point T, take the point u so that Tu
may be equal to TV ; then take uV, such as shall be to vG as DC2 to PC2 . And
because Qr9 is to PvG as DC2 to PC2 (by the conic sections), we shall have Qv2
-= Pi X V. Add the rectangle n.Pv to both sides, and the square of the chord of
the arc PQ, will be equal to the rect angle VPv ; 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 nV to
rG, which is the same with the ratio of DC2 to PC2 , will become the ratio of
PV to PG, or PV 2DC2 to 2PC : and therefore PY will be equal to . And therefore
the force by which the body P revolves in the ellipsis will be reciprocally as
2 DC2 ry X PF2 (by Cor. 3, Prop. VI) ; that is (because 2DC2 X PF2 is I O
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 sim ilar ellipses will be equal (by Corol. 3 and S, 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 ; :hat is, as
those lesser axes dirtily, and the ordinates to the same point % f the common
axes in versely ; and therefore (because of the equality of the direct and
inverse ratios) in the ratio of equality. SCHOLIUM. If the ellipsis, by having
its centre removed to an infinite distance, de generates into a parabola, the
body will move in tin s parabola ; and the 116 THE MATHEMATICAL PRINCIPLES
[BOOK I 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, hav ing 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 di minishing 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 ordi- nates are augmented or
diminished in any given ratio, or their inclination is any way changed, the
periodic time remaining the same, the forces di rected to any centre placed in
the abscissa are in the several ordinatee augmented or diminished in the ratio
of the distances from the centre SECTION III. Of the motion of bodies in
eccentric conic sections. PROPOSITION XL PROBLEM VI. 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 diame ter DK of the ellipsis in E, and the ordinate Qv in x ; and com plete
the parallelogram d.rPR, It is evident that EP is equal to the greater
semi-axis AC : for drawing HI froln the other focus H of the ellipsis parallel
to EC, because CS, CH are equal, ES, El 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 perpendicu lar to SP, and putting L for the princi al latus rectum
of the ellipsis (or for III. OF NATURAL PHILOSOPHY. 117 L X ^R t0 L X Py aS ^R
t0 PV that 1S&gt; US PE or AC to PC ; and L X Pv to GvP as L to Gy ;
and GvP to Qi&gt;2 as to CD- ; and by (Corol. 2, Lem. VII) the points
Q, and P coinciding, Qv* is to Q,r- in the ratio of equality ; and Q,.r 2 or
Qv2 is to Q,T2 as EP2 to PF2 , that is, as CA2 to PF2 , or (by Lem. XII) as CD2
to CB2 . And com pounding all those ratios together, we shall have L X QR to
Q,T2 as AC X L X PC2 X CD2 , or 2CB2 X PC2 X CD2 to PC X Gv X CD2 X CB2 , or as
2PC to Gv. But the points Q and P coinciding, 2PC and Gr are equal. And
therefore the quantities L X QR and Q,T2 , proportional SP2 to these, will be
also equal. Let those equals be drawn into-p^B"? and L SP2 X QT2 X SP2
will become equal to --^p . And therefore (by Corol. 1 and 5, Prop. VI) the
centripetal force is reciprocally as L X SP2 , that is, re ciprocally in the
duplicate ratio of the distance SP. Q.E.I. The same otherwise. 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 paral lel 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 in- PE3 tersect in E, will be as ^T3 ,
(by Cor. 3, Prop. VII.) ; that is, if the point S is the focus of the ellipsis,
and therefore PE be given as SP2 recipro cally. 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.
PROPOSITION XII. PROBLEM VII. Suppose a body to move in an hyperbola ; it is
required to find lite 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 con jugate
diameters ; PF a perpendicular to the diameter KD ; and Qv an ordinate to the
diameter GP. Draw SP cutting the diameter DK in E, and the ordinate Qv in x,
and complete the parallelogram QRP.r. It is evident that EP is equal to the
semi-transverse axis AC ; for drawing HE, from the other focus H of the
hyperbola, parallel to EC, because CS, TH are equal, ES El will be also equal ;
so that EP is the half difference J1S THE MATHEMATICAL PRINCIPLES [Book I .of
PS, PI; that is (be cause 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
Q,T perpen dicular to SP; and put ting L for the principal latus rectum of the
hy perbola (that is, for 2BC2 \ .... -Tp- ) 7 we shall have L X QR to L X Pv as
QR to Pv, or Px to Pv, that is (because of the similar tri angles Pxv, PEC), as
PE to PC, or AC to PC. And L X Pv will be to Gv X Pv as L to Gv; and (by the
properties of the conic sections) the rec tangle G? P is to Q,v2 as PC2 to CD2
; and by (Cor. 2, Lem. VII.), Qv2 to Qa* the points Q and P coinciding, becomes
a ratio of equality ; and Q,.r 2 or Qv2 is to Q,T2 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 X Q,R to Q,T2 as AC X L X PC2 X CD2 , or 2CB2
X PC 2 X CD2 to PC X Gv X CD2 X CB2 , or as 2PC to Gv. But the points P and Q,
coinciding. 2PC and Gv are equal. And therefore the quantities L X Q,R arid
Q.T2 , propor tional to them, will be also equal. Let those equals be drawn
into SP2 sp2 x o/r2 ^, and we shall have L X SP2 equal to ^^ . And therefore
(by Cor. 1. and 5, Prop. VI.) the centripetal force is reciprocally as L X SP2
. hat is, reciprocally in the duplicate ratio of the distance SP. Q,.E.I. TJie
same otherwise. Find out the force tending from the centre C of the hype rbola.
This will be proportional to the distance CP. But from thence (by Cor. 3, Prop.
PE3 VII.) the force tending to the focus S will be as -^-^ th; (t is, because
PE is given reciprocally as SP-. Q,.E.I. SEC. III.] OF NATURAL PHILOSOPHY. 119
And the same way may it be demonstrated, that the body having its cen tripetal
changed into a centrifugal force, will move in the conjugate hy perbola. LEMMA
XIII. The latus rectum of a parabola belonging to any vertex is quadruple the
distance of that vertex from the focus of thejigurc. This is demonstrated by
the writers on the conic sections. LEMMA XIV. Tlie perpendicular, let fall from
the focus of a parabola on its tangent, is a mean proportional between the.
distances of the focus from the poini of contact, andfrom 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 fo-~ M A s o cus on the tangent : join AN, and because of the equal lines
MS and SP, MN and NP, MA and AC, 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 j 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, SN- will be as PS. COR. 3.
And the concourse of any tangent PM, with the right line SN. drawn from the
focus per] endicular on the tangent, falls in the right line AN that touches
the parabola in the principal vertex. PROPOSITION XIII. PROBLEM VIII. 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 thatfigure. 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 QH
parallel IS!. and Q,T perpendicular to SP, as also Qv parallel to the tan gent,
and mating the diame ter PG in v, and the distance 120 THE MATHEMATICAL
PRINCIPLES [BOOK I. SP in x. Now. because of the similar triangles Pxv, SPM,
and of the equal sides SP, SM of the one, the sides Px or Q,R and Pv of the
other will be also equal. But (by the conic sections) the square of the
ordinate Q,y is equal to the rectangle under the latus rectum and the segment
Pv of the diameter ; that is (by Lem. XIII.), to the rectangle 4PS X Pv, or 4PS
X Q,R ; and the points P and Q, coinciding, the ratio of Qv to Q,.r (by Cor. 2,
Lem. VII.,) becomes a ratio of equality. And therefore Q,#2 , in this case,
becomes equal to the rectangle 4PS X Q,R. But (because of the similar triangles
Q#T, SPN), Q^2 is to QT2 as PS2 to SN2 , that is (by Cor. 1, Lem. XIV.), as PS
to SA ; that is, as 4PS X QR to 4SA x QR, and therefore (by Prop. IX. Lib. V.,
Elem.) QT* and 4SA X QR are SP2 SP2 X QT2 equal. Multiply these equals by ^-^-,
and ^5 -will become equal to SP2 X 4SA : and therefore (by Cor. 1 and 5, Prop.
VL), the centripetal force is reciprocally as SP2 X 4SA ; that is, because 4SA
is given, recipro cally 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 fo cus 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 be ing given ; and two orbits,
mutually touching one the other, cannot be de scribed 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 describe I: 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 QT2 . the conic sections, whose principal latus rectum is the quantity
Tjfr in its ultimate state, when thelineoke 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. PROPOSITION XIV.
THEOREM VI. Tf several bodies revolve about one common centre, and the
centripetal force is reciprocally in tlie duplicate ratio of the distance of
places from the centre ; I say, that the principal latera recta of tfieir
orbits are in the duplicate ratio of the areas, which the bodies by radii drawn
to the centre describe it\ the same time. SEC. HI. OF NATURAL PHILOSO1 HY. For
(by Cor 2, Prop. XIII) the latus rectum QT*. L is equal to the quantity-^-in
its ultimate state when the points P and Q, coincide. But the lineola QR in a
given time is as the gen erating centripetal force ; that is (by supposiQT2
tion), reciprocally as SP2 . And therefore-^-^ is as Q.T2 X SP2 ; that is, the
latus rectum L is in the duplicate ratio of the area QT X SP. Q.E.D. C?OR.
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 X SP, described in a given time, mul tiplied by the periodic time.
PROPOSITION XV. THEOREM VII. The same things being supposed, J say, that the
periodic times in ellip ses 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. o. 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 ses quiplicate 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. PROPOSITION XVI. THEOREM VIII. 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
commonfocus ; I say, that the velocities oj the bodies are in a ratio
compounded of the ratio of the perpendiculars inversely, and the, subduplicate
ratio of the principal latera recta direct!]). 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 SY2 quantity -y . For that
velocity is as the infinitely small arc PQ de- 122 THE MATHEMATICAL PRINCIPLES
[BOOK I. scribed in a given moment of time, that is (by Lem. "VII), as
the tangent PR ; that is (because of the proportionals PR to Q,T, and SP to SP
X Q,T SY), as ~y ; or as SY reciprocally, and SP X Q,T directly ; but SP X 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 ve locities. 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 distan ces inversely, and the subduplicate ratio
of the principal latera recta di rectly. For those perpendiculars are now the
distances. COR. 3. Arid 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 dis tance from the centre, in the subduplicate ratio of the principal
latus rec tum 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), recipro cally 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 perpen
dicular 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 dis tance. 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, For (by Cor. 2 of this Prop.) the velocitv at the
vertex of a parabola is ir SEC. III.] OF NATURAL PHILOSOPHY. 123 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. S. 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 princi pal 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; there fore, ex czqiiO) 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. PROPOSITION XVII. PROBLEM IX.
Supposing the centripetal force to be reciprocally proportional to the squares
of the distances of places from the centre, and that the abso lute quantity of
that force is known ; it is required to determine t/te 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 go with a given ve locity 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 ; arid is therefore given. Let this latus rectum be L ; the focus S
of the conic L24 THE MATHEMATICAL PRINCIPLES [BOOK I 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 II is placed, is given by position. Let fall SK
perpendicular on PH, and erect the conjugate semi-axis BC ; this done, we shall
have SP2 2KPH + PH2 = SH2 = 4CH2 = 4BH2 4BC2 = SP + PH2 L X SiM SP2 + 2SPH +
PH2 L x SP + PH. Add on both sides 2KPH SP2 PH2 + L X SP + PH, and we shall
have L X SP + PH =2SPH f 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 w r ith 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 pas sing 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. Por 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 di rection
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 4US.
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 ig 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 Bourse. For bv compounding the proper motion oi SEC.
IV.] OF NATURAL PHILOSOPHY. 125 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. SCHOLIUM. If a body P, by means of a
centripetal force tending to any given point R, move in the perimeter of any given
conic sec tion whose centre is C ; and the law of the centripetal force is
required : draw CG parallel to the radius RP, and meet ing the tangent PG of
the orbit in G ; and the force required (by Cor. 1, and CG3 Schol. Prop. X.,
and Cor. 3, Prop. VII.) will be as - SECTION IV. Of the finding of elliptic,
parabolic, and hyperbolic orbits, from ttu. focus given. LEMMA XV. Iffrom the
two foci S, II, 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, thai is, to the axis in which the foci are situated, the other, SV, is
bisected in T by t/ie 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. Be cause 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 perpen dicular TR will touch the same ; and the contrary.
Q.E.D. 126 THE MATHEMATICAL PBINCIP, -ES [BOOK 1 PROPOSITION XVIII. PROBLEM X.
From a focus and the principal axes given, to describe elliptic and hy perbolic
trajectories, which shall pass through given points, and touch right lines
given by position. Let S be the common focus of the figures ; AB A 33 the
length of the principal axis of any trajectory ; r p T~* P a point through which
the trajectory should \ /R pass ; and TR a right line which it should touch. /
\ About the centre P, with the interval AB SP, \ S ~~yf if the orbit is an
ellipsis, or AB {- SP, if the y&gt; G ^ 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 -f- 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. PROPOSITION XIX. PROBLEM XI. 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 de scribed, 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 YG,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 /KS SEC. IV.] OF NATURAL PHILOSOPHY. 127
(by Cor. 3, Lem. XIV) because ST is equal to TV. and STR a light an gle, it
will touch the right line TR. Q.E.F. PROPOSITION XX. PROBLEM XII. About a given
focus to describe any trajectory given in specie which shah pass through given
points, and touch right lines given by position. CASE 1. About the focus S it
is re- uired to describe a trajectory ABC, pass ing through two points B, C.
Because the trajectory is given in specie, the ratio of the principal axis to
the distance of the foci GAS H 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 perpendicu lar SG ; which cut in A and a, so that GA may be to AS, and Ga
to aS, as KB to BS ; and with the axis 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 Ga to aS, then by division we shall
have Ga GA, or Aa to 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 tht- 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, St, which produce to V, v, so
that TV, tv may be equal to TS, tS. Bisect Vv in O, and j erect the indefinite
perpendicular OH, and cut I. the right line VS infinitely produced in K and V
k, so that VK be to KS, and VA* to A~S, as the principal axis of the tra
jectory to be described is to the distance of its foci. On the diameter K/J
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 Kk
in X, and joining HX, HS, HV, Hv, because VK is to KS as VA- to A*S ; and by composition,
as VK -f- V/c to KS + kS ; and by division, as VA* VK to kS 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 12S THE
MATHEMATICAL PRINCIPLES [BOOK 1. K S 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. Arid seeing VH, vH are equal to the principal axis, and VS, vS
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 Jet 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& to SAr, as the principal axis of the ellipsis to be described
to the distance of its foci ; and on the diameter KA: describing a circle, cut
the H right line VR produced in H ; then with the foci S, H, and principal axis
equal to R VH, describe a trajectory : I say, the thing .--- is done. For VH is
to SH as VK to SK, V" "1 and therefore as the principal axis
of the trajectory which was to be de scribed to the distance of its foci (as
appears from what we have demon strated 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 r 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 an gles 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 cut ting 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 pyq. Then with the foci S,
H, and B 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 SEC.
IV.] OF NATURAL PHILOSOPHY. 129 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 simi lar 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 per
pendicularly bisected by the rght line TR, the said figure touches the right
line TR. Q.E.F. LEMMA XVI. From three given points to draw to afonrth 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
princi pal 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.
130 THE MATHEMATICAL PRINCIPLES [BOOK 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 oi Tactions restored by Vieta. PROPOSITION XXL PROBLEM XIII. About a
given focus to describe a trajectory that shall pass through given points and
touch right Hues 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 -f SP ; or PH SP, if it be an hyperbola),
the trajectory is given. Q.E.I. SCHOLIUM. When the trajectory is an hyperbola,
I do not comprehend its conjugate hyperbola under the name of tins 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, I), 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 pro duced let fall the perpendiculars SG, BH, and in GS
produced indefinitely E take GA to AS, and Ga to aS, as HB is to BS ; then A
will be the vertex, and Aa the principal axis of the tra jectory ; which,
according as GA is greater than, equal to, or less than SEC. V.] OF NATURAL
PHILOSOPHY. 131 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 GP as the
point A ; in the second, going oft* to an infinite distance ; in the third,
falling on the other side of the line GP. For if on GF the perpendiculars CI,
DK are let fall, TC will be to HB as EC to EB ; that is, as SO to SB ; and by
permutation, 1C 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. Where fore 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 Conies, Prop. XXV., Lib. VIII. SECTION V. How
the orbits are to be found when neither focus is given. LEMMA XVII. Iffrom 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 ang 7ei, each line to each side ; the rectangle PQ, X PR of
those on the opposite sides AB, CD, will be to the rectangle PS X PT of those
on tie 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 I ^^ p ; T the other sides ; as PQ and PR to the side AC, and s |
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^
IQ I3 those parallel sides will be one of the diameters of the 1L 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 XXI1L, Book III., of Apollonius s Conies) 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 X PR are equal ; and therefore the rectangle PQ X PR is to the rectangle
A^ B, that Is, to the rectangle PS X PT in a given ratio. Q.E.D 132 THE MATHEMATICAL
PRINCIPLES [BOOK I CASE 2. Let us next suppose that the oppo site sides AC and
BD of the trapezium are not parallel. Draw Be/ parallel to AC, and meeting as
well the right line ST in /, as the conic section in d. Join Cd cutting PQ in
r, and draw DM parallel to PQ, cutting Cd in M, and AB in N. Then (because of
the similar triangles BTt, DBN), Et or PQ is to Tt as DN to NB. And ^^ Q N so
Rr is to AQ or PS as DM to AN. Wherefore, by multiplying the antece- dents by
the antecedents, and the consequents by the consequents, as the rectangle PQ X
Rr is to the rectangle PS X Tt, so will the rectangle N i)M be to the rectangle
ANB ; and (by Case 1) so is the rectangle PQ X Pr to the rectangle PS X Pt :
and by division, so is the rectangle PQ X PR to the rectangle PS X PT. Q.E.D.
CASE 3. Let us suppose, lastly, the four lines ?Q, PR, PS, PT, not to be
parallel to the sides AC, AB, but any way inclined to them. In their place draw
Pq, Pr, parallel to AC ; and Ps, Pt parallel to AB ; and because the angles of
the triangles PQ&lt;/, PRr, PSs, PTt are given, the ratios of IQ to Pq,
PR to Pr, PS to P*, PT to Pt will b? also given; and therefore the compound ed
ratios Pk X PR to P? X Pr, and PS X PT to Ps X Pt are given. But from what we
have demonstrated before, the ratio of Pq X Pi to Ps X Pt is given ; and
therefore also the ratio of PQ X PR to PS X PT. Q.E.D. LEMMA XVIII. The s niL
things supposed, if the rectangle PQ X PR of the lines drawn to the two
opposite sides of the trapezium is to the rectangle PS X 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 pas sing 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 c 1 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 ^ B 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 X bn to bf X bd, SEC. V.] OF NATURAL PHILOSOPHY 133 so (by
Lem. XVII) pq X pr to ps X pt ; and so (by supposition) PQ x PR to PS X PT. And
because of the similar trapezia bkAf, 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 ~Dnbd, DRPT, are similar, and consequently their diagonals D6, 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 com mon 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 X 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 X 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 (onger cut, but
only touch, the curve in those coinciding points. SCHOLIUM. 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 hap pens 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 *he
four points. If the two opposite an gles of the trapezium taken together are
equal c 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 X PR under two of the lines drawn PQ and PR, is equal to the rectangle PS X
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 X PR, under one pair of the lines drawn, is to the rectangle PS X
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 tw 134 THE MATHEMATICAL PRINCIPLES [BOOK 1.
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 A BCD, we may substitute a quadrilateral figure whose two
opposite sides cross one another like diago nals. 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.LEMMA XIX. To
find a point P from which if four right lines PQ, PR, PS, PT an drawn to as
many other right lines AB, CD, AC, BD, given by posi tion, each to each, at
given angles, the rectangle PQ X PR, under any two of the lines drawn, shall be
to the rectangle PS X PT, under the other tivo. in a given ratio. Suppose the
lines AB, CD, to which the two right lines PQ, PR, containing one of the rect
angles, 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 oi PQ X PR to PS X PT, the ratio of PR to PT will be given
; and ad ding 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 tan gent ; 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 dia meter to which BG and
FG are ordinates. Let this AG meet the locus SEC. V.J OF NATURAL PHILOSOPHY. in
H, and AH will be its diameter or latus trans- versum. to which the latus
rectum will be as BG2 to AG X GH. If AG nowhere meets the locus, the line AH
being infinite, the locus will be a par abola ; and its latus rectum
corresponding to the diameter AG will be -.-7^ AC* 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 Prob
lem of the ancients concerning four lines, begun by Euclid, and carried on by
Apollonius ; and this not an analytical calculus, but a geometrical com
position, such as the ancients required. LEMMA XX. If the two opposite
angularpoints 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 meet- ing tlie two other sides PS, PQ of the parallelogram, indefinitely
pro duced 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, F 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 X DF will be to the rect angle DG X DH in a given ratio. But PQ is
to DE (or IQ) as PB to HB, and con sequently 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 com
pounding those ratios, the rectangle PQ X PR will be to the rectangle PS X PT
as the rectangle DE X DF is to the rectangle DG X DH. and consequently in
"a given ratio. But PQ and PS are given, and there fore the ratio of
PR to PT is given. Q.E.D. 136 THE MATHEMATICAL PRINCIPLES 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 X DF is to the
rectangle DG X DH in a given rati) ; and so the point D (by Lem. XVIII) will
lie in a conic section pass ing through the points A., B, C, P, as its locus.
Q.E.I). COR. 1. Hence if we draw BC cutting PQ in r and in PT take Pt to Pr in
the same ratio which PT has to PR ; then Et 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 Bt. COR. 2. And, vice versa, if Bt is a tangent, and the lines BD, CD
meet in any point D of a conic section, PR will be to PT as Pr to Pt. And, on
the contrary, if PR is to PT as Pr to Pt, 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 hve points A, B, C, P, O ; and let the right line BD
cut them in the points D, d, and the right line Cd cut the right line PQ, in q.
Therefore PR is to PT as Pq to PT : whence PR and Pq are equal one to the
other, against the supposition. LEMMA XXI. 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 lints 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 giren 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 immo vable point P. Join ON, BN, CP, BP,
and from the point P draw the right lines PT, PR meeting BD, CD in T and R, C
and making the angle BPT c jual to the given angle BNM, and the angle CPR SEC.
V.J OF NATURAL PHILOSOPHY. 137 equal to the given angle CNM. Wherefore since
(by supposition) the an gles MBD, NBP are equal, as also the angles MOD, NCP,
take away the angles NBD and NOD 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: wheiefore
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 an gle 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 suc
cessively on two immovable points /?, N. Through these points ??, N, draw the
right line nN : this line nN 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 de monstrated before, the point D will be also placed in a conic section
pass ing through the same five points B, C, A, p, P, when the point M is per
petually 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. QE.D. PROPOSITION
XXII. PROBLEM XIV. To describe a trajectory that shall pass through Jive given
points. Let the five given points be A, B, C, P, D. c From any one of them, as
A, to any other s v two as B, C, which may be called the poles, draw the right
lines AB, AC, and parallel to those the lines TPS, PRO, 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 138
THE MATHEMATICAL PRINCIPLES IBOOK L 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 Pt, Pr, proportional to PT, PR ;
and if through their extremities, t, r, and the poles B, C, the right lines
lit, Cr are drawn, meeting in d, that point d will be placed in the trajectory
required. For (by Lena. XX) that point d is placed in a conic section passing
through the four points A, B, C, P ; and the lines R/ , TV 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. The same otherwise. Of the given
points join any three, as A, B, C ; and about two of them 15, 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. C 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
???, of the legs BL, CL; or BM7 CM, may always fall in that indefinite right
line MN ; and the intersection, which is now supposed to be d, of the legs BA
^A, or BD; CD, will describe the trajectory required, PADc/B. For (by Lem. XXI)
the point d will be placed in a conic section passingthrough 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 coin cide 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 tan gent to the trajectory in any given point B. Let the point d come to co
incide with the point B, arid the right line Bt/ Avill 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. SCHOLIUM. The former of
these constructions will be- c come something more simple by joining , and in
that line, produced, if need be, aking Bp to BP as PR is to PT ; and t rough p
draw the indefinite right inc j0e parallel to S PT, and in that line pe taking
always pe equal to Pi , and draw the right lines Be, Cr SEC. Y.J OF NATURAL
PHILOSOPHY. 139 to meet in d. For since Pr to Pt, PR to PT, pB to PB, pe to Pt,
are all in the same ratio, pe and Pr 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. PROPOSITION
XXIII. PROBLEM XV. 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, 1., P, the three other given
points. Jo n BC. and draw IS paral lel 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
Pr, Pt proportional to PR, PT respectively ; and draw Cr, Bt their point of
concourse d will (by Lem. XX) always fall on the trajectory to be described.
The same otherwise. 1 et tl e angle CBH of a given magnitude re volve about the
pole B; as also the rectilinear rad : us 1C, 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 Ikie; 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 there fore the constructions there
set down will become the same with the con structions 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-
ont 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 mannr: 140 THE
MATHEMATICAL PRINCIPLES [BOOK I X IT that HA may be to IA as the rectangle un
der a mean proportional between CG and GP, and a mean proportional between BH
and HD is to a rectangle under a mean pro portional between GD and GB, and a
mean proportional betweeen PI and 1C, and A will be the point of contact. For
if HX, a par allel 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 AP in a ratio that is compounded out of the
ratio of the rec tangle XHY to the rectangle BHD, or of the rectangle CGP to
the rec tangle DGB; and the ratio of the rectangle BHD to the rectangle PIC.
But after the point of contac.t 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. PROPOSITION XXIV. PROBLEM XVI. 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 indefi nite 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 tan gents 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 pioportional 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 pro portional between IX and IY, the rectangle XIY or IZ2 , will (by the
pro perties of the conic sections) be to LP2 as the rectangle CID is to the
rect angle CLD, that is (by the construction), as SI is to SL2 ; and therefore
SEC. V.] OF NATUKAL PHILOSOPHY. 141 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 con structions where it does not
are also known ; and therefore, for brevity s sake, I omit any farther
demonstration of them. LEMMA XXII. To transform figures into other figures of
the same kind. Suppose that any figure HGI is to be transformed. Draw, at
pleasure, two par allel 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 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 Qd 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 con tinual 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
distinction 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 Oa (by which the parallelogram OABa is
completed) the new ordinate radius. I say, then, that if the point G is placed
in a right line given by posi tion, 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 J42 THE MATHEMATICAL PRINCIPLES [BOOK 1. 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 ana lytical 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 Od to
OD, dg to DG, and AB to AD ; and there- OA X AB OA X dg fore AD is equal to , ,
and DG equal to 7 . Now if the ad ad 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 dimenv v xu- ,. OA X AB . OA X dg sion, by writing this equation .
m place of AD, and -. - in place of DG, a new equation will be produced, in
which the new ab scissa 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 tAvo 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 ap
proach 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 denned. This Lemma is of use in the solution of the more difficult
Problems ; for thereby we maj 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 con- SEC. V.] OF NATURAL PHILOSOPHY. 143
course 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 construc tion of plane problems, may be transformed into
a right line and a circle PROPOSITION XXV. PROBLEM XVII. 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 hi 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 parallelogra n fiikl, let
the right lines hi, ik, kl be BO cut in c, d, e, that he 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, he2 to the rectan gle ahb, and ic2 to id2 , and ke2 to kd2 ,
and el2 to the rectangle alb, are all in the same ratio ; and therefore he to
the square root of ahb, ic to id, ke to kdj 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 ante cedents hi + kly to the sum of all the consequents
^/ahb -\- ik : *Jalb, 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 //, /, or without taem, the points c, d,
e, must be taken 144 THE MATHEMATICAL PRINCIPLES BOOK I.J Cither between the
points, h, i, k, /, or without them. If one of the points a, b, falls between
the points h, i, and the other xvithout the points h, I, the Problem is
impossible. PROPOSITION XXVI. PROBLEM XVIII. 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; / xs o 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, Al l\ ik and hi, 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?
being equal to Op, q will be the other point through which the conic sec tion
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. LEMMA XXIII. 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
%- ,.--- I \ triangle EFC will be given in kind. Let E K cT^"^ 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 SEC. V.] OF NATURAL PHILOSOPHY.
145 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 po tiition) 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. LEMMA XXIV. 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 wkiJt 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 touch ing 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 G Q 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, orEAto 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 tli?
nature of the conic sections, LI (or CK) is to CD as CD to CH ; and therefore
(ex aquo pertnrhatfy 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 cequo pertnrbot, ) 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. LEMMA XXV. Iffour sides of a parallelogram indefinitely
produced touch any conic section, and are cut by a fifth tangent ; I say, that,
taking those seg ments of any two conterminous sides that terminate in opposite
angles 10 146 THE MATHEMATICAL PRINCIPLES [BooK 1. 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 Uie other segment, Let the four sides ML, IK, KL, MI, of the
parallelogram MLJK touch the conic section in A, B, C, I) ; and let the fifth
tangent FQ cut those sides in F, Q, H, and E : and taking the segments ME, KQ
of the sides Ml, KJ, or the segments KH, MF of the sides KL, ML, 1 s/.y, 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 El 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 X ME, as also
the rectangle KH X ME 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 Kl. MI in q and e, the rectangle KQ X
ME will be equal to the rectangle K&lt;/ X Me, and KQ will be to Me as
Kq to ME, and by division ns Q? to Ee. COR. 3. Hence, also, if E&lt;?,
eQ, 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&lt;/ is to Ee as KQ to Me, the same right line will pass through the
middle of all the lines Eq, eQ, MK (by Lem. XXIII), and the middle point of the
right line MK is the centre of the section. PROPOSITION XXVII. PROBLEM XIX. To
describe a trajectory that may touch jive 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 quadri lateral tained figure under
ABFE con- any four of them ; and (by Cor. 3, Lem. XXV) the right line MN draAvn
through the points (,f SEC. V.] OF NATURAL PHILOSOPHY. 147 bisection will pass
through the centre of the trajectory. Again, bisect in P and Q, the diagonals
(if I may so call them) Bl), GF of the quadrila teral figure EC OF contained
under any other four tangents, and the right line PQ, drawn through the points
of bisection will pass through the cen tre 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 tan gent BG 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 de scribed. Let this cut any other two tangents GCD, FJ)E, in
L and K. Through the points G and K, F and L, where the tangents not parallel,
CL, FK meet the parallel tangents CF, KL, draw GK, FL meeting in K ; and the
right line OR drawn and produced, will cut the parallel tan gents GF, KL, in
the points of contact. This appears from Gor. 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. SCPIOLTUM. 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, ami 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 manmr. In the construction and figure of Lem. XXI,
let those , legs BP, CP, of the moveable angles PEN, ^ PCN, by the concourse of
which the trajec- \ tory was described, be made parallel one to the other : and
retaining that position, let them revolve about their poles I , C, in that
figure. In the mean while let the other legs GN, 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 con- 148 THE MATHEMATICAL
PRINCIPLES [Book I trary will hajpen if those legs meet in the remotest point
L. Whence ii the centre of the trajectory is given, the axes will be given ;
and those be- ing 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 f mr given points. For if two of the given
points are made the poles C, 13, 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 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 CP; 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 to ich ano her conic
section of the same kind with the former, arid havin^ its axes parallel to the
axes of the former. But I hasten to things of greater use. LEMMA XXVI. To place
1ht lit rev angles of a triangle, given both in kind and magni tude, in,
respect of as many rigid lines given by position, -provided th\] are not all
parallel among themselves, in such manner tfia t jic spiral 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 1) 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 Rubles BAG, ABC, ACB respectively. But those segments are
to be de scribed t wards such sides of the lines DE, DF; EF; that the letters 3
EC. V.I OF NATURAL PHILOSOPHY. 1411 DRED may turn round about in the same order
with the letters I1ACB : 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 th se segmerts 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 Ga to AB as GP is to PQ ; and about the centre G, with the interval
Ga, describe a circle that may cut the first circle DGE in a. Join aD cutting
the second circle DFG in b, as well as aE cutting the third circle EMF in c.
Complete the figure ABCdef similar and equal to the figure a&cDEF : I say,
the thing is done. For drawing Fc meeting D in n, and joining aG; bG, QG, QD.
PD, by construction the angle EaD is equal to the angle CAB, and the angle acF
equal to the angle ACB; and therefore the triangle aiic equiangular to the
triangle ABC. Wherefore the angle anc or FnD is equal to the angle ABC, and
conse- &lt; uently to the angle F/&gt;D ; and there fore 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 GaD 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 GbD at the circum ference, and
therefore equal to the angle Gba. Upon which account the triangles GPQ, Gab,
are similar, and Ga is to ab as GP to PQ. ; that is (by construction), as Ga 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 I), E, F, of the triangle DEF do respectively touch the sides ab, ar, be
of the triangle afjc/ the figure AECdef may be completed similar and equal to
the figure afrcDEFj 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, arid by having the sides DE,
DF placed i&gt;t 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. THE MATHEMATICAL PRINCIPLES [BOOK 1. PROPOSITION
XXVIII. PROBLEM XX. To describe a trajectory giren 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, EF, 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. LEMMA XXVII. To describe a trapezium given in kind, the
angles whereof may be , placed, in respect offour right lines given by
position, that are neither all paralhl among themselves, nor converge to one
common point, ////// 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 FCHI, and whose angle
/, equal to the given angle F, may touch the right line ABC ; and (lie other
angles g, h, i, equal to the other given angles, G, H, I, may touch the other
lines AD, BD, CE, re spectively. Join FH, and upon FG. FH, FI describe J% 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.
Bnrf&gt;, 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 .ibout 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 cir cles, 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 SEC. V.J OF NATURAL PHILOSOPHY. 15] may be the same as of the
letters A, B, C ; and about the centre R with the interval RF describe a fourth
circle FNc cutting (lie third circle FVI in c. Join Fc1 cut ting the first
circle in a, and the second in / . Draw aG, &H, cl, and let the figure ABC/4f/ii
be made similar to the figure w^cFGHI; and the trapezium fghi will be that
which was required to be de scribed. For let the two first circles FSG, FTH cut
one the other in K ; join PK, Q,K, RK, "K, 6K, cK, and produce QP to
L. The angles FaK, F6K, FcK at the circumferences are the halves of the angles
FPK, FQK, FRK, at the centres, and therefore equal to LPK, LQ.K, LRK, the
halves of those angles. Wherefore the figure PQRK is iquiangular and similar to
the figure 6cK, and consequently ab is to be res PQ, to Q,R, that is, as AB to
BC. But by construction, the angles Air, /B//,/C?, are equal to the angles FG,
F&H, Fcl. And therefore the figure ABCfghi may be completed similar to the
figure abcFGHl. vVliich done a trapezium fghi will be constructed similar to
the trapezium FGHI, and which by its angles/, 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 in
creased 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 iL to M, so as LM may be to iL 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 M* say, the thing is done. For let MO- cut the right line AB in
Q, and AD the right line KL iu II ^52 THE MATHEMATICAL PRINCIPLES [BOOK I. S,
arid draw AP parallel to BD, and meeting iL in P, and -M to Lh (g\ to hi, Mi to
Li, GI to HI, AK to BK) and AP to BL, will be in the same ratio. Cut DL in 11,
so as DL to RL may be in that same ratio; and be cause ffS to g~M, AS to AP.
and DS to DL are proportional; therefore (ex ceqit.o) as gS to LA, so will AS
be to BL, and DS to RL ; and mixtly. BL RL to Lh BL, as AS DS to gS AS. That
is, BR is to Eh as AD is to Ag, and therefore as BD to gQ. And alternately BR
is to BD as 13/i 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 Mi to Li, that is, as GI to HI, it is manifest that the
lines FI, fi, are similarly cut in G and H, g and //.. Q.E.F. In the
construction of this Corollary, after the line LK is drawn cutting CE in i, we
may produce iE to V, so as EV may be to Ei as FH to HI, arid then draw V/~
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 iX to Y so as iY
may be equal to IF, and then draw Yf parallel to BO. Sir Christopher Wren and
Dr. Wallis have long ago given other solu tions of this Problem. PROPOSITION
XXIX. PROBLEM XXI. 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 po sition, 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 an gles/, g, h, i,
may touch the right lines given by posi tion AB, AD, BD, CE, severally
according to their order. And then about bins trapezium describe a trajectory,
that trajectory will be similar to the curve line FGHI. SCHOLIUM. This problem
may be likewise constructed in the following manner. Joining FG, GH, HI, FI,
produce GF to Y, and join FH, IG, and make SEC. VI OF NATURAL PHILOSOPHY. 153
El the angles CAK. DAL equal to the angles PGH, 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 CHI, 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 OA.KMC, ALKA, DALND may be carried round in the same order as the
letters FGHIF ; and draw MN meeting the right v line CE in L Make the angle iEP
equal to the angle IGF, and let PE be to Ei as FG to GI ; and through P draw
PQ/ that may with the right line ADE contain an angle PQE equal to the angle
FIG, and may meet the right line AB in /, and join fi. But PE and PQ arc- to be
drawn towards those sides of the lines CE, PE, that the circular order of the
letters PEtP and PEQP may be the same as of the letters FGHIF ; and if upon the
line/i, in the same order of letters, and similar to the trapezium FGHI, a
trapezium /^//.i 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 deter mine the motions of bodies in
the orbits so found. SECTION VI. How the motions are to be found in given,
orbits. PROPOSITION XXX. PROBLEM XXII. 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 X 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 propor tional to it. Bisect AS in G, and erect the
perpendicular GH equal to 3M, and a circle described about th j centre H, with
the interval HS, will cut the parabola in the place P required. For letting
fall PO perpendic ular on the axis, and drawing PH, there will be AG2 -f- GH2
(=.= HP2 -_ AO^TAGJ* + PO GH|2 ) = AO2 + PO2 2CA &gt; ?G!I f PO A G S
154 THE MATHEMATICAL PRINCIPLES [BOOK I AG* + GH2 . Whence 2GH X PO ( AO2 + PO2
2GAO) = AOJ PO2 -f | PO2 . For AO2 write AO X ; then dividing all the terms by
2PO; and multiplying them by 2AS, we shall have ^GH X AS (= IAO the area APO
SPO)| = to the area APS. But GH was 3M, and therefore ^GH X AS is 4AS X M.
Wherefore the area cut off APS is equal to the area that was to be cut off 4AS
X M. Q.E.D. Con. 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
xelocity 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 ver tex 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, LEMMA XXVIII. 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 mov cable 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 num ber 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 in- 8FC. Vl.J OF NATURAL PHILOSOPHY.
155 terscctions is not to be found but by an equation of two dimensions, fo
which the other intersection may be also found. Because there may b(- 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
bi&gt; 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
se" f ions with the curves of the third order, because they may amount
to six, (\,me out together by equations of six dimensions ; and the inter
sections 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 ne cessarily
happen, we might reduce all solid to plane Problems, and those higher than
solid to solid Problems. But here i speak of curves irreduci ble in power. For
if the equation by which the curve is defined may bo reduced to a lower power,
the curve will not be one single curve, but com posed 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 urder 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 iitfinitum.
Wherefore the innumerable inter sections of a right line with a spiral, since
this is but one simple curve and not reducible to more curves, require
equations infinite in r- .imber 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 intersec tions 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 equa
tion will return to its first form ; and consequently one and the same equation
will exhibit all the intersections, and will therefore have an infi nite 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 an^ such equation. 1 56 THE
MATHEMATICAL PRINCIPLES [BOOK 1 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 peri meter cannot be universally exhibited by any finite
equation. But here I speak of ovals that are not touched by conjugate figures
running out in infinitvm. 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 ol 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.
PROPOSITION XXXI. PROBLEM XXIII. Tofind 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 A PB ; 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, de scribe 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 ALL 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 SEC. VI. OF NATURAL PHILOSOPHY. 157
SQ, OQ. Let OQ meet the arc EFG in F, and upon OQ let fall the perpendicular
Sll. The area APS is as the area AQS, that is, as tlie difference between the
sector OQA and the triangle OQS, or as the difLience of the rectangles *OQ, X
AQ, and -J.OQ X SR, that is, because . &gt;,_ is given, as the
difference between the arc AQ and the right line Sll : ai.;l 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 Sii to GF sine of the arc AQ) as GK, the
difference between the arc C 1 and tlie sine of the arc AQ. Q.E.D. SCHOLIUM.
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 jo. 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 a iules as the time in which tlie
body described the arc A/?, 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
-fD 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 1OQ H- E to the radius, and an angle G, that may be to the
angle NAOQ E -f 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 f- E 4- G to the radius; and an angle I to
the angle N AOQ E G -f- H, as the 58 THE MATHEMATICAL PRINCIPLES jB(OK 1. length
L is to the same length L diminished by the cosine of the angle AOQ -f- 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 AOy equal to the
angle AOQ -f- E 4- G + I -\-} &c. and from its cosine Or and the
ordinatejor, which is to its sine qr as the lesser axis of the ellipsis to the
greater, \\ e shall have p the correct place of the body. When the angle N AOQ,
-f 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 F, and N AOQ E G +
H come out negative. But the infinite series AOQ -f- E -f- G -|- I +, &c.
converges so very fast, that it will be scarcely ever needful to pro ceed
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 rij;ht i ne SP, that cuts off an
area APS near the truth. Join OP, and from A and P to the asymptote T A S draw
AI, PK parallel to the other asymptote ; and by the table of loga rithms 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 2A, or
2A 2A PS, 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 par ticular calculus that
follows is better fitted for as tronomical purposes. Supposing AO, OB, OD, to
be the semi-axis of the ellipsis, and L its latus rec tum, and D the difference
betwixt the lesser semi- SEC. VII.] OF NATURAL PHILOSOPHY. J 59 axis OD, and
-,L 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 J), and AO 4- 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 rec tangle 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 thr 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 suf ficient 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 oi 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 mav 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. SECTION VII. Concerning the
rectilinear ascent and descent of bodies, PROPOSITION XXXII. PROBLEM XXIV.
Supposing that the centripetal force is reciprocally proportional to tht square
of tlie 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. I 160 THE MATHEMATICAL
PRINCIPLES [BOOK I Prop. XIII) describe some conic section whose focus is A
placed in the centre of force. Suppose that conic sec tion to be A RPB 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 I) PC pass
through the falling body, making right angles with the axis; and drawing DS,
PS, the area ASD will c be proportional to the area ASP, and therefore also to
the time. The axis AB still reaiaining the same, let the breadth of the
ellipsis be perpetually diminished, and s the area ASD will always remain
proportional to the time. Suppose that breadth to be diminished in, in fruit um
; 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
AC1 . 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 perpendic ularly
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/P, SPy13, are severally to the sev eral areas CSD, CBED,
SDEB, in the given ratio of the heights CP, CD, and the area SP/B is
proportional to the time in which the body P will move through the arc P/B. the
area SDEB will be also proportional to that time. Let the latus rectum of the
hyperbola RPB be diminished in infitiitum, 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, A- 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 per pendicular 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
ver tex 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 if to the time in which that
body P or C will descend to the centre S or B Q.K.T fl.l OF NATURAL PHILOSOPHY.
PROPOSITION XXXIII. THEOREM IX. The tilings above found being supposed. I say,
thai the, velocity of a Jai ling 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 iAB, the principal semi-diameter of the
Let AB, the common dia meter 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 (pro duced, if need be) in T; and let SY be
perpendicular to this line, and BQ to this di ameter, and suppose the latus
rectum of the figure RPB to be L. Prom 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 L X SP to SY2 Por
by the properties of the conic sections ACB is to CP2 as 2AO to L. 2CP5 X AO
and therefore rrrr; is equal to L. Therefore those, velocities an o-- ACB to
each other in the subduplicate ratio of CP3 X AO X SP ACB toSY~. More over, by
the properties of the conic sections, CO is to BO as BO to TV.? and (by
composition or division) as CB to BT. Whence (by division cs composition) BO or
+ CO will be to BO as CT to BT, that is, AC CP2 X AO X SP ACB" is
equal will be to AO as CP to BQ; and therefore to ~AO X BC * ^ W suPPose GV,
tne 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 11 162 THE MATHEMATICAL PRINCIPLES [BOOK 1 a body describing a
circle about the centre B, at the distance BC, in thr BQ2 X AC X SP
subduplicate ratio of --r-^-^-to SY2 , that is (neglecting the ra- X Jo tios of
equality of SP to BC, and BQ,2 to SY2 ), in the subduplicate ratio of AC to AO,
or iAB. 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. PROPOSITION XXXIV. THEOREM X. 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 cen tre S, in any
place P, is equal to the velocity of a body uniformly describing a circle about
the c same centre S at half the interval SP. Let the breadth CP of the parabola
be diminished in itifiiiitirni, so as the parabolic arc P/B may come to
coincide with the right line CB, the centre S s with the vertex B, and the
interval SP with the interval BC, and the proposition will be manifest. Q.E.D.
PROPOSITION XXXV. THEOREM XL The same things supposed, I say, that the area of
the figure DES, de scribed by the indefinite radius SD, is equal to the area
which a body with a radius equal to h df the latus rectum of the figure DES, by
uniformly revolving about the centre S, may describe in the same tijiw. 1 JD/
AJ SEC. ni: OF NATURAL PHILOSOPHY. For suppose a body C in the smallest moment
of time describes in fal ling the infinitely little line Cc. while another body
K, uniformly revolv ing about the centre S in the circle OK/r, describes the
arc KA:. Erect the perpendiculars CD, cd, meeting the figure DES in D, d. Join
SD, Sf/. SK. SA*; and draw Del 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 Cc to Dd, and TD to TS as CD to S Y ;
ex aquo TC will be to TS as CD X Cc to SY X Dd. 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 X Cc
to S Y X Vd, Farther, the velocity of the descending body in C IF, 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 Pi-op. XXXIII) ; and this
velocity is to the velocity of a body describing the circle OKA: in the
subduplicate ratio of SK to SC (by Cor. 6, Prop IV) ; and, ex aqnnj the first velocity
to the last, that is, the little line Cc to the arc K/r, in the subduplicate
ratio of AC to SC, that is, in the ratio of AC to CD. Wherefore CD X Cc is
equal to AC X KA*, and consequently AC to SK as AC X KA: to SY X IW. and thence
SK X KA: equal to SY X Drf, and iSK X KA: equal to SY X DC/, that is, the area
KSA* equal to the area SDrf. Therefore in every moment of time two equal
particles, KSA" and SDrf, of areas are generated, which, if their
magnitude is diminished, and their number increased in iiifinif t-w, obtain the
ratio cf 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 X Cc to SY X Df/ as TC to TS, that is, as
2 to 1 ; and that therefore |CD X Cc is equal to i SY X Vd. But the veloc ity
of the falling body in C is equal to the velocity w r ith which a circle may be
uniformly described at the interval 4SC (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 Cc to the arc KA, is (by Cor. 6, Prop. IV) in the
subduplicate ratio of SK to iSC ; that is, in the ratio of SK to *CD. Wherefore
iSK X KA: is equal to 4CD X Cc, and therefore equal to SY X T)d ; that is, the
area KSA* is equal to the area SIW, as above. Q.E.D. 164 THE MATHEMATICAL
PRINCIPLES [BOOK 1. PROPOSITION XXXVI. PROBLEM XXV. 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. M a given
PROPOSITION XXXVII. PROBLEM XXVI. 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 oif 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 \ H D the centre S at the given
interval SG, take GA to AS. 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 ver tex 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/vK ; 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& be made equal to the
segments SEIS, SEDS. and (by Prop. XXXV) the body G will describe SEC. VII.] OF
NATURAL PHILOSOPHY. 165 the space GO in the same time in which the body K may
describe t*he arc Kk. Q.E.F. PROPOSITION XXXVIII. THEOREM XII. Supposing that
the centripetal force is proportional to the altitude or distance ofplaces from
the centre, I say, that the times and velocities offalling 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 A. 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 ve locity CD. This is demonstrated the same
way from Prop. X, as Prop. XXX11 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 re volving bodies
are equal (by Cor. 3; Prop. IV). PROPOSITION XXXIX. PROBLEM XXVIT. Supposing a
centripetal force of any kind, and granting the quadratnres of curvilinear
figures ; it is required to find the velocity of a bod)/, 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 D 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 cur vilinear area ABGE.
Q.E.I. In EG take EM reciprocally proportional to E 366 THE MATHEMATICAL
PRINCIPLES [BOOK 1 a right line whose square is equal to the area ABGE, and let
VLM be 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 descend ing 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 VY, and the area ABGE as YY +
2VI -f II; and by division, the area DFGE as 2VI -f LI, and therefore ^ will be
as--^r that is. if we take the first ratios of those quantities when just
nascent, the 2YI length DF is as the quantity -|yrr an(i therefore also as half
that quantity 1 X Y But the time in which the body in falling describes the
very line DE, is as that line directly and the velocity Y 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 I X
V are just nascent, as -jy==r-. 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 (he 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. Lern. IV), the whole time in which the
line AE is described will be as the whole area ATYME. 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 o DF as that uniform
force to the other force in the place D. Complete the rectangle PDRQ,, and cut
iff the area. ABFD equal to that rectangle. Then A will be the place SEC. VII.
I OF NATURAL PHILOSOPHY. 10; from whence the other body fell. For com pleting
the rectangle DRSE, since the area ABFD is to the area DFGE as VV to 2VI, and
therefore as 4V 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 incre ment 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 or- dinates DF, DR, and consequently as the
nascent areas DFGE, DRSE : therefore, ex aq-uo, 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 DFge, if
the place e is below the place D, or diminished by the same area DFg-e, 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 recipro cally
proportional to the square root of PQRD -f- or T)Fge, and taking the time in
which the body has described the line De to the time in which another body has
fallen with an uniform force from P, and in falling ar rived at D in the
proportion of the curvilinear area DLme to the rectan gle 2PD X 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 -f ^DE; or 2PD to 2PD -f- 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 X 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 De as the area DLME
to the area DLme ; and, ex aquo, the first mentioned of these times is to the
last as the rectangle 2PD X DL to the area DLrae. 163 THE MATHEMATICAL
PRINCIPLES [BoOK I SECTION VIII. Of the invention of orbits wherein bodies will
revolve, being acted upon by any sort of centripetal force. PROPOSITION XL.
THEOREM XIII. // 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 amj one case of equal altitudes, t/ieir velocities will be also equal
at all equal altitudes. Let a body descend from A through D and E, to the
centre (j : and let another body move from V in the curve line VIK&. From
the centre C, with any distances, describe the concentric circles DI, EK,
meeting the right line AC in I) and E; and the curve VIK in I and K. Draw 1C
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 ; K / and
imagine the bodies in D and I to have equal velocities. Then because the
distances CD and CI are equal, the centri petal forces in D and I will be also
equal. Let those forces be k) expressed by the equal lineoke DE and IN ; and
let the force IN (by Cor. 2 of the Laws of Motion) be resolved into two others,
NT and IT. r l hen the force NT acting in the direction 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/j. 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 X 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
holies in E and K are also equal, and by the same reasoning they will always be
found equal in any subsequent equal dis tances. Q..E.D. SEC. VI11.J OF NATURAL
PHILOSOPHY. 169 By the same reasoning, bodies of equal velocities and equal
distances from the centre will be 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 An , 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 v/ P 11
A", 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.
PROPOSITION XLI. PROBLEM XXVTII. Supposing a centripetal force of any kind, and
granting the quadra tures of curvilinear figures, it is required to find as
well the trajecto ries 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 VIKAr. Let R, there be given the
circle VR, described from the centre C with any interval CV; and from the same
centre de scribe any other circles ID, KE cut ting the trajectory in I and K,
and the right line CV in D and E. Then draw the right line CNIX cutting the c
circles KE, VR in N and X, and the right line CKY meeting the circle VJi 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 anothe body
is to fall, so as in the place D to acquire a ve locity 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 THE MATHEMATICAL PRINCIPLES [BOOK 1 trill 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 1C : that is (if there be
given any quantity Q, and the altitude 1C be called A), as -T-. This quantity
call Z, and suppose the magnitude of Q, to oe such that in some case v/ABFD may
be to Z as IK to KN, and then in all cases V 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 V ABFD ZZ to Z, or as IN to KN; and therefore A x KN Q. x IN \vill be
equal to . Therefore since YX X XC is to A X KN ZZ Q. X IN x CX2 as CX2 , to
AA, the rectangle XY X XC will be equal to-AAv/ABFD ZZ. Therefore in the
perpendicular DF let there be taken continually I)//, IV Q ax ex2 equal to , =.
respectively, and 2 v/ ABFD ZZ 2AA V ABFD ZZ let the curve lines ab, ac, the
foci of the points b and c, be described : and from the point V let the
perpendicular Va be erected to the line AC, cut ting off the curvilinear areas
VD&a, VDra, and let the ordi nates Es:? E#, be erected also. Then because
the rectangle D& X IN or DbzR is equal to half the rectangle A X KN, or to
the triangle ICK ; and the rectangle DC X IN or Dc.rE is equal to half the
rectangle YX X XC, or to the triangle XCY; that is, because the nascent
particles I)6d3, ICK of the areas VD/&gt;#, VIC are always equal; and
the nascent particles Dc^-E, XCY of the areas VDca, VCX are always equal :
therefore the generated area VD6a will be equal to the generated area VIC, and
there fore proportional to the time; and the generated area VDco- 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 pro portional to
it VD/&gt;; and thence will be given the altitude of the body CD or CI
; and the area VDca, and the sector VCX equal there o, together with its angle
VCL But the angb 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 1C drawn through the centre falls perpendicularly
upon the trajectory VTK; which comes to pass when the right lines IK and NK
become equal; that is, when the area ABFD ig C nl to ZZ. SEC. OF NATURAL
PHILOSOPHY. 171 VI1LJ COR. 2. So also the angle KIN, in which the trajectory at
any place cuts the line 1C. may be readily found by the given altitude 1C 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 T RT meeting the axis CV
indefinitely pro duced in the point T ; and then joining C CR there be drawn
the right line CP, Qequal 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 direc
tion 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 VI\ S be an hyberbola, the body will descend to the cen tre ; but
if it be an ellipsis, it will ascend perpetually, and go farther and farther
off in infinilum. And, on the contrary, if a body endued with any velocity goes
off from the place V, and according as it begins either to de scend 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 increas ing 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 ourve, the invention of which, as
being easy enough, for brevity s sake I omit. PROPOSITION XLII. PROBLEM XXIX.
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 Ihe three preceding
propositions; and let the body go off from the place I in the direction of the
little line, IK, with the same ve locity as another body, by falling with an
uniform centripetal force from the place P, may acquire in I); and let this
uniform force be to the force with which the body 1.72 THE MATHEMATICAL
PRINCIPLES [BOOK 1. 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 Ck, describe the circle
ke, meeting the right line PD in e, and let there be erected the lines eg, ev,
ew, ordi- nately applied to the curves BF*, 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* 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 1K; 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 Dbve, there is given both the altitude of the body Ce or Ck,
and the area Dcwe, with the sector equal to it XCy, the angle 1CA:, 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 Bame. 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.
SECTION IX. Of the motion of bodies in moveable orbits ; and of the motion of
the apsides. PROPOSITION XLIII. PROBLEM XXX. Ft is required to make a body move
in a trajectory that revolves about the centre offorce 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 Cp, equal to CP, making the angle VC/?
proportional to the angle VCP ; and the area which the line Cp describes will
be to the area VCP, which the line CP describes at the same time, ns the
velocity of the describing line Cp to the velocity of the describing line CP ;
that is, as the angle VC/? to the angle VCP, therefore in a given ratio, and
therefore proportional to the time. Since, then, the area described by the line
Cp 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 SEC. L\.] OF
NATURAL PHILOSOPHY. 173 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^ equal to the angle PC/?, and the line Cu
equal to CV, and the figure uCp equal to the figure VCP; and the body being al
ways in the point p} will move in the perimeter of the revolving figure nCp,
and will describe its (revolving) arc up in the same time the* the other body P
describes the similar and equal arc VP in the quiescov.t fig ure YPK. 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 pom* p de scribes in an immovable plane,
and the Problem will be solved. O/E.K. PROPOSITION XLIV. THEOREM XIV. The
difference of the forces, by which two bodies may be madi, to KMVG equally, one
in a quiescent, the other in the same orbit revolving, i 1 in a triplicate
ratio of their common altitudes inversely. Let the parts of the quiescent or
bit 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 pC, and produce
it to m, so that mr may be to kr as the angle VC/? to the /2\- angle VCP.
Because the altitudes of the bodies PC and pV, KG and kC} are always equal, it
is manifest that the increments or decrements of the lines PC and pC 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, pC, and the
other, transverse to the former, hath a direction perpendicular to the lines PC
and pC ; the mo tions 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 mo tion of the line pC to the angular motion of the line PC ; that is,
as the angle VC/? to the angle VCP. Therefore, at the same time that the bodv
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 ; arid 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 pC ; and by its
transverse motion will acquire a distance from the line 174 THE MATHEMATICAL
PRINCIPLES [BOOK J. 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/? 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 ex
piration of that time, will be found in the place m. These things will be so,
if the bodies jo and P are equally moved in the directions of the lines pC and
PC, and are therefore urged with equal forces in those directions. I: ut if we
take an angle pCn that is to the angle pCk as the angle VGj0 to the angle VCP,
and nC be equal to kG, 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 nCp is greater than the angle kCp, that is, if the orbit npk,
move either in cmiseqnentia, or in antecedenticij with a celerity greater than
the double of that with which the line CP moves in conseqnentia ; and with a
less force if the orbit moves slower in antecedent-la. And ihj 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 Cn or Ck suppose a circle described
cutting the lines mr, tun pro duced in s and , and the rectangle mn X nit will
be equal to the rectan- *//? n ^* */?? ^ "le mk X ins, and therefore
mn will be equal to . But since mt the triangles pCk, pCn, in a given time, are
of a given magnitude, kr and mr. a id their difference mk, and their sum ms,
are reciprocally as the al titude pC, and therefore the rectangle mk X ms is
reciprocally as the square of the altitude pC. But, moreover, mt is directly as
|//z/, that is, as the altitude pC. These are the first ratios of the nascent
lines ; and hence r - that is, the nascent lineola mn. and the difference of
the forces mt proportional thereto, are reciprocally as the cube of the
altitude pC. Q.E.D. COR. I. Hence the difference of the forces in the places P
and p, or K and /.*, 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 de scribes the arc PK, as the nascent line m,n to the versed sine of the
nascent mk X ms rk2 arc RK, that is, as to ^g, or as mk X 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 VQ?, as GG FF to FF. And,
therefore, if from the centre C, with any distance CP or Cp, there be described
a circular sector equal to the whole area VPC, which the body OEC. IX.l OF
NATURAL PHILOSOPHY. 175 revolving in an immovable orbit has by a radius drawn
to the centre de- bribed 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 or bit, 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 pCk
are to one another as the times in which they are described. COR. 2. If the
orbit YPK be an ellipsis, having its focus C, and its highest apsis Y, and we
suppose the the ellipsis upk similar and equal to .. it, so that pC may be
always equal / to PC, and the angle YC/? be to the ; angle YCP in the given
ratio of G \ to F ; and for the altitude PC or pC \ we put A, and 2R for the
latus rec- /t\ turn of the ellipsis, the force with * which a body may be made
to re volve in a movable ellipsis will be as FF RGG RFF - + - -rg , and vice
versa. /Y A. A. Let the force with which a body may revolve in an immovable
ellipsis be expressed by the quantity , and the -. 7 force in V will be FF But
the force with which a body may revolve in a circle at the distance CY, with
the same velocity as a body revolving in an ellipsis has in Y, is to the force
with which a body revolving in an ellip sis is acted upon in the apsis Y, as
half the latus rectum of the ellipsis to the RFF semi-diameter CY of the
circle, and therefore is as , =- : and tlu RFF which is to this, as GG FF to
FF, is as - ~py^~~ ~ : and this force (by Cor. 1 cf this Prop.) is the
difference of the forces in Y, with which the body P revolves in the immovable
ellipsis YPK, 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
CY as -r-, to ^TF- the same AJ CYJ R C^ ( "* R P^ T* 1 difference in
every altitude A will be as - 3 : . Therefore to the FF force -T-: , by which
the body may revolve in an immovable ellipsis VPK 176 THE MATHEMATICAL
PRINCIPLES [BOOK I. idd the excess -:-= , and the sum will be the whole force
-r-r -\- A AA RGG RFF, .-5 by which a body may revolve in the same time in the
motA. able ellipsis upk. COR. 3. In the same manner it will be found, that, if
the immovable or bit 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 VCjo be continually
to the angle TCP as G to F ; the forces with which bodies may revolve in the
imFFA FFA movable and movable ellipsis, in equal times, will be as ^ and -p~
RGG RFF + .-3 respectively. A 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 Y, 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 VFF V be called -f-=Trri , 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 VCjD 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 -f- VRGG VRFF ~ A* 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 -p
length, perpendicular to the line CV given by po sition, and CP be drawn, and
Cp equal to it, mak ing the angle VC/? having a given ratio to the an gle VCP,
the force with which a body may revolve in the curve line Vjo/r, which the point
p is con tinually describing, will be reciprocally as the cube C of the
altitude Cp. For the body P, by its vis in ertia 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 Cp, and
(by what was just demonstrated) the SEC. IX..J OF NATURAL PHILOSOPHY. 177 body
will deflect from the rectilinear motion into the curve line Ypk. But this
curve ~Vpk is the same with the curve VPQ found in Cor. 3, Prop XLI, in which,
I said, hodies attracted with such forces would ascend obliquely. PROPOSITION
XLV. PROBLEM XXXL 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/?, and X
for the difference of the altitudes CV CP : and the force w r ith which a body
moves in an ellipsis revolving about its p-p T? C* f ^ T? F* F focus C (as in
Cor. 2), and which in Cor. 2 was as -r-r -\ -.-3 , FFA + RGG RFF , that is as,
-^ , by substituting T X for A, will be- A RGG RFF + TFF FFX come as -p . In
like manner any other cen tripetal force is to be reduced to a fraction whose
denominator is A 3 , and the numerators are to be made analogous by collating
together the homo logous terms. This will be made plainer by Examples. EXAMPLE
1. Let us suppose the centripetal force to be uniform, A3 and therefore as 3
or, writing T X for A in the numerator, as T3 3TTX + 3TXX X3 =-. Ihen collating
together the correspon- A3 dent 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
-f- TFF to T3 as FFX to 3TTX -f 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/? to the angle VCP, as 1 to v/3.
Therefore since the body, in an immovable 178 THE MATHEMATICAL PRINCIPLES [BOOK
I ellipsis, in descending from the upper to the lower apsis, describes an
angle, if I may so speak, of ISO deg., the other body in a movable ellipsis,
and there fore in the immovable orbit we are treating of, will in its descent
from 180 the upper to the lower apsis, describe an angle VCjt? of ^ deg. And
this \/o 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 quies
cent 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 180 force in an
orbit nearly circular, will always describe an angle of deg/, or v/o 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 in- finitwn. EXAM. 2.
Suppose the centripetal force to be as any power of the alti- An tude A, as,
for example, A n 3 , or-r^ ; where n 3 and n signify any inA. dices 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 Tn &gt;/XT n T + ^XXTn 2
, &c. And conferring these terms with the terms of the other numerator RGG
RFF + TFF FFX, it becomes as RGG RFF 4- TFF to Tn , so FF to ?/.Tn r + ? ~^ XTn
2 , &c. And taking the last ratios where the orbits approach to circles, it
becomes as RGG to T 1 , so FF to nT-1 T , or as GG to T" , so FF to
?*Tn ; and again, GG to FF, so Tn l to nT" 1 , that is, as 1 to n ;
and therefore G is to F, that is the angle VCp to the angle VCP, as 1 to ^/n.
Therefore since the angle VCP, described in the de scent of the body from the
upper apsis to the lower apsis in an ellipsis, is of 180 deg., the angle VC/?,
described in the descent of the body from the upper apsis to the lower apsis in
an orbit nearly circular which a body de scribes with a centripetal force
proportional to the power A n 3 , will be equal ISO to an angle of - deg., and
this angle being repeated, the body will re- \/ti turn 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, A 4 or -p, n will be equal
to 4, and ^/n equal to 2 ; and thereLre the angle IX.] OF NATURAL PHILOSOPHY.
IT9 ISO between the upper and the lower apsis will be equal to 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 infiuitum. This appears also
from Prop. X. For a body acted on by this centripetal force will re volve in an
immovable ellipsis, whose centre is the centre of force. If the 1 A 2 centripetal
force is reciprocally as the distance, that is, directly as or A A" ji
will be equal to 2 ; and therefore the angle between the upper and lower 180
apsis will be - deg., or 127 deg., 16 min., 45 sec. ; and therefore a body re
v/2 volving 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 , and, therefore, directly as -r-fp or as Ts&gt; n wil* ^e et l ual f
\&gt; an(1 4 A^- A 1 Of)- deg. will be equal to 360 deg. ; and
therefore the body parting from v/ n the upper apsis, and from thence perpetually
descending, will arrive at the lower apsis when it has completed one entire
revolution ; and thence as cending 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 alti
tude, and b and c for any given numbers, suppose the centripetal force 6Ara +
cA" b into T X&gt; -f- c into T X to be as r^ that is, as A3
A3 or (by the method of converging series above-mentioned) as bTm + cTn
m6XT" - 1 //cXTn mm m vvrpm un n ~~2--0A.A1 ^ t-XXT" 2 , fcc.
T$~ ~ and comparing the terms of the numerators, there will arise RGG II FF -f
TFF to ^Tm + cT" as FF to mbTm i " - + 2 " m bXT"
- * + "^p cXTn - .fee. And tak- ing the last ratios that arise when
the orbits come to a circular form, there will come forth GG to 6Tm l -f cTn 1
as FF to mbTm l + ncT" J ; and again, GG to FF as 6Tm + cTn to mbTn 1
-f ncTn \ This proportion, by expressing the greatest altitude CV or T
arithmeti cally by unity, becomes, GG to FF as b -{- c to mb -\- ?/c, and
therefore as I (80 THE MATHEMATICAL PRINCIPLES [BOOK 1 tub ~h nc to - y7 Whence
G becomes to P, that is, the angle VCjo to the anf) ~T~ C gle VCP. as 1 to
&gt;/- . - -. And therefore since the angle VCP between the upper and
the lower apsis, in an immovable ellipsis, is of 180 deg., thr angle VC/?
between the same apsides in an orbit which a body describes b A m I c A n with
a centripetal force, that is. as - r , will be equal to an angle of A. ISO v/
1~TT~; deg. And y tne same reasoning, if the centripetal force be as - 73 , the
angle between the apsides will be found equal to fi f* 18o V - - deg. After the
same manner the Problem is solved in nib &gt;ic more difficult cases.
The quantity to which the centripetal force is pro portional must always be
resolved into a converging series whose denomi nator is A*. 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 -f 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 contra riwise. 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., MS any number as m to another as n, and the altitude
called A ; the force nn will be as the power A HSii 3 of the altitude A; the
index of which power is - 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. XLL But if it should, at its part
ing from the lower apsis, begin to ascend never so little, it will ascend in
irtfimtifm, 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 tripli cate
ratio of the altitude, the body at its parting from the apsis, will either
descend to the centre, or ascend in iiiftnitum, according as it descends or
Ascends at the beginning of its motion. But if the force in its recess from
"SEC. IX.J OF NATURAL PHILOSOPHY. ISi 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 \\ that is, if m should be to n as 8, or 4, or 2, or H to 1. and therefore
---3, be g\ 3,or TV~3, or i 3, or mm 3 I - 3 ; then the force will be as A~ ?
or AT "~ 3j or A*~~ 3j or A"" G that is. it will be
reciprocally as A 3 C4 or A 3 T ^ or A 3 4 or A 3 "" If the
body after each revolution returns to the same apsis, and the apsis nn _
remains unmoved, then m will be to n as 1 to 1, and therefore A" will
be equal to A 2 , or - ; and therefore the decrease of the forces will AA 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 or ? n n i_6 _ 3 9 _ 3
o or ^ or i to 1, and therefore Amm 3 is equal to A 9 or A4 or A _ 3 1 6 _ 3
l_l or A ; and therefore the force is either reciprocally as A fl or 3 613 A 4
or directly as A or A . 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 u as 363 deg. to 360 deg. or as 121 to 120,
and therefore Amm will be equal to 2 9_ 5_ 2_ JJ A " and therefore the
centripetal force will be reciprocally as ^T4"6TT&gt; or
reciprocally as A 2 ^ 4 ^ very nearly. Therefore the centripetal force
decreases in a ratio something greater than the duplicate ; but ap proaching
59f times nearer to the duplicate than the triplicate. COR. 2. Hence also if a
body, urged by a centripetal force which is re ciprocally 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 182 THE MATHEMATICAL PRINCIPLES
[BOOK I oe as -r-r- ; and the foreign force subducted as cA, and therefore the
remain- A.A. ^ c^4 ing force as -^ ; 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 be 1 c tween the apsides is equal to 180 &lt;*/- 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 -3 }y j ; A or T
being equal to 1 ; and then l8(Vl~4c will be 18&lt;Vfff Jf 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 eccen trical 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 per pendicular 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 performer 1 in
curve superficies. SECTION X. Of the motion of bodies in given superficies, and
of the reciprocal motion offnnependulous bodies. PROPOSITION XLVI. PROBLEM
XXXII. Any kind of centripetal force being supposed, and the centre offorce,
atfft any plane whatsoever in which the body revolves, being given, and tint
quadratures of curvilinear figures being allowed; it is required to de termine
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. SEC. X.J OF NATURAL PHILOSOPHY-
183 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 PQ,R the trajectory
itself which is required to be found, described in that given plane. Join CQ,
Q.S, and if in Q,S we take SV proportional to the centripetal force with which
the body is attracted to wards 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 aMracting 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 c f the other force TV,
coinciding with the position of the plane itself, at tracts the body directly
towards the given point C in that plane ; ad t icreftre causes the body to move
in this plane in the same manner as if the force S F 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 PQ.R 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 e contra. Q..E.I. PROPOSITION XLV1L THEOREM XV. Supposing the
centripetal force to be proportional to t/ie 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 andforwards alternately, will complete ttieir 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 PQ,R is attracted to wards the centre S, is as the distance SO. ;
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 dis tance CQ,. Therefore the forces with which bodies found in
the plane PQ,R are attracted towaitis the point O, are in proportion to the
distances equal to the forces with which the same bodies are attract-ed every
way to wards 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. n* they THE
MATHEMATICAL PRINCIPLES [BOOK I. would do in free spaces about the centre S ;
and therefore (by Cor. 2, Prop. X, ai d Cor. 2, Prop. XXXVIII.) they will in
equal times either describe ellipses m 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. SCHOLIUM. 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 de scribed 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 m those superficies. If those
bodies reciprocate to and fro with an oblique ascent and descent, their motions
will be performed in planes passing through tiie 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 thp motion in those
curve lines. PROPOSITION XLVIII. THEOREM XVI. If a wheel stands npon the
outside of a globe at right angles thereto, and revolving about its own axis
goes forward in a great circle, the length of lite 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 w~e 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 sn,m of the diameters of
the globe and the wheel to the semi-diameter of the globe. PROPOSITION XLIX.
THEOREM XVII. ff a wheel stand upon the inside of a concave globe at right
angles there to, 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 toncJied the
globe, imll 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, :if;d the given point P in the peri
meter of the wheel may describe in thf SEC. X.I OF NATURAL PHILOSOPHY. s 185 H
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 cf this path
AP will be to twice the versed sine of the arc |PB as 20E 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 perpen dicular VF. Let PH, VH,
meeting in H, touch the circle in P and V, and let PH cut YF in G, and to VP
let fall the perpendiculars GI, HK. From the centre C with any interval let
there be described the circle wow, cutting the right line CP in nt the
perimeter of the wheel BP in o, and the curvilinear path AP in m ; and from the
centre V with the interval Vo let there be described a circle cutting VP
produced in q. Because the wheel in its progress always revolves about the
point of con tact 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 nmn 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
Pnn-mq, and the figure PFGVI, the ultimate ratio of the evanescent lined ae
Pra, P//, Po, P&lt;y, 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 &lt; iSS THE MATHEMATICAL PRINCIPLES [BOOK 1. the same as of the
lines PV, PF, PG, PI, respectively. But since VF is perpendicular to OF, 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 V HG, 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 asCB to 2CE so PI to PV, and so is Pq to Pm. Therefore the
decrement of the line VP, that is, the increment of the line BY VP to the
increment of the curve line AP is in a given ratio of CB to 2CE, and therefore
(by Cor. Lena. IV) the lengths BY YP and AP, generated by those increments, arc
in the same ratio. But if BY be radius, YP is the cosine of the angle BYP or
-*BEP, and therefore BY YP is the versed sine of the same angle, and therefore
in this wheel, whose radius is ^BV, BY YP will be double the versed sine of the
arc ^BP. Therefore AP is to double the versed sine oi the arc ^BP as 2CE to CB.
Q.E.D. The line AP in the former of these Propositions we shall name the cy
cloid 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 lenc th of the part PS will be
to the length PV (which is the double of the sine of the angle YBP, 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 dumeter of the wheel BY as 2CF toCB. PROPOSITION L. PROBLEM XXXIII. To
cause a pendulous body to oscillate in a given cycloid. Let there be given
within the globe QYS de- scribed with the centre C, the cycloid QRS, bi sected
in R, and meeting the superficies of the globe with its extreme points Q and S
on either hand. Let there be drawn CR birxcting 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 SlCC. X.J OF NATURAL PHILOSOPHY. 187 semi-cycloids AQ, AS, that, as often
as the pendulum parts from the per pendicular 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 m V,
and let 0V be drawn j 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 per pendiculars PB, TVV,
cut off from CV the lengths VB, VVV equal the diameters of the wheels OA, OR.
Therefore TP is to VP (which is dou ble the sine of the angle VBP when ^BV is
radius) as BYV to BV, or AO -f-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 there fore 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. PROPOSITION LI. THEOREM XVIII. 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 thisforce alone acting upon it,
the body T oscillate (in the manner above de scribed] in the perimeter of the
cycloid QRS ; / say, that all the oscil lations, how unequal soever in
tfiemselves, 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 I ,aws) 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 rhread makes to it is totally employed, producing no
other effect ; but the 3ther part TX, impelling the body transversely or
towards X, directly accelerates the motion in the cycloid. Then it is plain
that the accelera tion of the body, proportional to this accelerating force,
will bo every 188 THE MATHEMATICAL PRINCIPLES [BOOK 1 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 there fore two pendulums APT, Apt, be unequally drawn aside from
the perpendicular AR, and let fall together, their accelerations will be always
as the arcs to be de scribed TR, tR. But the parts described at the beginning
of the motion are as the accelerations, thai is, as the wholes that are to be
described at the be ginning, 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 de
scent was accelerated : it is plain that the velocities of their ascent and de
scent 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 pendu lums 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
PROPOSITION LIL PROBLEM XXXIV. 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 centripe tal force proportional to the distance of
the places from the centre tend to the centre G, and it be in the peri meter
HIK equal to the centripetal force in the perime ter of the globe Q,OS 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
th&lt; SEC. X.J OF NATURAL PHILOSOPHY. 189 forces which act upon the
bodies are equal at the be ginning, 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 M 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 ordinato LI to the radius GK. or as
v/SR2 Til2 to SR. Hence, since in unequal oscillations there are described in
equal time arcs proportional to the en tire 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 Q.OS 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 X V.
Therefore the lineola HY, which is as this accelerated force CO X 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 v/GH X CO X V Whence the time of an entire oscillation in the
cycloid Q,RS (it being as the semi-periphery HKM, w rhich 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 v/GH X CO X V inversely ; that is,
because GH and SR are equal, as Vnr, . or (by Cor. Prop. L,) as UU X/-TTVT- X V
AO X 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. t90 THE MATHEMATICAL PRINCIPLES [Bo^K 1. 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 QJR.S as
1 to AR V AC COR. 2. Hence also follow what Sir Christopher Wren and M. Huygevs
have discovered concerning the vulgar cycloid. For if the diameter of the globe
be infinitely increased, its sphacrical 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 de scribe,
by the motions of nails fixed in their perimeters, cycloids without the globe ;
and pendulums, in mines and deep caverns of the Earth, must oscil late 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 sim ple ratio of
the same. PROPOSITION LIII. PROBLEM XXXV. 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 SEC. X. OF NATURAL PHILOSOPHY. 191 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,
docs not at all change its motion ; whereas the other force TY directly
accelerates or retards its mction in the curve STRQ. Wherefore since that force
is as the space to be described TR, the acceler ations or retardations of the
body in describing two proportional parts (u greater arid 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.l). 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 uni
form 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 there fore 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 ma chine 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. PROPOSITION LIV. PROBLEM XXXYI. 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/R given in a plane
passing through the centre of force C. Join CS, and lei 192 THE MATHEMATICAL PRINCIPLES
[BOOK 1 Q it be divided into innumerable equal parts, and let Dd be one of
those parts. From the centre C, with the intervals CD, Cd, let the circles DT,
dt be de scribed, meeting the curve line ST*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 Tt is
as the length of that lineola, that is, as the secant of the angle /TC
directly, and the velocity inversely. Lei, the ordinate DN, proportional to
this time, be made perpendicular to the right line CS at the point D, and
because Dd is given, the rectangle Dd X DN, that is, the area DNwc?, will be
proportional to the same time. Therefore if PN/?, 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 SQPJN D 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. PROPOSITION LV.
THEOREM XIX. If a body move in any curve superficies, whose axis passes through
the centre offorce, and from the body a perpendicular be let fall iipon 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
perpendic ularly from the body to the axis ; OP a line parallel and equal
thereto drawn from the given point O in the axis ; AP the orthogra phic
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 ; TO 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
SEC. X.] OF NATURAL PHILOSOPHY. 193 the superficies, and therefore with which
it is again repelled by the super ficies towards M ; PTF a right line parallel
to the axis and passing through the body, and OF, IH right lines let fall
perpendicularly from the points G and I upon that parallel PHTF. I say, now.
that the area AGP, de scribed 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 re solved 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 di rection 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 wei e acted on by the forces FG, HI alone ; that is, the same as
,f 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 de
scribed 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. PROPOSITION LVI. PROBLEM XXXVII. Granting the
quadratures of curvilinear figures, and supposing that there are given both the
law of centripetal force tending to a given cen tre, 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 offfrom 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
di rection of a line given by position, and turn into the trajectory sought
STR, whose ortho graphic projection in the plane BDO is AP. And from the given
velocity of the body in the altitude SC, its velocity in any other al titude TC
will be also given. With that velocity, in a given moment of time, let the body
describe the particle Tt of its trajectory, and let P/? be the projection of
that particle described in the plane AOP. Join Op, and a little circle being
described upon the curve superficies about the centre T 13 194 THE MATHEMATICAL
PRINCIPLES [BOOK I with the interval TV let the projection of that little
circle in the plane AOP be the ellipsis pQ. And because the magnitude of that
little circle T/, and TN or PO its distance from the axis CO is also given, the
ellipsis pQ, will be given both in kind and magnitude, as also its position to
the right line PO. And since the area PO/? is proportional to the time, and
therefore given because the time is given, the angle POp will be given. And
thence will be given jo the common intersection of the ellipsis and. the right
line Op, together with the angle OPp, in which the projection APp of the tra
jectory cuts the line OP. But from thence (by conferring Prop. XLI, with Us 2d
Cor.) the mariner of determining the curve APp 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 iven the
several points T of the trajectory. Q.E.I. SECTION XL f f the motions of bodies
tending to each other with centripetal forces. I have hitherto been treating of
the attractions of bodies towards an im movable 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 ; BO 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 mutu ally , 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. PROPOSITION LVII. THEOREM XX. Two bodies attracting each
other mutually describe similarfigures about their common centre of gravity,
and about each other mutually. For the distances of the bodies from their
common centre of gravity are leciprocally as the bodies; and therefore in a
given ratio to each other: *nd thence, bv composition of ratios, in a given
ratio to the whole distance SEC. XL] OF NATURAL PHILOSOPHY. 195 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 ot these distances are similar. Q.E.D.
PROPOSITION LVIll. THEOREM XXI. 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 un moved ajigure similar and equal to the figures ivhich 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 lot there be continually drawn sp, sq, equal and
parallel to SP, TQ, ; and the ;urve 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 PQ,V 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 Mo tion) 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 im movable in s, the other movable in p,
similar and equal to the bodies S arid P. Then let the right lines PR and pr
touch the curves PQ, and pq ki 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 to wards 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 5. in the same given ratio, these forces would in equal
times attract 196 THE MATHEMATICAL PRINCIPLES |BoOK 1 the bodies from the
tangents PR, pr to the arcs PQ, pq, through the in tervals 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 becomr similar to the curve PQV, in which
the first force obliges the body P i( revolve ; and their revolutions would be
completed in the same timeg But because those forces are not to each other in
the ratio of CP to sp, bu; (by reason of the similarity and equality of the
bodies S and s, P and / and the equality of the distances SP, sp) mutually
equal, the bodies ii equal times will be equally drawn from the tangents; and
therefore tLV the body p may be attracted through the greater interval rq,
there is re quired a greater time, which will be in the subduplicate ratio of
the inter vals ; because, by Lemma X, the spaces described at the very
beginning ol 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 n 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 ivhich the body P describes round the movable body S. Q.E.I) CASE 2.
Suppose now that the common centre of gravity, together with the space in which
the bodies are moved among themselves, proceeds uni formly in a right line ;
and (by Cor. 6, of the Laws of Motion) all the mo tions in this space will be
performed in the same manner as before ; and therefore the bodies will describe
mutually about each other the same fig ures 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 ol gravity, and round each other mutually
concentrical ellipses ; and, vice versa, if such figures are described, the
forces are proportional to the dis tances. 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 sec tions having their focus in the centre about which
the figures are described. And, vice versa, if such figures are described, the
centripetal forces are re ciprocally 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- &gt;EC. XL] OF NATURAL PHILOSOPHY. 197
PROPOSITION LIX. THEOREM XXII. 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 bwlies 1? re volving round the other S remaining unmoved, and describing
a fig ure similar and equal to those which the bodies describe about each other
mutuallyr , in a subduplicate ratio of the other body S to the sii/rn of the
bodies S -f 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 ody 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. PROPOSITION LX. THEOREM XXIII. If two bodies S and
P, attracting each other with forces reciprocally pro portional 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, propor tionals 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 4- 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 4- 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 -f- P and S to S 4-
P. And inversely the principal axis of the ellipsis de scribed 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 be tween S 4- P and
S. Q.E.D. PROPOSITION LXI. THEOREM XXIV. 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 sameforces by a
third body placed in their common centre of gravity ; and the law of the 198
THE MATHEMATICAL PRINCIPLES [BOOK I attracting Jones will be the sam# 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 di rectly between them ; and therefore are the same as
if they proceeded from an intermediate body. QJG.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 . ther 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 di rectly 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 man ner from that distance compounded with given q-uantities
; 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 at tracted 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. PROPOSITION LXII. PROBLEM
XXXVIII. To determine the motions of two bodies which attract each other with
forces reciprocally proportional to the squares of the distance between them,
aflid 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 begin ning 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 deter mined (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. PROPOSITION LXIII. PROBLEM
XXXIX. To determine the motions of two bodies attracting each other with forces
reciprocally proportional to the squares of their distance, and going offfrom
given places in, given directions with given velocities. The motions of the
bodies at the beginning being given, there is given SEC. XL] OF NATURAL
PHILOSOPHY. 1 % 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 subse quent
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 velo city, 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 com pound 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.
PROPOSITION LXIV. PROBLEM XL. Supposingforces with which bodies mutually
attract each other to in crease in a simple ratio of their distances from the
centres ; it is ro- quired 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 L). These, by Cor. 1, Theor. XXI, will S describe ellipses having
their centres in D, the magnitudes of which ellipses are known by Prob. V. J-
-- \- ? L Let now a third body S attract the two former T and L with the
accelerative forces ST, SL, and let it be attract ed 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 there fore (by Cor. 1, Prop. X, and Cor. l,and 8, Prop. IV) they will
cause those bodies to describe ellipses as before, but with a swifter motion.
The re maining accelerative forces SD and DL, by the motive forces SD X Tand SD
X 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 situ
ations 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 200 THE
MATHEMATICAL PRINCIPLES [BoOK I. 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 X T and SD X L is
proportional to the distance OS, 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, at tracted by the motive forces SD X T and SD X 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. Arid
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. PROPOSITION LXV.
THEOREM XXV. 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 keepirg a certain proportion of distances from each
other. However, in the following crises the orbits will not much dif fer 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 rvery one of the bodies proportional to each. And because (by Cor. 4, ol the
I aws) the common centre of gravity of them all is either at rest, 01 iSEC. XL]
OF NATURAL PHILOSOPHY. 20 i moves uniformly forward in a right line, suppose
the lesser bodies so small that the groat 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 intro duced 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 an swer
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 ofanother 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 fhe 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 mo tions
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 bod/ one of the conic sec tions (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 thos 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 in- finitum. COR 1. In the second Case, the nearer the very
great body approaches to ^0^ THE MATHEMATICAL PRINCIPLES [CoOK I the system of
two or more revolving bodies, the greater will the pertur bation be of the
motions of the parts of the system among themselves; be cause 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 uccelerative 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 some where, 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. *.
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. PROPOSITION LXVL THEOREM XXVI. Tf 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 tivo revolving bodies will, by
radii drawn to the innermost and greatest, describe round thai body areas more
propor tional to the times, and a figure more approaching to that of an ellip
sis 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 lhat 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,
&lt;&gt;r very much more or very much less agitated, by the
attractions. This appears plainly enough from the demonstration of the second
Corollary of tl.e foregoing Proposition; but it may be 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 SEC. XI.J
OF NATURAL PHILOSOPHY. 203 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 E C square of SK to the square of SP, and SL will be the accelerative
attrac tion 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 resolv ed (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, cceteris 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 I 1 ; 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,
cceteris 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, cceterisparibus ; arid 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 th* THE MATHEMATICAL PRINCIPLES [BOOK 1
elliptical figure before-mentioned upon two accounts ; first because that force
is not directed from P to T ; and, secondly, because it is not recipro cally
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 remain ing 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 mo tions 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 re main 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 pertur bation 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 rema ning per fectly 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 dif ferent 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,
there fore, 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 nrirb greater
nor much less than the attraction SK. Q.E.D. SK-C. XL] OF NATURAL PHILOSOPHY.
205 COR. 1. Hence it may be easily collected, that if several less bodies P 8,
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 (accord ing 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 attrac tions 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 arid 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 cwtecedentia. Such is the force NM.
This force in the passage of the body P frcm C to A is directed in consequentia
to its motion, and therefore accelerates it; then as far as D in atttecedentia,
and retards the motion; then in, con sequentia 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 ccBteris paribuSj moves more swiftly in the conjunction
and opposition than in the quadratures. COR. 4. The orbit of the body P,
cc&teris 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, cceteris paribus, goes farther from the body T at the quadratures
than at the conjunction and opposition. This is said, E C_ L B 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 206 THE
MATHEMATICAL PRINCIPLES [BOOK. 1 the body P is retained in its orbit, is
increased at the quadratures by tho 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 sim ple ratio of the radius TP directly, and the duplicate
ratio of the periodi cal 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 diminish ed ; and, therefore, supposing this radius increased or
diminished, the peri odical time will be increased more or diminished less than
in the sesquipli- cate 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
diminshed by turns ; and the periodical time will be increased and diminished
in a ratio com pounded of the sesquiplicate ratio of the radius, and of the
subduplicate oi that ratio in which the centripetal force of the central body T
is dimin ished 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 for
wards 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 compound ed of the force LM
and the centripetal force with which the body T at tracts 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, be cause 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 for
wards. In the places between the syzygies and the quadratures, the motion SEC.
Xl.J OF NATURAL PHILOSOPHY. 207 of the line of the apsides depends upon both
&lt; f these causes conjuncdy, so that it either goes forwards or backwards
in proportion to the excess ol 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
arid the foregoing IE 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, dec., disposed about the orbit ESE. For by the actions of these
bo dies 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. IJut 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 ap sis 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 di minished 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 there- THE MATHEMATICAL PRINCIPLES [BOOK I. fore 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 ec centricity 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 FAB are in the quadratures, the ratio of
that increase and de crease 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 7 o 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 ra tio ; 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 aug mented, and
increases the eccentricity of the ellipsis ; and in the passage from the
syzygies to the quadratures it is perpetually decreasing, and di minishes 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 mo tions as to latitude
; and that the force NM, when the nodes are in the gyzygies, 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 dis turbs tliem 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 SEC. XL] OF NATURAL PHILOSOPHY. 209 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 ii C L E wnat 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 dimin ished. 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
diminish ed 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 incli nation
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 attrac tion is made
towards S in its passage from the node C through the con junction 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
perturba tion can be produced in the motion as to latitude, are quiescent : in
the in termediate 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 atitecedentia, each revolution. COR. 12. All
the errors described in these corrollaries arc a little greater 14 210 THE
MATHEMATICAL PRINCIPLES BOOK L 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 varia tions
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 dis tances, will be greater in
this case, than in the other where the body S re volves 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
con- junctly ; that is, if both the distance PT, and the absolute force of the
body 8 be given, as ST 3 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 con
tinue 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 dis tant
body S, those forces NM, ML, and their eifl ts, will be (by Cor. 2 and 6, Prop
IV) reciprocally in a duplicate ratio c/f the periodical time. And thence,
also, if the magnitude of the bodv 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 men tioned. COR. 15.
And because if the orbits ESE and PAB, retaining their fig ure, proportions,
and inclination to each other, should alter their magni tude ; arid 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 propor tion : it follows, that all the effects will
be similar and proportional, arid the times of those effects proportional also
; that is, that all the linear er rors will be as tne 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, arid distances of the bodies he any how
changed, we may. from the errors and times of those errors in SEC. XI.] OF
NATURAL PHILOSOPHY. 2 \\ 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, Lein. 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 re
volves about S at a great distance, the angular errors of the body P, ob served
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 in versely. 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
inc/case or diminution be very great indeed. COR. 17. Sines the line LM becomes
sometimes greater and sometimes less than the radius PT, let the mean quantity
of the force LM be expressed E C sa --::-..::::::; 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 re tained 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 cequo, the mean force LM is to the force by which the body P
is retain ed 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. 212 THE MATHEMATICAL PRINCIPLES [BoOK I
Con. IS. By tlie same laws by which the body P revolves about the body T, let
us suppose many fluid bodies to move round T at equal dis tances 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, perform ing 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 spherical body T, consisting of some matter not
fluid, to be enlarged, and to extend its If on every side as far as that
annulus, and that a channel were cut all round its circumference contain ing
water j 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 revolv- .ng 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 uni formly 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 A\ater 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 compares SEC. XI. OF NATURAL
PHILOSOPHY. 213 will oscillate, and the nodes will go backward, for the globe,
as \ve 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, and by that endea vour
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 B after the
quadratures ; and, again, the greatest motion of roclination 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 an nulus,
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 cen
tripetal 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 al tered ;
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 quadra tures, 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 down wards 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 impe diments 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 214 THE MATHEMATICAL PRINCIPLES [BOOK I 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 consti tution of the globe. That is, if the
globe retains unalterably the same poles, and the motion (of the nodes) be in.
antecedetitia, there is a redundance oi the matter near the equator; but if in
conseqnentia, a deficiency. Sup pose 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 oi 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 sec ond 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 intersec tions 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 inclina tion 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 direct ed, 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 oi 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 evagatior XL] OF NATURAL PHILOSOPHY. 2 In of the poles be corrected,
unless by placing that mountain ei . er in one ol the poles; in which case, by
Cor. 21, the nodes of the equator will go for wards ; 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 anew 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. PROPOSITION LXV1I. THEOREM XXVII. The
same laics of attraction being supposed, I say, that the exterior body S does,
by radii dra.cn 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 cen &gt; .-. than, it can describe round the innermost
and greatest body T by ra Hi 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 S(i of the bodies T and P, than it is
to the . reatest body T ; and which is more in a reciprocal propor tion 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. PROPOSITION LXVIII. THEOREM
XXVIII. 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 propor tional 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 tnore 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 suf ficient 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 ellip* 216 THE MATHEMATICAL PRINCIPLES -
[BOOK 1 ses about that quiescent common centre. This appears from Cor. 2, Pro])
LVIII, compared with what was demonstrated in Prop. LX1V, and LXY Now this
accurate elliptical motion will be disturbed a little by the dis tance of the
centre of the two bodies from the centre towards which tht third body S is
attracted. Let there be added, moreover, a motion to the Bommon 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 I is, when the innermost and greatest body T is at tracted 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 bodi

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