Principia Mathematica (Motte Translation, 1848) — part 12 of 45

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, SP/B, are severally to the several 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 iyifinitum, 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, and the right line SD with the right line BD. And therefore the area BDEB will be proportional to the time in which ih.Q body C, by its per- pendicular descent, describes the line CB. Q.E.L

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 to the time in which that body P or C will descend to the centre S or

a.E.i.

A-i

B.

5EC.

YIL]

OF NATURAL PHILOSOPHY.

161

PROPOSITION XXXIII. THEOREM IX.

The things above found being supposed. I say, that the velocity of a fal- 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 rectangidar hijperbola, to hKB, the principal semi-diameter of the figure. Let AB, the common dia- t

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. From Cor. 9, Prop.

XYI, 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 iL X SP to SY^.

For by the properties of the conic sections ACB is to CP^ as 2A0 to L,.

2CP^ X AO

is equal to L. Therefore those velocities are

T

0--

A^

and therefore

ACB

CP2 X AO X SP

toSY^. More-

to each other in the subduplicate ratio of a PR

over, by the properties of the conic sections, CO is to BO as BO to TO, and (by composition or division) as CB to BT. Whence (by division or composition) BO — or + CO will be to BO as CT to BT, that is, AC

will be to AO as CP to BQ, ; and therefore-

CP2 X AO X SP

ACB

is equal to

BQ^ X AC X SP

Now suppose CP, the breadth of the figure RPB, to

AO X BC

be diminished in infiyiitum, 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 pernendicularly in the line CB will be to the velocity of

11

162

THE MATHEMATICAL PRINCIPLES

[Book L

n^ body describing a circle about the centre B, at the distance BC, in the

BQ,^ X AC X SP subduplicate ratio of — ^ — ~Wn ^^ ^^^^ ^^ ^^ (neglecting the ra- tios of equality of SP to BC, and BQ,^ to SY^), in the subduplicate ratio of AC to AO, or UB. 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 XXXIY. 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 tohich a body 7nay xmiformly describe a circle about the centre B at half the interval BC. For (by Cor. 7, Prop. XYI) 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

infinitum, 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 half the latus rectum of the figure DES, by uniformly revolving about the centre S, 'tnay describe in the same time.

Sec. VII.1

OF NATURAL PHILOSOPHY.

163

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^, describes the arc KA:. Erect the perpendiculars CD, cd, meeting the figure DES in D, d. Join SD, 8d, SK. Sk, and draw J)d 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 Go to Bd, and TD to TS as CD to SY ; ex (cquo TC will be to TS as CD X Cc to SY X D(/. But (by Cor. 1, Prop. XXXITI) 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 D^. Farther, the velocity of the descending body in C is 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 Prop. XXXIII) ; and this velocity is to the velocity of a body describing the circle 0K^^ in the subduplicate ratio of SK to SC (by Cor. 6, Prop IV) ; and, ex mqiio^ the first velocity to the last, that is, the little line Cc to the arc KA*, 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 K^' to SY X Drf, and thence SK X K^ equal to SY X Do?, and ^SK X K/^ equal to iSY X D«^, that is, the area KS/i: equal to the area SDc?. Therefore in every moment of time two equal particles, KS^^ and SDc?, of areas are generated, which, if their magnitude is diminished, and their number increased in infimtwm^ obtain the ratio of 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 D^ as TC to TS, that is, as 2 to 1 ; and that therefore |CD X Cc is equal to h. SY X D(/. But the veloc- ity of the falling body in C is equal to \ the velocity with which a circle may be \ uniformly described at the interval iSC (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 K^, is (by Cor. 6, Prop. IV) in the subduplicate ratio of SK to iSC ; that is, in the ratio of SK to IG). Wherefore ^SK X K/^ is equal to ICD X Cc, and therefore equal to ^SY X Dc?; that is, the area KSA: is equal to the area SDc?, as above. Q.E.D.

164

THE MATHEMATICAL PRINCIPLES

[Book 1.

PROPOSITION XXXVI. PROBLEM XXY.

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 a, centre at the beginning, describe the semi-circle ADS, as likewise the semi-circle OKH eqnal thereto, about the centre c S. From any place C of the body erect the ordinate CD. O 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. aE.F. K

a given

PROPOSITION XXXYII. 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 off 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

c

s

t)

"h

'^

G

s

^^ 1

\ ^^^^-. /

\

\ /k

H

ii

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 HA'K ; 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. VIL

OF NATURAL PHILOSOPHY.

165

the space GC in the same time in which the body K may describe the arc Kk. Q.E.F.

PROPOSITION XXXVIII. THEOREM XII.

i^apposing that the centripetal force^ is proportional to the altitude or distance of places from the centre, I say, that the times and velocities of falling bodies, and the spaces tchich 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. XXXll was demonstrated from Prop. XL

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 XXVII.

Supposing a centripetcd force of any kind, and granting the quadra- tures of cnrvilinear figures ; it is required to find the velocity of a body, ascending or descending in a right line, in the severed places through lohich it passes ; as cdso the time in which it luill 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 perpendiculor 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

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

166 THE MATHEMATICAL PRINCIPLES [BoOK I.

a right line whose square is equal to the area ABGE^ and let YLM 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 ABTYME.

aE.L

For in the right line AE let there be taken the very small line DE of a given length, and let Dl^F 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 Y and Y + I, the area ABFD will be as YY, and the area ABGE as YY + 2YI -[- II ; and by division, the

T)FC*F 9VT -I- TT

area DFGE as 2YI + 11, and therefore ^^^ will be as— — p-tp; ;

Dili Dht

that is, if we take the first ratios of those quantities when just nascent, the

2YI

length DF is as the quantity fyFry and therefore also as half that quantity

I X Y ^„ * But the time in which the body in fallino; describes the very DE -^ °

small 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 Y are just nascent, as— Y^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 the 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. Lem. lY), 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 jolace from whence a body ought to fall, so as that, w^hen urged by any known uniform centripetal force (such as gravity is vulgarly supposed to be), it may acquire in the place D a velocit}^ 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, w^hich may be to DF as that uniform force to the other force in the place D. Complete the rectangle PDRQ., and cut off the area ABFD equal to that rectangle. Then A will be the place

Sec. YIL]

OF NATURAL PHILOSOPHY.

167

from whence the other body fell. For com- pleting the rectangle DRSE, since the area ABFD is to the area DFGE as YV to 2VI, and therefore as ^Y 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 PQ.RD to the area DPtSE 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 cequo, the' whole areas ABFD, PQ,RD 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 DFg^e, if the place e is below the place D, or diminished by the same area J)¥ge, if it be higher, is to the right line whose square is equal to the rectangle PQ,RD alone.

Cor. 3. The time is also known by erecting the ordinate em, recipro- cally proportional to the square root of PQ,RD -f- or — DP^^-e, 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 -h ^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 whicli 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 cequo, the first mentioned of these times is to the last as the rectangle 2PD X DL to the area DLwe.

168 THE MATHEMATICAL PRINCIPLES [BoOK I.

SECTION YIIL

Of the invention of orbits loherein bodies loill 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 any one case of equal altitudes, tJieir velocities will be also equal at all equal altitudes.

Let a body descend from A through D and E, to the centre C ; and let another body move from Y in the curve line YIKA:. From the centre C. with any distances, describe the concentric circles DI, EK, meeting the right line AC in D and E, and the curve YIK in I and K. Draw IC 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 ; 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 fv expressed by the equal lineola3 DE and IN ; and let the force IN (by Cor. 2 of the Laws of Motion) be resolved into two others, NT and IT. Then the force NT acting in the direction of the 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 A:. 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 bodies 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. VIIL] 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 eiFected 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 A" — \ 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 ^/ P'^ — A^^, and therefore will be given. For by Prop. XXXIX, the velocity of a body ascending and descending in a right line is in tha't very ratio.

PROPOSITION XLI. PROBLEM XXVIIL

Supposing a centripetal force of any kind, and granting the quadra- tures of cnrviliiiear figures, it is required to find as well the trajecto- ries in lohich 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 YIKA:. Let r, there be given the circle YR, 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, YR in N and X, and the right line CKY meeting the circle VR in Y. Let the points I and K be indefinitely near ; and let the body go on from Y through I and K to A; / and let the point A be the place from whence another 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,

iT'O THE MATHEMATICAL PRINCIPLES [BoOK I.

will be as tlie velocity, and therefore as the right line whose square is equal to the area ABFD, and the triangle lOK proportional to the time will be given, and therefore KN will be reciprocally as the altitude IC ; that is (if there be given any quantity Gi, and the altitude IC be called

A), as -Y' This quantity — call Z, and suppose the magnitude of Q, to

be such that in some case ^/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 KW, and by division ABFD — ZZ to ZZ as IN^ to KN^, and there-

a

fore V ABFD — ZZ to Z, or — as IN to KN ; and therefore A X KN

Q, X IN

will be equal to ^=^. Therefore since YX X XC is to A X KN

v^ABFD— ZZ

Q, X IN X CX^

as CX^, to AA, the rectangle XY X XC will be equal to

AA^/ABFD— ZZ.

Therefore in the perpendicular DF let there be taken continually Db, Dc

a a X CX2 ...

equal to ^= =, __ ■ respectively, and

2 ^ ABFD — ZZ 2AA ^ ABFD — ZZ

let tiie 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 YT)ba, YDca, and let the ordinates E^r, Er, be erected also. Then because the rectangle D5 X IN or D^zE 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 DbzEi, ICK of the areas l"Dba, YIC are always equal; and the nascent particles Dc^E, XCY of the areas YDca, YCX are always equal ; therefore the generated area YD^a will be equal to the generated area YIC, and there- fore proportional to the time ,• and the generated area YDca- is equal to the generated sector YCX. If, therefore, any time be given during which the body has been moving from Y, there will be also given the area pro- portional to it YD6a ; and thence will be given the altitude of the body CD or CI ; and the area YDca, and the sector YCX equal thereto, together with its angle YCI. But the angle YCI, 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 IC drawn through the centre falls perpendicularly upon the trajectory YIK ; which comes to pass when the right lines IK and NK become equal ; that is, when the area ABFD is equal to ZZ.

Sec. VIIL]

OF NATURAL PPIILOSOPHY.

171

Cor. 2. So also the angle KIN, in which the trajectory at any place cuts the line IC. may be readily found by the given altitude IC 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 Y, there be described a conic section VRS ; and from any point v 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, equal 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 CY ; that body will proceed in a trajectory YPQ, which the point P will always touch ; and therefore if the conic section YRS 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 iyifinitu'in. And, on the contrary, if a body endued with any velocity goes off from the place Y, and according as it begins either to de- scend obliquely to the centre, or ascends obliquely from it, the figure YRS be either an hyperbola or an ellipsis, the trajectory may be found by increas- ing or diminishing the angle YCP in a given ratio. And the centripetal force becoming centrifugal, the body will ascend obliquely in the trajectory YPQ, which is found by taking the angle YCP proportional to the elliptic sector YRC, 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 curve, the invention of which, as being easy enough, for brevity's sake I omit.

PROPOSITION XLII. PROBLEM XXIX.

The Imo 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

the 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 centripet?J force

from the place P, may acquire in

D ; and let this uniform force be

to the force with which the body

172 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, eiv, ordi- nately applied to the curves BF^", abv, aciu. From the given rectangle PDRQ, and the given law of centripetal force^ by which the first body is acted on, the curve line BFg- 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 IK, KN ; and thence, by the construction of Prob. XXYIII, there is given the quantity Q, with the curve lines abv, acw ; and therefore, at the end of any time T)bve, there is given both the altitude of the body Ce or Ck, and the area Dcz/;e, with the sector equal to it XCi/, the angle ICk, and the place k, in which the body will then be found. Q.E.L

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 same.

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 'inotion of bodies in moveable orbits ; and of the ^notion of the

apsides.

PROPOSITION XLIII. PROBLEM XXX.

It is required to make a body m.ove in a trajectory that revolves about the centre of force in the same manner as another body in the same trajectory at rest.

In the orbit YPK, given by position, let the body P revolve, proceeding from V towards K. From the centre C let there be continually drawn Gp, equal to CP, making the angle VCjo proportional to the angle YCP ; and the area which the line Cp describes will be to the area YCP, which the line CP describes at tbe same time, as the velocity of the describing /^ x^^.^^^/^ line C^ to the velocity of the describing line CP ; that is, as the angle YC;? to the angle YCP, 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. IX.]

OF NATURAL PHILOSOPHY.

173

revolve with tlie 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 NCu equal to the angle PC/>, and the line Gu equal to Cy, and the figure uCp equal to the figure TCP, and the body being al- ways in the point jo, will move in the perimeter of the revolving figure uCp^ and will describe its (revolving) arc up in the same time that the other body P describes the similar and equal arc VP in the quiescent 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 point p de- scribes in an immovable plane, and the Problem will be solved. Q..E.F.

PROPOSITION XLIV. THEOREM XIY.

The differejice of the forces, by which two bodies may be made to move equally, one in a quiescent, the other in the same orbit revolving, is in a triplicate ratio of their common altitiides inversely. Let the parts of the quiescent or- bit YP, 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

^0, and produce it to m, so that mr

may be to kr as the angle VC^ to the /2^V

UX

m}

m-

angle VCP. Because the altitudes of the bodies PC and pC, KC and ^"C, are always equal, it is manifest that the increments or decrements of the lines PC and pG 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, pG, and the other, transverse to the former, hath a direction perpendicular to the lines PC and pG ; 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 pG to the angular motion of the line PC ; that is, as the angle 'YGp to the angle YCP. Therefore, at the same time that the body 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 ; and 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 pG ; and by its transverse motion will acquire a distance from the line

174 THE MATHEMATICAL PRINCIPLES [BoOK I.