Principia Mathematica (Motte Translation, 1848) — part 8 of 45

Cor. 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 first 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 whole remaining force by which the first body is urged will tend to the other 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 Avhole force compounded of them all, if they are many, is directed to another (immovable or moveable) centre. The same thing ob- tains, when the other body is moved by any motion whatsoever ; provided that centripetal force is taken, which 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 the other as the squares of the 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. I.), 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- meters are as the radii, the forces will be as the squares of any arcs de- scribed in the same time applied to the radii of the circles. Q.E.D. Cor. 1. Therefore, since those arcs are as the velocities of the bodies

lOp THE MATHEMATICAL PRINCIPLES [BoOK I.

the centripetal forces are in a ratio compounded of the duplicate ratio of the velocities directly, and of the simple ratio of the radii inversely.

Cos.. 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 the contrary.

Cor. 4. If the periodic times and the velocities are both in the subdu- plicate 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 R° of the radius R, and therefore the velocity reciprocally as the power R" ^ of the radius, the centripetal force will be reciprocally as the power R^"""^ 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 earthy 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 fiven (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 iiijiriitum, 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 Y. PROBLEM I.

There being given, in any places, the velocity with which a body de- scribes a given figure, by vieans of forces directed to so^ne commo7i centre: to find that centime.

Let the three right lines PT, TGIY, VR touch the figure described in as many points, P, Q, R, and meet in T and Y. 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 Q,B 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, YE 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

no 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 YI. THEOREM Y,

In a space void of resistance, if a body revolves in any orbit about an im- ?novable 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 of force : th£. centripetal force in the middle of the arc tvill 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, and the square of

the time inversely. Q,.E.D.

And the same thing may also be easily demonstrated by Corol. 4,

Lem. 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, Q,R is

drawn parallel to the distance SP, meeting

the tangent in R ; and Q,T is drawn perpen- ^^S

dicular to the distance SP ; the centripetal force will be reciprocally as the

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 SQ,P, 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

SY^ X Q.P^ solid z^-:^ ; 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 QT are equal.

Sec. II.] OF NATURAL PHILOSOPHY. ill

Cor. 3. If the orbit is either a circle, or touches or cuts a circle concen- trically, that is, contains with a circle the least angle of contact or sec- tion, having the same curvature and 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

GIP2 SY2 X PY. ForPVis^^.

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 APGt 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 be 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 QT^

tion. That is, we are to find, by computation, either the solid ^^

or the solid SY2 X PV, reciprocally proportional to this force. Examples of this we shall give in the following Problems.

PROPOSITION VII. PROBLEM II.

If a body revolves i?i the mrcumference of a circle; it is proposed to find 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 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 circleln L, and the tangent PZ in R. And, because of the similar triangles ZQR, ZTP, VPA, we shall have RP^, that is, GIRL to QT^ as AV^ to PV^. And

,^ ^ QRL X PV2 . ^ SP2

therefore -j^^ is equal to QT^. Multiply those equals by -^^,

and the points P and Q, coinciding, for RL write PV ; then we shall have "KT~ — "^ Q R * ^^^ therefore (by Cor. 1 and 5, Prop. VI.)

112 THE MATHEMATICAL PRINCIPLES [BoOK 1.

the centripetal force is reciprocally as Ttt^— ; that is (because AV*

is given), reciprocally as the square of the distance or altitude SP, and the cube of the chord PV conjunctly. Q..E.L

The same othenvise. On the tangent PR produced let fall the perpendicular SY ; and (be- cause of the similar triangles SYP, YPA). we shall have AY to PY as SP

SP X PY SP2 X PY^

to SY, and therefore -^ = SY, and —-^ =- SY~ X PY.

And therefore (by Corol. 3 and 5, Prop. YI). the centripetal force is recip-

SP^ X PY^ rocally as ^^li — 5 that is (because AY is given), reciprocally as SP'

X PY^. Q.E.L

Cor. 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 Y, the cen- tripetal force wdll 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 APTY revolves about the centre of force S rj, /^ R \

is to the force by which the same body P may re- A ^ /V

volve in the same circle, and in the same periodic 1 \ y^ )\

time, about any other centre of force R, as RP- X \ \ ,.- -3 '/^

SP to the cube of the right line SG, which from H;;^ / ^

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 RP X PT^ to SP^ X PY^- that is, as

gps yy pys SP X RP' to p^,-^ — ; or (because of the similar triangles PSG,TP>^

to SG^

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 RP^, 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 the second centre of force R, m^eeting 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.

\P

Sec. IL]

OF NATURAL PHILOSOPHY.

113

PROPOSITION YIIL PROBLEM III.

If a body moves in the semi-circumference VQ^K\ 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 P.

the semi-diameter CA be drawn, cutting the parallels at right angles in M and N, and I'oin CP. Because of the similar triangles CPM, PZT, and RZa, Ave shall have CP^ a to VW as PR2 to aT^: and, from the na- ture of the circle, PR^ is equal to the rect- angle Q,R X RN + GIN,, or, the points P, Q coinciding, to the rectangle QR X 2PM. Therefore CP^ is to PM^ as QR X 2PM to QT^; and QT^ 2PM' , aT^ X SP2 2VW X SP^ , , ,^ , ^

Q,K ^ ~~CP^' ^^^ OR ^ CP2 • ^^^ therefore (by

Corol. 1 and 5, Prop. VL), the centripetal force is reciprocally as

2PJVP X sp2 2SP^

p,-p^ ; that is (neglecting the given ratio -^Tdf)? reciprocally as

PMl 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 in an hyperbola, or parabola, by a centripetal force which is reciprocally as 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, <^c., ill a given angle; it is proposed to find the law of the centripetal force tendiyig to the 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 SPRQT will

be

given m specie.

Q.T Q.T^

Therefore the ratio is also given, and ^ is as QT, that is (be«

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 QPR

8

114

THE MATHEMATICAL PRINCIPLES

[Book 1.

(by Lemma XI) will be changed in tlie duplicate ratio of PR or QT. Therefore the ratio ^ -r. remains the same as before^ that iS; as SP. And

QT^ X SP^ QR

aR

is as SP^, and therefore (by Corol. 1 and 5, Prop. VI) the

centripetal force is reciprocally as the cube of the distance SP. Q,.E.L

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 SP^ is as SY^ x PY, that is (by Corol. 3 and 5, Prop. YI) reciprocally as the centripetal force.

LEMMA XIL

All parallelograms circiimscribed ahoiut any co7iJ2igate 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 the 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, GIT perpendiculars to those diameters ; Qio an ordinate to the diame- ter GP ; and if the parallelogram Gli;PR be completed, then (by the properties of the conic sections) the rec- tangle PvG will be to Qiv^ as PC^ to CD2; and (because of the similar triangles Qi^T, PCF), Qiv^ to QCT^ as PC^ to PF^- and, by com- position, the ratio of P^G to Q,T^ is compounded of the ratio of PC^ to

QT^

0D2 and of the ratio of PC^ to PF^, that is, vG to -^ as PC^ ' Vv

to— --p^^ -. Pnt GIR for Vv, and (by Lem. XII) BC X CA for CD

:X PF; also (the points P and Gt coinciding) 2PC for vG] and multiply-

Sec. II.] OF NATURAL PHILOSOPHY. 116

QJY2 y PQ2

ing the extremes and means together, we shall have ^-^ equal to

2BC^ X CA^

■ —— , Therefore (by Cor. 5, Prop. VI), the centripetal force is

reciprocally as p^^ ; that is (because 2BC^ X CA^ is given), re- ciprocally as-T^TT," that is, directly as the distance PC. Q.EI.

The same otherioise.

In 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 vN, such as shall be to vG as

DC2 to VC\ And because Qiv'' is to VvG as DC^ to PC^ (by the conic

sections), we shall have Q.^-^ = Vv X v^. Add the rectangle iil^v to both

sides, and the square of the chord of the arc PQ, will be equal to the rect-

ano-le NVv : and therefore a circle which touch.es the conic section in P,

and passes through the point Q., will pass also through the point V. Now

let the points P and Gt meet, and the ratio of tiN to vG, which is the same

with the ratio of DC^ to PC^, will become the ratio of PY to PG, or PY

2D 02 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

2D02

—^ X PF2 (by Cor. 3, Prop. YI) ; that is (because 2DC2 x PF^ is

given) directly as PC. Q,.E.L

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 Corel. 3 and S, Prop. lY) ; 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 ; that is, as those lesser axes directly, and the ordinates to the same point of 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 this 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 ordinates auo^mented or diminished in the ratio of the distances from the centre.

SECTION III.

Of the viotioyi 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 ^^^^^ ~ ~~~~--^ >B

SP cutting the diame- ter DK of the ellipsis in E, and the ordinate Giv in X ; and com- plete the parallelogram Q.2:PR. It is evident i that EP is equal to the greater semi-axis AC : for drawing HI from the other focus H of the ellipsis parallel to EC, because CS, CH are equal, ES, EI 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 Q,T perpendicu- lar to SP, and putting L for the princi al latus rectum of the ellipsis (or for

Sec. III.] OF NATURAL PHILOSOPHY. 117

— ), we shall have L X QR to L X Pv as Q.R to Vv, that is, as PE

or AC to PC ; and L X Pv to GvP as L to Gz? ; and GvV to Q.v'^ as PC^ to CD-; and by'(Corol. 2, Lem. VII) the points Q and P coinciding, QjP' is to Q..i'2 in the ratio of equality ; and Qiar^ or Q.^'^ is to Q,T- as EP^ to PF-, that is, as CA^ to PF^ or (by Lem. XII) as CD^ to CB^. And com- pounding all those ratios together, we shall have L X Q.R to Q.T- as AC X L X PC2 X CD2, or 2CB^ X PC^ x CD^ to PC X G?; X CD^ X CB"^, or as 2PC to Gv. But the points Q and P coinciding, 2PC and Gv are equal. And therefore the quantities L X Q.R and Q,T-, proportional

SP2

to these, will be also equal. Let those equals be drawn into-^^^, and L

X SP^ will become equal to ^y^ . And therefore (by Corel. 1 and

5, Prop. VI) the centripetal force is reciprocally as L X SP-, that is, re- ciprocally in the duplicate ratio of the distance SP. Q,.E.I.

The same othenoise.

Since the force tending to the centre of the ellipsis, by which the body P may revolve in that ellipsis, is (by Corel. 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-

PE^

tersect in E, will be as ^^ (by Cor. 3, Prop. VII.) ; that is, if the point

S is the focus of the ellipsis, and therefore PE be given as SP^ 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 XIL PROBLEM VII.

Suppose a body to move in an hyperhola ; it is required to find the 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 Qjv an ordinate to the diameter GP. Draw SP cutting the diameter DK in E, and the ordinate Qjv in x, and complete the parallelogram QRP:r. It is evident that EP is equal to the semi-transverse axis AC ; for drawing HI, from the other focus H of the hyperbola, parallel to EC, because CS, CH are equal, ES, EI will be also equal ; so that EP is the half difference

118

THE MATHEMATICAL PRINCIPLES

[Book I.

of PS, PI; that is (be- cause of tlie parallels IH, PR, and the equal angles IPR, HPZ), of PS, PH, the difference of which is equal to the whole axis 2AC. Draw aT perpen- dicular to SP; and put- ting Jj for the principal latus rectum of the hy- perbola (that is, for

-— ^- J , we shall have L

X QR to L X Pi^ as OK to Fv, or Vx to Fv, that is (Ibecause of the similar tri- angles Fxv, PEC), as PE i^ to PC, or AC to PC. And L X Pv Avill be to G^; X Pv as L to Gv ; and (by the properties of the conic sections) the rec- tangle GvF is to Q.v'^ as PC'^ to CD2 ; and by (Cor. 2, Lem. VII.), Giv^ to Gix^, the points Q and P coinciding, becomes a ratio of equality ; and Gix^ or Qv^ is to QT^ as EP^ to PF^, that is, as CA^ to PF^ or (by Lem. XIL) as CD'^ to CB^ : and, compounding all those ratios together, we shall have L X QtR to Q,T^ as AC X L X PC-^ X CD2, or 2CB-^ X PC^ X CD"^ to PC X Gv X CD^ X CB-, 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 and Q,T^, propor- tional to them, will be also equal. Let those equals be drawn into

^Y-n; ^^^ we shall have L X SP^ equal to ^^ . And therefore (by

Cor. 1 and 5, Prop. VI.) the centripetal force is reciprocally as L X SP=^, that is, reciprocally in the duplicate ratio of the distance SP. Q.E.I.

The same otherwise.

Find out the force tending from the centre C of the hyperbola. This will be proportional to the distance CP. But from thence (by Cor. 3, Prop.

PE^

VII.) the force tending to the focus S will be as -Hp^, that 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 the figure. This is demonstrated by the writers on the conic sections.

LEMMA XIT.

The perpendicular, let fall from the focus of a parabola on its tangent, is a 'nieani proportional betioeen the distances of the focus from the point of contact, and from 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 AO, 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 ,• therefore PS is to SN as SN to SA. Q.E.D.

Cor. 1. PS2 is to SN^ 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 perpendicular on the tangent,, falls in the right line AN that touches the parabola in the principal vertex.

PROPOSITION^ xin. 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 that figure.

Retaining the construction of the preceding Lemma, let P be the body in the perimeter of the parabola ; and from the place 4, into which it is next to succeed, draw Q.R parallel and Q,T perpendicular to SP, as also 0,17 parallel to the tan- gent, and meeting the diame- ter PG in V, and the distance''

M

A

120 THE MATHEMATICAL PRINCIPLES [BoOK I.

SP in X. Now, because of the similar triangles Vxv, SPM, and of the equal sides SP, SM of the one, the sides Vx or Q,R and Vv of the other will be also equal. But (by the conic sections) the square of the ordinate Qiv is equal to the rectangle under the latus rectum and the segment Vv of the diameter ; that is (by Lem. XIIL), to the rectangle 4PS X Vv, or 4PS X Q.R ; and the points P and Q. coinciding, the ratio of Qv to Qix (by Cor. 2, Lem. VII.,) becomes a ratio of equality. And therefore Q.^:*-, in this case, becomes equal to the rectangle 4PS X Q.R. But (because of the similar triangles QjjcT, SPN), Q^^ ig to QT^ as PS^ to SN^, that is (by Cor. 1, Lem. XIY.), as PS to SA ; that is, as 4PS X QR to 4SA X QR, and therefore (by Prop. IX. Lib. Y., Elem.) QCV^ and 4SA X ^R are

equal. Multiply these equals by ^-^, and ^^ will become equal

to SP^ X 4SA : and therefore (by Cor. 1 and 5, Prop. VL), the centripetal force is reciprocally as SP-^ X 4SA ; that is, because 4S A is given, recipro- cally in the duplicate ratio of the distance SP. Q.E.L

Cor. L 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 described ; 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

the conic sections, whose principal latus rectum is the quantity -^y^ in its

ultimate state, when the lineolse PR, Q.R 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 XIY. THEOREM YL

If several bodies revolve about one common centre, and the centripetal

force is reciprocally in tJie duplicate ratio of the distance of places

fro'tn the centre ; I say, that the principal latera recta of tlieir orbits

are in the duplicate ratio of the areas, which the bodies by radii drawn

Uo the centre describe in the sa/me time.

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ff

Q. '

Sec. III.] OF NATURAL PHILOSOPHY.' 121

For (by Cor. 2, Prop. XIII) the latus rectum

L is equal to the quantity-^r-j^-in its ultimate

state when the points P and Q. coincide. But the lineola Q,R in a given time is as the gen- erating centripetal force ; that is (by supposi-

tion), reciprocally as SP^. And therefore-^-^

is as QT^ X SP^ ; that is, the latus rectum L is in the duplicate ratio of the area QT X SP. a.E.D.

Cor. 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 subdu- plicate ratio of the latiis rectum, and the ratio of the periodic time. For the whole area is as the area Q,T X SP, described in a given time, mul- tiplied by the periodic time.

PROPOSITION XY. THEOREM VII.