Principia Mathematica (Motte Translation, 1848) — part 9 of 45

The same things being supposed, 1 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. -3, 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 Avill 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 XVL THEOREM VIII.

The same things being supposed, and right lines being draivn to the bodies that shall touch the orbits, and perpendiculars being let fall on those tangents froin the confimon focus ; I say, that the velocities of the bodies are in a ratio compounded of the ratio of the perpendicidars inversely, and the subduplicate ratio of the principal latera recta directly. 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 — ^ — . For that velocity is as the infinitely small arc PQ, de-

122 THE MATHEMATICAL PRINCIPLES [BoOK L

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

SY), as ^v^ ; or as ibY reciprocally,

and SP X QT directly ; but SP X QT is as the area described in the given time, that is (by Prop. XIY), 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. And 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 velocity at the vertex of a parabola is in

Sec. hi.

OF NATURAL PHILOSOPHY.

123

this ratio, and (by Cor. 6 of this Prop, and Prop. lY) 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. 8. 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. lY), 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 cequo, 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 XYII. 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 knoivn. ; it is required to determine the line lohich a body ivill describe that is let go from a given place loith a given velocity in the direction of a given right li7ie. 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 inp; and if from S you suppose perpendiculars let fall on those tangents, the principal latus rectum of the conic section (by Cor. 1, Prop. XYI) 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 ; and is therefore given. Let this latus rectum be L ; the focus S of the conic

124 THE MATHEMATICAL PRINCIPLES [BoOK I.

section is also given. Let the angle RPH be tlie complement of the angle RPS to two right ; and the line PH, in which the other focus H is placed, is given by position. Let fall SK perpendicular on PH, and erect the conjugate semi-axis BC ; this done, we shall have SP- — 2KPH + PH^ = SW = 4CH2 = 4BH2 — 4BC2 = SP + Pff— L X SP~+TH =

SP2 + 2SPH + PH2 — L X SP + PH. Add on both sides 2KPH —

SP2_PH2 + L X SP + PH, and we shall haveL X SP + PH = 2SPH

• 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 with the line SP; and therefore the figure will be an ellipsis, Avhich 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. For if the body, in these cases, revolves in a conic section so found, it is demonstrated in Prop. XI, XH, and XHI, 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 4DS. 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 is 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. XVL), 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 course. For by compounding the proper motion of

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125

the body \yith 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, Ave 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- ino^ the tano-ent 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 ^p

SECTION lY.

Of the finding of elliptic, parabolic, and hyperbolic orbits, from the focus given.

LEMMA XV.

If from the two foci S, H, 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, that is, to the axis in which the foci are situated, the other, SV, is bisected in T by the perpendicular TR let fall upon it ; that perpendicular TR loill somerohere touch the conic section : and, vice versa, if it does touch it, HV icill 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 PRINCIPLES [BoOK I.

PROPOSITION XYIII. PROBLP^M X.

From a focus and the prwcipal axes given, to describe elliptic and hy- perbolic trajectories, lohich shall pass through given pjoints, and touch right lines given by position.

Let S be the common focus of the figures; ABA ;g

the length of the principal axis of any trajectory ; _ "^ ~^

P a point through which the trajectory should \ /R

Hi/

pass ; and TR a right line which it should touch. X\ About the centre P^ with the interval AB — SP, X if the orbit is an ellipsis, or AB r SP, if the ^ G' '"^

orbit is an hyperbola, describe the circle HG. On the tangent TR let fall the perpendicular ST, and produce the same to Y, so that TY may be equal to ST; and about Y 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 -- 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 XL

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 Y, so as TY may be equal to ST. After the same manner another circle fg is to be de- scribed, if another point jo is given ; or another point vl is to be found, if another tangent tr is given; then draw the right line IF, which shall touch the two circles FG,/g', if two points P, p are given ; or pass through the two points Y, v, if two tangents TR, tr, are given : or touch the circle FG, and pass through the point Y, 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

Sec. IV.] OF natural philosophy. 127

(by Cor. 3, Lem. XIV) because ST is equal to TV, and STR a right 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 shall pass through given points, and touch right lines give?! by position. Case 1. About the focus S it is re- r ^.

uired to describe a trajectory ABC, pass- i

mg through two points B, C. Because the ^| --^

trajectory is given in specie, the ratio of the I

principal axis to the distance of the foci GAS BE

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 Aa, 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 G« to aS, then by division we shall have Ga — GA, or Aa to aS — 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 the points B, C, as is manifest from the conic sections. Case 2. About the focus S it is required to y describe a trajectory which shall somewhere fx;---.. ^^

touch two right lines TR, tr. From the focus i ^""'--.„_^ /"'X on those tangents let fall the perpendiculars \ / \ X'^':^-;.,., l ST, S^, which produce to V, v, so that TV, tv ^r7r'r~-^:;:^^ may be equal to TS, ^S. Bisect Yv in O, and j tV'%^-'-"^'" ^ erect the indefinite perpendicular OH, and cut L;>:^-"'"\ the right line VS infinitely produced in K and V R k, so that VK be to KS, and Yk to kS, as the principal axis of the tra- jectory to be described is to the distance of its foci. On the diameter K^ 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 V^ to kS ; and by composition, as VK + Yk to KS -- kS ; and by division, as Yk — YK to 7jS — 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. W^herefore VH, the principal axis of the described trajectory, has the same ratio to SH, the distance of the foci, as

l^S

THE MATHEMATICAL PRINCIPLES

[Book 1.

II

,./-^'

N

R

... ' .->, _

\

V T

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. And seeing YH, ?jH are equal to the principal axiS; and VS, v% are perpendicularly bisected by the right lines TR^ tr^ it is evident (by Lem. XY) that those right lines touch the described trajectory. GI.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 let fall the perpendicular ST, which produce to Y, so that TY may be equal to ST ; join YR, and cut the right line YS indefinitely produced in K and k^ so that YK may be to SK, and YA: to SA^, as the principal axis of the ellipsis to be described to the distance of its foci ; and on the diameter \Uc describing a circle, cut the right line YR produced in H ; then with the foci S, H, and principal axis equal to YH, describe a trajectory : I say, the thing is done. For YH is to SH as YK to SK, V

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 YRS 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 aph^ described with the principal axis ah^ and foci 5, li. On the tangent TR let fall the perpendicular ST, Avhich /' ,,'''',,--"'" produce to Y, so that TY may be A^:^---""' equal to ST ; and making the an- '"''

gles lisq^ shq, equal to the angles YSP, SYP, about §' as a centre, with an interval which shall be to ab as SP to YS, describe a circle ting the figure aph 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 YSH equal to the angle psq. Then with the foci S, H, and b principal axis AB, equal to the distance YH, describe a conic section : I say, the thing is done ; for if sv is drawn so that it shall be to

and cut-

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 ah to pq. Wherefore vh and ah 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 ah 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 jo^A, this figure passes through the point P ; and because VH is equal to its axis, and VS is per- pendicularly bisected by the right line TR, the said figure touches the right line TR. Q.E.P.

LEMMA XVI.

From, three given points to draw to a fourth point that is not given three rigid lines lohose 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 b 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.L

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.

9

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[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 ApoUonius's Book of Tactions restored by Yieta.

PROPOSITION XXL PROBLEM XIII.

About a given focus to describe a trajectory that shall pass through given points and touch right lines 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 "J^

be equal to ST, and YH will be equal

to the principal axis. Join SP, HP, and

SP Avill 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 Avhich 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 -j- 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 this 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, D, 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 j^ fake 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 GF as the point A ; in the second, going off to an iniinite distance ; in the third, falling on the other side of the line GF. For if on GF the perpendiculars CI, DK are let fall, IC will be to HB as EC to EB ; that is, as SC to SB : and by permutation, IC to SC as HB to SB, or as GA to SA. And, by the like aro-ument, 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. YIII.

SECTION Y. Hoiv the orbits are to be found token neither focus is given.

LEMMA XYII. If from any poiiit V 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 giveji angles, 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 the other two opposite sides AC, BD, in a given ratio. Case 1. Let us suppose, first, that the lines drawn c to one pair of opposite sides are parallel to either of (^\s. p 't the other sides; as PQ. and PR to the side AC, and ^T PS and PT to the side AB. And farther, that one I pair of the opposite sides, as AC and BD, are parallel I

betwixt themselves ; then the right line which bisects ^^J;

those parallel sides will be one of the diameters of the ^

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 PCIK (by Prop. XYIL, XIX., XXL and XXIIL, Book ILL, of Apollonius's Conies) will be to the rectangle AQ,B in a given ratio. But Q,K 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 AQB, that s, to the rectangle PS X PT in a given ratio. Q.E.D.

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Case 2. Let us next suppose that the oppo- site sides AC and BD of the trapezium are not parallel. Draw Bd parallel to AC, and meeting as well the right line ST in t, 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. I'hen (because of the similar triangles BT^, DBN), B^ or PGl is to T^ as DN to NB. And ^~ Q

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 T^, so will the rectangle IN DM be to the rectangle ANB; and (by Case 1) so is the rectangle PQ, X Pr to the rectangle PS X P^^ : and by division, so is the rectangle pa 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 P^-, Pr, parallel to AC ; and P^, Vt ^j parallel to AB ; and because the angles of the triangles PQ^', PRr, PS^, PT^ are given, the ra- tios of PGl to P^, PR to Pr, PS to P^, PT to Ft will be also given ; and therefore the compound- ed ratios PQ X PR to P^ X Pr, and PS X ^T to P^ X Vt are given. But from what we have demonstrated before, the ratio of Pq X Pr to Fs X P^ is given ; and therefore also the ratio of PQ, X PR to PS X PT. aE.D.

LEMMA XVIIL

The s i.7}ie things supposed, if the rectangle PQ, X PR of the lines drawn to the tivo 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 lohence those lines are drawn, will he 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

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 ^^

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 hkkf, PQAS, as hk to hf^ so PQ, to PS. Wherefore by dividing the terms of the preceding proportion by the correspondent terms of this^ we shall have hn to hd as PR to PT. And therefore the equiangular trapezia J)rihd^ DRPT^ are similar, and consequently their diagonals D6, DP do coincide. Wherefore h 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. Gl.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, AO^ 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 PS^, and the right lines AB, CD, which before did cut the curve in the points A and B, C and D, can no longer 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 the four points. If the two opposite an- ^^^- '"

gles of the trapezium taken together are equal c\C^\ 3^---

to two right angles, and if the four lines PQ, \ "''C.^j^jJ/^J\\ PR, PS, PT, are drawn to the^sides thereof at W-y-^'^X^^'S^ right angles, or any other equal angles, and the ^\ )/ \ \

rectanscle PQ X PR under two of the lines y \ LL

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 two

134 THE MATHEMATICAL PRINCIPLES [BoOK L

PQj 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 i??. wfiniiwm.