Principia Mathematica (Motte Translation, 1848) — part 33 of 45

Cor. 1. Hence if from P and M, the extreme points of a least arc PM, on the line Q.^- joining the quadratures we let fall the perpendiculars PK, Mk, and produce the same till they cut the line of the nodes N/?. in D and d, the horary motion of the nodes will be as the area MPDc?, and the square of the line AZ conjunctly. For let PK, PH, and AZ, be the three said sines, viz., PK the sine of the distance of the moon from j;he quadra-

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429

ture, PH the sine of the distance of the moon from the node, and AZ the sine of the distance of the node from the sun ; and the velocity of the node will he as the solid content of PK X PH X AZ. But PT is to PK as PM to Kk ; and, therefore, because PT and PM are given, Kk will he as PK. Likewise AT is to PD as AZ to PH, and therefore PH is as the rectangle PD X AZ ; and, by compounding those proportions, PK X PH is as the solid content KAr X PD X AZ, and PK X PH X AZ as Kk X PD X AZ^ ; that is, as the area VDdM and AZ^ conjunctly. d.E.D. CoR. 2. In any given position of the nodes their mean horary motion is half their horary motion in the moon's syzygies ; and therefore is to 16" 35'" 16'\ 36\ as the square of the sine of the distance of the nodes from the syzygies to the square of the radius, or as AZ^ to AT^. For if the moon, by an uniform motion, describes the semi-circle QAq, the sum of all the areas PDc/M, during the time of the moon's passage from Q, to M, will make up the area QMdFt, terminating at the tangent Q,E of the circle ; and by the time that the moon has arrived at the point n, that sum will make up the whole area 'EGiAn described by the line PD : but when the moon proceeds from n to q, the line PD will fall without the circle, and describe the area 7iqe, terminating at the tangent qe of the circle, which area, because the nodes were before regressive, but are now progressive, must be subducted from the former area, and, being itself equal to the area Q.EN, will leave the semi-circle NQAn. While, therefore, the moon de- scribes a semi-circle, the sum of all the areas FDdM will be the area of that semi-circle ; and while the moon describes a complete circle, the sum of those areas will be the area of the whole circle. But the area PDg^M, when the moon is in the syzygies, is the rectangle of the arc PM into the radius PT ; and the sum of all the areas, ever]/ one equal to this area, in the time that the moon describes a complete circle, is the rectangle of the whole circumference into the radius of the circle ; and this rectangle, being double the area of the circle, will be double the quantity of the former sum.

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If, therefore, the nodes went on with that velocity uniformly continued which they acquire in the moon's syzygies, they would describe a space double of that which they describe in fact ; and, therefore, the mean motion, by which, if uniformly continued, they would describe the same space with that which they do in fact describe by an unequal motion, is but one-half of that motion which they are possessed of in the moon's syzygies. Where- fore since their greatest horary motion, if the nodes are in the quadratures, is 33" 10'''' 33''\ 12\ their mean horary motion in this case will be 16" 35'" 16'''. 36". And seeing the horary motion of the nodes is every where as AZ=^ and the area PDc?M conjunctly, and, therefore, in the moon's syzygies, the horary motion of the nodes is as AZ^ and the area PDc^M conjunctly, that is (because the area PDc^M described in the syzygies is given), as AZ^, therefore the mean motion also will be as AZ^ : and, there- fore, when the nodes are without the quadratures, this motion will be to 16" 35'" 16^ 36". as AZ^ to AT^. Q.E.D.

PROPOSITION XXXI. PROBLEM XII.

To find the horary motion of the nodes of the moon in an elliptic orbit. Let Qipmaq represent an ellipsis described with the greater axis Q,^', and the lesser axis ab ; QA^'B a circle circumscribed ; T the earth in the com- mon centre of both ; S the sun ; p the moon moving in this ellipsis ; and

Book IIL] of natural philosophy. 431

pm an arc which it describes in the least moment of time ; N and n the nodes joined by the line N?i ; joK and mk perpendiculars upon the axis Q^, produced' both ways till they meet the circle in P and M, and the line of the nodes in D and d. And if the moon, by a radius drawn to the earth, describes an area proportional to the time of description, the horary motion of the node in the ellipsis will be as the area/iDc^m and AZ^ conjunctly.

For let PF touch the circle in P, and produced meet TN in F ; and pf touch the ellipsis in p, and produced meet the same TN in/, ^nd both tangents concur in the axis TQ, at Y. And let ML represent the space which the moon, by the impulse of the above-mentioned force 3IT or 3PK, would describe with a transverse motion, in the meantime while revolving in the circle it describes the 'arc PM ; and ml denote the space which the moon revolving in the ellipsis would describe in the same time by the im- pulse of the same force 31 T or 3PK ; and let LP and Ip be produced till they meet the plane of the ecliptic in G and g, and FG andyg" be joined, of which FG produced may cut pf pg; and TQ, in c, e, and R respect- ively : and/g- produced may cut TQ. in r. Because the force 3IT or 3PK in the circle is to the force 3IT or 3/?K in the ellipsis as PK to pK, or as AT to «T, the space ML generated by the former force will be to the space ml generated by the latter as PK to pK ; that is, because of the similar figures FYKp and FYRc, as FR to cR. But (because of the similar triangles PLM, PGF) ML is to FG as PL to PG, that is (on ac- count of the parallels hk, PK, GR), as pi to pe, that is (because of the similar triangles plm, cpe), as Zm to ce ; and inversely as LM is to Im, or as FR is to cR, so is FG to ce. And therefore if /g' was to ce as/y to cY, that is, as /r to cR (that iS; as fr to FR and FR to cR conjunctly, that is, as/T to FT, and FG to ce conjunctly), because the ratio of FG to ce, expunged on both sides, leaves the ratios /§' to FG and/T to FT, fg would be to FG as/T to FT; and, therefore, the angles which FG and/§* would subtend at the earth T would be equal to each other. But these angles (by what we have shewn in the preceding Proposition) are the motions of the nodes, while the moon describes in the circle the arc PM, in the ellipsis the arc pm ; and therefore the motions of the nodes in the circle and in the ellipsis would be equal to each other. Thus, I say, it

ceX fY would be, if fg was to ce as/Y to cY, that is, iifg was equal to— ^ *

But because of the similar triangles /g-p, cep, fg is to ce as/t? to cp ; and

CP ^ f'D

therefore /g- is equal to- — -— ; and therefore the angle which /^ sub- tends in fact is to the former angle which FG subtends, that is to say, the motion of the nodes in the ellipsis is to the motion of the same in the

ce X fv ce "K fY circle as this fg or — -^^to the former /g- or — ^ ? ^^ ^^? as^ X

432 THE MATHEMATICAL PRINCIPLES [BoOK III.

cY to/Y X cp, or as//) to/Y, and cY to cp ; that is, if ph parallel to TN m'eet FP in h, as ¥h to FY and FY to FP ; that is, as ¥h to FP or Djt? to DP, and therefore as the area Dpmd to the area DPMcT. And, therefore, seeing (by Corol. 1, Prop. XXX) the latter area and AZ^ con- junctly are proportional to the horary motion of the nodes in the circle, the former area and AZ^ conjunctly will be proportional to the horary motion of the nodes in the ellipsis. Q,.E.D.

Cor. Since, therefore, in any given position of the nodes, the sum of all the areas pDdm, in the time while the moon is carried from the quadra- ture to any place m, is the area fnpGlEd terminated at the tangent of the ellipsis GtE ; and the sum of all those areas, in one entire revolution, is the area of the whole ellipsis ; the mean motion of the nodes in the ellip- sis will be to the mean motion of the nodes in the circle as the ellipsis to the circle ; that is, as Ta to TA, or 69 to 70. And, therefore, since (by Corol 2, Prop. XXX) the mean horary motion of the nodes in the circle is to 16'^ 35'" 1&\ 36\ as AZ^ to AT^ if we take the angle 16'' 2V" 3'^ 30\ to the angle 16" 35'" 16^\ 36\ as 69 to 70, the mean horary mo- tion of the nodes in the ellipsis will be to 16" 21'" 3^ 30\ as AZ^ to AT 2 ; that is, as the square of the sine of the distance of the node from the sun to the square of the radius.

But the moon, by a radius drawn to the earth, describes the area in the syzygies with a greater velocity than it does that in the quadratures, and upon that account the time is contracted in the syzygies, and prolonged in the quadratures ; and together with the time the motion of the nodes is likewise augmented or diminished. But the moment of the area in the quadrature of the moon was to the moment thereof in the syzygies as 10973 to 11073 ; and therefore the mean moment in the octants is to the excess in the syzygies, and to the defect in the quadratures, as 11023, the half sum of those numbers, to their half difference 50. Wherefore since the time of the moon in the several little equal parts of its orbit is recip- rocally as its velocity, the mean time in the octants will be to the excess of the time in the quadratures, and to the defect of the time in the syzy- gies arising from this cause, nearly as 11023 to 50. But, reckoning from the quadratures to the syzygies, I find that the excess of the moments of the area., in the several places above the least moment in the quadratures, is nearly as the square of the sine of the moon's distance from the quad- ratures : and therefore the difference betwixt the moment in any place, and the mean moment in the octants, is as the difference betwixt the square of the sine of the moon's distance from the quadratures, and the square of the sine of 45 degrees, or half the square of the radius ; and the in- crement of the time in the several places between the octants and quad- ratures, and the decrement thereof between the octants and syzygies, is in the same proportion. But the motion of the nodes, while the moon de- scribes the several littV i»mn1 r^ovf. r.-f -'?ct r^vhit iq accelerated or retarded

Book III.] of natural philosophy. 433

in the duplicate proportion of the time ; for that motion, while the moon describes PM, is (cceteris paribus) as ML, and ML is in the duplicate proportion of the time. Wherefore the motion of the nodes in the syzy- gies, in the time while the moon describes given little parts of its orbit^ is diminished in the duplicate proportion of the number 11073 to the num- ber 11023; and the decrement is to the remaining motion as 100 to 10973 ; but to the whole motion as 100 to 11073 nearly. But the decre- ment in the places between the octants and syzygieS; and the increment in the places between the octants and quadratures, is to this decrement nearly as the whole motion in these places to the whole motion in the syzygies, and the difference betwixt the square of the sine of the moon's distance from the quadrature, and the half square of the radius, to the half square of the radius conjunctly. Wherefore, if the nodes are in the quadratures, and we take two places, one on one side, one on the other, equally distant from the octant and other two distant by the same interval, one from the syzygy, the other from the quadrature, and from the decrements of the motions in the two places between the syzygy and octant we subtract the increments of the motions in the two other places between the octant and the quadrature, the remaining decrement will be equal to the decre- ment in the syzygy, as will easily appear by computation ; and therefore the mean decrement, which ought to be subducted from the mean motion of the nodes, is the fourth part of the decrement in the syzygy. The whole horary motion of the nodes in the syzygies (when the moon by a ra- dius drawn to the earth was supposed to describe an area proportional to the time) was 32" 42'" 7'^^. And we have shewn that the decrement of the motion of the nodes, in the time while the moon, now moving with greater velocity, describes the same space, was to this motion as 100 to 11073; and therefore this decrement is 17'" 43'\ 11^ The fourth part of which 4'" 25'^ 48^ subtracted from the mean horary motion above found, 16" 21'" 3^ 30^ leaves 16" 16'" 37^^. 42\ their correct mean ho- rary motion.

If the nodes are without the quadratures, and two places are considered, one on one side, one on the other, equally distant from the syzygies, the sum of the motions of the nodes, when the moon is in those places, will be to the sum of their motions, when the moon is in the same places and the nodes in the quadratures, as AZ^ to AT^. And the decrements of the motions arising from the causes but now explained will be mutually as the motions themselves, and therefore the remaining motions will be mu- tually betwixt themselves as AZ^ to AT^ ; and the mean motions will be as the remaining motions. And, therefore, in any given position of the nodes, their correct mean horary motion is to 16" 16'" 37^". 42\ as AZ^ to AT^ ; that is, as the square of the sine of the distance of the nodes from the syzygies to the square of the radius.

28

434

THE MATHEMATICAL PRINCIPLES

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PROPOSITION XXXII. PROBLEM XIII.

e mean motion of the nodes of the moon.

motion is the sum of all the mean horary motions

To find th

The yearly mean throughout the course of the year. Suppose that the node is in N, and that, after every hour is elapsed, it is drawn back again to its former place; so that, notwithstanding its proper motion, it may constantly re- main in the same situation with respect to the fixed stars ; while in the mean time the sun S, by the motion of the earth, is seen to leave the node,

and to proceed till it completes its appa- rent annual course by an uniform motion. Let Aa represent a given least arc, which the right line TS always drawn to the sun, by its intersection with the circle NA??., describes in the least given moment of time; and the mean horary motion (from what we have above shewn) will be as AZ^, that is (because AZ and ZY are proportional), as the rectangle of AZ into ZY, that is, as the area AZYa / and the sum of all the mean horary motions from the beginning will be as the sum of all the areas «YZA, that is, as the area NAZ. But the greatest AZYa is equal to the rectangle of the arc Ka into the radius of the circle ; and therefore the sum of all • these rectangles in the whole circle will be to the like sum of all the greatest rectangles as the area of the whole circle to the rectangle of the whole circumference into the ra- dius, that is, as 1 to 2. But the horary motion corresponding to that greatest rectangle was 16" 16'" 37'^^. 42"'. and this motion in the complete course of the sidereal year, 365"^. 6'\ 9', amounts to 39^ 38' 7" 50'", and therefore the half thereof, 19° 49' 3" 55'", is the mean motion of the nodes corresponding to the whole circle. And the motion of the nodes, in the time while the sun is carried from N to A, is to 19° 49' 3" 5B'" as the area NAZ to the whole circle.

Thus it would be if the node was after every hour drawn back again to its former place, that so, after a complete revolution, the sun at the year's end would be found again in the same node which it had left when the year begun. But, because of the motion of the node in the mean time, the sun must needs meet the node sooner ; and now it remains that we compute the abbreviation of the time. Since, then, the sun, in the course of the year, travels 360 degrees, and the node in the same time by its greatest motion would be carried 39° 38' 7" 50'", or 39,6355 degrees ; and the mean motion of the node in any place N is to its mean motion in its quadratures as AZ" to AT 2 : the motion of the sun will be to the motion of the node

Book III] of natural philosophy. 435

in N as 360 AT- to 39,6355 AZ'- ; that is, as 9,032764.6 AT^ to AZ^ Wherefore if we suppose the circumference NA?i of the whole circle to be divided into little equal parts, such as Act, the time in which the sun would describe the little arc Aa, if the circle was quiescent, will be to the time of which it would describe the same arc, supposing the circle together with the nodes to be revolved about the centre T, reciprocally as 9,0827646 AT^ to 9;0S27646AT'- + AZ- ; for the time is reciprocally as the velocity with which the little arc is described; and this velocity is the sum of the velocities of both sun and node. If, therefore, the sector NTA represent the time in which the sun by itself, without the motion of the node, would describe the arc NA, and the indefinitely small ,part ATa of the sector represent the little moment of the time in which it would describe the least arc Aa ; and (letting fall ciY perpendicular upon Nu) if in AZ we take dZ of such length that the rectangle of dZ into ZY may be to the least part ATa of the sector as AZ^ to 9,0827646 AT ^ 4- AZ% that is to say, that dZ may be to lAZ as AT^ to 9,0827646 AT ^ -{- AZ^ ; the rectangle of dZ into ZY will represent the decrement of the time arising from the motion of the node, while the arc Aa is described ; and if the curve NdGn is the locus where the point d is always found, the curvilinear area Nt/Z will be as the whole decrement of time while the whole arcNA is described ; and, therefore, the excess of the sector NAT above the area Nc/Z will be as the whole time. But because the motion of the node in a less time is less in proportion of the time, the area AaYZ must also be di- minished in the same proportion; which may be done by taking in AZ the line eZ of such length, that it may be to the length of AZ as AZ^ to 9,0827646 AT 2 + AZ^ ; for so the rectangle of eZ into ZY will be to the area AZYa as the decrement of the time in which the arc Aa is de- scribed to the whole time in which it would have been described, if the node had been quiescent : and, therefore, that rectangle will be as the de- crement of the motion of the node. And if the curve NeF?i is the locus of the point e, the whole area NeZ, which is the sum of all the decrements of that motion, will be as the whole decrement thereof during the time in which the arc AN is described ; and the remaining; area NAlG will be as the remaining motion, which is the true motion of the node, during the time in which the whole arc NA is described by the joint motions of both sun and node. Now the area of the semi-circle is to the area of the figure NeF/i found by the method of infinite series nearly as 793 to 60. But the motion corresponding or proportional to the whole circle was 19° 49' 3" 55'" ; and therefore the motion corresponding to double the figure NeFw is 1"^ 29' 58" 2'", which taken from the former motion leaves 18° ID' 5" 53'", the whole motion of the node with respect to the fixed stars in the interval between two of its conjunctions with the sun ; and this motion sub- ducted from the annual motion of the sun 360°. leaves 341° 40' 54" 7"',

436 THE MATHEMATICAL PRINCIPLES [BoOK HI.

the motion of the sun in the interval between the same conjunctions. But as this motion is to the annual motion 360°, so is the motion of the node but just now found 18° 19' 5" 53'" to its annual motion, which will there- fore be 19° IS' 1" 23'" ; and this is the mean motion of the nodes in the sidereal year. By astronomical tables, it is 19° 21' 21" 50'". The dif- ference is less than ^^-^ part of the whole motion, and seems to arise from the eccentricity of the moon's orbit, and its inclination to the plane of the ecliptic. By the eccentricity of this orbit the motion of the nodes is too much accelerated ; and, on the other hand, by the inclination of the orbit, the motion of the nodes is something retarded, and reduced to its just velocity.

PROPOSITION XXXIII. PROBLEM XIY.

To find the true motion of the nodes of the moon.

In the time which is as the area NTA— N^Z (in the preceding Fig.) ^ that motion is as the area NAe, and is thence given ; but because the cal- culus is too difficult, it will be better to use the following construction of '^ the Problem. About the centre C,

with any interval CD,- describe the circle BEFD ; produce DC to A so as AB may be to AC as the mean motion to half the mean true motion when the nodes are in their quadratures (that is, as 19° IS' 1" 23'" to 19° 49' 3" B5'" ; and therefore BC to AC as the difference of those motions 0° 31' 2" 32'" to the latter motion 19° 49' 3" bb'", that is, as 1 to 38j-). Then through the point D draw the indefinite line Gg, touching the circle in D ; and if we take the angle BCE, or BCF, equal to the double distance of the sun from the place of the node, as found by the mean motion, and drawing AE or AF cutting the perpendicular DG in G, we take another angle which shall be to the whole motion of the node in the interval be- tween its syzygies (that is, to 9° 11' 3") as the tangent DG to the whole circumference of the circle BED, and add this last angle (for which the angle DAG may be used) to the mean motion of the nodes, while they are passing from the quadratures to the syzygies, and subtract it from their mean motion while they are passing from the syzygies to the quadratures, we shall have their true motion ; for the true motion so found will nearly agree with the true motion which comes out from assuming the times as the area NT A — No?Z, and the motion of the node as the area NAe ; as whoever will please to examine and make the computations will find : and this is the semi -menstrual equation of the motion of the nodes. But there is also a menstrual equation, but which is by no means necessary for find-

Book III.] _ of natural philosophy. 437

ing of the moon's latitude ; for since the variation of the inclination of the moon's orbit to the plane of the ecliptic is liable to a twofold inequality, the one semi-menstrual, the other menstrual, the menstrual inequality of this variation, and the menstrual equation of the nodes, so moderate and correct each other, that in computing the latitude of the moon both may be neglected.

Cor. From this and the preceding Prop, it appears that the nodes are quiescent in their syzygies, but regressive in their quadratures, by an hourly motion of 16" 19'" 26'\ ; and that the equation of the motion of the nodes in the octants is 1° 30' ; all which exactly agree with the phae- nomena of the heavens.

SCHOLIUM.

Mr. Machhi, Astron,, Prof. Gresh., and Dr. Heyiry Pemberton, sepa- rately found out the motion of the nodes by a different method. Mention has been made of this method in another place. Their several papers, both of which I have seen, contained two Propositions, and exactly agreed with each other in both of them. Mr. Machines paper coming first to my hands, I shall here insert it.

OF THE MOTION OF THE MOON'S NODES.

"PROPOSITION I.

" The mean motion of the sun from the node is defined by a geometric mean proportional between the mean m^oiion of the sun and that mean motion with lohich the sun recedes with the greatest swiftness from the node in the quadratures.

" Let T be the earth's place, N?z the line of the moon's nodes at any given time, KTM a perpendicular thereto, TA a right line revolving about the centre with the same angular velocity with which the sun and the node recede from one another, in such sort that the angle between the quiescent right line N?z and the revolving line TA may be always equal to the distance of the places of the sun and node. Now if any right line TK be divided into parts TS and SK, and those parts be taken as the mean horary motion of the sun to the mean horary motion of the node in the quadratures, and there be taken the right line TH, a mean propor- tional between the part TS and the whole TK, this right line will be pro- portional to the sun's mean motion from the node.

" For let there be described the circle NKnM from the centre T and with the radius TK, and about the same centre, with the semi-axis TH

438

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[Book III.

and TN, let there be described an ellipsis NH?iL ; and in the time in Trhich the sun recedes from the node through the arc Na, if there be drawn the right line Tba, the area of the sector NT« will be the exponent of the sum of the motions of the sun and node in the same time. Let, there- fore, the extremely small arc aA be that which the right line Tba, revolv- ing according to the aforesaid law, will uniformly describe in a given particle of time, and the extremely small sector TAa will be as the sum of the velocities with which the sun and node are carried two different ways in that time. Now the sun's velocity is almost uniform, its ine- quality being so small as scarcely to produce the least inequality in the mean motion of the nodes. The other part of this sum, namely, the mean quantity of the velocity of the node, is increased in the recess from the syzygies in a duplicate ratio of the sine of its distance from the sun (by Cor. Prop. XXXI, of this Book), and, being greatest in its quadratures with the sun in K, is in the same ratio to the sun's velocity as SK to TS, that is, as (the difference of the squares of TK and TH, or) the rectangle KHM to TH^ But the ellipsis NBH divides the sector ATa, the expo- nent of the sum of these two velocities, into two parts ABha and BTb, proportional to the velocities. For produce BT to the circle in /3, and from the point B let fall upon the greater axis the perpendicular BG, which being produced both ways may meet the circle in the points F and f; and because the space ABba is to the sector TB6 as the rectangle AB/3 to BT^ (that rectangle being equal to the difference of the squares of TA and 1'B, because the right line AjS is equally cut in T, and unequally in B), therefore when the space ABba is the greatest of all in K, this ratio will be the same as the ratio of the rectangle KHM to HT^. But the greatest mean velocity of the node was shewn above to be in that very

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439

ratio to the velocity of the sun ; and therefore in the quadratures the sec- tor ATa is divided into parts proportional to the velocities. And because the rectangle KHM is to HT^ as FB/to BG^ and the rectangle AB/3 is equal to the rectangle FBf, therefore the little area ABba, where it is greatest, is to the remaining sector TBb as the rectangle AB/3 to BG^. But the ratio of these little areas always was as the rectangle ABfj to BT^ ; and therefore the little area ABba in the place A is less than its correspondent little area in the quadratures in the duplicate ratio of BG to BT, that is, in the duplicate ratio of the sine of the sun's distance from the node. And therefore the sum of all the little areas ABba, to wit, the space ABN, will be as the motion of the node in the time in which the sun hath been going over the arc NA since he left the node ; and the remaining space,- namely, the elliptic sector NTB, will be as the sun's mean motion in the same time. And because the mean annual mo- tion of the node is that motion which it performs in the time that the sun completes one period of its course, th* mean motion of the node from the sun will be to the mean motion of the sun itself as the area of the circle to the area of the ellipsis; that is, as the right line TK to the right line TH, which is a mean proportional between TK and TS ; or, which comes to the same as the mean proportional TH to the right line TS.

'^ PROPOSITION II.

" The mean motio7i of the mooii^s nodes being given, to find their trne motion,

" Let the angle A be the distance of the sun from the mean place of the node, or the sun's mean motion from the node. Then if we take the ancrle B, whose tangent is to the tangent of the angle A as TH to TK, that is,

440 THE MATHEMATICAL PRINCIPLES [BoOK 111.

in the sub-duplicate ratio of the mean horary motion of the sun to the mean horary motion of the sun from the node, when the node is in the quadrature, that angle B will be the distance of the sun from the node's true place. For join FT, and, by the demonstration of the last Propor- tion, the angle FTN will be the distance of the sun from the mean place of the node, and the angle ATN the distance from the true place, and the tangents of these angles are between themselves as TK to TH.

" Cor. Hence the angle FTA is the equation" of the moon's nodes ; and the sine of this angle, where it is greatest in the octants, is to the radius as KH to TK + TH. But the sine of this equation in any other place A is to the greatest sine as the sine of the sums of the angles FTN + ATN to the radius ; that is, nearly as the sine of double the distance of the sun from the mean place of the node (namely, 2FTN) to the radius.

"SCHOLIUM.

" If the mean horary motion of the nodes in the quadratures be 16" 16'" 37K4:2\ that is, in a whole sidereal year, 39° 38' 7" 50'", TH will be to TK in the subduplicate ratio of the number 9,0827646 to the num- ber 10,0827646, that is, as 18,6524761 to 19,6524761. And, therefore, TH is to HK as 1 8,6524761 to 1 ; that is, as the motion of the sun in a sidereal year to the mean motion of the node 19° 18' 1" 23|"'.

" But if the mean motion of the moon's nodes in 20 Julian years is 386° 50' 15", as is collected from the observations made use of in the theory of the moon, the mean motion of the nodes in one sidereal year will be 19° 20' 31" 58'". and TH will be to HK as 360° to 19° 20' 31" 58'" ; that is, as 18,61214 to 1 : and from hence the mean horary motion of the nodes in the quadratures will come out 16" 18'" 48'\ And the ■ greatest equation of the nodes in the octants will be 1° 29' 57'

'// V

PROPOSITION XXXIV. PROBLEM XV.

To find the horary variation of the inclination of the moon^s orbit to the

plane of the ecliptic.

Let A and a represent the syzygies ; Q, and q the quadratures ; N and n the nodes ; P the place of the moon in its orbit ; p the orthographic projection of that place upon the plane of the ecliptic ; and rnTl the mo- mentaneous motion of the nodes as above. If upon Tw we let fall the perpendicular PG, and joining pG we produce it till it meet Tl in g, and join also Vg, the angle PGjo will be the inclination of the moon's orbit to the plane of the ecliptic when the moon is in P ; and the angle Fgp will be the inclination of the same after a small moment of time is elapsed ; and therefore the angle GP^ will be the momentaneous variation of the inclination. But this angle GPg is to the angle GT^ as TG to PG and Vp to PG conjunctly. And; therefore, if for the moment of time we as-

Book III]

OF NATURAL PHILOSOPHY.

441

m I

smne an hour, since the angle GTg* (by Prop. XXX) is to the angle 33" 10'" 33^ as IT X PG X AZ to AT^ the angle GP^ (or the horary va- riation of the inclination) will be to the angle 33" 10'" 33^ as IT X AZ

X TG X ^ to AT 3. a.E.L

And thus it would be if the moon was uniformly revolved in a circular orbit. But if the orbit is elliptical, the mean motion of the nodes will be diminished in proportion of the lesser axis to the greater, as we have shewn above ; and the variation of the inclination will be also diminished in the same proportion.

Cor. 1. Upon N/2 erect the perpendicular TP, and let joM be the horary motion of the moon in the plane of the ecliptic; upon Q.T let fall the perpendiculars pK, MA^, and produce them till they meet TF in H and h ; then IT will be to AT as K/j to M/? ; and TG to Hjt? as TZ to AT ; and,

therefore, IT X TG will be equal to ^ , that is, equal to

TZ

the area UpM/i multiplied into the ratio ^r^ : and therefore the horary

variation of the inclination will be to 33" 10"^ 33^"". as the area HjoMA

TZ P»

multiplied into K7a X ^ X p^ to AT^.

Cor. 2. And, therefore, if the earth and nodes were after every hour drawn back from their new and instantly restored to their old places, so as their situation might continue given for- a whole periodic month together, the whole variation of the inclination durino; that month would be to 33"

442 THE MATHEMATICAL PRINCIPLES [BoOK 111.

10'" 33'^ as the aggregate of all the areas UpMh, generated in the time of one revolution of the point p (with due regard in summing to their proper

Vp

signs H ), multiplied into AZ X TZ X p^ to Mp X AT^ ; that is, as

Vp

the whole circle GlAqa multiplied into AZ X TZ X p^ to M^ X AT=^,

that is, as the circumference Q^Aqa multiplied into AZ X TZ X p^^ to

2Mp X AT 2.

Cor. 3. And, therefore, in a given position of the nodes, the mean ho- rary variation, from which, if uniformly continued through the whole month, that menstrual variation might be generated, is to 33" 10'" 33'^ as

AZ X TZ X p^ to 2AT^ or as Fp X — TJ^— ^^ ^^ X ^AT; that

is (because Fp is to PG as the sine of the aforesaid inclination to the ra-

AZ X TZ dius, and — —r-^^ to 4AT as the sine of double the angle AT?i to four

times the radius), as the sine of the same inclination multiplied into the sine of double the distance of the nodes from the sun to four times the square of the radius.

Cor. 4. Seeing the horary variation of the inclination, when the nodes are in the quadratures, is (by this Prop.) to the angle 33" 10"' 33'\ as IT

Fp . IT X TG Pd ■ _ ,