be neglected.COR. From this and the preceding Prop, it appears that the nodes arequiescent in their syzygies, but regressive in their quadratures, by anhourly motion of 16" 19 " 26iv. : and that the equation of the motion ofthe nodes in the octants is 1 30 ;all which exactly agree with the phaBnomenaof the heavens.SCHOLIUM.Mr. Machin, Astron., Prof. Gresh.. and Dr. Flenry Pemberton, separately found out the motion of the nodes by a different method. Mentionhas been made of this method in another place. Their several papers, bothof which I have seen, contained two Propositions, and exactly agreed witheach 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.1 The mean motion of the sir/i from the node is defined by a geometricmean proportional between the mean motion of the sun and that meanmotion, with which the sun recedes with the greatest swiftness from thenode in the quadratures." Let T be the earth s place, Nn the line of the moon s nodes at anyaiven time, KTM a perpendicular thereto, TA a right line revolvingabout the centre with the same angular velocity with which the sun andthe node recede from one another, in such sort that the angle between thequiescent right line Nra and the revolving line TA may be always equalto the distance of the places of the sun and node. Now if any right lineTK be divided into parts TS and SK, and thost parts be taken as themean horary motion of the sun to the mean horary motion of the node inthe quadratures, and there be taken the right line TH, a mean proportional between the part TS and the whole TK, this right line will be proportional to the sun s mean motion from the node." For let there be described the circle NKnM from the centre T andwith the radius TK, and about the same centre, with the semi-axis TH438 THE MATHEMATICAL PRINCIPLES [BOOK IIINand TNr let there be described an ellipsis NHwL ; and in the time inwhich the sun recedes from the node through the arc N0, if there be drawnthe right line Tba, the area of the sector NTa will be the exponent of thesum of the motions of the sun and node in the same time. Let, therefore, the extremely small arc aA. be that which the right line T/w, revolving according to the aforesaid law, will uniformly describe in a givenparticle of time, and the extremely small sector TAa will be as the sumof the velocities with which the sun and node are carried two differentways in that time. Now the sun s velocity is almost uniform, its inequality being so small as scarcely to produce the least inequality in themean motion of the nodes. The other part of this sum, namely, the meanquantity of the velocity of the node, is increased in the recess from thegyzygies in a duplicate ratio of the sine of its distance from the sun (byCor. Prop. XXXI, of this Book), and, being greatest in its quadratureswith 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 rectangleKHM to TH 2. But the ellipsis NBH divides the sector AT, the exponent of the sum of these two velocities, into two parts ABba and BTb,proportional to the velocities. For produce BT to the circle in 0, andfrom 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 andf; and because the space ABba is to the sector TBb as the rectangle ABto BT2(that rectangle being equal to the difference of the squares of TAnnd TB, because the right line A3 is equally cut in T, and unequally inB), therefore when the space ABba is the greatest of all in K, this ratiowill be the same as the ratio of the rectangle KHM to HT2. But thegreatest mean velocity of the node was shewn above to be in that veryBOOK III.] OF NATURAL PHILOSOPHY. 439ratio to the velocity of the sun ; and therefore in the quadratures the sector ATa is divided into parts proportional to the velocities. And becausethe rectangle KHM is to HT2 as FB/ to BG 2, and the rectangle AB(3 isequal to the rectangle FB/, therefore the little area ABba, where it isgreatest, is to the remaining sector TB6 as the rectangle AB/3 to BG2But the ratio of these little areas always was as the rectangle AB# toBT 2; and therefore the little area ABba in the place A is less than itscorrespondent little area in the quadratures in the duplicate ratio cf BGto BT, that is, in the duplicate ratio of the sine of the sun s distancefrom the node. And therefore the sum of all the little areas ABba, towit, the space ABN, will be as the motion of the node in the time inwhich 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 diesun s mean motion in the same time. And because the mean annual motion of the node is that motion which it performs in the time that the suncompletes one period of its course, the mean motion of the node from thesun will be to the mean motion of the sun itself as the area of the circleto the area of the ellipsis; that is, as the right line TK to the right lineTH, which is a mean proportional between TK and TS ; or, which comesto the same as the mean proportional TH to the right line TS.< PROPOSITION II.u The rmean motion of t/ie -moon s nodes being given, to find their truemotion." Let the angle A be the distance of the sun from the mean place of thenode, or the sun s mean motion from the node. Then if we take the angleB, whose tangent is to the tangent of the angle A as TH to TK, that ia,440 THE MATHEMATICAL PRINCIPLES [BOOK JI1.in the sub-duplicate ratio of the mean horary motion of the sun to themean horary motion of the sun from the node, when the node is in thequadrature, that angle B will be the distance of the sun from the node strue place. For join FT, and, by the demonstration of the last Proportion, the angle FTN will be the distance of the sun from the mean placeof the node, and the angle ATN the distance from the true place, and thetangents of these angles are between themselves as TK to TH." COR. Hence the angle FTA is the equation of the moon s nodes ; andthe sine of this angle, where it is greatest in the octants, is to the radiusas KH to TK + TH. But the sine of this equation in any other placeA 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 ofthe 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" 37iv. 42V. that is, in a whole sidereal year, 39 38 7" 50", TH willbe to TK in the subduplicate ratio of the number 9,0827646 to the number 10,0827646, that is, as 18,6524761 to 19,6524761. And, therefore.TH is to HK as 18,6524761 to 1; that is, as the motion of the sun in asidereal year to the mean motion of the node 19 18 1" 231 "." But if the mean motion of the moon s nodes in 20 Julian years is386 50 15", as is collected from the observations made use of in thetheory of the moon, the mean motion of the nodes in one sidereal year willbe 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 motionof the nodes in the quadratures will come out 16" 18 " 48iv. And thegreatest equation of the nodes in the octants will be 1 29 57"."PROPOSITION XXXIV. PROBLEM XV.Tofind the horary variation of the inclination of the moon s orbit to theplane of the ecliptic.Let A and a represent the syzygies ; Q and q the quadratures ; N andn the nodes ; P the place of the moon in its orbit; p the orthographicprojection of that place upon the plane of the ecliptic ; and mTl the momentaneousmotion of the nodes as above. If upon Tm we let fall *;hcperpendicular PG, and joining pG we produce it till it meet T/ in g, andjoin also Pg~, the angle PGp will be the inclination of the moon s orbit tothe plane of the ecliptic when the moon is in P ; and the angle Pgp willbe the inclination of the same after a small moment of time is elapsed;and therefore the angle GPg- will be the momentaneous variation of theinclination. But this angle GPg- is to the angle GTg as TG to PG andPp to PG conjunctly. And, therefore, if for the moment of time we as71Bnme an hour, since the angle GTg* (by Prop. XXX) is to the angle 3310 " 33iv. as IT X PG X AZ to AT 3, the angle GP^ (or the horary variation of the inclination) will be to the angle 33" 10 " 33iv. as IT X AZX TG X to AT 3. Q.E.I.And thus it would be if the moon was uniformly revolved in a circularorbit. But if the orbit is elliptical, the mean motion of the nodes willbe diminished in proportion of the lesser axis to the greater, as we haveshewn above ; and the variation of the inclination will be also diminishedin the same proportion.COR. 1. Upon N/i erect the perpendicular TF, and let pM. be the horarymotion of the moon in the plane of the ecliptic; upon Q.T let fall theperpendiculars pK, MA*, and produce them till they meet TF in H and h ;then IT will be to AT as Kk to Mjt? ; and TG to Up as TZ to AT ; and,KA* X H# x T7therefore, IT X TG will be equal to -=, that is, equal toT7the area HpWi multiplied into the ratio ^ : and therefore the horaryvariation of the inclination will be to 33" 10" 33iv. as the area HpMATZ Pmultiplied into AZ X ,T~ X ^ to AT 3.MJD PGCOR. 2. And, therefore, if the earth and nodes were after every hourdrawn back from their new and instantly restored to their old places, so astheir situation might continue given for a whole periodic month together,the whole variation of the inclination durinor that month would be to 33442 THE MATHEMATICAL PRINCIPLES [BOOK III10 " 33iv. as the aggregate of all the areas H/?MA. generated in the time otone revolution of the point p (with due regard in summing to their properPsigns + -*), multiplied into AZ X TZ X 5^ to Mjo X AT 3; that is, asPpthe whole circle QAqa multiplied into AZ X TZ X *, to Mp X AT3,that is, as the circumference QAqa multiplied into AZ X TZ X -^ to2Mj0 X AT2.COR. 3. And, therefore, in a given position of the nodes, the mean horary variation, from which, if uniformly continued through the wholemonth, that menstrual variation might be generated, is to 33" 10 " 33iv. asPD AZ x TZAZ X TZ X ~~ to 2AT2, or as Pp X - LT^7p"to PG X 4AT; that1 VJT - A. 。is (because Pp is to PG as the sine of the aforesaid incHnation to the ra-AZ X TZdius, and - -- to 4AT as the sine of double the angle ATu to fourtimes the radius), as the sine of the same inclination multiplied into thesine of double the distance of the nodes from the sun to four times thesquare of the radius.COR. 4. Seeing the horary variation of the inclination, when the nodesare in the quadratures, is (by this Prop.) to the angle 33" 10 " 33iv. as ITX AZ X TG Xpto AT 3, that is, as *, X j~ to 2AT, thatis, as the sine of double the distance of the moon from the quadraturesPpmultiplied into .y^to twice the radius, the sum of all the horary variations during the time that the moon, in this situation of the nodes, passesfrom the quadrature to the syzygy (that is, in the space of 177} hours) willbe to the sum of as many angles 33" 10 " 331V. or 5878 , as the sum of allthe sines of double the distance of the moon from the quadratures multi-Ppplied into p^ to the sum of as many diameters ; that is. as the diameterPpmultiplied into =~ to the circumference; that is, if the inclination be 51 , as 7 X i-fU* to 22> or as 27S to 1000a And>therefore; *he wholevariation, composed out of the sum of all the horary variations in theaforesaid time, is 103", or 2 43".B-OGX 11I.J OF NATURAL PHILOSOPHY. 443PROPOSITION XXXV. PROBLEM XVI.To a given time to find the inclination of the mooiis orbit to the plantof the ecliptic.Let AD be the sine of the greatest inclination, and AB the sine of theleast. Bisect BD in C ; and round the centre C, with the interval BC,describe the circle BGD. In AC take CE in the same proportion to EBB。 HA ECas EB to twice BA. And if to the time given we set off the angle AEGequal to double the distance of the nodes from the quadratures, and uponAD let fall the perpendicular GH, AH will be the sine of the inclinationrequired.For GE2is equal to GH 2 + HE2 = BHD + HE2 = HBD 4- HE2__ BH3 = HBD + BE 2 2BH X BE = BE2 + 2EC X BH = SECX AB + 2EC X BH=2EC X AH; wherefore since 2EC is given. GE2will be as AH. Now let AEg- represent double the distance of the nodesfrom the quadratures, in a given moment of time after, and the arc G^, onaccount of the given angle GE^-, will be as the distance GE. But HA isto GO- as GH to GC, and, therefore, HA is as the rectangle GH X G^, orGH x GE, that is, as ^ X GE2, or 7^ X AH: that is, as AH andljr_tithe sine of the angle AEG conjunctly. If, therefore, in any one case. AHbe the sine of inclination, it will increase by the same increments as thebine of inclination doth, by Cor. 3 of the preceding Prop, and therefore willalways continue equal to that sine. But when the point G falls uponCither point B or D, AH is equal to this sine, and therefore remains alwaysequal thereto. Q.E.D.In this demonstration I have supposed that the angle BEG, representingdouble the distance of the nodes from the quadratures, increaseth uniformly ;for I cannot descend to every minute circumstance of inequality. Nowsuppose that BEG is a right angle, and that Gg is in this case the horary increment of double the distance of the nodes from the sun ; then, byCor. 3 of the last Prop, the horary variation of the inclination in the samecase will be to 33" 10" 33iv. as the rectangle of AH, the sine of the inclination, into the sine of the right angle BEG, double the distance of thenodes from the sun, to four times the square of the radius ;that is, as AH,THE MATHEMATICAL PRINCIPLES [Bc-OK )lLthe sine of the mean inclination, to four times the radius; that is, seeingthe mean inclination is about 5 S, as its sine 896 to 40000, the quadruple of the radius, or as 224 to 10000. But the whole variation corresponding to BD, the difference of the sines, is to this horary variation asthe diameter BU to the arc G%, that is, conjunctly as the diameter BD tothe semi- circumference BGD, and as the time of 2079 T。 hours, in whichthe node proceeds from the quadratures to the syzyffies, to one hour, thatis as 7 to 11, and 2079 T。 to 1. Wherefore, compounding all these proportions, we shall have the whole variation BD to 33" 10" 33iv. as 224 X7 X 2079 T。 to 110000, that is, as 29645 to 1000; and from thence thatvariation BD will come out 16 23i".And this is the greatest variation of the inclination, abstracting fromthe situation of the moon in its orbit: for if the nodes are in the syzygies,the inclination suffers no change from the various positions of the moon.But if the nodes are in the quadratures, the inclination is less when themoon is in the syzygies than when it is in the quadratures by a differenceof 2 43", as we shewed in Cor. 4 of the preceding Prop. ; and the wholemean variation BD, diminished by 1 21 i", the half of this excess, becomes15 2", when the moon is in the quadratures: and increased by the same,becomes 17 45" when the moon is in the syzygies. If, therefore, themoon be in the syzygies, the whole variation in the passage of the nodesfrom the quadratures to the syzygies will be 17 45"; and, therefore, if theinclination be 5 17 20", when the nodes are in the syzygies, it will be 459 35" when the nodes are in the quadratures and the moon in the syzygies. The truth of all which is confirmed by observations.Now if the inclination of the orbit should be required when the moon isin the syzygies, and the nodes any where between them and the quadratures,let AB be to AD as the sine of 4 59 35" to the sine of 5 17 20", andtake the angle AEG equal to double the distance of the nodes from thequadratures ; and AH will be the sine of the inclination desired. To thisinclination of the orbit the inclination of the same is equal, when the moonis 90 distant from the nodes. In other situations of the moon, this menstrual inequality, to which the variation of the inclination is obnoxious inthe calculus of the moon s latitude, is balanced, and in a manner took off,by the menstrual inequality of the motion of the nodes (as we saidbefore), and therefore may be neglected in the computation of the saidlatitude.SCHOLIUM.By these computations of the lunar motions I was willing to shew thatby the theory of gravity the motions of the moon could be calculated fromtheir physical causes. By the same theory I moreover found that the annual equation of the mean motion of the moon arises from the variousBOOK III.] OF NATURAL PHILOSOPHY 445dilatation which the orbit of the moon suffers from the action of the sunaccording to Cor. 6, Prop. LXVI. Book I. The force of this action isgreater in the perigeon sun, and dilates the moon s orbit; in the apogeonsun it is less, and permits the orbit to be again contracted. The moonmoves slower in the dilated and faster in the contracted orbit ; and theannual equation, by which this inequality is regulated, vanishes in theapogee and perigee of the sun. In the mean distance of the sun from theearth it arises to about 11 50"; in other distances of the sun it is proportional to the equation of the sun s centre, and is added to the meanmotion of the moon, while the earth is passing .from its aphelion to itsperihelion, and subducted while the earth is in the opposite semi-circle.Taking for the radius of the orbis niagnus 1000, and 16} for the earth seccentricity, this equation, when of the greatest magnitude, by the theoryof gravity comes out 11 49". But the eccentricity of the earth seems tobe something greater, and with the eccentricity this equation will be augmented in the same proportion. Suppose the eccentricity 16}^, and thegreatest equation will be 11 51".Farther ;I found that the apogee and nodes of the moon move fasteiin the perihelion of the earth, where the force of the sun s action is greater,than in the aphelion thereof, and that in the reciprocal triplicate proportion of the earth s distance from the sun ; and hence arise annual equations of those motions proportional to the equation of the sun s centre.Now the motion of the sun is in the reciprocal duplicate proportion of theearth s distance from the sun ; and the- greatest equation of the centrewhich this inequality generates is 1 56 20", corresponding to the abovementionedeccentricity of the sun, 16}. But if the motion of the sunhad been in the reciprocal triplicate proportion of the distance, this inequality would have generated the greatest equation 2 54 30"; and therefore the greatest equations which the inequalities of the motions of themoon s apogee and nodes do generate are to 2 54 30" as the mean diurnal motion of the moon s apogee and the mean diurnal motion of itsnodes are to the mean diurnal motion of the sun. Whence the greatestequation of the mean motion of the apogee comes out 19 43", and thegreatest equation of the mean motion of the nodes 9 24". The formerequation is added, and the latter subducted, while the earth is passingfrom its perihelion to its aphelion, and contrariwise when the earth is inthe opposite semi-circle.By the theory of gravity I likewise found that the action of the sunupon the moon is something greater when the transverse diameter of themoon s orbit passeth through the sun than when the same is perpendicular upon the line which joins the earth and the sun ; and therefore themoon s orbit is something larger in the former than in the latter case.And hence arises another equation of the moon s moan motion, depending446 THE MATHEMATICAL PRINCIPLES [BOOK IIIupon the situation of the moon s apogee in respect of the sun, which is inits greatest quantity when the moon s apogee is in the octants of the sun,and vanishes when the apogee arrives at the quadratures or syzygies ; andit is added to the mean motion while the moon s apogee is passing fromthe quadrature of the sun to the syzygy, and subducted while the apogeeis passing from the syzygy to the quadrature. This equation, which Ishall call the semi-annual, when greatest in the octants of the apogee,arises to about 3 45", so far as I could collect from the phenomena : andthis is its quantity in the mean distance of the sun from the earth. Butit is increased and diminished in the reciprocal triplicate proportion ofthe sun s distance, and therefore is nearly 3 34" when that distance isgreatest^ and 3 56" when least. But when the moon s apogee is withoutthe octants, it becomes less, and is to its greatest quantity as the sine ofdouble the distance of the moon s apogee from the nearest syzygy or quadrature to the radius.By the same theory of gravity, the action of the sun upon the moon issomething greater when the line of the moon s nodes passes through thesun than when it is at right angles with the line which joins the sun andthe earth ; and hence arises another equation of the moon s mean motion,which I shall call the second semi-annual ; and this is greatest when thenodes are in the octants of the sun, and vanishes when they are in thesyzygies or quadratures ; and in other positions of the nodes is proportional to the sine of double the distance of either node from the nearestsyzygy or quadrature. And it is added to the mean motion of the moon,if the sun is in antecedentia, to the node which is nearest to him, and