drawn to the earth, describes in a circular orbit.We have aboveshown that the areawhich the moon describes by a radiusdrawn to the earthis proportional tothe time of description, excepting in sofar as the moon smotion is disturbedby the action of thesun ; and here wepropose to investigate the HI equality of the moment, or horary increment of that area ormotion so disturbed. To render the calculus more easy, we shall supposethe orbit of the moon to be circular, and neglect all inequalities but thatonly which is now under consideration ; and, because of the immense distance of the sun, we shall farther suppose that the lines SP and ST areparallel. By this moans, the force LM will be always reduced to its meanquantity TP, as well as the force TM to its mean quantity 3PK. Theseforces (by Cor. 2 of the Laws of Motion) compose the force TL ; andthis force, by letting fall the perpendicular LE upon the radius TP, isresolved into the forces TE, EL ;of which the force TE, acting constantlyin the direction of the radius TP, neither accelerates nor retards the description of the area TPC made by that radius TP ; but EL, acting onthe radius TP in a perpendicular direction, accelerates or retards the description of the area in proportion as it accelerates ->r retards the moon.BOOK lit.] OF NATURAL PHILOSOPHY.That acceleration of the moon, in its passage from the quadrature C to theconjunction A, is in every moment of time as the generating accclerative3PK X TKforce EL, that is, as.5. Let the time be represented by themean motion of the moon, or (which comes to the same thing) by the angleCTP, or even by the arc CP. At right angles upon CT erect CG equalto CT ; and, supposing the quadrantal arc AC to be divided into an infinitenumber of equal parts P/?, &c., these parts may represent the like infinitenumber of the equal parts of time. Let fall pic perpendicular on CT, anddraw TG meeting with KP, kp produced in F arid /; then will FK beequal to TK, and K/v be to PK as P/>to T/?, that is, in a given propor-3PK X TKtion ; and therefore FK X K&, or the area FKkf, will be as -~^pp >that is, as EL: and compounding, the whole area GCKF will be as thesum of all the forces EL impressed upon the moon in the whole time CP ;and therefore also as the velocity generated by that sum, that is, as the acceleration of the description of the area CTP, or as the increment of themoment thereof. The force by which the moon may in its periodic timeCADB of 27 1. 7h. 43 be retained revolving about the earth in rest at thedistance TP, would cause a body falling in the time CT to describe thelength ^CT, and at the same time to acquire a velocity equal to that withwhich the moon is moved in its orbit. This appears from Cor. 9, Prop,IV., Book I. But since K.d, drawn perpendicular on TP, is but a thirdpart of EL, and equal to the half of TP, or ML, in the octants, the forceEL in the octants, where it is greatest, will exceed the force ML in theproportion of 3 to 2 ; and therefore will be to that force by which the moonin its periodic time may be retained revolving about the earth at rest as100 to | X 178721, or 11915; and in the time CT will generate a velocity equal to yylfs parts of the velocity of the moon ; but in the timeCPA will generate a greater velocity in the proportion of CA to CT orTP. Let the greatest force EL in the octants be represented by the areaFK X Kk, or by the rectangle |TP X Pp, which is equal thereto; andthe velocity which that greatest force can generate in any time CP will beto the velocity which any other lesser force EL can generate in the sametime as the rectangle |TP X CP to the area KCGF ; but the velocitiesgenerated in the whole time CPA will be one to the other as the rectangle2-TP X CA to the triangle TCG, or as the quadrantal arc CA to theradius TP ; and therefore the latter velocity generated in the whole timewill be T T$TJ parts of the velocity of the moon. To this velocity of themoon, which is proportional to the mean moment of the area (supposingthis mean moment to be represented by the number 11915), we add andsubtract the half of the other velocity ; the sum 11915 + 50, or 11965,will represent the greatest moment of the area in the syzygy A : and theTHE MATHEMATICAL PRINCIPLES [BOOK IIIdifference 11915 50, or 11865, the least moment thereof in the quadratures. Therefore the areas which in equal times are described in the syzygiesand quadratures are one to^the other as 11965 to 11865. And if tothe least moment 11865 we add a moment which shall be to 100, the difference of the two former moments, as the trapezium FKCG to the triangleTCG, or, which comes to the same thing, as the square of the sine PK tothe square of the radius TP (that is, as Pd to TP), the sum will representthe moment of the area when the moon is in any intermediate place P.But these things take place only in the hypothesis that the sun and theearth are at rest, and that the synodical revolution of the moon is finishedin 27 1. 7h. 43 . But since the moon s synodical period is really 29a. 12h.4 T, the increments of the moments must be enlarged in the same proportion as the time is, that is, in the proportion of 1080853 to 1000000.Upon whicli account, the whole increment, which was TTITTT parts of themean moment, will now become TY| 3- parts thereof; and therefore themoment of the area in the quadrature of the moon will be to the momentthereof in the syzygy as 11023 50 to 11023 + 50; or as 10973 to11073; and to the moment thereof, when the moon is in any intermediateplace P, as 10973 to 10973 -f Pd ; that is, supposing TP = 100.The area, therefore, which the moon, by a radius drawn to the earth,describes m the several little equal parts of time, is nearly as the sum ofthe number 219,46, and the versed sine of the double distance of the moonfrom the nearest quadrature, considered in a circle which hath unity for itsradius. Thus it is when the variation in the octants is in its mean quantity.3ut if the variation there is greater or less, that versed sine must be augnentedor diminished in the same proportion.PROPOSITION XXVIL PROBLEM VI11.From the horary motion of the moon to find its distance from the earth.The area which the moon, by a radius drawn to the earth, describes inevery, moment of time, is as the horary motion of the moon and the squareof the distance of the moon from the earth conjunctly. And therefore thedistance of the moon from the earth is in a proportion compounded of thesubduplicate proportion of the area directly, and the subduplioate proportion of the horary motion inversely. Q.E.T.COR. 1 . Hence the apparent diameter of the moon is given ; for it is reciprocally as the distance of the moon from the earth. Let astronomerstry how accurately this rule agrees with the phenomena.COR. 2. Hence also the orbit of the moon may be more exactly definedfrom the phaenomena than hitherto could be done.BOOK III,"! OF NATURAL PHILOSOPHY. 423PROPOSITION XXVIII. PROBLEM IX.To find the diameters of the orbit, in which, without ec*.t itricity, themoon would move.The curvature of the orbit which a body describes, if attracted in linesperpendicular to the orbit, is as the force of attraction directly, and thesquare of the velocity inversely. I estimate the curvatures of lines compared one with another according to the evanescent proportion of the sinesor tangents of their angles of contact to equal radii, supposing those radiito be infinitely diminished. Blit the attraction of the moon towards theearth in the syzygies is the excess of its gravity towards the earth abovethe force of the sun 2PK (see Pig. Prop. XXV); by which force the accelerativegravity of the moon towards the sun exceeds the accelerative gravityof the earth towards the sun, or is exceeded by it. But in the quadraturesthat attraction is the sum of the gravity of the moon towards the earth,and the sun s force KT, by which the moon is attracted towards the earth.AT + CT 178725And these attractions, putting N for-^-> are nearly as T^--and - + - or as 178725N X CT* - 2000AT*X CT, and 17S725N X AT2 + 1000CT 2 X AT. For if the accelerative gravity of the moon towards the earth be represented by the number178725, the mean force ML, which in the quadratures is PT or TK, anddraws the moon towards the earth, will be 1000, and the mean force TM inthe syzygies will be 3000 ; from which, if we subtract the mean force ML,there will remain 2000, the force by which the moon in the syzygies isdrawn from the earth, and which we above called 2 PIC. But the velocityof the moon in the syzygies A and B is to its velocity in the quadraturesC and D as CT to AT, and the moment of the area, which the moon bya radius drawn to the earth describes in the syzygies, to the moment of thatarea described in the quadratures conjunctly ; that is, as 11073CT to10973AT. Take this ratio twice inversely, and the former ratio once directly, and the curvature of the orb of the moon in the syzygies will be tothe curvature thereof in the quadratures as 120406729 X 17S725AT 2 XCT2 X N 120406729 X 2000AT 4 X CT to 122611329 X 178725AT2X CT2 X N + 122611329 X 1000CT 4 X AT, that is, as 2151969ATX CT X N 24081AT 3 to 2191371AT X CT X N + 12261CT 3.Because the figure of the moon s orbit is unknown, let us, in its stead,assume the ellipsis DBCA, in the centre of which we suppose the earth tobe situated, and the greater axis DC to lie between the quadratures as thelesser AB between the syzygies. But since the plane of this ellipsis is rerolvedabout the earth by an angular motion, and the orbit, whose curvature we now examine, should be described in a plane void of such motion424 THE MATHEMATICAL PRINCIPLES [BOOK IIIwe are to consider the figure which the moon,while it is revolved in that ellipsis, describes iuthis plane, that is to say, the figure Cpa, theseveral points p of which are found by assumingany point P in the ellipsis, which may representthe place of the moon, and drawing Tp equalto TP in such manner that the angle PT/? maybe equal to the apparent motion of the sun fromthe time of the last quadrature in C ; or (whichcomes to the same thing) that the angle CTpmay be to the angle CTP as the time of thesynodic revolution of the moon to the time otthe periodic revolution thereof, or as 29 1. 12h. 44 to 27d. 7 1. 43 . If, therefore, in this proportion we take the angle CTa to the right angle CTA,and make Ta of equal length with TA, we shall have a the lower and Cthe upper apsis of this orbit Cpa. But, by computation, I find that thedifference betwixt the curvature of this orbit Cpa at the vertex a, and thecurvature of a circle described about the centre T with the interval TA, isto the difference between the curvature of the ellipsis at the vertex A, andthe curvature of the same circle, in the duplicate proportion of the angleCTP to the angle CTp ; and that the curvature of the ellipsis in A is tothe curvature of that circle in the duplicate proportion of TA to TC ; andthe curvature of that circle to the curvature of a circle described about thecentre T with the interval TC as TC to TA ; but that the curvature ofthis last arch is to the curvature of the ellipsis in C in the duplicate proportion of TA to TC ; and that the difference betwixt the curvature of theellipsis in the vertex C* and the curvature of this List circle, is to the difference betwixt the curvature of the figure Cpa, at the vertex C, and thecurvature of this same last circle, in the duplicate proportion of the angleCTp to the angle CTP ;all which proportions are easily drawn from thesines of the angles of contact, and of the differences of those angles. But,by comparing those proportions together, we find the curvature of the figureCpa at a to be to its curvature at C as AT 3,- rWoVoCT2AT to CT 3 -r_i_6_8_2_4_AT 2 X CT ; where the number yVYVYo represents the differenceof the squares of the angles CTP and CTp, applied to the square of thelesser angle CTP ; or (which is all one) the difference of the squares of thelimes 27. 7h- 43 , and 29 1. 12h. 44 , applied to the square of the time27(1.7h. 43 .Since, therefore, a represents the syzygy of the moon, and C its quadrature, the proportion now found must be the same with that proportion ofthe curvature of the moon s orb in the syzygies to the curvature thereof inthe quadratures, which we found above. Therefore, in order to find thBOOK III.] OF NATURAL PHILOSOPHY. 425proportion of CT to AT, let us multiply the extremes and the means, an(?the terms which come out, applied to AT X CT, become 2062,79CT 42151969N x CT 3 + 368676N X AT X CT 2 + 36342 AT 2 X CT2 -362047N X AT2 X CT + 2191371N X AT 3 + 4051,4AT 4 = 0.Now if for the half sum N of the terms AT and CT we put 1, and x fortheir half difference, then CT will be = 1 + x, and AT = 1 x. Andsubstituting those values in the equation, after resolving thereof, wre shallfind x = 0,00719 ; and from thence the semi-diameter CT = 1,00719, andthe semi-diameter AT = 0,99281, which numbers are nearly as 70^, and692V- Therefore the moon s distance from the earth in the syzygies is toits distance in the quadratures (setting aside the consideration of eccentricity) as 09 2^ to 70^ ; or, in round numbers, as 69 to 70.PROPOSITION XXIX. PROBLEM X.To find the variation of the moon.This inequality is owing partly to the elliptic figure of the moon s orbit,partly to the inequality of the moments of the area which the moon by aradius drawn t。) the earth describes. If the moon P revolved in the ellipsisDBCA about the earth quiescent in the centre of the ellipsis, and by theradius TP, drawn to the earth, described the area CTP, proportional tothe time of description ; and the greatest semi-diameter CT of the ellipsiswas to the least TA as 70 to 69; the tangent of the angle CTP would beto the tangent of the angle of the mean motion, computed from the quadrature C, as the semi-diameter TA of the ellipsis to its semi-diameter TC,or as 69 to 70. But the description of the area CTP, as the moon advances from the quadrature to the syzygy, ought to be in such manner accelerated, that the moment of the area in the moon s syzygy may be to themoment thereof in its quadrature as 11073 to 10973; and that the excessof the moment in any intermediate place P above the moment in the quadrature may be as the square of the sine of the angle CTP ; which we mayeffect with accuracy enough, if we diminish the tangent of the angle CTPin the subduplicate proportion of the number 10973 to the number 11073,that is, in proportion of the number 68,6877 to the number 69. Uponwhich account the tangent of the angle CTP will now be to the tangentof the mean motion as 68,6877 to 70 ; and the angle CTP in the octants,where the mean motion is 45, will be found 44 27 28", which subtracted from 45, the angle of the mean motion, leaves the greatest variation 32 32". Thus it would be, if the moon, in passing from the quadrature to the syzygy, described an angle CTA of 90 degrees only. Butbecause of the motion of the earth, by which the sun is apparently transferred in consequentia^ the moon, before it overtakes the sun, describes anangle CTtf, greater than a right angle, in the proportion of the time of thesynodic revolution of the moon to the time of its periodic revolution, thai126 THE MATHEMATICAL PRINCIPLES [BOOK IIIis, in the proportion of 29 1. 12h. 44 to 27(l. 7X 43 . Whence it comes tcpass that all the angles about the centre T are dilated in the same proportion ; and the greatest variation, which otherwise would be but 3232", now augmented in the said proportion, becomes 35 10".And this is its magnitude in the mean distance of the sun from theearth, neglecting the differences which may arise from the curvature ofthe orbis magnns, and the stronger action of the sun upon the moon whenhorned and new, than when gibbous and full. In other distances of thesun from the earth, the greatest variation is in a proportion compoundedof the duplicate proportion of the time of the synodic revolution of themoon (the time of the year being given) directly, and the triplicate proportion of the distance of the sun from the earth inversely. And, therefore, in the apogee of the sun, the greatest variation is 33 14", and in itsperigee 37 11", if the eccentricity of the sun is to the transverse semi-diameter of the orbis magnus as 16} to 1000.Hitherto we have investigated the variation in an orb not eccentric, inwhich, to wit, the moon in its octants is always in its mean distance fromthe earth. If the moon, on account of its eccentricity, is more or less removed from the earth than if placed in this orb, the variation may besomething greater, or something less, than according to this rule. But Ileave the excess or defect to the determination of astronomers from thephenomena.PROPOSITION XXX. PROBLEM XI.To find the horary motion of the nodes of ihe moon in a circular orbit.Let S represent the sun, T the earth, P the moon, NP/A the orbit, of thrmoon, Njo/? the orthographic projection of the orbit upon the plane of thecliptic : N. n the nodes. nTNm the line of the nodes produced indetiBOOK III.] OF NATURAL PHILOSOPHY. 427nitely ; PI, PK perpendiculars upon the lines ST, Qq ; Pp a perpendicular upon the plane of the ecliptic; A, B the moon s syzygies in the planeof the ecliptic; AZ a perpendicular let fall upon Nil, the line of thenodes ; Q, g the quadratures of the moon in the plane of the ecliptic, andpK a perpendicular on the line Qq lying between the quadratures. Theforce of the sun to disturb the motion of the moon (by Prop. XXV) istwofold, one proportional to the line LM, the otlier to the line MT, in thescheme of that Proposition ; and the moon by the former force is drawntowards the earth, by the latter towards the sun, in a direction parallel tothe right line ST joining the earth and the sun. The former force LMacts in the direction of the plane of the moon s orbit, and therefore makesno change upon the situation thereof, and is upon that account to be neglected ; the latter force MT, by which the plane of the moon s orbit is disturbed, is the same with the force 3PK or SIT. And this force (by Prop.XXV) is to the force by which the moon may, in its periodic time, be uniformly revolved in a circle about the earth at rest, as SIT to the radius ofthe circle multiplied by the number 178,725, or as IT to the radius thereof multiplied by 59,575. But in this calculus, and all that follows. Iconsider all the lines drawn frorri the moon to the sun as parallel to theline which joins the earth and the sun ; because what inclination there isalmost as much diminishes all effects in some cases as it augments themin others : and we are now inquiring after the mean motions of the nodes,neglecting such niceties as are of no moment, and would only serve to render the calculus more perplexed.Now suppose PM to represent an arc which the moon describes in theleast moment of time, and ML a little line, the half of which the moon,by the impulse of the said force SIT, would describe in the same time ; andjoining PL, MP, let them be produced to m and /, where they cut the planeof the ecliptic, and upon Tm let fall the perpendicular PH. Now, sincethe right line ML is parallel to the plane of the ecliptic, and therefore cannever meet with the right line ml which lies in that plane, and yet boththose right lines lie in one common plane LMPm/, they will be parallel,and upon that account the triangles LMP, ImP will be similar. Andseeing MPra lies in the plane of the orbit, in which the moon did movewhile in the place P, the point m will fall upon the line N//, which passesthrough the nodes N, n, of that orbit. And because the force by which thehalf of the little line LM is generated, if the whole had been together, andit once impressed in the point P, would hav^ generated that whole line,and caused the moon to move in the arc whoso chord is LP ;t at is to say,would have transferred the moon from the plane MPwT into the planeLP/T; therefore th* angular motion of the nodes generated by that forcewill be equal to the angle mTL But n.l is to raP as ML to MP ; andsince ML3, because of the time given, is also given, ml will be as the rectan428 THE MATHEMATICAL PRINCIPLES [BOOK III.gle ML X mP, that is, as the rectangle IT X mP. And if Tml is a rightano-le, the angle mTl will be as 7T^m and therefore as T^m that is (be-ITxPH*cause Tm and mP, TP and PH are proportional), as FFp~ and, therefore, because TP is given, as IT X PH. But if the angle Tml or STNis oblique, the angle mTl will be yet less, in proportion of the sine of theangle STN to the radius, or AZ to AT. And therefore the velocity ofthe nodes is as IT X PH X AZ, or as the solid content of the sines of thethree angles TPI, PTN, and STN.If these are right angles, as happens when the nodes are in the quadratures, and the moon in the syzygy, the little line ml will be removed toan infinite distance, and the angle mTl will become equal to the anglemPl. But in this case the angle mPl is to the angle PTM, which themoon in the same time by its apparent motion describes about the earth,as 1 to 59,575. For the angle mPl is equal to the angle LPM, that is, tothe angle of the moon s deflexion from a rectilinear path; which angle, ifthe gravity of the moon should have then ceased, the said force of the sunSIT would by itself have generated in that given time : and the anglePTM is equal to the angle of the moon s deflexion from a rectilinear path;which angle, if the force of the sun 31T should have then ceased, the forcealone by which the moon is retained in its orbit would have generated inthe same time. And. these forces (as we have above shewn) are the one tothe other as I to 59,575. Since, therefore, the mean horary motion of themoon (in respect of the fixed stars) is 32 56" 27 "12^-iv. the horary motionof the node in this case will be 33" 10" 331V. 12V. But in other cases thehorary motion will be to 33" 10 " 33iv. 。2。 as the solid content of the sinesof the three angles TPI, PTN, and STN (or of the distances of the moonfrom the quadrature, of the moon from the node, and of the node from thesun) to the cube of the radius. And as often as the sine of any angle ischanged from positive to negative, and from negative to positive, so oftenmust the regressive be changed into a progressive, and the progressive intoa regressive motion. Whence it comes to pass that the nodes are progressive as often as the moon happens to be placed between either quadrature, and the node nearest to that quadrature. In other cases they areregressive, and by the excess of the regress above the progress, they aremonthly transferred in antecedentia.COR. 1. Hence if from P and M, the extreme points of a least arc PM,on the line Qq joining the quadratures we let fall the perpendiculars PKMA", and produce the same till they cut the line of the nodes Nw in D anad, the horary motion of the nodes will be as the area MPDd, and thesquare of the line AZ conjunctly. For let PK, PH, and AZ, be the threesaid sines, viz., PK the sine of the distance of the moon from the quadraBOOK III.] OF NATURAL PHILOSOPHY.Q421*