自然哲学的数学原理-47

ture, PH the sine of the distance of the moon from the node, and AZ theof the distance of the node from the sun ; and the velocity of the nodewill be as the solid content of PK X PH X AZ. But PT is to PK asPM to KA;; and, therefore, because PT and PM are given, Kk will be asPK. Likewise AT is to PD as AZ to PH, and therefore PH is as therectangle PD X AZ ; and, by compounding those proportions, PK X PHis as the solid content Kk X PD X AZ; and PK X PH X AZ as KAX PD X AZ 2; that is, as the area PDrfM and AZ 3conjunctly. Q.E.I).COR. 2. In any given position of the nodes their mean horary motion ishalf their horary motion in the moon s syzygies ; and therefore is to 16"35 " 16iv. 36V. as the square of the sine of the distance of the nodes fromthe syzygies to the square of the radius, or as AZ 2 to AT2. For if themoon, by an uniform motion, describes the semi-circle QA</, the sum of allthe areas PDdM, during the time of the moon s passage from Q, to M, willmake up the area QMc/E. terminating at the tangent Q,E of the circle ;and by the time that the moon has arrived at the point //, that sum willmake up the whole area EQ,Aw described by the line PD : but when themoon proceeds from n to q, the line PD will fall without the circle, anddescribe the area nqe, terminating at the tangent qe of the circle, whicharea, because the nodes were before regressive, but are now progressive,must be subducted from the former area, and, being itself equal to the areaQ.EN, will leave the semi-circle NQAn. While, therefore, the moon describes a semi-circle, the sum of all the areas PDdM will be the area ofthat semi-circle ; and while the moon describes a complete circle, the sumof those areas will be the area of the whole circle. But the area PDc^M,when the moon is in the syzygies, is the rectangle of the arc PM into theradius PT ; and the sum of all the areas, every one equal to this area, inthe time that the moon describes a complete circle, is the rectangle of thewhole circumference into the radius of the circle; and this rectangle, beingdouble the area of the circle, will be double the quantity of the former sum130 THE MATHEMATICAL PRINCIPLES [BOOK 1IJIf, therefore, the nodes went on with that velocity uniformly continuedwhich they acquire in the moon s syzygies, they would describe a spacedouble of that which they describe in fact; and, therefore, the mean motion,by which, if Uniformly continued, they would describe the same space withthat which they do in fact describe by an unequal motion, is but one-halfof that motion which they are possessed of in the moon s syzygies. Wherefore since their greatest horary motion, if the nodes are in the quadratures,is 33" 10 " 33iv. 12V. their mean horary motion in this case will be 16"35 " 16iv. 36V. And seeing the horary motion of the nodes is every whereas AZ 2 and the area PDdM conjunctly, and. therefore, in the moon ssyzygies, the horary motion of the nodes is as AZ 2 and the area PDdMconjunctly, that is (because the area PDdNL described in the syzygies isgiven), as AZ2, therefore the mean motion also will be as AZ2; and, therefore, when the nodes are without the quadratures, this motion will be to16" 35 " I6 v. 36V. as AZ 2 to AT 2. Q.E.D.PROPOSITION XXXI. PROBLEM XII.To find the horary motion of the nodes of the moon in an elliptic orbitLet Qjpmaq represent an ellipsis described with the greater axis Qy, amthe lesser axis ab : QA^B a circle circumscribed ; T the earth in the common centre of both ; S the sun ; p the moon moving in this ellipsis ; andBOOK I1I.J OF NATURAL PHILOSOPHY. 431pm an arc which it describes in the least moment of time; N and n tlwnodes joined by the line N//, ; pK and ink perpendiculars upon the axisQ,</,produced both ways till they meet the circle in P and M, and the line ofthe nodes in D and cl. And if the moon, by a radius drawn to the earth,describes an area proportional to the time of description, the horary motionof the node in the ellipsis will be as the area pDdm and AZ2conjunctly.For let PF touch the circle in P, and produced meet TN in F; arid pjtouch the ellipsis in p, and produced meet the same TN in /, and bothtangents concur in the axis TQ, at Y. And let ML represent the spacewhich the moon, by the impulse of the above-mentioned force 3IT or 3PK,would describe with a transverse motion, in the meantime while revolvingin the circle it describes the arc PM ; and ml denote the space which themoon revolving in the ellipsis would describe in the same time by the impulse of the same force SIT or 3PK ; and -let LP and Ip be produced tillthey meet the plane of the ecliptic in G and g, and FG and /"^be joined,of which FG produced may cut pf, pa; and TQ, in c, e, and R respectively ; and/0" produced may cut TQ in r. Because the force SIT or 3PKin the circle is to the force SIT or 3/?K in the ellipsis as PK to /?K, oras AT to T, the space ML generated by the former force will be to thespace ml generated by the latter as PK top"K ;that is, because of thesimilar figures PYK/? and FYRc, as FR to cR. But (because of thesimilar triangles PLM, PGF) ML is to FG as PL to PG. that is (on account of the parallels L/r, PK, GR), as pi to pe, that is (because of thesimilar triangles plm, cpe), as lm to ce ; and inversely as LM is to lm, oras FR is to cR, so is FG to ce. And therefore if fg was to ce as/// tocY, that is, as fr 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 FGto ce, expunged on both sides, leaves the ratiosfg to FG and/T to FT,fg would be to FG as/T to FT; and, therefore, the angles which FGand/- would subtend at the earth T would be equal to each other. Butthese angles (by what we have shewn in the preceding Proposition) are themotions of the nodes, while the moon describes in the circle the arc PM,in the ellipsis the arc jt?w; and therefore the motions of the nodes in thecircle and in the ellipsis would be equal to each other. Thus, I say, itcex /Ywould be, if fg was to cc as/Y to cY, that is, if/,r was oqual to ^ .But because of the similar triangles/?/?, cep, fg is to cc as//? to cp ; anJtherefore/?- is equal to -; and therefore the angle which fg subtends in fact is to the former angle which FG subterds. that is to say, themotion of the nodes in tl;^ ellipsis is to the motion of the same in thecircle aa this/^ or- to the forrer/o- or , that is, as//? X432 THE MATHEMATICAL PRINCIPLES [HOOK 111.cY to/ Y X cp, or as//? to/ Y, and cY to cjo ; that is;if ph parallel toTN meet FP in h, as FA to FY and FY to FP ; that is, as Fh to FPor DJO to DP, and therefore as the area Dpmd to the area DPMc?. And,therefore, seeing (by Corol. 1, Prop. XXX) the latter area and AZ2 conjunctlyare proportional to the horary motion of the nodes in the circle,the former area and AZ2conjunctly will be proportional to the horarymotion of the nodes in the ellipsis. Q.E.D.COR. Since, therefore, in any given position of the nodes, the sum of allthe areas />Drfm, in the time while the moon is carried from the quadrature to any place tn, is the area mpQ&d terminated at the tangent of theellipsis Q,E ; and the sum of all those areas, in oTne entire revolution, isthe area of the whole ellipsis ; the mean motion of the nodes in the ellipsis will be to the mean motion of the nodes in the circle as the ellipsis tothe circle; that is, as Ta to TA, or 69 to 70. And, therefore, since (byCorol 2, Prop. XXX) the mean horary motion of the nodes in the circleis to 16" 35"7 16iv. 36V. as AZ2 to AT2, if we take the angle 16" 21 "3iv. 30V. to the angle 16" 35 " 16iv. 36V. as 69 to 70. the mean horary motion of the nodes in the ellipsis will be to 16" 21 " 3iv. 30V. as AZ2 toAT2;that is, as the square of the sine of the distance of the node fromthe sun to the square of the radius.But the moon, by a radius drawn to the earth, describes the area in thesyzygie-s with a greater velocity than it does that in the quadratures, andupon that account the time is contracted in the syzygies, and prolonged inthe quadratures ; and together with the time the motion of the nodes islikewise augmented or diminished. But the moment of the area in thequadrature of the moon was to the moment thereof in the syzygies as10973 to 11073 ; and therefore the mean moment in the octants is to theexcess in the syzygies. and to the defect in the quadratures, as 1 1023, thehalf sum of those numbers, to their half difference1 50. Wherefore sincethe time of the moon in the several little equal parts of its orbit is reciprocally as its velocity, the mean time in the -octants will be to the excessof the time in the quadratures, and to the defect of the lime in the syzygies arising from this cause, nearly as 11023 to 50. But, reckoning fromthe quadratures to the syzygies, I find that the excess of the moments ofthe 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 quadratures : and therefore the difference betwixt the moment in any place,and the mean moment in the octants, is as the difference betwixt the squareof the sine of the moon s distance from the quadratures, and the squareof the sine of 45 degrees, or half the square of the radius ; and the increment of the time in the several places between the octants and quadratures, and the decrement thereof between the octants and syzygies, is inthe same proportion. But the motion of the nodes, while the moon describes the several little equal parts of its orbit, is accelerated or retardedBOOK III.] OF NATURAL PHILOSOPHY. 433in the duplicate proportion of the time ; for that motion, while the mooudescribes PM, is (cceteris parilms) as ML. and ML is in the duplicateproportion of the time. Wherefore the motion of the nodes in the syzygj-es, in the time while the moon describes given little parts of its orbit,is diminished in the duplicate proportion of the number H07. J to the number 11023: and the decrement is to the remaining motion as 100 to10973 ; but to the whole motion as 100 to 11073 nearly. But the decrement in the places between the octants and syzygies, and the increment inthe places between the octants and quadratures, is to this decrement nearlyas 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 distancefrom the quadrature, and the half square of the radius, to the half squareof 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 distantfrom the octant and other two distant by the same interval, one from thesyzygy, the other from the quadrature, and from the decrements of themotions in the two places between the syzygy and octant we subtract theincrements of the motions in the two other places between the octant andthe quadrature, the remaining decrement will be equal to the decrement in the syzygy, as will easily appear by computation ; and thereforethe mean decrement, which ought to be subducted from the mean motionof the nodes, is the fourth part of the decrement in the syzygy. Thewhole horary motion of the nodes in the syzygies (when the moon by a radius drawn to the earth was supposed to describe an area proportional tothe time) was 32" 42" ? iv. And we have shewn that the decrement ofthe motion of the nodes, in the time while the moon, now moving withgreater velocity, describes the same space, was to this motion as 100 to1.1073; and therefore this decrement is 17 " 43iv. 11 v. The fourth partof which 4 " 25iv. 48V. subtracted from the mean horary motion abovefound, 16" 21 //; 3iv. 30V. leaves 16" 16 " 37iv. 42V. their correct mean horary 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, thesum of the motions of the nodes, when the moon is in those places, will beto the sum of their motions, when the moon is in the same places and thenodes in the quadratures, as AZ2 to AT2. And the decrements of themotions arising from the causes but now explained will be mutually asthe motions themselves, and therefore the remaining motions will be mutually betwixt themselves as AZ2 to AT2; and the mean motions will beas the remaining motions. And, therefore, in any given position of thenodes, their correct mean horary motion is to 16" 16 " 37iv. 42V. as AZ2to AT2; that is, as the square of the sine of the distance of the nodesfrom the syzygies to the square of the radius.28434 THE MATHEMATICAL PRINCIPLES [BOOK IIIPROPOSITION XXXII. PROBLEM XIII.Tofind the mean motion of the nodes of the moon.The yearly mean motion is the sum of all the mean horary motionsthroughout the course of the year. Suppose that the node is in N, andthat, after every hour is elapsed, it is drawn back again to its formerplace; so that, notwithstanding its proper motion, it may constantly remain in the same situation with respect to the fixed stars; while in themean time the sun S, by the motion of the earth, is seen to leave the node,and to proceed till it completes its apparent annual course by an uniform motion.Let Aa represent a given least arc, whichthe right line TS always drawn to thesun, by its intersection with the circleNA?/, describes in the least given momentof time; and the mean horary motion(from what we have above shewn) will beas AZ 2, that is (because AZ and ZY areproportional), as the rectangle of AZ into ZY. that is, as the areaAZYa ; and the sum of all the mean horary motions from the beginningwill be as the sum of all the areas oYZA, that is, as the area NAZ. Butthe greatest AZYa is equal to the rectangle of the arc Aa into the radiusof the circle ; and therefore the sum of all these rectangles in the wholecircle will be to the like sum of all the greatest rectangles as the area ofthe whole circle to the rectangle of the whole circumference into the radius, that is, as 1 to 2. But the horary motion corresponding to thatgreatest rectangle was 16" 16 " 37iv. 42V. and this motion in the completecourse of the sidereal year, 365d. 6". 9 , amounts to 39 38 7" 50", andtherefore the half thereof, 19 49 3" 55",is the mean motion of thenodes 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" 55" asthe area NAZ to the whole circle.Thus it would be if the node was after every hour drawn back again toits former place, that so, after a complete revolution, the sun at the year send would be found again in the same node which it had left when theyear begun. But, because of the motion of the node in the mean time, thesun must needs meet the node sooner ; and now it remains that we computethe abbreviation of the time Since, then, the sun, in the course of theyear, travels 360 degrees, and the node in the same time by its greatestmotion would be carried 39 > 38 7" 50 ",or 39,6355 degrees ; and the meanmotion of the node in any place N is to its mean motion in its quadraturesas AZ 2 to AT- the motion of the sun will be to the motion of the nodaBOOK III.] OF NATURAL PHILOSOPHY. 43oin N as 360AT2 to 39,6355AZ2; that is, as 9,OS27646AT 2 to AZ .Wherefore if we suppose the circumference NA/* of the whole circle to hedivided into little equal parts, such as Aa, the time in which the sun woulddescribe the little arc Aa, if the circle was quiescent, will be to the time ofwhich it would describe the same arc, supposing the circle together withthe nodes to be revolved about the centre T, reciprocally as 9,0827646AT2to 9,082764 6AT2 + AZ 2;for the time is reciprocally as the velocitywith which the little arc is described, and this velocity is the sum of thevelocities of both sun and node. If, therefore, the sector NTA representthe time in which the sun by itself, without the motion of the node, woulddescribe the arc NA, and the indefinitely small part ATa of the sectorrepresent the little moment of the time in which it would describe the leastarc Aa ; and (letting fall aY perpendicular upon N//) if in AZ we takec/Z of such length that the rectangle of dZ into ZY may be to the leastpart AT of the sector as AZ 2 to 9,OS27646AT 2 -f AZ 2, that is tosay, that dZ may be to |AZ as AT2 to 9,0827646AT 2-f AZ 2;therectangle of dZ into ZY will represent the decrement of the time arisingfrom the motion of the node, while the arc Aa is described ; and if thecurve NdGn is the locus where the point d is always found, the curvilineararea Ne/Z will be as the whole decrement of time while the whole arcNAis described; and; therefore, the excess of the sector NAT above the areaNrfZ will be as the whole time But because the motion of the node in aless time is less in proportion of the time, the area AaYZ must also be diminished in the same proportion : which may be done by taking in AZ theline eZ of such length, that it may be to the length of AZ as AZ 2 to9,OS27646AT 2 -f AZ 2; for so the rectangle of eZ into ZY will be tothe area AZYa as the decrement of the time in which the arc Aa is described to the whole time in which it would have been described, if thenode had been quiescent ; and, therefore, that rectangle will be as the decrement of the motion of the node. And if the curve NeFn is the locus ofthe point e, the whole area NeZ, which is the sum of all the decrements ofthat motion, will be as the whole decrement thereof during the time inwhich the arc AN is described ; and the remaining area N Ae will be as theremaining motion, which is the true motion of the node, during the timein which the whole arc NA is described by the joint motions of both sunand node. Now the area of the semi-circle is to the area of the figureNeFn found by the method of infinite series nearly as 793 to o-。 But themotion corresponding or proportional to the whole circle was 19 49 3"55 "; and therefore the motion corresponding- to double the figure NeF//is t 29 58" 2 ", which taken from the former motion leaves 18 19 5"53 ", the whole motion of the node witn respect to the fixed stars in theinterval between two of its conjunction? with the sun ; and this motion subducted from the annual motion of the sun 360C, leaves 341 40 54" 7 ",4o6 THE MATHEMATICAL PRINCIPLES [BOOK 111.the motion of the sun in the interval between the same conjunctions. Butas this motion is to the annual motion 360, so is the motion of the nodebut just now found 1S 19 5" 53 " to its annual motion, which will therefore be 19 IS I" 23 "; and this is the mean motion of the nodes in thesidereal year. By astronomical tables, it is 19 21 21" 50 ". The difference is less than 3- j^- part of the whole motion, and seems to arise fromthe eccentricity of the moon s orbit, and its inclination to the plane of theecliptic. By the eccentricity of this orbit the motion of the nodes is toomuch accelerated ; and, on the other hand, by the inclination of the orbit,the motion of the nodes is something retarded, and reduced to its justvelocity.PROPOSITION XXXIII. PROBLEM XIV.To find the true motion, of the nodes of the moon.In the time which is as the areaNTA NrfZ (in the preceding Fig.)that motion is as the area NAe, andis thence given ; but because the calculus is too difficult, it will be betterto use the following construction ofthe Problem. About the centre C,with any interval CD, describe the circle BEFD ; produce DC to A so asAB may be to AC as the mean motion to half the mean true motion whenthe nodes are in their quadratures (that is, as 19 18 I" 23 " to 19 49 3"55 "; and therefore BC to AC as the difference of those motions G Jl 2"32 " to the latter motion 19 49 3" 55 ", that is, as 1 to 38 T。). Thenthrough the point D draw the indefinite line Gg, touching the circle in.I); and if we take the angle BCE, or BCF, equal to the double distanceof the sun from the place of the node, as found by the mean motion, anddrawing AE or AF cutting the perpendicular DG in G, we take anotherangle which shall be to the whole motion of the node in the interval between its syzygies (that is, to 9 IV 3") as the tangent DG to the wholecircumference of the circle BED, and add this last angle (for which theangle DAG may be used) to the mean motion of the nodes, while they arepassing from the quadratures to the syzygies, and subtract it from theirmean motion while they are passing from the syzygies to the quadratures,we shall have their true motion ; for the true motion so found will nearlyagree with the true motion which comes out from assuming the times asthe area NTA NrfZ, and the motion of the node as the area NAe ;aswhoever will please to examine and make the computations will find : andthis is the semi -menstrual equation of the motion of the nodes. But thereis also a menstrual equation, but which is by no means necessary for findBOOK III.] OF NATURAL PHILOSOPHY. 437ing of the moon s latitude ;for since the variation of the inclination of themoon s orbit to the plane of the ecliptic is liable to a twofold inequality,the one semi-menstrual, the other menstrual, the menstrual inequality ofthis variation, and the menstrual equation of the nodes, so moderate andcarrect each other, that in computing the latitude of the moon both may