Projectivle

Proposition 31 Problem 12

Find the horary motion of the nodes of the moon in an elliptic orbit

Let:

• Qjpmaq represent an ellipsis with the greater axis Qq, and the lesser axis ab QAqB a circle circumscribed
• T is the earth in the common centre of both
• S the sun
• p is the moon moving in this ellipsis
• pm is an arc which it describes in the least moment of time
• N and n are the nodes joined by the line Nn
• pK and mk are perpendiculars upon the axis Qq produced both ways till they meet the circle in P and M, and the line of the nodes in D and d

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 pDdm and AZ2 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 f, and 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 impulse of the same force 3IT or 3PK; and let LP and lp be produced till they meet the plane of the ecliptic in G and g, and FG and fg be joined, of which FG produced may cut pf, pg, and TQ, in c, e, and R respectively; and fg produced may cut TQ in r. Because the force 3IT or 3PK in the circle is to the force 3IT or 3pK in the ellipsis as PK to pK, or as AT to aT, 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 PYKp 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 account of the parallels Lk, PK, GR), as pl to pe, that is (because of the similar triangles plm, cpe) as lm to ce; and inversely as LM is to lm, or as FR is to cR, so is FG to ce. And therefore if fg was to ce as fy to cY, that is, as fr to cR (that is, as fr to FR and FR to cR conjunctly, that is, as fT to FT, and FG to ce conjunctly), because the ratio of FG to ce, expunged on both sides, leaves the ratios fg to FG and fT to FT, fg would be to FG as fT to FT; and, therefore, the angles which FG and fg 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 would be, if fg was to ce as fY to cY, that is, fg was equal to c e × f Y c Y {\displaystyle \scriptstyle {\frac {ce\times fY}{cY}}}.

But because of the similar triangles fgp, cep, fg is to ce as fp to cp; and therefore fg is equal to c e × f p c p {\displaystyle \scriptstyle {\frac {ce\times fp}{cp}}}; and therefore the angle which fg subtends 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 circle as this fg or c e × f p c p {\displaystyle \scriptstyle {\frac {ce\times fp}{cp}}} to the fromer fg or c e × f Y c Y {\displaystyle \scriptstyle {\frac {ce\times fY}{cY}}}, that is, as fp × {\displaystyle \scriptstyle \times } cY to fY × {\displaystyle \scriptstyle \times } cp, or as fp to fY, and cY to cp; that is, if ph parallel to TN meet FP in h, as Fh to FY and FY to FP; that is, as Fh to FP or Dp to DP, and therefore as the area Dpmd to the area DPMd.

Therefore, seeing (by Corol. 1, Prop. XXX) the latter area and AZ² conjunctly are proportional to the horary motion of the nodes in the circle, the former area and AZ² conjunctly will be proportional to the horary motion of the nodes in the ellipsis. Q.E.D.

Corollary

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 quadrature to any place m, is the area mpQEd terminated at the tangent of the ellipsis QE.

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 ellipsis 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.

Therefore, since (by Corol 2, Prop. XXX) the mean horary motion of the nodes in the circle is to 16″ 35‴ 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 to AT²; 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.

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 reciprocally 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 syzygies 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 quadratures; 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 increment of the time in the several places between the octants and quadratures, and the decrement thereof between the octants and syzygies, is in the same proportion.

But the motion of the nodes, while the moon describes the several little equal parts of its orbit, is accelerated or retarded in the duplicate proportion of the time; for that motion, while the moon describes PM, is (cæteris paribus) as ML, and ML is in the duplicate proportion of the time. Wherefore the motion of the nodes in the syzygies, 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 number 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 decrement 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 decrement 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 radius drawn to the earth was supposed to describe an area proportional to the time) was 32″ 42‴ 7iv. 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.

Therefore, this decrement is 17‴ 43iv.11v. The fourth part of which 4‴ 25iv.48v. subtracted from the mean horary motion above found, 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, 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 AZ² to AT². 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 mutually betwixt themselves as AZ² to AT²; 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‴ 37iv.42v. as AZ² to AT²; that is, as the square of the sine of the distance of the nodes from the syzygies to the square of the radius.