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 Proportion, 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‴ 37iv.42v. 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 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 a sidereal year to the mean motion of the node 19° 18′ 1″ 23⅔‴.
"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‴ 48iv. And the greatest equation of the nodes in the octants will be 1° 29′ 57″.“
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 mTl the momentaneous motion of the nodes as above. If upon Tm we let fall the perpendicular PG, and joining pG we produce it till it meet Tl in g, and join also Pg, the angle PGp will be the inclination of the moon's orbit to the plane of the ecliptic when the moon is in P; and the angle Pgp will be the inclination of the same after a small moment of time is elapsed; and therefore the angle GPg will be the momentaneous variation of the inclination. But this angle GPg is to the angle GTg as TG to PG and Pp to PG conjunctly. And, therefore, if for the moment of time we assume
