THE MATHEMATICAL PRINCIPLES [BOOK III difference 11915 — 50, or 11865, the least moment thereof in the quadra tures. Therefore the areas which in equal times are described in the syzygies and quadratures are one to^the other as 11965 to 11865. And if to the least moment 11865 we add a moment which shall be to 100, the dif ference of the two former moments, as the trapezium FKCG to the triangle TCG, or, which comes to the same thing, as the square of the sine PK to the square of the radius TP (that is, as Pd to TP), the sum will represent the 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 the earth are at rest, and that the synodical revolution of the moon is finished in 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 propor tion as the time is, that is, in the proportion of 1080853 to 1000000. Upon whicli account, the whole increment, which was TTITTT parts of the mean moment, will now become TY£