424 THE MATHEMATICAL PRINCIPLES [BOOK III we are to consider the figure which the moon, while it is revolved in that ellipsis, describes iu this plane, that is to say, the figure Cpa, the several points p of which are found by assuming any point P in the ellipsis, which may represent the place of the moon, and drawing Tp equal to TP in such manner that the angle PT/? may be equal to the apparent motion of the sun from the time of the last quadrature in C ; or (which comes to the same thing) that the angle CTp may be to the angle CTP as the time of the synodic revolution of the moon to the time ot the periodic revolution thereof, or as 29'1. 12h. 44' to 27d. 7'1. 43'. If, there fore, 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 C the upper apsis of this orbit Cpa. But, by computation, I find that the difference betwixt the curvature of this orbit Cpa at the vertex a, and the curvature of a circle described about the centre T with the interval TA, is to the difference between the curvature of the ellipsis at the vertex A, and the curvature of the same circle, in the duplicate proportion of the angle CTP to the angle CTp ; and that the curvature of the ellipsis in A is to the curvature of that circle in the duplicate proportion of TA to TC ; and the curvature of that circle to the curvature of a circle described about the centre T with the interval TC as TC to TA ; but that the curvature of this last arch is to the curvature of the ellipsis in C in the duplicate pro portion of TA to TC ; and that the difference betwixt the curvature of the ellipsis in the vertex C* and the curvature of this List circle, is to the dif ference betwixt the curvature of the figure Cpa, at the vertex C, and the curvature of this same last circle, in the duplicate proportion of the angle CTp to the angle CTP ; all which proportions are easily drawn from the sines of the angles of contact, and of the differences of those angles. But, by comparing those proportions together, we find the curvature of the figure Cpa at a to be to its curvature at C as AT3,- rWoVoCT2AT to CT3 -r _i_6_8_2_4_AT2 X CT ; where the number yVYVYo represents the difference of th°e° squares of the angles CTP and CTp, applied to the square of the lesser angle CTP ; or (which is all one) the difference of the squares of the limes 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 quadra ture, the proportion now found must be the same with that proportion of the curvature of the moon's orb in the syzygies to the curvature thereof in the quadratures, which we found above. Therefore, in order to find th«