BOOK lit.] OF NATURAL PHILOSOPHY. That acceleration of the moon, in its passage from the quadrature C to the conjunction A, is in every moment of time as the generating accclerative 3PK X TK force EL, that is, as —.5 . Let the time be represented by the mean motion of the moon, or (which comes to the same thing) by the angle CTP, or even by the arc CP. At right angles upon CT erect CG equal to CT ; and, supposing the quadrantal arc AC to be divided into an infinite number of equal parts P/?, &c., these parts may represent the like infinite number of the equal parts of time. Let fall pic perpendicular on CT, and draw TG meeting with KP, kp produced in F arid /; then will FK be equal to TK, and K/v be to PK as P/> to T/?, that is, in a given propor3PK X TK tion ; and therefore FK X K&, or the area FKkf, will be as - ~^pp '> that is, as EL: and compounding, the whole area GCKF will be as the sum of all the forces EL impressed upon the moon in the whole time CP ; and therefore also as the velocity generated by that sum, that is, as the ac celeration of the description of the area CTP, or as the increment of the moment thereof. The force by which the moon may in its periodic time CADB of 27'1. 7h. 43' be retained revolving about the earth in rest at the distance TP, would cause a body falling in the time CT to describe the length ^CT, and at the same time to acquire a velocity equal to that with which the moon is moved in its orbit. This appears from Cor. 9, Prop, IV., Book I. But since K.d, drawn perpendicular on TP, is but a third part of EL, and equal to the half of TP, or ML, in the octants, the force EL in the octants, where it is greatest, will exceed the force ML in the proportion of 3 to 2 ; and therefore will be to that force by which the moon in its periodic time may be retained revolving about the earth at rest as 100 to