Page:Newton's Principia (1846).djvu/435

PH the sine of the distance of the moon from the node, and AZ the sine of the distance of the node from the sun; and the velocity of the node will be as the solid content of PK $\scriptstyle \times$ PH $\scriptstyle \times$ AZ. But PT is to PK as PM to Kk; and, therefore, because PT and PM are given, Kk will be as PK. Likewise AT is to PD as AZ to PH, and therefore PH is as the rectangle PD $\scriptstyle \times$ AZ; and, by compounding those proportions, PK $\scriptstyle \times$ PH is as the solid content Kk $\scriptstyle \times$ PD $\scriptstyle \times$ AZ, and PK $\scriptstyle \times$ PH $\scriptstyle \times$ AZ as Kk $\scriptstyle \times$ PD $\scriptstyle \times$ AZ²; that is, as the area PDdM and AZ² conjunctly.   Q.E.D.

Cor. 2. In any given position of the nodes their mean horary motion is half their horary motion in the moon's syzygies; and therefore is to 16″ 35‴ 16^iv.36^v. as the square of the sine of the distance of the nodes from the syzygies to the square of the radius, or as AZ² to AT². For if the moon, by an uniform motion, describes the semi-circle QAq, the sum of all the areas PDdM, during the time of the moon's passage from Q to M, will make up the area QMdE, terminating at the tangent QE of the circle; and by the time that the moon has arrived at the point n, that sum will make up the whole area EQAn described by the line PD: but when the moon proceeds from n to q, the line PD will fall without the circle, and describe the area nqe, terminating at the tangent qe of the circle, which area, because the nodes were before regressive, but are now progressive, must be subducted from the former area, and, being itself equal to the area QEN, will leave the semi-circle NQAn. While, therefore, the moon describes a semi-circle, the sum of all the areas PDdM will be the area of that semi-circle; and while the moon describes a complete circle, the sum of those areas will be the area of the whole circle. But the area PDdM, when the moon is in the syzygies, is the rectangle of the arc PM into the radius PT; and the sum of all the areas, every one equal to this area, in the time that the moon describes a complete circle, is the rectangle of the whole circumference into the radius of the circle; and this rectangle, being double the area of the circle, will be double the quantity of the former sum.