Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/219

 as a bae, uppose the wheel GEF to move forward, revolving about its axis, and in the mean time by its point A decribing the cycloid ALI. which done, take GK to tEe perimeter GEFG of the wheel, in the ratio of the time in which the body, proceeding from A, deribed the arc AP, to the time of a whole revolution in the ellipis. Erect the perpendicular KL meeting the cycloid in L, then LP drawn parallel to KG will meet this ellipis in P the required place of the body.

For about the centre O with the interval OA decribe the emi-circle AQB, and let LP, produced, if need be, meet the arc AQ in Q, and join SQ, OQ. Let OQ meet the arc EFG in F upon OQ let fall the perpendicular SR. The area APS is as the area AQS, that is, as the difference between the ector OQA and the triangle OQS, or as the difference of the rectangle $$\scriptstyle \frac 12 OQ \times AQ$$, and $$\scriptstyle \frac 12 OQ \times SR$$, that is, becaue $$\scriptstyle \frac 12 OQ$$ is given, as the difference between the arc AQ and the right line SR; and therefore (becaue the equality of the given ratios SR to the ine of the arc AQ, OS to OA, OA to OG, to GF, and by diviion. AQ - SR to GF - ine of the arc AQ) as GK the difference between the arc GF and the ine of the arc AQ. Q. E. D.

But ince the decription of this curve is difficult, a olution by approximation will be preferable. Firt let there be found a certain angle B which may be to an angle of 57,29578 degrees,