NEWTON S PRINCIPIA. 



75 



verse axis, the focal distance, and the eccentricity, and 

 with a generating circle whose radius is the distance of 

 this perpendicular from the centre. A parallel to the cy 

 cloid s axis, at the point whose abscissa is to the periphery 

 of the generating circle in the proportion of the given 

 time to the periodic time, cuts the ellipse at the place 

 required. This solution requires a construction beside that 

 of the curve described ; but a cycloid may be described 

 which shall cut the ellipse directly at the point required. 

 If a circle is described on A B the transverse axis, and its 

 quadrant A k is cut in O, in the given ratio of the times 

 in which the elliptical area is to be cut ; and then a cycloid 

 is described, whose ordinate P M is always a fourth pro 

 portional to the arch O Q, the rectangle of the two axes 

 and the distance between the foci ; or to O Q, A B x 

 2 . C F, and 2 . C S, this cycloid cuts the ellipse in the 

 point required, P. The equation to this curve G P is simple 



enough, and the construction easy ; for the ordinate is in a 

 given proportion to the arc Q O of the quadrant. As, 

 however, an arithmetical approximation by means of series 

 is required in practice, Sir Isaac Newton gives two me 

 thods, both of great elegance and efficiency. 



It may be proper here to note the names given by astro 

 nomers to the lines and angles in ihe ellipse connected 

 mainly with the investigation of this problem. The sun 

 being in the focus S, and P the planet s place, the aphelion 



