322 Mr DA VIES on the Equations of Loci 



the lengths of arcs in two circles are equal, the angles subtended by them 

 are inversely as their radii. Hence -^ MON = s 



sinr 



Now, cos = cos R r cos p sin R r sin p cos (% \, or 



I sin r J 



do 



sin R r sin p / 



sin r + cos 4> + cos R r cos p 

 sinR 



Again, cos p = cos cos R + r + sin (p sin R r cos (# 6), or 



y = fl + cos- 1 os ~ cos cos -- r (2.; 



sin sin R rb r 



Equating the values of x given in equations (1.) and (2.), we obtain the 

 general equation of the epicycloid ; viz. 



/, sin r , ,_i cosd) + cosRrcosp ,cosp cos <b cos R -+- r 



v 5 cos ' 2. COS 2 ' -= (3.) 



sinR + rsinp sin^sinR + r 



This equation is the only one in which the variables are separated that 

 I have been able to obtain ; but as the properties of the curve are neither 

 numerous nor important, I have not been very solicitous to protract my in- 

 quiries on the subject. Perhaps, indeed, it is well for science that this curve 

 offered so little to tempt inquiry in the early period of mathematico-philo- 

 sophical history, as it is probable that the celestial machinery of PTOLEMY 

 and TYCHO would not then have engaged the attention of astronomers till 

 a much later period than they did. The complex character of the inquiries to 

 which that system led in its geometrical details, induced astronomers to wish 

 to hope to search for some simple method of explaining the motions of 

 the heavenly bodies. The " harmony " of the elliptic, contrasted with the 

 " discord " of the epicycloid motions, speedily obtained for the former the 

 preference of reflecting minds, even before any satisfactory reason had been 

 given why it should be the true system. Indeed, the epicycloid system of- 

 fered a closer and simpler interpretation of the phenomena, so long as the 

 fundamental dogma of the Ptolemaic astronomy was admitted ; and certain- 

 ly upon the face of the inquiry this was the most obvious and natural hypo- 

 thesis. Had, therefore, the epicycloid been a curve capable of gratifying 

 KEPLER'S thirst for analogies and harmonies with respect to properties 



