ON THE AREA OF THE CYCLOID*. 



To the Editor of the Cambridge and Dublin Mathematical Journal. 



SIR, The determination of the area of the Cycloid, so 

 easily effected by modern analysis, was regarded by the geo- 

 metricians of the seventeenth century as a problem of no smalt 

 difficulty. Mersenne was the first who attempted a solution : 

 he was however unsuccessful. It was proposed by him in 

 despair to Koberval in 1628, who also failed in his attempt 

 at that time. About seven years afterwards, however, Koberval 

 overcame the difficulty, and communicated his good fortune 

 to Mersenne. 



In a letter to Descartes, Mersenne made mention of Ro- 

 berval's discovery of the area of the Cycloid as a great feat 

 in geometry, simply stating the result obtained by Roberval, 

 without giving any clue to the method. Descartes, solving 

 the problem himself with little difficulty, communicated his 

 method in reply to Mersenne, with some supercilious remarks 

 about the supposed difficulty of the problem. Fermat and other 

 mathematicians of that day exercised their ingenuity in the 

 same question. A solution of the problem by pure geometry, 

 which was some time ago communicated to me by Mr R. L. 

 Ellis of Trinity College, possesses so great a superiority over 

 any of the geometrical methods of these early mathematicians 

 which I have seen, that I think it may be acceptable to those 

 readers of your Journal who take an interest in the history 

 of mathematics. 



" The motion of the generating circle may be resolved into 

 two uniform motions, a motion of translation parallel to the 



* Cambridge and Dublin Mathematical Journal, Vol. ix. p. 263, 1854. 



