VOL. XXVI.] PHILOSOPHICAL TRANSACTIONS. 407 



which may be of the same length with it : hence, from the elements of curves, 

 j''^ -|- ^2 __ ^2 _|_ j^2 . ^\^Q^ \jy ^Y\Q foregoing lemma, 



(mm — ran) z + 2mnu , • (nn — mm) u + Qmnz , ^ , . , , - , , 



<t = ^^- ', , and y = '^ , ; and the mteerrals of these 



mm -f ran ^ mm + nn o 



(mm — nn) z + Qmnu , (nn— mm) u + ^mnz * • i i 



are x = ^^ ; , and y = ^^^ ; . And thus the co-or- 



mm + nn ^ mm + nn 



dinates x and y of one of the required curves become known : in like manner 

 from this one may be found a second ; from the second, a third ; and so on, as 

 far as we please, a. e. i. 



I add no examples now, as a fitter opportunity will occur hereafter, by apply- 

 ing this method to several problems of this kind, and illustrating it by examples. 

 And I have more than once so plainly pointed out the solution, that it might 

 easily have been deduced from what is subjoined to the solution of a particular 

 case of this problem, in which the proposed curve is algebraical, as exhibited 

 in the Philos. Trans. N° 289. So that Mr. John Bernoulli, the learned pro- 

 poser of the problem, may see that the solution is obtained from the common 

 rules of the inverse method of fluxions, though he insinuated in his private 

 letters to Dr. Cheyne, that it could not be exhibited by our theorems in the 

 Philos. Trans. N° 284. And as I perceive from the Leipsic Acts of August 

 J 705, that our solution did not please that learned man, though more than suf- 

 ficient to the purpose, I have therefore published the foregoing solution, which 

 cannot be liable to any objection. 



I shall now notice some things which I cannot approve in Mr. Bernoulli's 

 own solution of his problem. As first, that he has applied it only to alge- 

 braical curves. Secondly, that it is mechanical, and depends wholly on what 

 he calls creeping motion. As to Mr. Huygens, he is deserving of immortal 

 honour, for his invention of the motion of evolution, from whence both he 

 himself and others have derived curious geometrical theorems. But neither 

 Leibnitz's motion of traction, nor Bernoulli's creeping motion, will ever be 

 comparable to Huygens's motion of evolution, till those ingenious men, as 

 Huygens has done, shall reduce the curves generated by their motions to the 

 laws of geometry. But since neither of them has yet performed this, the solu- 

 tion of problems, depending on curves produced by their motions, can only be 

 reputed as mechanical. 



An Account of a New Island raised near Sant-Erini in the Archipelago. By 

 Dr. W. Sherard, Consul at Smyrna, &c. N°314, p. 67. 



On the 12th of May, 1707, an island began to rise up, a musket-shot distant 

 from the island of Sant-Erini, which continually increasing from day to day in 



