VOL. XXIX.j PHILOSOPHICAL TRANSACTIONS. 143 



method, I not only perceived, after his book of Principia, and your book were 

 published ; but I also acknowledged as much in the Acta Erudit. and on other 

 occasions; for, I judged that this became my own candour as well as his merit; 

 therefore I usually call them both by the common name of Analysis infinitesi- 

 malis, which is more extensive than the Tetragonistica ; for, as Vieta's and 

 Descartes's methods are called Analysis speciosa, though here is some difference 

 between them ; so perhaps Mr. Newton's method and mine may differ in some 

 things." Here also Mr. Leibnitz allows that when Mr. Newton's principles of 

 philosophy came abroad, he understood thereby the affinity that there was 

 between the methods, and therefore called them both by the common name of 

 the infinitesimal method, and thought himself bound in candour to acknow- 

 ledge this affinity; and there is still the same obligation upon him in point of 

 candour. And besides this acknowledgment, he here gives the preference to 

 Mr. Newton's method in antiquity. For he represents that as the common 

 analysis in species was invented by Vieta and augmented by Cartes, which made 

 some differences between their methods ; so Mr. Newton's method and his own 

 might differ in some things. And then he goes on to enumerate the differ- 

 ences by which he had improved Mr. Newton's method, as we mentioned above. 

 And this subordination of his method to Mr. Newton's, which he then 

 acknowledged to Dr. Wallis, he ought still to acknowledge. 



In enumerating the differences and improvements which he had added to Mr. 

 Newton's method, he names in the second place differential equations; but the 

 letters which passed between them in the year 1676, show that Mr. Newton 

 had such equations at that time, and that Mr. Leibnitz had them not. He 

 names in the third place exponential equations ; but these equations are owing 

 to his correspondence with the English. Dr. Wallis, in the interpolation of 

 series, considered fractional and negative indices of powers. Mr. Newton in- 

 troduced into his analytical computations, the fractional, surd, negative, and inde- 

 finitive indices of powers; and in his letter of Oct. 24, 1676, represented to 

 Mr. Leibnitz, that his method extended to the resolution of affected equations 

 involving powers whose indices were fractional or surd. Mr. Leibnitz, in his 

 answer dated June 21, 1677, mutually desired Mr. Newton to tell him what he 

 thought of the resolution of equations involving powers whose indices were 

 undetermined, such as were these a:' -\- y" = xy, xf -\- y^ ■=. x -{• y. And these 

 equations he now calls exponential, and represents to the world that he was the 

 first inventor of them, and magnifies the invention as a great discovery. But 

 he has not yet made a public acknowledgment of the light which Mr. Newton 

 gave him into it, nor produced any one instance of the use that he has been 

 able to make of it where the indices of powers are fluents. And since he has 



