21< DISSERTATION SECOND. [part n. 



other, was to possess the most direct access to the knowledge 

 of those properties. They observed also, that when an 

 equation of this kind was deduced from the general equation, 

 it admitted of being brought to great simplicity, and of being 

 resolved much more readily than the other. In effect, it 

 assumed the form of a simple equation ; but, in order to make 

 this deduction in the readiest and most distinct way, the in- 

 troduction of new symbols, or of a new algorithm, was 

 necessary, the invention of which could cost but little to the 

 creative genius of the men of whom I now speak. They 

 appear, as has been already shown, to have made their dis- 

 coveries separately ; — Newton first, — Leibnitz afterwards, at 

 a considerable interval, yet the earliest, by several years, in 

 communicating his discoveries to the world. 



Thus, though there had been for ages a gradual approach 

 to the new analysis, there were in that progress some great 

 and sudden advances, which elevated those who made them 

 to a much higher level than their predecessors. A great 

 number of individuals co-operated in the work ; but those 

 who seem essential, and in the direct line of advancement, 

 are Euclid, Cavalieri, Descartes, Newton and Leibnitz. If 

 any of the others had been wanting, the world would have 

 been deprived of many valuable theorems, and many colla- 

 teral improvements, but not of any general method essential 

 to the completion' of the infinitesimal analysis. 



The views, however, of this analysis, taken by the two 

 inventors, were not precisely the same. Leibnitz, consider- 

 ing the differences of the variable quantities as infinitely 

 small, conceived that he might reject the higher powers of 

 those differences without any sensible error; so that none 

 of those powers but the first remained in the differential 

 equation finally obtained. The rejection, however, of the 

 higher powers of the differentials, was liable to objection, for 

 it had the appearance of being only an approximation, and 



