ject. i] DISSERTATION SECOND. 39 



is known by the name of Taylor's Theorem, and expresses 

 the value of any function of a variable quantity in terms of 

 the successive orders of increments, whether finite or in* 

 finitely small. If any one proposition can be said to com- 

 prehend in it a whole science it is this : for from it almost 

 every truth and every method of the new analysis may be 

 deduced. It is difficult to say, whether the theorem does 

 most credit to the genius of the author, or the power of the 

 language which is capable of concentrating such a vast body 

 of knowledge in a single expression. Without an acquaint- 

 ance with algebra, it is impossible, I believe, to conceive the 

 manner in which this effect is produced. 



By means of its own intrinsic merit, and the advantageous 

 display of it made in the works now enumerated, the new 

 analysis, long before the expiration of the period of which I 

 am here treating, was firmly established all over Europe. 

 It did not, however, exist everywhere in the same condition, 

 nor under the same form ; with the British and Continental 

 mathematicians, it was referred to different origins ; it was 

 in different states of advancement ; the notation and some of 

 the fundamental ideas were also different. The authors 

 communicated little with one another, except in the way of 

 defiance or reproach ; and, from the angry or polemical tone 

 which their speculations often assumed, one could hardly 

 suppose, that they were pursuing science in one of its most 

 abstract and incorporeal forms. 



Though the algorithm employed, and the books consulted 

 on the new analysis, were different, the mathematicians of 

 Britain and of the Continent had kept pace very nearly with 

 one another during the period now treated of, except in one 

 branch, the integration of differential or of fluxional equa- 

 tions. In this, our countrymen had fallen considerably be- 

 hind, as has been already explained ; and the distance be- 

 tween them and their brethren on the Continent continued 



