JOURNAL 



Elisha Mitchell Scientific Society, 



ON THE FUNDAMENTAL PRINCIPIvES OF THE 

 DIFFERENTIAL CALCULUS. 



BY WILLIAM CAIN, C. E., Mem. Am. Soc. C. E. 



There are probably no students of the infinitesimal cal- 

 culus, who have seen its varied applications, that are not 

 impressed with its immense scope and power, "constitut- 

 ing, as it undoubtedly does," says Comte, "the most loftv 

 thought to which the human mind has as yet attained." 



It was not to be expected that a science of reasoning, 

 involving so many new and delicate relations between 

 infinitely small quantities, should appear perfect, in its 

 logical development, from the beginning, even with such 

 men as Newton and Leibnitz as its creators. For a longr 

 time mathematicians were more concerned in extendi nor 

 the usefulness of the transcendental analysis than in "rig- 

 orously establishing the logical bases of its operations," 

 though it has given rise at all times to a great deal of con- 

 troversy, which has been of. great aid to those geometers 

 who concerned themselves particularly with establishing it 

 upon a logical basis. Of this number none are more 

 prominent than the French author, Duhamel. He pro- 

 ceeded by a rigorous use of "the method of limits," whose 

 thorough comprehension he regarded as so important that 



