6 JOURNAL OF THE 



he devoted the first half of his Differential Calculus to its 

 numerous applications. 



In the United States, after the appearance of Bledsoe's 

 "Philosophy of Mathematics" in 1867, calling especial 

 attention to Duhamel's elegant treatment and contrasting 

 it with the false logic of various other schools, there have 

 appeared a few good elementary books, nearly free from 

 errors, though sometimes showing a trace of them; thus 

 illustrating the tenacious grip of errors induced by early 

 vicious training. With these fairly good books have ap- 

 peared some as bad as have ever been written, from a log- 

 ical stand-point, as well as others, where ingenious soph- 

 istry has done its utmost to try and blind the student (and 

 possibly the author) to the false logic involved. The 

 English as a rule have followed in the lead of Newton, 

 perpetuating his error that a variable can reach its limit, 

 and they have occasionally introduced a number of errors 

 from the Leibnitz school, whose teachings still pervade most 

 of Germany, the place of its birth. 



If the above is true as to the persistent perpetuation of 

 false logic in the treatment of the first principles of the 

 calculus, it would seem that no apology was needed for a 

 critical review of those first principles, particularly as no 

 matter what school is followed in learning the calculus the 

 scientific student will be sure to come across the teachings 

 of various schools in the applications and thus should be 

 prepared to take them at their true worth and modify them 

 in statement or otherwise when necessary. 



Although a good deal of old ground is gone over, it was 

 essential to do so to bring out the points criticized in strong 

 relief The grouping of subjects is intended to be such as 

 to enable the beginner in the calculus to see at once its 

 truth and to catch on to its true spirit. The methods of 

 Newton and of Leibnitz, with criticism, is given in fine 

 print to avoid confusion, and can be omitted the first read- 

 ing, without detriment to the rest, if preferred. 



