114 MEMOIR OF AUGUSTUS DE MORGAN. 



1842. defective demonstration of the possibility of expanding 

 <f> (x + h) in whole powers of h, had taken deep root in 

 elementary works ; it was the sacrifice of the clear and 

 indubitable principle of limits to a phantom, the idea that 

 an Algebra without limits was purer than one in which 

 that notion was introduced. But, independently of the 

 idea of limit being absolutely necessary even to the proper 

 conception of a convergent series, it must have been 

 obvious enough to Lagrange himself that all application 

 of the science to concrete magnitude, even on his own 

 system, required the theory of limits. Some time after 

 the publication of the first numbers of this work, four 

 different treatises appeared in the French language, all of 

 which rejected the doctrine of series, and adopted that of 

 limits. I have, therefore, no occasion to argue further 

 against the former method, which has been thus abandoned 

 in the country which saw its birth, and will certainly lose 

 ground in England when it is no longer maintained by a 

 supply from abroad of elementary treatises written upon 

 its principles.' 



The doctrine of series in opposition to that of limits 

 was practically overthrown before the completion of the 

 work, and the new principle had engaged the attention of 

 Mathematicians. As might be expected, a volume embody- 

 ing them, important in its bearing upon metaphysical 

 as well as mathematical thought, excited great interest 

 in the minds of the few who could enter into the question. 

 Of these Dr. Whewell, who had written on it in 1838, was 

 one of the most pronounced. But with one exception the 

 ideas of cotemporary thinkers must be gathered from the 

 letters. 1 A full review of the subject, if it were within 

 my power to make it, would not be in place here, and an 

 imperfect one would be useless. But some of the bearings 

 of the principles developed in my husband's Differential 

 Calculus were thus referred to by Mr. John Stuart Mill, 



1 See next Section. 



