INFINITELY SMALL QUANTITIES. 115 



thirty years later, in an admirable article ' On Berkeley's 1842. 

 Life and Writings : ' l 



It is difficult to read without parti pr is 'The Analyst,' and 

 the admirable rejoinder to its assailants, entitled 'A Defence 

 of Free-thinking in Mathematics,' and not to admit that 

 Berkeley made out his case. It was not until later that the 

 Differential Calculus was placed on the foundation it now stands ^j j ' * 

 on the conception of a limit, which is the true basis of all Differential 

 reasoning respecting infinitely small quantities, and properly Calculus - 

 apprehended, frees the doctrine from Berkeley's objections. 

 Nevertheless, so deeply did those objections go into the heart of 

 the subject, that even after the false theory had been given up, 

 the true one was not (so far as we are aware) worked out com- 

 pletely in language open to no philosophical objection by any 

 one who preceded the late Professor De Morgan, who combined 

 with the attainments of a mathematician those of a philosopher, 

 logician, and psychologist. Though whoever had mastered the 

 idea of a limit could see, in a general way, that it was adequate 

 to the solution of all difficulties, the puzzle arising from the 

 conception of different orders of differentials quantities infinitely 

 small, yet infinitely greater than other infinitely small quantities 

 had not (to our knowledge) been thoroughly cleared up, and 

 the meaning that lies under those mysterious expressions 

 brought into the full light of reason by any one before Mr. De 

 Morgan. 



My husband died shortly before this was written. He 



* Philosophy. 



had, as his letters show, a sincere respect and regard for 

 the writer, though they had met only on one occasion, and 

 lie had corresponded with Mr. James Mill, his father. But 

 though, as was truly said, his mathematical reasoning 

 had deprived Berkeley of an argument drawn from the 

 mystery of infinitely small numbers, his sympathies were 

 in many ways more on the side of Bishop Berkeley than 

 on that of Mr. Mill. 2 The works of Berkeley had been, as 



1 Fortnightly Review, Nov. 1871. 



2 I am aware that the principles of Berkeley's philosophy have 

 been found by some thinkers to lead to a pantheistic materialism. 

 Much depends upon words, but more on the minds of those who use 

 the n, and a spiritual pantheism must be a near approach to truth. 

 When the words spiritual, material, theistic, pantheistic, and atheistic 



i 2 



