jkct. i] DISSERTATION SECOND. 31 



under this aspect, can, by the common rules of algebra, have 

 the quantities so separated, that on each side of the sign of 

 equality there shall be but one variable quantity with its flux- 

 ion ; and when this is done, the integration is reduced to one 

 of the cases already enumerated. 



When such a separation cannot be made, the problem is 

 among the most difficult which the infinitesimal analysis pre- 

 sents, at the same time that it is the key to a vast number of 

 interesting questions both in the pure and mixed mathema- 

 tics. The two Bernoullis applied themselves strenuously to 

 the elucidation of it ; and to them we owe all the best and 

 most accurate methods of resolving such questions which ap- 

 peared in the early history of the calculus, and which laid 

 the foundation of so many subsequent discoveries. This is 

 a fact which cannot be contested ; and it must be acknow- 

 ledged also, that, on the same subject, the writings of the 

 English mathematicians were then, as they continue to be at 

 this day, extremely defective. Newton, though he had 

 treated of this branch of the infinitesimal analysis with his 

 usual ingenuity and depth, had done so only in his work on 

 Fluxions, which did not see the light till several years after 

 his death, when, in 1736, it appeared in Colson's translation. 

 But that work, even had it come into the hands of the public 

 in the author's lifetime, would not have remedied the defect 

 of which I now speak. When the fluxionary equation could 

 not be integrated by the simplest and most elementary rules. 

 Newton had always recourse to approximations by infinite 

 series, in the contrivance of which he indeed displayed great 

 ingenuity and address. But an approximation, let it be ever 

 so good, and converge ever so rapidly, is always inferior to an 

 accurate and complete solution, if this last possess any tolerable 

 degree of simplicity. The series which affords the approxi- 

 mation cannot converge always, or in all states of the varia- 

 ble quantity ; and its utility, on that account, is so much limit- 



