36 DISSERTATION SECOND. [>aht u. 



an elementary treatise on the new analysis, the Quadra- 

 ture of Curves is therefore imperfect, and not calculated, 

 without great study, to give to others any portion of the 

 power which the author himself has exerted. The pro- 

 blem of finding fluents, though it be that on which the 

 whole quadrature of curves depends, is entirely kept out 

 of view, and never once proposed in the course of a 

 work, which, at the same time, is full of the most elabo- 

 rate and profound reasonings. 



Newton had a great fondness for the synthetical method, 

 which is apparent even in the most analytical of his works. 



In his Fluxions, when he is treating of the quadrature 

 of curves, he says, " After the area of a curve has been 

 found and constructed, we should -consider about the de- 

 monstration of the construction, that, laying aside all alge- 

 braical calculation, as much as may be, the theorem may 

 be adorned and made elegant, so as to become fit for 

 public view." ' This -is followed by two or three exam- 

 ples, in which the rule here given is very happily illustrated. 

 When the analysis of a problem requires, like the quad- 

 rature of curves, the use of the inverse method of flux- 

 ions, the reversiou of that analysis, or the synthetical de- 

 monstration, must proceed by the direct method, and there- 

 fore may admit of more simplicity than the others, so as, 

 in the language of the above passage, to be easily adorn- 

 ed and made elegant. 



The book of Fluxions is, however, an excellent work, en- 

 tering very deeply into the nature and spirit of the calcu- 

 lus, — illustrating its application by well chosen examples, 

 — and only failing, as already said, by having recourse, 

 for finding the fluents of fluxionary equations, too exclu- 

 sively to the method of series, without treating of the ca- 

 ses in which exact solutions can be obtained. 



1 Newton's Fluxions, Colson's Translation, p. 116, § 107. 



