REPORT ON CERTAIN BRANCHES OF ANALYSIS. 187 



general relations which exist between the speculative and physi- 

 cal sciences, but merely to point out the distinction between the 

 ultimate objects of our reasonings in the one class and in the 

 other : in the first, we merely regard the results of the science 

 itself, and the logical accuracy of the reasoning by which they 

 are deduced from assumed first principles ; and all our conclu- 

 sions possess a necessary existence, without seeking either for 

 their strict or for their approximate interpretation in the nature 

 of things : in the second, we found our reasonings equally upon 

 assumed first principles, and we equally seek for logical accu- 

 racy in the deduction of our conclusions from them ; but both 

 in the principles themselves and in the conclusions from them, 

 we look to the external world as furnishing by interpretation 

 corresponding principles and corresponding conclusions ; and 

 the physical sciences become more or less adapted to the ap- 

 plication of mathematics, in proportion to the extent to which 

 our assumed first principles can be made to approach to the 

 most simple and general facts or principles which are discovei'- 

 able in those sciences by observation or experiment, when di- 

 vested of all incidental and foreign causes of variation ; and 

 still more so, when the causes of such variation can be di- 

 stinctly pointed out, and when their extent and influence are 

 reducible to approximate at least, if not to accurate estimation. 



The first principles, therefore, which form the foundation 

 of our mathematical reasonings in the physical sciences being 

 neither arbitrary assumptions nor necessary truths, but really 

 forming part of the series of propositions of which those sci- 

 ences are composed, can never cease to be more or less the 

 subject of examination and inquiry at any point of our re- 

 searches : they form the basis of those interpretations which 

 are perpetually required to connect our mathematical with the 

 corresponding physical conclusions ; and even supposing the 

 immediate appeal to them to be superseded, as will frequently 

 be the case, by other propositions which are deducible from 

 them, they still continue to claim our attention as the proposi- 

 tions which terminate those physical and logical inquiries at 

 which our mathematical reasonings begin. But in the abstract 

 sciences of geometry and algebra, those principles which are 

 the foundation of those sciences are also the proper limits of 

 our inquiries ; for if they are in any way connected with the phy- 

 sical sciences, the connexion is arbitrary, and in no respect af- 

 fects the truth of our conclusions, which respects the evidence 

 of their connexion with the first principles only, and does not 

 require, though it may allow, the aid of physical interpretation. 



It is true that there exists a connexion between physical and 



