166 On the Application of algebraic Functions 



cos A sin B n -, . n , , 



, , &c. ; and it we suppose that the same quantities 



are changed into the equivalent values expressed in terms of 



A B 

 -Q-, — , &c. ; we shall obtain the functional equations of Le- 



gendre, between the sides of the triangle and the angular 



ABC 

 quantities - --, — , — . It must be acknowledged, however, 



that this is some abatement in the simplicity of the equations. 

 Yet it seems difficult to conceive any other way in which the 

 ratios of the angles of a triangle to a right angle, can find their 

 way into the same equations with the length of the sides. It 

 deserves to be particularly noticed that there is no trigonome- 

 trical equation corresponding to the case 



C = <Kc,A,B),_ 

 which is a mere symbol without meaning, or rather an absur- 

 dity, marking an impossibility in the course of investigation 

 pursued, and from which no certain conclusion can be drawn. 

 But it may be questioned whether, in sound philosophy, the 

 algebraic analysis can be applied to investigate a really funda- 

 mental principle in any science. Analysis treats of number, 

 and of measured magnitude, and of these alone. Some pre- 

 vious discussion founded on the peculiar nature of the mag- 

 nitudes we have to consider, seems therefore to be necessary ; 

 their elementary properties must be explored at least to a 

 certain extent; in order to enable us to compare and measure 

 them, without which they cannot be brought within the scope 

 of analysis. The excellence of the modern method of mathe- 

 matical investigation consists in reducing the first principles 

 in every case to the least number possible, not in dispensing 

 with any one that is essential. This may be illustrated by 

 referring to what so often happens in physical science. When 

 observation and experiment have failed in discovering suffi- 

 cient data, the analyst has recourse to the most probable hy- 

 potheses for obtaining the requisite number of equations. We 

 are not now to discuss the proper use of such hypothetical in- 

 vestigations in promoting physical knowledge ; our intention is 

 merely to show that without sufficient data analysis is a use- 

 less instrument. Fundamental principles there must be, either 

 legitimately obtained, or assumed hypothetically, or mixed up 

 in a latent manner in the process of investigation. Whether 

 there be any general character that distinguishes the principles 

 inherent in the nature of things, which we should in vain en- 

 deavour to deduce from a foreign source ; and whether the 

 difficulty about parallel lines in geometry do not belong to 

 that class: are points which we do not undertake to settle. 



The 



