XXX 11 MATHEMATICAL INVESTIGATIONS. 



UPON MATHEMATICAL INVESTIGATIONS IN GENERAL. 



Mathematical truths may be attained in two essentially different methods 

 by synthesis or by analysis, by composition or by resolution. In synthe- 

 sis, we ascend from particular cases to general ones; in analysis, we descend 

 from general cases to particulars. By synthesis we pass from the simplest 

 or admitted truths, by combination and comparison, to more complicated 

 phenomena. Analysis seeks to refer back such phenomena to their fun- 

 damental relations, or to deduce special properties from the general con- 

 ditions. 



The analysis of a phenomenon presupposes, then, an accurate compre- 

 hension of all its elements. So far as these last stand in relations of cause 

 and effect to the whole and its parts, or so far as such relations exist be- 

 tween the parts themselves, they may be expressed by equations. Thus 

 the operations which are necessary in analysis become independent of con- 

 crete phenomena, and are governed only by the laws of abstract quantities 

 as included by algebra in the widest sense of the word. Algebra, then, is 

 not analysis itself, but only its instrument, " instrument precieux et neces- 

 saire sans doute, parce qu'il assure et facilite noire marche, mats qui n'a par 

 lui meme aucune vertu propre ; qui ne dirige point Vesprit, mais que V esprit 

 doit diriger comme tout autre instrument " (Poinsot, Theorie nouvelle de la 

 rotation, prSs a T Acad., 1834). Ordinarily the higher branches of algebra, 

 with which numberless really analytical investigations are connected, are 

 designated as analysis. More properly, all investigations which rest upon 

 equations of condition may be termed analytical investigations. 



Synthetic investigation rests mainly upon geometrical conceptions, and 

 attains to the knowledge of phenomena through concrete conditions, which 

 latter may be designated as space relations and processes. Hence the usual 

 division into analytical and geometrical methods, even in applied mathe- 

 matics. "We have thus with equal appropriateness an analytical geometry 

 as also a geometrical analysis. When pure geometry (in distinction from 

 analytical) makes use of the symbols and operations of algebra, it is only 

 to express with corresponding generality and more concisely than in words 

 truths attained to by abstraction, and independent of the dimensions of the 

 auxiliary figure ; or so to formulate such truths that they may be applied 

 in analytical investigation. Accordingly, such use of algebraic formulae 

 has as little effect upon the synthetic process as from the above it would 

 eeem essential to the analytic treatment. In either case, algebra is but the 

 instrument, the method lies back of and directs it. 



If analytical formula and operations are entirely excluded from the 

 more complicated geometrical investigations, we are at once restricted to 

 general laws of metrical relation. There remains only the faculty of 



