1843.] Treatment of Geometry as a branch of Analysis. 1 15 



with, and which must either be the subject of postulate, axiom or de- 

 monstration. 



6. To his striking presentation of the 32d Prop. Legendre added 

 the " plusieurs theoremes sur la proportionalite des cotes des figures," 

 yet notwithstanding the intense interest excited by the publication, 

 the violent discussions to which it gave rise, and the eminent indivi- 

 duals who enlisted themselves on one or other side, it has often appear- 

 ed to me singular, that no attempt should have been made to develope 

 the whole system of elementary geometry in a concinnous form on the 

 same principles.* Independent of its utility as an introduction to the 

 methods of analysis, the young mathematician would be benefited by 

 seeing grouped together those truths which are nearest related in 

 affinity : he would, in the language of Decandolle, have those nearest 

 in books which are nearest in the order of nature. 



7. The only principle on which it would be necessary to base such 

 an attempt would be this : that every defined geometrical figure is the 

 representation of certain invariable formulae of calculation, the numbers 

 involved in such calculations being represented by the ratios of sides, 

 angles, areas, and the other concomitants of the figure, either inter 

 se, or to any homogeneous natural constants that may exist. 



8. A triangle then considered analytically will represent a set of 

 formulae expressing the relation between its sides and angles. If 

 according to the usual notation these be a, b, c, A, B, C, we have 



F | a, b, c, A, B, C, k| =0 



The letter K, introduced into the formula, stands for the constants 

 which may be furnished by nature. There are, however, no linear con- 

 stants, but there is an angular one — the right angle ; it follows, there- 

 fore that K can only be a function of the right angle. When therefore 

 the formula assumes a numerical shape, it must be written 

 F f« a AB C\ 

 \V 7' K' K' K J 

 These are ail the ratios necessary to be taken into account as 



b . a a A A B 



- 1S - -j- r and 5* = is -^ 7? » «c. 



c c b B K K 



* While this has been passing through the press, I have met with in Lacroix an 

 allusion to "M. Corancez qui dans un Memoire fonde sur des principes analogues 

 est parvenu aux theoremes les plus importans de la Geometrie Elementaire." 



