356 Hemarks on the Lo(jic of Elementary Geometry. 



analytiic method of proving Umt tli« nagks m a triangle ave 

 equal to two right angles, when he rests it on the possrbvlity 

 that two triangles may exist, of which two angles are equal 

 each to each. Problems are the practice resulting from theory, 

 and shoukl be considered separately. Avhen the theory is fully 

 established. 



The only thin? \rhich the- strictest k)gic requires in regard ta 

 construction is this: — that every thing done should be hwii- 

 cated as possible ;^ this wiil generally be best effected by a 

 corollary from tte theorem on which the performance depends. 



In other respects the arrasngement ought to be of the most 

 natural order possible, using the word in the sense indicated in 

 Natural systems of Botany as distinguished from artificial 

 system. This is an exceeding advantage, both in regard to the 

 right apprehension of the subjects presented, and in regard to 

 facility of retaining and recalling the information acquired. 

 The arrangement may become to the mind a sort of formula, 

 by means of which the whole trutlis relating to any object are 

 at once presented and brought under review, and it seems even 

 better, if it were necessary, to deviate from logical perfection in 

 reasoning, so far as to prefer an inferior or more clumsy method 

 of demonstration, rather tharrto deviate from the strictness of 

 a natural arrangement. It cannot be doubted that every one 

 who uses geometrical truths as the means of foirtlier investiga- 

 tion, does form some such natural airangement to himself, 

 which really is his guide in his progress. But tlie- advantages 

 of having this originally done well and logieally, are evidently 

 great; there may be some difficulty ia managing it, but the 

 difficulty must give way before endeavours directed by this 

 conviction, — that the positions stated are true, and that the 

 reasons of their truth may always be discovered without inter- 

 ference with objects of a different character. 



In conformity with tl^ese principles, the theorems ought to 

 be arranged according to the natural relations of their subjects, 

 and not according to the methods employed in the investiga- 

 tions. Thus the application of the doctrine of proportion, as 

 it is called, ought not to be the ground of sub-division, as in 

 Euclid and many other systems. 



4. Our fourth inquiry will be, respecting the nature and 

 demonstration of Theorems. 



As the investigation of relations belonging to another science 

 ought to have no place in a system of Geometry, and as sub- 

 jects in the system ought to be arranged naturally, so ought 

 the theorems relating to one subject be kept uninterrupted by 

 any argument relating to subjects of a different character. 

 Thus the arrangement in Legendre is defective, when in Book 

 VI. Thcor. 21, while investigating the relations of solids. 



