366 Remarks on the Logic of Elementary Geometry. 



the coincidence of limits or boundaries. We cannot conceive 

 the equality of straight lines, &c. independent of the concep- 

 tion of coincident limits, or without entertaining the position 

 that the objects may be conceived to coincide ; and there is, 

 therefore, nothing illogical in the idea of supraposition. In 

 fact, the method of ascertaining such truths by conceiving the 

 application of one object to another, instead of being objection- 

 able, in many cases is the most natural and advantageous. 



In regard, for instance, to the four theorems in which we 

 demonstrate the equality of certain parts of triangles, it seems 

 preferable in all of these to employ the method of supraposition 

 rather than to restrict it to one case, and then derive the other 

 three from that one ; the necessity of having recourse to this 

 method is established as clearly by adopting it in one case as by 

 doing so in all. It must be made the source of our knowledge 

 on the subject; this cannot be avoided, and it does not seem of 

 any use to avoid if it were possible. There is an advantage in 

 adopting uniformity of proceeding in all the cases, insomuch as 

 wc show more distinctly the necessary variation in the nature of 

 the result from the variation of the conditions on whicii it rests, 

 and this seems to me to prepare-the pupil for discovering in his 

 subsequent proceedings, whether one or an other of these rules 

 is most conveniently applicable. 



' In conformity with positions which I have endeavoured to 

 illustrate, I would propose to arrange a course of Elementary 

 Geometry as follows ; 



I 



