» 



Remarks on the Logic of Elementary Geometry. 355 



are required as introductory to the separate compartments of 

 the science may be derived. It may then leave these lemmas 

 to form separate introductions to the subjects with which they 

 are related ; or, on the other hand, it may contain some of 

 these and not others, or may contain all of them. 



The extent, therefore, is nothing more than a matter of 

 general convenience, and appears to be settled by a certain 

 common understanding that the particular branch ot the science 

 for which an elementary work should make the most special pro- 

 vision, is the mensuration of figures and solids, of which the 

 elements are planes and circles ; while other branches of the 

 science, as Trigonometry, Conic Sections, &c., have generally 

 special lemmas introductory to their subjects'. 



Such appearing to be the general understanding on the' 

 subject, there is no great use in disturbing it. Perhaps the 

 best proceeding is to fix a term or limit for the system, and to 

 construct it of those theorems which lead to that term most 

 naturally. We may, therefore, define an elementary treatise 

 on Geometry, to be a series of theorems leading to the men- 

 suration of figures, bounded by straight lines or circular arcs : 

 of which the terminating theorem should be this — " That a 

 circular circumference includes the greatest surface under a 

 perimeter of given length ;" or, if the system extend to the 

 mensuration of solids, the term will be this — " That the 

 spherical surface includes the greatest capacity within a super- 

 ficies of given extent. 



All theorems not subservient to these mensurations, ought 

 logically to be excluded, but the system may be rendered 

 more useful by attaching as appendices to the different books or 

 sections into which the system is divided, ample collections of 

 all those theorems which are of interest or are of importance 

 as introductory to other branches of Geometrical Science, as 

 Trigonometry, Conic Sections, &c. &c. 



3. The next inquiry Avhich occurs, is as to the arrangement 

 of the Theorems. 



Logical precision and accuracy in argument are indispens- 

 able J and, therefore, nothing of a merely geometrical character, 

 can be assumed as true which has not been the subject of defi- 

 nition or demonstration. But it is by no means so indispensiblc, 

 that nothing should be supposed, to be done, the method of 

 doing which has not been shown. Euclid has followed this 

 rule only thus far, that no construction should be used, the 

 method of performing which has not been shown. The ride in 

 its more general form cannot be carried into effect. Euclid 

 deviates from it in the very first of his Theorems, where he makes 

 it part of the hypothesis, that two angles arc equal. 

 Legendre's postulate is no more illogical than this, in his 



