354 Bcmarlis on the Logic of Elementary Geometry. 



or quantity in tlie general or algebraic sense, ought to be 

 considered as claiming a place among our demonstrations. 

 Sucli relations of quantity, in the general sense, ought to be 

 considered as known and determined, as being portions of 

 another science, prior to their application to the relations of 

 extension, just as they may logically be considered as de- 

 termined and as requiring no demonstration, prior to their 

 application to the relations of chemical agents. We ought to 

 he concerned with the application alone of such truths, if they 

 enter at all into our consideration ; and this application will 

 aflbrd simply a syllogistic statement, that any relation true of 

 magnitude or quantity generally, or in its algebraic sense, is 

 true of extension and its modes or parts. 



The principle stated above ought to exclude such treaties as 

 the 5th book of Euclid and various lemmas which are some- 

 times interspersed among the theorems and sometimes included^ 

 in the demonstrations. In regard to proportions we should 

 have the application of principles alone and not the proof of 

 them. We require nothing more than a simple test of the 

 existence of equality of ratio among four geometrical quan- 

 tities. It may also be considered as excluding the first 10 

 theorems of the 2nd book of Euclid, or these are or ought to 

 become instances of applications of the relations of magnitude 

 generally to geometrical magnitudes; in which we have to 

 exhibit to the eye the occurrence of these relations as existing 

 between portions of superficies . The 5th book, in the form in 

 which it was originally given by Euclid, was probably intended 

 to be of a similar character. 



2. Such being the nature of the materials in a system, we 

 have to inquire what ought to be its extent when intended to be 

 Elementary. In regard to this it is clear, that if it were to 

 occupy the same place in science as a system of chemistry or 

 geology, it ought to include all that is known in regard to 

 extension and its modes and relations. But as the science is 

 subdivided into compartments, all of them having certain com- 

 mon principles, an lilementary work which is to be introductory 

 to ail of these compartments should contain only the principles 

 common to all. If, however, it is to be introductory to one or 

 more of these compartments in a special degree, it will extend 

 to the principles common to these. 



The extent or term therefore of an Elementary treatise on 

 Geometry is to be determined, not as has been stated, by the 

 order of those equations which express the relations it includes 

 and demonstrates, but by the nature of those subjects to which 

 it is to lead. If intended to be introductory to all the different 

 branches of Geometry, it ought to contain the truths applicable 

 to all, or those alone from which the different lemmas which 



