tlemarhs on the Locj'ic of Elementary Geometry. 361 



tlins to present to us a deduction from at least two strictly 

 geometrical hypotheses. The others may in contradistinction 

 be called axiomic theorems. In regard lo them the following 

 is the circumstance of chief interest: If there be a set of truths 

 relating to quantity generally, and such that any one may 

 serve as a hypothesis from which the others may be tbrived, 

 it is a matter of importance to ascertain the best metliod of 

 applying them to geometrical magnitudes. It is sufficient, if 

 we show that one of the series is true of a set of geometrical 

 magnitudes: such a theorem might be geometric, but the 

 tDthers relative to the same set of truths would be axiomic. It 

 is of no importance logically with which one of the series wc 

 commence. We can be influenced in our choice only by con- 

 venience and propriety of arrangement. The theorems relating 

 to proportion or the equality of two quotients from a series of 

 this character. It matters nothing whether we call this 

 selected one, by means of which the others are rendered ap- 

 plicable, a definition of proportion or a theorem, our business 

 with it ia to show that it forms a connecting link betweeci 

 quantity in the general or algebraic sense and geometrical 

 quantities, and we have to choose that one which most readily 

 connects them. 



Thus, having established the relation between algebraic pro 

 ducts and the rectangles contained by straight lines, we might; 

 infer that the containing sides have all the relations and pro- 

 perties of proportional quantities, when two rectangles are equal ; 

 and thus we have a test of the ratios of geometric magnitudes. 

 This is a method, however, which would not answer our pur- 

 pose in a natural arrangement, because it would interpose 

 relations of superficies, among those relations of lines only, 

 which ought to precede them. In the comparison of geome- 

 trical quantities prior to the introduction of ratios generally, 

 our attention is directed only to the relations of equality and 

 difference, — and the adoption of this course is indispensable, — 

 since the relations of ratios of quantities or the relations of 

 vjuotients, could not in any case be established, without establish- 

 ing relations of equality as a first step. It hence follows, that 

 that property of proportion in which wc are called to the con- 

 templation of equality or excess, is naturally the most easily 

 available. It seems to me, among the most remarkable and 

 most interesting facts in the history of geometry, that the 

 discovery of a property of this character appears to have been 

 among the earliest objects and earliest results of geometrical 

 investigation. 



It does by no means seem a very obvious course of pro- 

 ceeding tliat we should investigate the erjuality of quotients by 

 considering the equality or diiference of quantities. This, 



2.1 



