98 REPLIES TO CRITICISMS.&quot; 



cation of the sciences. Such a union as that presented in 

 their works, adopted merely for the sake of convenience, is, 

 in fact, the indication of incomplete development; and has 

 been paralleled in simpler sciences which have afterwards 

 outgrown it. Two conclusive illustrations arc at hand. The 

 name Geometry, utterly inapplicable by its meaning to the 

 science as it now exists, was applicable in that first stage 

 when its few truths were taught in preparation for land- 

 measuring and the setting-out of buildings; but, at a com 

 paratively early date, these comparatively simple truths 

 became separated from their applications, and were embodied 

 by the Greek geometers into systems of theory.* A like puri 

 fication is now taking place in another division of the science. 

 In the Geometric Descriptive of Monge, theorems were mixed 

 with their applications to projection and plan-drawing. But, 

 since his time, the science and the art have been segregating ; 

 and Descriptive Geometry, or, as it may be better termed, 

 the Geometry of Position, is now recognized by mathemati 

 cians as a far-reaching system of truths, parts of which are 

 already embodied in books that make no reference to derived 

 methods available by the architect or the engineer. To meet 

 a counter-illustration that will be cited, I may remark that 

 though, in works on Algebra intended for beginners, the 

 theories of quantitative relations, as treated algebraically, 

 are accompanied by groups of problems to be solved, the 

 subject-matters of these problems are not thereby made 

 parts of the Science of Algebra. To say that they are, is 

 to say that Algebra includes the conceptions of distances 

 and relative speeds and times, or of weights and bulks 

 and specific gravities, or of areas ploughed and days and 

 wages ; since these, and endless others, may be the terms of 



* It may be said that the mingling of problems and theorems in Euclid is not 

 quite consistent with this statement ; and it is true that we have, in this mingling, 

 a trace of the earlier form of the science. But it is to be remarked that these 

 problems are all purely abstract, and, further, that each of them admits of being 

 expressed as a theorem. 



