102 On the Use of Fund iunal Equations 



pressed by the equation C = 4) (A, B), which is therefore true. 

 This is quite logical. It would be a waste of time to examine 

 Mr. Leslie's objections to the universality of the law of homo- 

 geneity. They must have escaped him in a moment when his 

 eagerness to put down analysis (which he seems to consider 

 synonymous with mysticism) got the better of his natural quiet 

 apprehension of intellectual truth. They will be found suffi- 

 ciently discussed in Baron Maurice's pa})er. But he brings 

 forward an objection of a different character, which probably 

 deserves greater attention than it has met with. He attempts 

 to deduce an absurdity by the employment of similar reason- 

 ing in a precisely similar case, and hence infers the fallacy of 

 the whole procedure. 



A triangle, says he, is determined by an angle and its in- 

 cluding sides ; or, these are all the variables which can enter 

 into the determination of the base. But the angle cannot en- 

 ter in virtue of the law of homogeneity. Hence c =■ <p{a, b). 

 Playfaii', Legendre, and Maurice, it is well known, have all 

 insisted in reply to this, that there exists a most obvious distinc- 

 tion betwixt the two cases. They have insisted that the dif- 

 ferent relations in which C and c stand with respect to their 

 respective standards of admeasurement, render necessarj' the 

 introduction of a specific reasoning suited to each case. It is 

 quite plain that the adoption of any linear unit is altogether 

 conventional, and the unit itself of course variable. A line 

 taken by itself, then, is of no definite magnitude. The right 

 angle, on the other hand, is a fixed and determinate quantity — 

 a quantity quite independent of every other; and consequently 

 every angle as a part of it is likewise fixed and determinate. 

 Now let us see what effect this distinction will have upon the 

 similar equations C = jp (c, A, B) 



c = \(/(C, o, b). 



There exist two reasons for dismissing c from the first of 

 these; a reason arising from the operation of the law of homo- 

 geneity, and a reason arising from its indeterrniiiateness. On 

 the other hand C is a determinate quantity ; — but does that de- 

 stroy the heterogeneity of the latter expression ? C is certainly 

 definite, but it is merely so as it is referable to an independent 

 unit. It is but a portion of that unit, and consequently still a 

 quantity of its own Jciiid. There is no other angle in \J/ with 

 which it may be connected, in order that it bear a relation to 

 the ratios of the lines a, b, c ; and the law of homogeneity 

 therefoi'e ordains its dismissal. Even allowing the distinc- 

 tion contended for, Mr. Leslie's reduction thus seems perfectly 

 valid. But the geometers I already quoted have not thought 

 it so. If the right angle, says Legendre, be termed 1, all 



angles 



