Remarh, on the Logic of Elementary Geometry. 365 



what needs denionstiation ; and the rule as depending on this 

 character is by no means generally observed. Perhaps the fol- 

 lowing is better, viz: that a corollary is a truth which is derived 

 from a theorem by a syllogistic statement only ; which implies 

 that no new truth is introdnced into the hypothesis, to be com- 

 bined with those existing in it; but that the conditions consti- 

 tuting the hypothesis or conclusion, vary in their relations among 

 themselves. Thns having demonstrated that the three interior 

 angles of a triangle are equal to two right angles, it is a corol- 

 lary according to this definition, that when these angles are 

 equal each to each, one of them is two-thirds of a right angle ;. 

 but it is not a corollary, to found upon this theorem the infer- 

 ence that the exterior angles of a figure are equal to four right 

 angles ; for in this case, both the data and conchision vary ab- 

 solutely ; and the position ought to be a theorem. Under the 

 same terra "corollary" will also be included, the case where the 

 conclusion is presented to us, in its nature identical \xith the 

 assertion of the theorem but differing in expression. So also in 

 accordance with this definition, a corollary may be the state- 

 ment of a practical result, the performance of which depends 

 on the truth developed in the theorem. 



6. Our last inquiry will be respecting the methods of demon- 

 stration 



There seems to bo some misconception in regard to this sub- 

 ject also, which it is of importance to correct. We may make 

 the demonstration of most theorems to be direct or indirect just 

 as we choose — there is involved in this distinction nothing more 

 than a difference of language ; and, therefore, there is logically 

 no reason of preference for the one beyond the other. Wc may 

 either make the truth we are demonstrating, to be an undeniable 

 inference from things demonstrated before, or we may shew the 

 denial of our position to be contradictory to truths demonstrated 

 before- That one of the two methods which is shortest and 

 most distinct in any given case is to be prefered, and there 

 ought to be no other ground of preference. In one series of 

 cases the indirect method is distinctly preferable, and that is, — 

 whenever the converse of a theorem is to be deduced from it. 

 In demonstrating- those also, which arc derived directly from 

 definitions, the object being to show that denying the theorem is 

 contradictory to the definition, recourse must in general be had 

 to the indirect method. 



Our whole decisions in regard J;o the ratio of quantities de- 

 pend, as I have said, on the establishment of ratios of equality 

 in certain cases, and, therefore, our method of proceeding ia 

 all cases will ultimately depend on the nature of those concep- 

 tions by which equality is suggested or ascertained. Now, ia 

 regard to geometrical niagnitudcs; this conception involves in it 



