364 Remarks on the Logic of Elementary GeameiTy. 



the ratio of two homologous sides, their sides are proportional ;" 

 this assertion, however, is not the converse of the theorem, 

 but the converse really is : — If the perimeter be as any two, or 

 rather as every two corresponding sides, the sides then are pro- 

 portional : which is true. The proposition, however, can 

 scarcely be said to be a geometric theorem, at least, i: is axio- 

 mic only, or is a corollary from the theorem whatever it may 

 be, which establishes the occurrence of the relations of pro- 

 portion among geometric quantities. — 2nd, If the hypothesis of 

 a theorem has its proper character of comprizing two or more 

 geometric truths, then the converse is at least two fold. There 

 are in fact three or more truths mutually dependant, of which 

 any two may constitute a hypothesis, to afford the other as a 

 conclusion, and the truth of them all is certain, if one be 

 demonstrated. Thus in an isosceles triangle, a line may be 

 drawn so as to fulfil these conditions, viz : being perpeudicular 

 to the base, besecting the base, and besecting the vertical 

 angle. There are here four conditions mutually related, and if 

 any two be made a hypothesis, the other two arise from it as 

 conclusion ; so that we have as many theorems as are afforded 

 by combining four things, two and two, or six theorems, each 

 converse to the others, and all are true ; but none would be 

 true, if only a single one of these four conditions be assumed 

 as hypothesis ; and it is at once seen that no such assertion in 

 this case forms the proper converse in regard to any of the 

 theorems. In all cases, therefore, we must have respect to the 

 number of conditions contained in the hypothesis, and esta- 

 blished in the conclusion, before we are entitled to say — we 

 have announced the converse of the proposition. Thus when 

 we assert the fact, that two triangles having respectively equal 

 sides, have also equal angles, we cannot assume as the con- 

 verse of this: — that those having respectively equal angles have 

 also equal sides, because the fact stated as the original conclu- 

 sion in this case is only partially the conclusion, it is rather a 

 deduction or a corollary from the real conclusion than the conclu- 

 sion itself, which is, that the triangles will coincide, or are in 

 every respect equal : it is, therefore, as much the conclusion, 

 that the areas are equal, or that perpendiculars from the angles 

 respectively in each are equal, as it is, that the angles are 

 equal ; take, therefore, the whole conclusion and make it the 

 hypothesis, and we have the real converse of the theorem, and 

 it is true. 



The only thing we have now to'settle in regard to the nature 

 of theorems, is the distinction between them and the corollaries, 

 A corollary is generally understood to be that which follows 

 from a theorem, without need of farther demonstration. Now 

 these terms are somewhat iudcfinitc, as it may not be easy to say 



