298 Dr. Adamson on Geometrical Demonstrations, 



there could scarcely, I apprehend, have been found any difficulty 

 in the demonstration, and that there is no necessity for any indi- 

 rectness of procedure in it, according to the common apprehen- 

 sion of the term indirect. 



Mr. Sylvester makes an inquiry also regarding a theorem, 

 which we may, if we choose, treat as subordinate to that above 

 alluded to. The data are, that straight lines are drawn from the 

 middle of an arc, so that the portions of these lines intercepted 

 between the chord and the opposite part of the circumference 

 shall be equal. The conclusion sought to be proved is^ that the 

 portions of the lines between the chord and the arc are equal ; 

 or, which is the same thing, that the sums of both portions are 

 equal. Now here are three positions, such that any one being 

 assumed as a portion of the hypothesis, the other two become 

 easy conclusions ; any case being converse to any other. Now 

 it may be stated as a general rule, that a converse is most easily 

 and elegantly established by an indirect demonstration. The 

 correct mode in such cases therefore is, to proceed directly with 

 that demonstration which is the most easy, and to take the others 

 indirectly as its converses. These easier cases are determined by 

 this principle, viz. that the constituents, whether lines or angles, 

 whose relations are assumed as data, are, as to their position, 

 origin, or termination, connected with points, the relations of 

 which to others are settled by theorems already known. When 

 the data are in regard to origin, position, or termination, other- 

 wise dispersed, the demonstration ought to follow as a converse 

 and be indirect. But it is not necessarily indirect, or is so in 

 few cases. 



The question put by Mr. Sylvester is, whether in the case 

 above mentioned this making of it to be a converse, or treating 

 it indirectly, can be avoided. This is easily settled. The position 

 assumed as hypothesis being, that the more distant portions of 

 the straight lines drawn from the middle of the arc are equal, it 

 will be seen at a glance that the chord of either half of the arc 

 is a mean proportional between the whole straight lines and their 

 nearer portions. This necessarily and directly affords the con- 

 clusions, that these wholes and portions are equal each to each. 



From this we can show directly, that if two triangles have 

 equal bases, equal vertical angles, and equal angle-bisectors at 

 that vertex, then the portions of their bases intercepted by the 

 angle-bisectors ar(; equal each to each. From this will follow 

 directly the conclusion, that if two angle-bisectors of a triangle 

 be equal, the triangle is isosceles, which is the theorem proposed. 



If these conclusions be correct, it will follow that the test of 

 necessary indirectness in demonstration proposed by Mr. Syl- 

 vester is not applicable, as being founded on a case where that 

 property of geometrical argument does not necessarily exist. 



