Mr. J. J. Sylvester on a simple Geometrical Problem, 369 



geometrical theorem, as simple a one as it is perliaps possible to 

 imagine : — " To prove that if from the middle of a circular arc 

 two chords be drawn, and u4e remoter segments of these chords 

 cut off by the line joining the end of the arc be equal, the 

 nearer segments will also be equal.'^ The analytical proof de- 

 pends upon the fact of the equation a.'^-\-fix = b'^ (where {a) is 

 the given length of each segment, and {b) the length of the chord 

 of half the given arc) having only one admissible root ; and if the 

 principle assumed or presumed to be true be valid, no other form 

 of pure geometrical demonstration than the recluctio ad ubsurdum 

 should be applicable in this case. For the converse case, where 

 the nearer segments are given equal, the reducing equation is 

 a{a-\-x)=b'^, indicating nothing to the contrary of the possibility 

 of there being a direct solution, which accordingly is easily shown 

 to exist. The indirect form of demonstration, it may be mentioned, 

 is sometimes liable to be introduced in a manner to escape notice. 

 As, for instance, if it should be taken for granted in the course of 

 an argument, that one triangle upon the same base and the same 

 side of it as another triangle, and having the same vertical angle, 

 must have its vertex lying on the same arc j this would seem to be 

 mme^/za^e/y true by virtue of the well-known theorem, that angles 

 in the same circular segment are equal, but in reality can only be 

 inferred from it indirectly by showing the impossibility of its 

 lying outside or inside the arc in question. To go one step fur- 

 ther, I believe it to be the case, that granted to be trvie all those 

 fundamental propositions in geometry which are presupposed in 

 the principles upon which the language of analytical geometry is 

 constructed, then that the reductio ad absurdum not only is of 

 necessity to be employed, but moreover in propositions of an 

 affirmative character, never need be employed except when as 

 above explained the analytical demonstration is founded on the 

 impossibility or inadmissibility of certain roots due to the degree 

 of the equation implied in the conditions of the question. If 

 this surmise turn out to be correct, we are furnished with a uni- 

 versal criterion for determining ivhen the use of the indirect method 

 of yeometrical proof should be considered valid and admissible and 

 when not^'. 



7 New Square, Lincoln's-Iun, 

 October 4, 1852. 



* If report may be believed, intellects capable of extending the bounds 

 of the planetary system and lighting \\\t new regions of the universe with 

 the torch of analysis, have been baffled by the difficulties of the elementary 

 problem stated at the outset of this j)ai)er, in consequence, it is to be ])re- 

 sumed, of seeking a form of geometrical demonstration of which the (jiies- 

 tion from its nature does not admit. If this be so, no better evidence could 

 be desired to evince the importance of such a criteriou than that suggested 

 in the text. 



Phil. Mag. S. 4. Vol. 't. No. 30. Nov. 185?. 2 B 



