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LVITI. On a simple Geometrical Problem illustrating a conjec- 

 tured Principle in the Theory of Geometrical Method. Bij J. J. 

 Sylvester, Barrisfer-ai-Law*. 



THE following theorem deserves attention as illustrating a 

 principle of geometrical method which will be presently- 

 adverted to. It is curious, also, from the fact of its solution 

 being by no means so obvious and self-evident as one would 

 expect from the extreme simplicity of its enunciation. It appeared, 

 and for the first time, it is believed, at the University of Cam- 

 bridge about a twelvemonth back, where it excited considerable 

 attention among some of the mathematicians of the place. The 

 proposition, as originally presented, was merely to prove that if 

 ABC be a triangle, and if AD and BE drawn bisecting the angles 

 at A and B and meeting the opposite sides in D and E be equal, 



then the triangle must be isosceles. It is particularly noticeable 

 that all the geometrical demonstrations yet given of this theorem 

 are indirect. Thus the first and simplest (communicated to me by 

 a promising young geometrician, Mr.B. L. Smith of Jesus College, 

 Cambridge), was the following : — Assume one of the angles at 

 DAB to be greater than the corresponding angleEBA; it can easily 

 be shown that, upon this supposition, D will be higher up from AB 

 than E ; so that if DFand EG be drawn parallel to AB, DE will be 

 abovcEG; it istheu easilyshown thatDF = AE,EG = BG,and con- 

 sequently DF and AF are each respectively less than EG and BG ; 

 and also DFA, which is the supplement of twice DAB, will be less 

 than EGB, which is the supplement of twice FBA ; from which 

 it is readily inferred, by an easy corollary to a proposition of 

 Euclid, that DA will be less than FB, whereas it should be equal 

 to it ; so that neither of the half angles at the base can be greater 

 than the other, and the triangle is proved to be isosceles. Another 

 and independent demonstration by the writer of this article is 

 less simple, but has the advantage of lending itself at once to a 

 considerable generalization of the theorem as proposed. Assu- 

 ming, as above, that DAB is greater than EBA, it is easily seen 

 that DE produced will cut BA at K on the side of it : also if 



* Communicated bv the Author. 



