1 72 On the Symbol V — I in Geometry. 



world. To this, therefore, I shall confine my remarks at the 

 present moment, and they shall be as brief as possible. 



A paper by Foncenex in the Turin Memoircs (if my me- 

 mory be correct, for 1778), is devoted to a discussion of the 

 notion of this symbol which is now so frequently adopted — 

 that it is the symbol of perpendicularity. Who the author 

 was whose views Foncenex combated, he does not state; but 

 very likely it was to the Abbe Sauri that he referred. The 

 doctrine was again broached by the emigrant priest Buee, in 

 the Philosophical Transactions for 1806 ; and "it was extended 

 shortly after by Francais so as to include essentially all the 

 interpretations which I have yet seen put upon it, in the An- 

 nates dcs Mathematiques. English speculatists should read 

 more than they usually do. 



I will not, of course, adopt the logical fallacy of undertaking 

 <c to prove a negative," but I may broadly state as my convic- 

 tion, that the appearance of the symbol \/ — 1 expresses im~ 

 'possibility r , and nothing more. Such a conclusion is, of course, 

 only inductive, but founded on this fact, that I have never met 

 with a case which was not in strict conformity with it, or 

 easily reducible to such a one. 



It is admitted on all hands, that when the data of a problem 

 are incompatible with each other, the quassitum of the problem 

 will always be of the form « + /3 V — 1 ; and conversely, that 

 whenever this form appears in the solution, the data of the 

 problem are incompatible with each other. This is not ex- 

 clusively the case in geometrical problems, but appertains to 

 all algebraic researches indiscriminately, whatever be the 

 subject-matter of the inquiry. The principle is fundamental 

 and universal. 



In algebra, the sole difference between a theorem and a 

 problem is this : — that in the theorem the equivalence of two 

 different expressions (or functions of different forms) is af- 

 firmed, whilst in the problem one side of that equality, subject 

 to assigned conditions, is demanded. The processes of solving 

 the proposition either as a theorem or a problem, have not the 

 slightest logical difference of character. Thus we may pro- 

 pose as a theorem to be proved, that a; being indeterminate, 



, A* A 2 * 2 



where A = {a — 1) - — (a - l) 2 + ... ; 



or we may propose to expand a* in a series of integer positive 

 powers of x. The actual process of investigation will be the 



