XIV. A proof of the Existence of a Root in every Algebraic Equation: 

 with an examination and extension of Cauchy's Theorem on Imaginary 

 Roots, and Remarks on the Proofs of the existence of Roots given by Argand 

 and by Mourey. By Augustus De Mokgan, F.R.A.S., of Trinity College, 

 Professor of Mathematics in University College, London. 



[Read Dec. 7, 1857.] 



To those teachers who value the logic of mathematics it has always been a subject of 

 regret that the fundamental proposition of the theory of equations — every algebraical equation 

 has as many roots as dimensions, and no more — is either to be taken on trust, or deferred to 

 a late period of the course. Every such proceeding is, in mathematics, a confession of 

 incompetency, either in the state of the subject or in the teacher. This confession I have 

 until now been obliged to make by deferring the proof of the theorem until it can be deduced 

 from Cauchy's theorem on the limits of imaginary roots, a theorem which incidentally brings 

 out the existence of the roots. Having been recently led to examine the first* of Sturm's 

 demonstrations of this theorem, in the first volume of Liouville's Journal, it struck me, from 

 the very fundamental character of this proof, that there must be some equally fundamental 

 demonstration of the existence of the roots, which would be the natural prefix to Sturm's 

 demonstration. Attentive examination proved my conjecture to be correct ; and at the same 

 time I found an addition to Cauchy's theorem, which makes it include roots derived from the 

 circuit itself, and also roots of the reciprocal of the function in hand. This I shall incorporate 

 with Sturm's proof in the present paper : joining with it the consideration of Argand's and 

 Mourey's proofs, which have points worthy of particular attention. 



The proof which I prefix to Sturm's demonstration depends upon a preliminary theorem, 

 which is one of combination and position. It takes no account of the meaning of 0, « , +, — , 

 but only postulates that + and - shall be separated either by or by « . All changes con- 

 sistent with this condition are to be held allowable. Then + + may become + + : but 

 + — must not become + — . Either or oo may open; that is, may become — 0, or 

 + 0, or [+ - 00 + -] &c. Again + may become + + or + oo + ; and so on. And 

 and 00 may come together, and either cross each other or recede from each other without 

 crossing ; having, after crossing or recession, either the same sign between them as before, or 

 a different sign. 



Theobem. In any number of signs, each of which is + or -, interspersed with the signs 

 and 00 , in any manner which satisfies the condition that either or oo always comes be- 

 tween + and — and between — and +, let k be the number of occurrences of + — » 

 and I the number of occurrences of — +. Then it is impossible that k —I should undergo 



* I mean the first by Sturm alone : the first in the memoir cited is by Sturm and Liouville jointly. 



