XXV. Oil the Geometrical Representation of the Roots of Algebraic Equations. 

 By the Rev. H. Goodwin, late Fellow of Cains College, and Fellow of 

 the Cambridge Philosophical Society. 



[Read April 27, 1846.] 



1. It is usual to distinguish t]ie roots of Algebraic Equations into three classes, viz , positive, 

 negative, and imaginary or impossible. Roots of all kinds may Jiowever be included under 

 one head, by considering them as composed of a modulus and a sign of affection, that sign 



of affection being some power of - 1 : thus if a be the modulus, positive roots will be expressed by 



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(- l)" . a, negative by (- l)'.o, and imaginary by (- l)".*?, and thus we may take (— lY-o 

 as the general expression for the root of an algebraic equation, and if reasoning could be con- 

 ducted by means of such a symbol it would not be necessary to distinguish between real and 

 imaginary roots, but all would come under the same view ; and speaking quite generally we 

 may say, that the root of an algebraic equation is a quantity with the negative affection developed 

 in any degree between zero and actual minus. 



This mode of considering roots of course coincides with the ordinary mode of representing 

 the root of an equation by a (cos Q + \/ — 1 sin 0), which symbol will be real and positive 

 if 9 = 0, real and negative if Q = it, and imaginary in other cases ; but what has been said 

 appears to point out more clearly the true connexion between the different species of roots, 

 and to remove in some degree the artificial character which at first sight attaches to the 

 representation of real roots under an imaginary form. 



2. We may also bring the roots of an equation under one view geometrically ; for 

 considering the positive and negative roots only, we should represent them by setting off 

 distances in opposite directions from a given point along a given line : now instead of a line 

 passing through the point which we take as origin conceive a plane drawn through it, then 



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all the roots will be represented by lines in this plane ; for the root (- 1)" . a or a (cos d + ^/ - \ sin Q) 

 will correspond to a line of length a and which is inclined at an angle Q to the line along which 

 positive roots are measured; the conjugate root a (cos - \/— 1 sin 0) will be a line similarly 

 .situated on the oppo.site side of the positive line. 



This is no new remark, but it has not, so far as I am aware, been followed into any of 

 its consequences; reflection upon it lias led me to consider whether it might not be developed 

 into a theory which should throw some light on the nature of Algebraic Equations, that is, whether 

 it would not be possible so to represent geometrically the changes of value of a function of 

 ■V, as to throw light upon the existence of the roots of the equation /(a) = 0. 



With this view I have composed the following Memoir, and though I am not aware of 

 any practical step in the Theory of Equations which can result from my investigations, yet I 

 think they tend to throw considerable light upon existing knowledge, and to give us as it were 

 the rationale of some familiar theorems. 



