262 Method of Solving Quadratic and Cubic Equations. 

 and we know thai A= -^p ; wherefore 



/-r — 3-, A /mi jm A /WW 



^A-fa-=y MP~"*MP = V MP ; 



which gives us this curious result — that if PM be the real 

 root of the equation, the other two roots are 



PM 



- =-p + V ratio of WM : PM. 



If W lies to the left of NT, then WN is negative and the roots 

 are impossible. If it lies to the right of K" then the roots are 

 real ; and of course the ratio WN to PM is easy to find with 

 the tangent scale ; for it is the difference of the cotangents of 

 the angles PNM and PWN, and may be read off at once by 

 sliding the cotangent scale so as to have its centre at W and 

 its edge parallel to NP; or, in fact, in any convenient 

 manner. 



It remains only to add, that a geometrical relation can 

 always be found for the impossible parts of the roots of all 

 equations given by the form % n + m%=0. Thus, in a parabola 

 the impossible part would be represented by the horizontal 

 distance of the line PN, measured along the axis of X from 

 the tangent to the curve drawn parallel to P^N, and would 

 thus be the square root of a line, not of a ratio. 



The same instrument may be used to take square roots of 

 any numbers, so as to find the square root of the ratio above 

 mentioned, by holding the cotangent protractor so that its 

 centre is at 0, and the division on its scale corresponding to 

 the square of which the root is required is on the axis of X. 



Then we shall have r—— = cot a = division on the scale. 



: . . PM 3 



But OM = PM 3 ; .'. division on the scale shows , that 



is to say PM 2 , whence PM gives the square root required. 

 The values of PM should be written off in divisions and de- 

 cimals along the branches OP and OQ, one being negative, 

 the other positive ; and the sides of the protractor, and also 

 the axes OX, 0X X , should be marked with their proper 

 signs. 



It will be observed throughout that T have treated lines and 

 areas and ratios of lines as representing numerical values. 

 Hence the equations above given have numerical values for x ; 

 and to make all the terms of the equations of the same order, 



