ON THE TRIGONOMETRY OP THE PARABOLA. 99 



Now finding the roots of these binomial factors by the ordinary methods, 

 we shall have, since M=:sec + tan <p, 



«==(sec^+tan <p) (multiplied successively into the n roots of unity) 1 

 and (sec ^ — tan (p) (multiplied successively into the n roots of unity). J 



We are thus enabled to exhibit the 2n roots when a >1. 

 Thus, let n=3, then the equation becomes 



^8— 2 sec 00' + 1=0, 

 and 



consequently the six roots are 



and 



(sec0 + tan0)M, =^ j> 



(sec0-tan0)|^l, =^ j- 



(63) 



By the same method we may exhibit the roots when a is less than 1, or 

 a=cos 0. 



XL. We might pursue this subject very much further, but enough has 

 been done to show the analogy which exists between the trigonometry of the 

 circle and that of the parabola. As the calculus of angular magnitude has 

 always been referred to the circle as its type, so the calculus of logarithms 

 may in precisely the same way be referred to the parabola as its type. 



The obscurities which hitherto have hung over the geometrical theory 

 of logarithms are, it is hoped, now removed. It is possible to represent 

 logarithms, as elliptic integrals usually have been represented, by curves de- 

 vised to exhibit some special property only ; and accordingly such curves, 

 while they place before us the properties they have been devised to represent, 

 fail generally to carry us any further. The close analogies which connect 

 the theory of logarithms with the properties of the circle will no longer appear 

 inexplicable. 



To devise a curve that shall represent one condition of a theory, or one 

 truth of many, is easy enough. Thus, if we had first obtained by pure ana- 

 lysis all the properties of the circle without any previous conception of its 

 form, and then proceeded to find a geometrical figure which should satisfy 

 all the conditions developed in the theory, we might hit upon several geome- 

 trical curves that would satisfy some of the established conditions, though 

 not all. That all lines passing through a fixed point and terminated both 

 ways by the curve shall be bisected in that point, would be satisfied as well 

 by an ellipse or an hyperbola as by a circle. That all the lines passing 

 through this point and terminated both ways by the curve shall be equal, 

 would be satisfied as well by the cusp of a cardioide as by the centre of a 

 circle ; but no curve but the circle will fulfil all the analytical conditions of 

 the theory of the circle. 



In the same way, no curve but the parabola will satisfy all the conditions 

 of the arithmetical theory of logarithms. 



The equilateral hyperbola gives a false analogy and leads into error, because 

 to base the properties of logarithms on those of the equilateral hyperbola 

 leads to the conclusion that negative numbers have real logarithms. 



H 2 



