34G BISHOP TEREOT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 



Ordinary algebra, however, has not provided any system of symbols by which 

 these inclined lines may be expressed, both as to length and position, but affords 

 s}Tnbols only for the two extreme cases CA and CB. This deficiency Mr Warren 

 has undertaken to supply in his Treatise on the Square Roots of Negative Quan- 

 tities, published in 1828; and has proposed a system of symbols, which, on the 

 same principle as justifies the use of - 1 as the coefficient designating the position 

 of CB, designate as coefiicients the position of all lines drawn from C, and making 

 angles with CA. 



. On some points, however, Mr Waeeen has been too sparing of his words, and 

 has thus apparently used the common symbols of algebra in a sense A'ery different 

 fi"om their ordinary acceptation. In the following paper I have endeavoured to 

 supply this deficiency of explanation ; and then to apply the system of symbols 

 so established to some important problems of gonioraetry to which, as far as I 

 know, it has not yet been applied. Dr Peacock, in his Treatises on Algebra, has 

 made a somewhat similar use of the coefficients of direction, though an-iving at 

 his conclusions by a different route. 



II. If from C (Fig. 1.) we draw any number of straight lines in the same 

 plane, such that CA, CAi, CA,,, &c., shall be continued proportionals, according to 

 EucLiDS definition; and make, at the same time, the angles ACA,, AjCA„, AjCA,, 

 &c., all equal ; then if we call CA=1 and CA, =a, CA^ will equal a', CA^=a\ and so 

 on. The several lines then are arithmetically represented as to their respective 

 lengths by the series 1, or a", a'^, a-, &c. But it is manifest that the several in- 

 dices which detennine the length of the several lines, designate, at the same time, 

 the angles which they make respectively with CA. Thus if a' makes with CA, 

 or unity, an angle a, a^ makes with CA an angle 2 3, «=> an angle 3 3, and so on. 

 And conversely the line which makes with CA an angle » 3 is properly represented 

 by a". If, instead of calling CA unity, we re- 

 present it by R or Rxl, then CAj^^R. a\ 

 CA^ =R . a", and so on. 



III. If, next, we assume that the several 

 lines CA, CAi, &c. are all equal, i. e., that they 

 are the consecutive radii of a circle making 

 equal angles with one another (as in Fig. 2.), 

 the first property, proportion, is not there- 

 by destroyed ; and we may still properly re- 

 present them (beginning with CA, ) by the 

 series a', a- . . . a". 



Now let n be a divisor of 2/-7r; or, a 

 being that angle which each line makes with 



AarA 



the succeeding, let n 3=2 ?■ tt, or 3=- 



Then from the last proposition we infer 



