346 BISHOP TERROT 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 
symbols 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 coefficients the position of all lines drawn from C, and making 
angles with CA. 
. On some points, however, Mr Warren has been too sparing of his words, and 
has thus apparently used the common symbols of algebra in a sense very different 
from 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 goniometry 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 arriving at 
his conclusions by a different route. 
Il. If from C (Fig. 1.) we draw any number of straight lines in the same 
plane, such that CA, CA,, CA:, &c., shall be continued proportionals, according to 
Euciip’s definition; and make, at the same time, the angles ACA,, A,CA,, A,CA,, 
&c., all equal; then if we call CA=1and 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°, a1, a2, &c. But it is manifest that the several in- 
dices which determine 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 3, a? makes with CA an angle 23, a an angle 33, 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 Rx1, then CA,=R. a’, 
CA,=R. a?, and so on. 
III. If, next, we assume that the several 
lines CA, CA, &c. are all equal, 2. 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', a2... a”. 
Now let » be a divisor of 277; or, 3 
being that angle which each line makes with 
Fig. 2. 

2 te X 
the succeeding, let »3=2r 7, or ga Then from the last proposition we infer 
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