19 ee 
348 BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 
Here CA and CB are, by Art. 1, symbolized by +1 and —1 respectively; 
therefore CD and CE must be symbolized by +/—1 and —./—1. It is, however, 
quite optional which direction from C we consider positive, whether in the hori- 
zontal or perpendicular line. 
V. It appears from the foregoing propositions, that if a line is presented to 
9 
us under the symbol a . 17"7, we know both its length and the angle 3 which 
it makes with a given line whose coefficient of direction we assume to be unity, 
9 
and which, therefore, we symbolize by a simply. The symbol a. jae therefore, 
represents the actual transference of position in 
space which a point would undergo by moving from Fig. 3. 
the one extremity of the line to the other, as from A B 
to C (Fig. 3.). Butit is clear, also, that ifa point be 
supposed to be removed from A to B, and then from 
B to C, the actual transference in space, though not 
the distance travelled, would be the same asifthe A B,C 
transference had been direct from A to C. There- 
fore the symbol which properly represents the one transference, must be symbo- 
lically equal to the sum of the two symbols which respectively represent the 
other two transferences, or AC x its coefficient of direction = AB x its coefficient 
of direction + BC into its coefficient of direction.* 
This fundamental proposition is given by Mr Warren as a definition, That 
the sum of any two lines making an angle with one another is the diagonal cf 
their parallelogram completed. Even in this startling form, it is only the general 
assertion of a proposition, particular cases of which are admitted, when we say 
(Fig.3.) that AB, + B,;C=AC, or that AC + CB, = AB,. 
By such assertions we really mean that if a point 
moves from A to B,, and then from B, to C, the 
whole transference in space will be represented by 
the sum AB,+B,C; and that if the point moves 
from A to C, and then from C to B,, the whole 
transference is expressed by the sum AC+CB,, 
which is the same thing as the arithmetical differ- 
ence AC—B,C. 
As examples to elucidate this proposition, let 
us take (Fig. 4.) an isosceles right-angled triangle 
Fig. 4, 

* This appears to be the view taken by Sir W. Hamizron, in the first of his series of papers on 
Symbolical Geometry, printed in the Cambridge and Dublin Mathematical Journal. He there says, — 
“ This symbolic sum of lines represents the total (or final) effect of all those successive rectilineal mo- — 
tions, or translations in space, which are represented by the several summands.” 

bi 
