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 + -/^ and - \/^. 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 



-— 

 us under the symbol a .1''' ^ 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, 



and which, therefore, we symbolize by « simply. The symbol a . 1" '' ", therefore, 

 represents the actual transference of position in 

 space which a point would undergo by moving from 

 the one extremity of the line to the other, as from A 

 to C (Fig. 3. ). But it is clear, also, that if a 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 as if the 

 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 dnection.* 



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 of 

 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.)thatAB, + BiC=AC,orthatAC + CBi=ABi. 

 By such assertions we really mean that if a point 

 moves from A to Bj, and then from B, to C, the 

 whole transference in space will be represented by 

 the sum ABi + 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 + CBi, 

 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 



* This appears to be the view taken by Sir W. Hamilton, 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." 



