226 THIRD REPORT — 1833. 



— , when used indef^fndently, and the sign cos d + v' — 1 sin d, 

 Avhen taken in its most enlarged sense, equally originate in the 

 generalization of the operations of algebra, and are equally in- 

 dependent of any previous definitions of the meaning and extent 

 of such operations, they are also equally the object of inter- 

 pretation, and are in this respect no otherwise distinguished 

 from each other than by the greater or less facility with which 

 it can be applied to them. 



Many examples* of their consistent interpretation may be 

 pointed out in geometry as well as in other sciences : thus, if 

 + a and — a denote two equal lines whose directions are op- 

 posite to each other, then (cos 9 + -v^ — 1 sin 9) a may denote 

 an equal line, making an angle 9 with the line denoted by + a ; 

 and consequently a V — V will denote a line which is perpen- 

 dicular to + a. This interpretation admits of very extensive 

 application, and is the foundation of many important conse- 

 quences in the application of algebra to geometry. 



The signs of operation + and — may be immediately inter- 

 preted by the terms addition and subtraction, when applied to 

 unaffected symbols denoting magnitudes of the same kind : if 

 they are applied to symbols affected with the sign — , these 

 signs, and the terms used to interpret them, become convertible. 

 Thus a -\- {— b) = a — b, and a — {— b) = a -\- b; or the al- 

 gebraical sum and difference of a and — b, is equivalent to the 

 algebraical difference and sum of a and b : but if they are applied 

 to lines denoted by symbols affected by the signs cos 9 + V —I. 

 sin 9, and cos 9' + -/ — 1 sin 9', the results will no longer de- 

 note the arithmetical (or geometrical) smn and difference of the 

 lines in question, but the magnitude and position of the dia- 

 gonals of the parallelogram constructed upon them, or upon 

 lines which are equal and parallel to e^ 

 them. Thus, if we denote the hne 

 A B by a, and the h ne A C at right 

 angles to it by 6 \/ — \, and if we 

 complete the parallelograms AB D C 



and AB C E, then a 4- 6 V - 1 will 



denote the diagonal A D, and a — b \/ —\ will denote the 

 other diagonal B C, or the equal and parallel line A E. 



It is easily shown that a -^ b */ '^^ - V'(«^ + 6^ (cos 9 



+ */ ^^\ sin 9), (where 9 = ~ t ^ ,^X ^"<^ a — b >/ — \ 



as i>/{<^ + h^) {cos 9 — s/^^\ sin 9} ; it follows, therefore, that 



* Peacock's Jlgebra. chap. xii. Art. 437, 447, 448, 449. 



