226 THIRD REPORT — 1833. 



— , when used indepefndently, and the sign cos 6 + V — I sm d, 

 when 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 fl) a may denote 

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

 and consequently a ^ — \ will denote a line which is perpen- 

 dicular to -1- a. This interpretation admits of very extensive 

 application, and is the foimdation of many important conse- 

 quences in the application of algebra to geometry. 



The signs of operation 4- 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 + {— h) — 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 6, and cos 9' + -/ — 1 sin 9', the results will no longer de- 

 note the arithmetical (or geometrical) su?n 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 

 them. Thus, if we denote the line 

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

 angles to it by 6 V — 1 , and if we 

 complete the parallelograms AB D C 

 and AB C E, then a + b \/ - i will 

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

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



It is easily shown that a + b \/ ^^ = V'Ca^ + *^ (cos 9 



cos" ^ a 



+ »/ —\ sin 9), (where 9 = — r ^ , J , and a — b i/ — I 



a: v'(«' + b^) {cos 9 — a/^^I sin 9} ; it follows, therefore, that 

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



