492 Prof. Donkin on the Geometrical Jnler}wetation 



algebra in plane geometry are now generally understood and 

 recognized. There are some important remarks, however, to 

 be made with reference to what follows. The operand, in 

 this system of interpretation, is always a directed line, that is, 

 a straight line in a determinate direction, and with a determi- 

 nate begi7ini7ig and end. It is convenient for our present pur- 

 pose to assume that all the lines considered shall have their 

 beginnings at the same point or origin. The operand, then, 

 is a radius vector drawn from a determinate origin in a deter- 

 minate plane. The symbol + denotes the operation of turn- 

 ing this line round the origin through a whole revolution, in 

 a determinate sense, which we will assume to be contrary to 

 that of the motion of the hands of a watch. It is especially 

 important to observe, that + does not represent the operation 

 of placing the line in a determinate direction (such as that of a 

 given fixed axis), but the operation of turning it round from 

 the direction it had atjirst (which may be any whatever) till it 

 comes into the same direction again. Then ( + )*, or — , re- 

 presents a semi-revolution ; ( — )*, or, as it is commonly written, 

 V — \, 'A quarter of a revolution ; and so on. And if a be a 

 numerical symbol, then ( + )*« represents the compound ope- 

 ration of altering the length of the operand line in the ratio 

 of fl to 1, and also turning it round through an angle equal 

 to 2«7r. 



The addition and subtraction of directed lines is to be per- 

 formed, as is well known, according to the rules for the com- 

 position of forces. Thus, if a, b be two radii vectores, a-f b 

 represents another radius vector, namely the diagonal (drawn 

 from the origin) of the parallelogram constructed on them ; 

 and a — b represents a third radius vector, namely the dia- 

 gonal (from the origin) of the parallelogram of which two sides 

 are a, and a line equal to b, but in the opposite direction. 

 The symbol cos 9 + V — \ .sin 9 represents a compound opera- 

 ration, namely (1) multiplying the operand by cos 9 without 

 altering its direction ; (2) multiplying it by sin fl and turning it 

 through a right angle ; (3) adding together the two lines so 

 obtained, or taking the diagonal of the parallelogram described 

 upon them ; which diagonal is obviously equal in length to 

 the original operand, but inclined to it at an angle S, so that 

 cos 6 + V —I .sin 9 represents the operation of turning through 

 an angle 9, and is equivalent to ( + )", \f^ = 2a'ji. Nothing 

 depends, as was before observed, on the direction of the ope- 

 rand line ; but in interpreting such expressions as /> + 5', where 

 p, q are any symbols of operation, of course it is assumed 

 that both refer to the same operand. In plane geometry, the 



