178 PROFESSOR DE MORGAN, ON THE 



of the kind, our difficulties recur in a circle; the means which we have 

 used to propound a possible method require the problem itself to be solved 

 before they can be successfully used. 



Let the object of contemplation be simple magnitude of any one kind, 

 as in the arithmetic of concrete quantity. The process which must precede 

 all others is what we call selecting one magnitude for consideration. 

 Previously to this step, we have no object under our perceptions, and 

 may write as the representative of this preceding state, and as the 

 recognition of its existence. This first magnitude we may call 1, and 

 the operation of transition from one state to the other we may denote by 



+ 1. The contemplation of simple existence, and of the possibility of ex- 

 pressing it by a spoken symbol, suggested the earliest definition of unity — 



M0NA2 ecrTi, icaO' r/v o eKaarov run bvrwv ev Xeyexai. If we represent 



our present state by (0 +1), we may consider that with respect to any other 

 possible magnitude our position is what it was when we denoted it by 0. 

 If we now denote it by 0', we may, as before, make the transition from 

 0' to 0' + 1, which implies that we have further taken into consideration a 

 new magnitude of the same amount. 



This result, (0 + 1) + 1, we may, if we please to consider it as 

 attained by one operation, signify by + 2 : and so on. Using the symbol 

 — to denote the process by which we retrace our steps, we have all 

 that is necessary to express addition and subtraction. The principle which 



1 wish here to enforce is, that addition is connected with the symbol in 

 a manner which requires us to imagine that we start from one magnitude, 

 as it were from a new 0, and renew* the process by which we passed from 

 the first to that magnitude. 



Let us now suppose that modified magnitude is under contemplation, 

 and let the simple symbol a denote a line measured in a given direction 

 from the zero point 0. In this zero of space, which admits of an infinite 

 number of positions, we seize more clearly than before that notion which, 

 as to simple magnitude, is not easily admitted as necessary, and may 

 seem rather fanciful : namely, that every magnitude attained may, as 



* Any one who doubts the justness of this fundamental position should add six to four on 

 his fingers, having previously refreshed his notions of six and four by the same process. 



