220 SCIENCE PROGRESS 



We must now at once distinguish between an operation and 

 a function. The former is the act of grouping numbers together 

 in a particular manner ; the latter is the result of such grouping 

 when it is performed. The former is a verb, while the latter 

 is a substantive ; and the proposed notation gives us the 

 power (which strangely enough we have not previously 

 possessed) of expressing any algebraic operation by itself apart 

 from the number or other subject to which it may be applied. 

 Thus, since o° = i, we may now reason directly about the 



operation a + bo -\- co l ) or the operation — — ; or the opera- 

 tion bo + sin o + log (i + o). The operation i + o is evi- 

 dently that of adding unity to the subject ; and the operation 

 bo is that of multiplying the subject by b ; and the operation 

 sin o is that of taking the sine of the subject. 



(2) An operation may be reversed, that is, may consist in 

 undoing rather than in doing. We denote this by writing a 

 minus sign as an index outside the square brackets. Thus 

 [1 -f 0] ~ denotes subtracting unity from the subject ; and 

 [bo] ~ denotes dividing the subject by b. Properly speaking, 

 there is no such thing as a negative number, any more than 

 there is a negative stone or a negative elephant. It is only 

 the operation, that is, the action we perform either in grouping 

 numbers or in measuring distances, which is negative — that 

 is, is reversed. But it is convenient to use negative numbers 

 in order to facilitate writing, so long as we clearly understand 

 that we do so only by convention. Thus for example : 



L J b Li +oj 1 — 



(3) An operation or reversed operation may be performed 

 upon another operation or reversed operation. Thus for 

 example : 



[a +bo + co'-] [p + 0] = a + bp + cp i + (b + 2cp) + co l 

 [a + bo + co 2 ] [p + o] _1 = a - bp + cp* + (b - 2cp) + co l 



[ r ^_ +0 ] [4o] [_4 ?+0 ]- , =a + 



{a + bo) 1 : -r{</o — a) =0 



(4) The performance of an operation upon itself any number 

 of times may be called operative volution, or iteration, and may 



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