FOUNDATION OF ALGEBRA. 181 



<f>(x - oo h) + + (p(x - h) + <px + (p(x + h) + + <j>(x + a — \h). 



The inverse operation, or rather the operation by which <p (x + ah) is 

 obtained from 2<£(# + ah), is either 2 {<p (x + a + \h) - <p (x + ah)\, 

 or 2 <p (x + a + 1 h) — 2 cj> {x + ah), and may be symbolized either by 

 A2 <p (x + ah) or 2A <p (x + ah). 



The proper way, however, of considering this class of extensions may 

 not be as a simple explanation of technical algebra, (though it might be 

 regarded in that point of view,) but as an extension of technical algebra 

 itself, in which new explanations of the direct and inverse unit process are 

 used co-ordinately with the one already established. If we agree to signify 

 by v°, v 1 , v 2 , &c. a new progression of operations, in which the zero and its 

 processes remain subject to the usual definitions, nothing prevents us from 

 supposing that the prescribed definitions of the unit process may remain 

 true if v° be made the unit, v 2 being derived from v ' by the same train of 

 operations as v 1 from v°, and so on. Neither is it impossible that the same 

 laws of convertibility and distribution may exist between compound opera- 

 tions, in which different units are employed, as are laid down in the pre- 

 scribed definitions relatively to the different unit processes suggested by 

 simple magnitudes. 



Let v° = <px, and 



V 1 = a a (px + «i (p {x + h) + a 2 (p (x + 2h) + 



where a , a x , &c. may be functions of h, but not of x. Technical algebra may 

 be carried to its full length under these explanations, and many deve- 

 lopements may be and have been simplified by their means. It is not my 

 intention here to write a treatise on this subject : my object is, to point out 

 that the logic of each and all of these explanations is the same ; no mode of 

 arriving at any one explanation differing from that of any other in the funda- 

 mental, and what ice may call the arithmetical, part of the subject. It is certain 

 that the discovery of inverse operations is not yet complete : this must be 

 reserved until such time as the branches, which adopt length modified by 

 direction as the explanation of simple symbols, are properly connected with 

 that technical algebra, in which various unit processes are used co-ordinately 

 with the same zero process. 



