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X. On the Real Nature of Symbolical Algebra. By D. F. Gregory, B. A., Triii. 



Coll. Cambridge. 



(Read 7th May 1838.) 



The following attempt to investigate the real nature of Symbolical Algebra, 

 as distinguished from the various branches of analysis which come under its do- 

 minion, took its rise from certain general considerations, to which I was led in 

 following out the principle of the separ^^,tion of symbols of operation from those 

 of quantity. I cannot take it on me to say that these views are entirely new, 

 but at least I am not aware that any one has yet exhibited them in the same 

 form. At the same time, they appear to me to be important, as clearing up in a 

 considerable degree the obscm-ity which still rests on several parts of the elements 

 of symbolical algebra. Mr Peacock is, I believe, the only writer in , this country 

 who has attempted to write a system of algebra founded on a consideration of 

 general principles, for the subject is not one which has much attraction for the 

 generality of mathematicians. Much of what follows will be found to agree with 

 what he has laid down, as well as with what has been written by the Abb^ 

 BuEE and Mr Warren ; but as I think that the view I have taken of the subject 

 is more general than that which they have done, I hope that the following pages 

 will be interesting to those who pay attention to such speculations. 



The light, then, in which I would consider symbohcal algebra, is, that it is 

 the science which treats of the combination of operations defined not by their na- 

 ture, that is, by what they are or what they do, but by the laws of combination 

 to which they are subject. And as many different kinds of operations may be 

 included in a class defined in the manner I have mentioned, whatever can be 

 proved of the class generally, is necessarily true of all the operations included 

 under it. This, it may be remarked, does not arise from any analogy existing in 

 the nature of the operations, which may be totally dissimilar, but merely from 

 the fact that they are all subject to the same laws of combination. It is true 

 that these laws have been in many cases suggested (as Mj- Peacock has aptly 

 termed it) by the laws of the known operations of number ; but the step which is 

 taken from arithmetical to symbolical algebra is, that, leaving out of view the 

 nature of the operations which the symbols we use represent, we suppose the ex- 

 istence of classes of unknown operations subject to the same laws. We are thus 

 able to prove certain relations between the different classes of operations, which, 



