REPORT ON CJERTAIN BRANCHES OF ANALYSIS. 198 



a sufficient basis for symbolical algebra considered under its 

 most general form ; that symbols, considered as representing 

 numbers, may represent every kind of concrete magnitude ; 



the indifference of the order of succession of different algebraical operations, 

 as so many theorems founded upon the ordinary principles and reasonings of 

 arithmetic. In order to show, however, the extraordinary vagueness of the 

 reasoning which is employed to establish these theorems, we will notice some 

 of them in detail : On repvesente, says he, les grandeurs qui doivent servir d'ac- 

 croissements, par des nomhres precedes du signs +. <?' l^s grandeurs qui doivent 

 servir de diminutions par des nomhres precedes du signe — . Cela pose, les signes 

 -\- et — places devant les nombres peuvent itre compares, suivani la remarque 

 qui en a He faite^, a. des adjectifs place's aupres de leurs suhstantifs. It is unques- 

 tionable, however, that in the most common cases of the interpretation of 

 specific magnitudes affected with the signs + and — , there is no direct refer- 

 ence either to increase or diminution, to addition or to subtraction. He sub- 

 sequently gives those signs a conventional interpretation, as denoting quan- 

 tities which are opposed to each other ; and assuming the existence of quan- 

 tities affected by independent signs, and denoting + A by a, and — A by 6, 

 he savs that 



4-«=-|-A + h=z — K 



— a = — A — b = + A; 



and therefore, 



-f(+A) = + A +(_A) = -A 



- (-h A) = - A - (- A) = -h A ; 



which he considers as a sufficient proof of the rule of the concurrence of 

 signs in whatever operations they may occur ; though it requires a very slight 

 examination of this process of reasoning to show that it involves several ar- 

 bitrary assumptions and interpretations which may or may not be consistent 

 with each other. In the proofs which he has given of the other fundamental 

 theorems which we have mentioned above, we shall find many other instances 

 of similar confusion both in language and in reasoning : thus, " subtraction 

 is the inverse of addition in arithmetic ; then therefore, also, subtraction is 

 the inverse of addition in algebra, even when applied to quantities affected 

 with the signs -\- and — , and whatever those quantities may be." But is 

 this a conclusion or an assumption? or in what manner can we explain in 

 words the process which the mind follows in effecting such a deduction ? 

 " If a and h be whole numbers, it may be proved that a 6 is identical with 

 h a : therefore, a b is identical with b a, whatever a and b may denote, and 

 whatever may be the interpretation of the operation which connects them." 

 But any attempt to establish this conclusion, without a previous definition 

 of the meaning of the operation of multiplication when applied to such quanti- 

 ties, will show it to be altogether impracticable. The system which he has fol- 

 lowed, not merely in the establishment of the fundamental operations, but 

 likewise in the interpretation of what he terms symbolical expressions and 

 symbolical equations, requires the introduction of new conventions, which are not 

 the less arbitrary because they are rendered necessary for the purpose of 

 making the results of the science consistent with each other : some of those 

 conventions I believe to be necessary, and others not ; but in almost every in- 

 stance I should consider them introduced at the wrong place, and more or 

 less inconsistently with the professed grounds upon which the science is 

 founded. 



' By Buee in the Philosophical Transactions, 1806. 



1833. o 



