i92 THIRD REPORT — 1833. 



If the authors of this attempt at algebraical reform had been 

 better acquainted with the more modern results of the science, 

 they would have felt the total inadequacy of the very limited 

 science of arithmetical algebi'a to replace it ; and they would 

 probably have directed their attention to discover whether any 

 principles were necessary to be assumed, which were not neces- 

 sarily deducible as propositions from arithmetic or arithmetical 

 algebra, though they might be suggested by them. As it was, 

 however, these speculations did not receive the consideration 

 which they really merited ; and it is very possible that the 

 attempt which was made by one of their authors to connect the 

 errors in reasoning, which he attacked, as forming part only of 

 a much more extensive class to which the human mind is liable 

 from the influence of prejudice or fashion, had a tendency to 

 divert men of an enlarged acquaintance with the results of 

 algebra from such a cautious and sustained examination of them 

 as was required for their refutation, or rather for such a correc- 

 tion of them as was really necessary to establish the science of 

 algebra upon its proper basis. 



I know that it is the opinion of many persons, even amongst 

 the masters * of algebraical science, that arithmetic does supply 



identical with each other : if we extract the square root on both sides, re- 

 jecting the negative value of the square root, we get in the first case 



50 — « = 10, 

 and in the second, 



a; — 50 = 10. 

 The first of these simple equations gives us a; = 40, and the second x = 60, both 

 of which satisfy the conditions of the problem proposed : the two roots which 

 are thus obtained, strictly by means of arithmetical algebra, show that the pro- 

 blem proposed is to a certain extent indeterminate. Mr. Frend and Baron 

 Maseres contended that multiple real roots, which are always the indication 

 of a similar indetermination in the problems which lead to such equations, 

 might be obtained by arithmetical algebra alone, and that all other roots were 

 useless fictions, which could lead to no practical conclusions. But it is very easy 

 to show, that incongruous and real, as well as negative and impossible roots, 

 may equally indicate the impossibility of the problem proposed : thus, if it 

 was proposed " to find a number the double of whose square exceeds three 

 times the number itself by 5," we shall find -J- and — 1 for the roots of the 

 resulting equation, both of which equally indicate the impossibility of the pro- 

 blem proposed, if by number be meant a whole positive number. 



* Cauchy, who has enriched analysis with many important discoveries, 

 and who is justly celebrated for his almost unequalled command over its lan- 

 guage, has made it the principal object of his admirable work, entitled Cours 

 d' Analyse de I'Ecole Royale Poly technique, to meet the diflBculties which pre- 

 sent themselves in the transition from arithmetical to symbolical algebra : and 

 though he admits to the fullest extent the essential distinction between them, 

 in the ultimate form which the latter science assumes, yet he considers the 

 principles of one as deducible from those of the other, and presents the rules 

 for the concurrence and incorporation of signs ; for the inverse relation of the 

 operations called addition and subtraction, multiplication and division ; for 



