REPORT ON CERTAIN BRANCHES OF ANALYSIS. 229 



swered. No further notice appears to have been taken of these 

 researches before the year 1828, when Mr. Warren's treatise on 

 the geometrical representation of the square roots of negative 

 quantities * was published. In this work Mr. Warren proposes 

 to give a geometrical representation to every species of quan- 

 tity ; and after premising definitions of addition, subtraction, 

 multiplication and division, involution and evolution, which are 

 conformable to the more enlarged sense which interpretation 

 would assign to those operations when applied to lines repre- 

 sented in position as well as in magnitude ; and after showing 

 in great detail the coincidence of the symbohcal results obtained 

 from such definitions with the ordinary results of arithmetical 

 and symbolical algebra, he proceeds to determine the meaning 

 of the diflferent symbolical roots of 1 and — 1, when applied 

 to symbols denoting lines, under almost every possible circum- 

 stance. The course which Mr. Warren has followed leads 

 almost necessarily to veiy embarrassing details, and perhaps, 

 also, to the neglect of such comprehensive propositions as can 

 only derive their authority from principles which make all the 

 results of algebra which are general in form independent of the 

 specific values and representation of the symbols : but at the 

 same time it must be allowed that his conclusions, when viewed 

 in connexion with his definitions, were demonstrably true ; a 

 character which could not be given to similar conclusions when 

 they were attempted to be derived by the mere aid of the arith- 

 metical definitions of the fundamental operations of algebra. 



This objection to the course pursued by Mr. Warren will 

 more or less apply to all attempts which are made to make the 

 previous interpretations of algebra govern the symbolical con- 

 clusions ; for though it is always possible to assign a meaning 

 to algebraical operations, and to pursue the consequences of 

 that meaning to their necessary conclusions, yet if the laws of 

 combination which lead to such conclusions are expressed 

 through the medium of general signs and symbols, they will 

 cease, when once formed, to convey the necessai-y limitations 

 of meaning which the definitions impose upon them. It is for 

 this reason that we must in all cases consider the laws of com- 

 bination of general symbols as being arbitrary and independent 

 in whatever manner suggested, and that we must make our in- 

 terpretations of the results obtained conformable to those laws, 

 and not the laws to the interpretations : it is for the same reason, 

 likewise, that our interpretations will not be necessary, though 



* A Treatise on the Geometrical Representation of the Square Roots of Ne- 

 gative Quantities, by the Rev. John Warren, M.A., Fellow and Tutor of Jesus 

 College Cambridge. J 828. 



