REPORT ON CERTAIN BRANCHES OF ANALYSIS, 189 



dation, and the names of the fundamental operations in one 

 science have been transferred to the other without any imme- 

 diate change of their meaning, yet it has generally been found 

 necessary subsequently to enlarge this very narrow basis of so 

 very general a science, though the reason of the necessity of 

 doing so, and the precise point at which, or the extent to which, 

 it was done, has usually been passed over without notice. The 

 science which was thus formed was perfectly abstract, in what- 

 ever manner we arrived at its fundamental conclusions ; and 

 those conclusions were the same whatever view was taken of 

 their origin, or in whatever manner they were deduced ; but a 

 serious error was committed in considering it as a science which 

 admitted of strict and rigorous demonstration, when it certainly 

 possessed no adequate principles of its own, whether assumed 

 or demonstrated, which could properly justify the character 

 which was thus given to it. 



There are, in fact, two distinct sciences, arithmetical and 

 symbolical algebra, which are closely connected with each 

 other, though the existence of one does not necessarily deter- 

 mine the existence of the other. The first of these sciences 

 would be, properly speaking, universal arithmetic : its general 

 symbols would represent numbers only ; its fundamental ope- 

 rations, and the signs used to denote them, would have the same 

 meaning as in common arithmetic ; it would reject the inde- 

 pendent use of the signs + and — , though it would recognise the 

 common rules for their incorporation, when they were preceded 

 by other quantities or symbols : the operation of subtraction 

 would be impossible when the subtrahend was greater than 

 the quantity from which it was required to be taken, and there- 

 fore the proper imjjossible quantities of such a scienee-would 

 be the negative quantities of symbolical algebra ; it would re- 

 ject also the consideration of the multiple values of simple 

 roots, as well as of the negative and impossible roots of equa- 

 tions of the second and higher degree : it is this species of al- 

 gebra which alone can be legitimately founded upon arithmetic 

 as its basis. 



Mr. Frend *, Baron Maseres, and others, about the latter 

 end of the last century, attempted to introduce arithmetical 



* The Principles of Algebra, by William Frend, 1796; and The true The- 

 ory of Equations, established on Mathematical Demonstration, 1799. The fol- 

 lowing extracts from his prefaces to these works will explain the nature of his 

 views : 



" The ideas of number are the clearest and most distinct of the human mind : 

 the acts of the mind upon them are equally simple and clear. There cannot 

 be confusion in them, unless numbers too great for the comprehension of the 



