ject. i.] DISSERTATION SECOND. 33 



representing the first by real magnitudes, and enunciating 

 the second in words, could not but make a great change on 

 the nature of mathematical investigation. The language of 

 mathematicks, whatever may be its form, must always con- 

 sist of two parts ; the one denoting quantities simply, and 

 the other denoting the manner in which the quantities are 

 combined, or the operations understood to be performed 

 on them. Geometry expresses the first of these by real 

 magnitudes, or by what may be called natural signs; a line 

 by a line, an angle by an angle, an area by an area, &c. ; 

 and it describes the latter by words. Algebra, on the oth- 

 er hand, denoies both quantity, and the operations on quan- 

 tity, by the same system of conventional symbols. Thus, 

 in the expression ;c 3 — rt.r 2 -}-& 3 = o, the letters «, b, x, 

 denote quantities, but the terms x 3 , a x 2 , 8tc. denote cer- 

 tain operations performed on those quantities, as well as 

 the quantities themselves ; x 2 is the quantity ,t raised to the 

 cube; and ax 2 the same quantity x raised to the square, 

 and then multiplied into a, &c. ; the combination, by addi_ 

 tion or subtraction, being also expressed by the signs + 

 and — . 



Now, it is when applied to this latter purpose that the 

 algebraick language possesses such exclusive excellence. 

 The mere magnitudes themselves might be represented by 

 figures, as in geometry, as well as in any way whatever ; 

 but the operations they are to be subjected to, if described 

 in words, must be set before the mind slowly, and in suc- 

 cession, so that the impression is weakened, and the clear 

 apprehension rendered difficult. In the algebraick expres- 

 sion, on the other hand, so much meaning is concentrated 

 into a narrow space, and the impression made by all the 

 parts is so simultaneous, that nothing can be more favour- 

 able to the exertion of the reasoning powers, to the continu- 

 ance of their action, and their security against errour. 



