analytical and' geometrical Methods of Investigation, log 
signification of the symbol =. It is true that its signification 
entirely depends on definition ; but, if the definition given of it 
in elementary treatises be adhered to, I believe it will be impos- 
sible to shew the justness and legitimacy of most mathematical 
processes. It scarcely ever denotes numerical equality. In its 
general and extended meaning,' * it denotes the result of certain 
operations. Thus, when from 
x. 
z 
+ 
I 2.34,5 
X 
1-2.3 
&C. 
x ~f — b &c. nothing is affirmed ; 
1 - 2 - 3 . 
% or z 1 is inferred — „ - r 1 
1 3-2 1 5* 8 
concerning a numerical equality; and all that is to be under- 
stood is, that x -f — j- -j- See. is the result of a certain 
3- 2 
See. 
operation performed on x = z ffy + - ■■■ ; y- - - 
XIII. It appears then, that according to the reversion of 
series, z, z', z", &c. must all be represented by the same series, 
proceeding according to the powers of x ; but, if a form for % be 
required, which shall in all cases afford us a means Of numeri- 
cally computing its value, such a form must involve certain 
arbitrary quantities. These arbitrary quantities are to be deter- 
mined by conditions which depend either on the original form 
of the equation between a; and z, or on the nature of the object 
to which the calculus is applied. 
Let now f 
V 1. 
mean-f x + 
X J 
3*2 
+ 
3- r 
-j- &c. 
* This is .consistent with what I advanced in the Phil. Trans, for 1801. p. 99, con- 
cerning the meaning of the symbols x -f , &c. It is beside my present purpose, to 
insist farther on the necessity of attaching precise notions to the symbols employed in 
calculation ; and the subject deserves a separate and ample discussion. 
f It is not so. easy to prove as it; may be imagined, that f 
X ‘ 
Vi — 
x 
x dr 
3*2 
5.8 
-p See.. 
