MR GREGORY ON THE REAL NATURE OF SYMBOLICAL ALGEBRA. 209 



when expressed between the symbols, are called algebraical theorems. And if 

 we can show that any operations in any science are subject to the same laws of 

 combination as these classes, the theorems are true of these as included in the 

 general case : Provided always, that the resulting combinations are all possible 

 in the particular operation under consideration. For it may very well, and does 

 actually happen, that, though each of two operations in a certain branch of science 

 may be possible, the complex operation resulting from their combination is not 

 equally possible. In such a case, the result is inapplicable to that branch of 

 science. Hence we find, that one family of a class of operations may have a more 

 general application than another family of the same class. To make my mean- 

 ing more precise, I shall proceed to apply the principle I have been endeavouring 

 to explain, by shewing what are the laws appropriate to the different classes of 

 operations we are in the habit of using. 



Let us take as usual F and /to represent any operations whatever, the na- 

 tures of which are unknown, and let us prefix these symbols to any other S3Tn- 

 bols, on which we wish to indicate that the operation represented by F or/ is to 

 be performed. 



I. We assume, then, the existence of two classes of operations F and/ con- 

 nected together by the following laws. 



(1.) F F (a) = F (a). (2.) ff{a) = F (a). 



(3.) F/(a) =/(a). (4.) /F (a) =f{a). 



Now, on looking into the operations employed in arithmetic, we find that there 

 are two which are subject to the laws we have just laid down. These are the 

 operations of addition and subtraction ; and as to them the peculiar symbols of 

 + and — have been affixed, it is convenient to retain these as the symbols of the 

 general class of operations we have defined, and we shall therefore use them in- 

 stead of F and/. As it is useful to have peculiar names attached to each class, 

 I would propose to caU this the class of circulating or reproductive operations, as 

 their nature suggests. 



Again, on looking into geometry, we find two operations which are subject 

 to the same laws. The one corresponding to -i- is the turning of a line, or rather 

 transferring of a point, through a circumference ; the other corresponding to — is 

 the ti-ansference of a point through a semicircumference. Consequently, whatever 

 we are able to prove of the general symbols -I- and — from the laws to which 

 they are subject, without considering the nature of the operations they indicate, 

 is equally true of the arithmetical operations of addition and subtraction, and of 

 the geometrical operations I have described. We see clearly from this, that there 

 is no real analogy between the natm-e of the operations + and — in arithmetic 

 and geometry, as is generally supposed to be the case, for the two operations can- 

 not even be said to be opposed to each other in the latter science, as they are ge- 



VOL. XIV. PART I. D d 



