MB GREGORY ON THE REAL NATURE OF SYMBOLICAL ALGEBRA. 213 



point to a rectilinear distance, or the tracing out of a straight line, will stand for 

 the result of the operation ; and then b {a (a;)) will he the transferring of a line to 

 a given distance from its original position. In order to effect this, the line must 

 be moved parallel to itself, the effect of which will he the tracing out of a paral- 

 lelogram. The effect will be the same if we suppose a to act on b (x), since in 

 this, as in the other case, the same parallelogram wUl be traced out : that is to 

 say, 



a(b{x)) = b(a(x)) 



or a and b are commutative operations. 



The binomial theorem, the most important in symbolical algebra, is a theo- 

 rem expressing a relation between distributive and commutative operations, index 

 operations, and circulating operations. It takes cognizance of nothing in these 

 operations except the six laws of combination we have laid down, and, as we 

 shall presently shew, it holds only of functions subject to these laws. It is con- 

 sequently true of aU operations which can be shewn to be commutative and dis- 

 tributive, though apparently, from its proof, only true of the operations of nmn- 

 ber. The difficulties attending the general proof of this theorem are weU known, 

 and much thought has been bestowed on the best mode of avoiding them. The 

 principles I have been endeavouring to exhibit appear to me to shew in a very 

 clear light the correctness of Euler's veiy beautiful demonstration. Starting 

 with the theorem as proved for integer indices, which he uses as a suggestive 

 form, he assumes the existence of a series of the same form when the index is 

 fractional or negative, which may be represented by / (w). He then considers 



what wUl be the form of the product / {x)xf (x). This form must depend only 



on the laws of combination to which the different operations in the expression are 

 subject. When a; is a. distributive and commutative function, and m and n inte- 

 ger numbers,, we know that f {x)xf {x) =f ix). Now integer numbers are 



771 7/i 7ft "T" W 



one of the families of the general class of distributive and peiToutative functions ; 

 and if we actually multiplied the expressions/ {x) and/ {x) together, we should, 



even in the case of integers, make use only of the distributive and permutative 

 properties. But these properties hold true also of fractional and negative quan- 

 tities. Therefore, in their case, the form of the product must be the same as when 

 the indices are integer numbers. Hence/ (x) x/ ix) =zf ix) whether in and 



n be integer or fractional, positive or negative, or generally if m and n be distri- 

 butive and permutative functions. 



The remainder of the proof follows very readUy after this step, which is the 

 key-stone of the whole, so that I need not dwell on it longer. I wUl only say, 

 that this mode of considering the subject shews cleai-ly, that not only must the 

 quantities under the vinculum be distributive and commutative functions, but 



