XII. " On the Laws of Operation, and the System atization of 

 Mathematics/' By ALEXANDER J. ELLIS, Esq., B.A., 

 F.C.P.S. Communicated by ARCHIBALD SMITH, Esq., 

 M.A. Received May 26, 1859. 



(Abstract.) 



The object of the following investigation is to give a firmer basis 

 to the calculus of operations, to assign the strict limits and con- 

 nexion of the mathematical sciences, and to found them upon purely 

 inductive considerations, without any metaphysical or a priori 

 reasoning. 



Starting with the indemonstrable but verifiable hypothesis, that 

 objects exist external to the subject, we recognize equality as exist- 

 ing between objects with common and peculiar properties, in respect 

 of their common properties. Operations, which, when performed on 

 equal objects, produce equal objects as their result, are recognized as 

 equal, in respect to the common properties considered in the equali- 

 ties of the objects. When one operation is performed on an object, 

 and another on the resultant object, the single operation by which the 

 first object is transformable into the last is regarded as the product 

 of the other two, the order of succession being important. When 

 the resultant object is the same as the original operand, the product 

 of the operations is termed unity. When two operations performed 

 on the same object produce different resultant objects, the operation 

 of transforming one of these resultant objects into the other, is re- 

 garded as the quotient of the two former'operations. Two opera- 

 tions are termed reciprocal when their product is unity. Hence the 

 quotient of two operations is the product of the one and of the reci- 

 procal of the other. When two objects are combined in any manner 

 so as to produce a third, and the two first are formable from any 

 fourth by two known operations, the single operation by which the 

 third object can be also formed from the fourth, is termed the same 

 combination of the two first operations. From this we gain the con- 

 ception of null or zero, as the operation of annihilating any object in 

 respect to any place. The product of a combination of two opera- 

 tions and a third operation, is the same combination of the products 



VOL. x. H 



