320 UNIVERSITY OF VIRGINIA PUBLICATIONS 



real power. It is not necessary to make the assumption, as did Tannery, 

 as to the limit of sin xjx for it can be demonstrated as is shown in the sequel. 

 2. The synthetic and purely analytical foundation and development 

 of the numbers of ordinary analysis proceeds from defining a group of 

 marks 



7 + 7 + /+...+/ (1) 



as a number called an integer. The direct operations of addition, multi- 

 plication and raising to powers follows as shorthand additions resulting, 

 for integers, in the associative, distributive and commutative forms 



a + 6= 6 + a, 



a + 6 + c= a + (6 + c), 



a+ i)} ^ &)= a^h + c; (2) 



ab= ba, 



a (be) = (ab) c, 



a{b -]- c)= ab -\- ac; 



and for exponents 



(ab)e = a)"', a'^¥ = [aby. (3) 



Two fundamental postulates are now made. First, if the symbol x be 

 defined to be a number then is the chain group of elements x 



re + :);+... +.T (4) 



defined to be a number. Conversely if the group (4) be defined as a num- 

 ber then is the element x defined as a number. Second, if x be defined as 

 a number then the product group 



.T X a; X ... X a; (5) 



is defined to be a number. Conversely, if the product group (5) be defined 

 as a number then is the element x defined to be a number. 



Definition. All numbers constructed from the mark 7 under the pos- 

 tulates (4) and (5) and which are subjected to obey the distributive, associa- 

 tive, commutative and exponent laws of the integers are thus defined to be 

 integral numbers. 



The integral numbers, or complex numbers of ordinary analysis, are 

 thus founded on twelve (if we include the definition of 7 as a number) axio- 

 matic postulates which are necessary and sufficient for the complete design 

 of these numbers. 



