George Ballard Mathews

“ This property is expressed by saying that is in close order when its elements are arranged in order of magnitude. Strange as it appears at first sight, is a countable aggregate; a theorem first proved by G. Cantor. To see this, observe that every element of R may be represented by a "reduced" couple , in which are prime to each other. If are any two reduced couples, we will agree that is anterior to if either (1) , or (2) but . This gives a new criterion by which all the elements of R can be arranged in the succession
which is similar to the natural scale. ”

“ An algebraic corpus is an aggregate, such that its elements are representable by symbols , &c., which can be combined according to the laws of ordinary algebra; every algebraic expression obtained by combining a finite number of symbols, by means of a finite chain of rational operations, being capable of interpretation as representing a definite element of the aggregate, with the single exception that division by zero is inadmissible. Since, by the laws of algebra, , and , every algebraic field contains , or, more properly, an aggregate which is an image of . ”

“ A more definite idea of number seems to have been gradually acquired by realizing the equivalence, as regards plurality, of different concrete groups, such as the fingers of the right hand and those of the left. ”

“ If are any two elements of , the equation is satisfied by putting . Thus the symbol is always interpretable as , and we may say that within subtraction is always possible; it is easily proved to be also free from ambiguity. On the other hand, is intelligible only if the absolute value of is a multiple of the absolute value of .
The aggregate has no first element and no last element. At the same time it is countable, as we see, for instance, by associating the elements with the natural numbers respectively, thus—
It is usual to write (or simply ) for and for ”

“ It can be proved that these operations are free from ambiguity and obey certain formal laws of commutation, &c., which will not be discussed here. Each of the three direct operations leads to an inverse problem which cannot be solved except under certain implied conditions. Let denote any two assigned natural numbers: then it is required to find natural numbers, such that
respectively. ”

“ The solutions, when they exist, are perfectly definite, and may be denoted by and ; but they are only possible in the first case when , in the second when is a multiple of , and in the third when a is a perfect th power. ”