MONISM IN ARITHMETIC. 13 



therefore, to comprise all these symbols under some one single sym- 

 bol, and to construct this latter symbol so that it will appear un- 

 equivocally from it by how much the number before the minus sign 

 is less than the number behind it. This difference, accordingly, is 

 written down and the minus sign placed before it. 



If the two numbers of such a differential form are equal, a totally 

 new sign must be invented for the expression of the fact, having 

 no relation to the signs which state results of counting. This in- 

 vention \vas not made by the ancient Greeks, as one might naturally 

 suppose from the high mathematical attainments of that people, but 

 by Hindu Brahman priests at the end of the fourth century after 

 Christ. The symbol which they invented they called tsip/ira, empty, 

 whence is derived the English cipher. The form of this sign has been 

 different in different times and with different peoples. But for the 

 last two or three centuries, since the symbolic language of arith- 

 metic has become thoroughly established as an international char- 

 acter, the form of the sign has been (French zero, German //#//). 



In calling this symbol and the symbols formed of a minus sign 

 followed by a result of counting, numbers, we widen the province of 

 numbers, which before was wholly limited to results of counting. 

 In no other way can zero and the negative numbers be introduced 

 into arithmetic. No man can prove that 7 minus 1 1 is equal to I 

 minus 5. Originally, both are meaningless symbols. And not until 

 we agree to impart to them a significance which allows us to reckon 

 with them as we reckon with real differences are we led to a state- 

 ment of identity between 7 minus n and i minus 5. It was a long 

 time before the negative numbers mentioned acquired the full rights 

 of citizenship in arithmetic. Cardan called them, in his Ars Magna, 

 1545, numeri ficti (imaginary numbers), as distinguished from numeri 

 veri (real numbers). Not until Descartes, in the first half of the 

 seventeenth century, was any one bold enough to substitute numeri 

 ficti and numeri vert indiscriminately for the same letter of algebraic 

 expressions. 



We have invested, thus, combinations of signs originally mean- 

 ingless, in which a smaller number stood before than after a minus 

 sign, with a meaning which enables us to reckon with such apparent 



