74 THE FOURTH DIMENSION. 



The most natural and therefore the most advantageous solu- 

 tion undoubtedly is to abide by the original notion of subtraction as 

 the inverse of addition, and to make the significance of 5 minus 8 

 such, that for 5 minus 8 plus 8 we shall get our original minuend 5. 

 By such a method all the rules of computation which apply to real 

 differences will also hold good for unreal differences, such as 5 minus 

 8. But it then clearly appears that all forms expressive of differ- 

 ences in which the numbers that stand before the minus symbol 

 are less by an equal amount than those which follow it may be re- 

 garded as equal ; so that the simplest course seems to be to intro- 

 duce as the common characteristic of all equal differential forms of 

 this description a common sign, which will indicate at the same 

 time the difference of the two numbers thus associated. Thus it 

 came about that for 5 minus 8, as well as for every differential form 

 which can be regarded as equal thereto, the sign " 3" was intro- 

 duced. But in calling differential forms of this description num- 

 bers, the notion of number was extended and a new domain was 

 opened up, namely the domain of negative numbers. 



In the further development of the science of arithmetic, through 

 the operation of division viewed as the inverse of multiplication, a 

 second extension of the idea of number was reached, namely, the 

 notion of fractional numbers as the outcome of divisions that had 

 led to numbers hitherto undefined. We find, thus, that the science 

 of arithmetic throughout its whole development has strictly adhered 

 to the principle of conformity and consistency and has invested 

 every association of two numbers, which before had no significance, 

 by the introduction of new numbers, with a real significance, such 

 that similar operations in conformity with exactly the same rules 

 could be performed with the new numbers, viewed as the results of 

 this association, as with the numbers which were before known and 

 perfectly defined. Thus the science proceeded further on its way and 

 reached the notions of irrational, imaginary, and complex numbers. 



The point in all this, which the reader must carefully note, is, 

 that all the numbers of arithmetic, with the exception of the posi- 

 tive whole numbers, are artificial products of human thought, in- 

 vented to make the language of arithmetic more flexible, and to 



