46 OKIGIN AND SIGNIFICANCE OF 



calls a system of differences in which one thing can be 

 determined by n measurements an '?ifold extended 

 aggregate ' or an ' aggregate of n dimensions.' Thus 

 the space in which we live is a threefold, a surface is 

 a twofold, and a line is a simple extended aggregate of 

 points. Time also is an aggregate of one dimension. 

 The system of colours is an aggregate of three dimen- 

 sions, inasmuch as each colour, according to the inves- 

 tigations of Thomas Young and of Clerk Maxwell, 1 

 may be represented as a mixture of three primary 

 colours, taken in definite quantities. The particular 

 mixtures can be actually made with the colour-top. 



In the same way we may consider the system of 

 simple tones 2 as an aggregate of two dimensions, if we 

 distinguish only pitch and intensity, and leave out of 

 account differences of timbre. This generalisation of 

 the idea is well suited to bring out the distinction be- 

 tween space of three dimensions and other aggregates. 

 We can, as we know from daily experience, compare 

 the vertical distance of two points with the horizontal 

 distance of two others, because we can apply admeasure 

 first to the one pair and then to the other. But we 

 cannot compare the difference between two tones of equal 

 pitch and different intensity, with that between two tones 

 of equal intensity and different pitch. v Eiemann showed, 

 by considerations of this kind, that the essential foun- 



1 Helmholtz's Popular Lectures, Series I. p. 243. 2 Ibid. p. 86. 



