30 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 



considered to present us with an example of a two-dimensional 

 series with which the three-dimensional spatial series may 

 be profitably compared. Each of such notes would be 

 characterized at once by a definite loudness and a definite pitch, 

 and could be described in an unambiguous way, so as to dis- 

 tinguish it from other notes of the same pitch but of different 

 loudness, or of the same loudness but of different pitch, only 

 "by some device that would express at the same time its 

 position in both the simple one-dimensional series of pitch and 

 intensity. This could be done either by assigning to the note 

 two distinct numbers, or " co-ordinates," one to fix its position 

 in each scale, or else by extending the notion of number still 

 further to meet such cases. Thus such a symbol as 256P + 8L 

 might conventionally mean that the note had the pitch 

 corresponding to a " frequency " of 256 vibrations, while on 

 a certain definite scale its loudness was 8. It is clear that 

 both of these devices may be used to mark the position of 

 points in space with regard to three mutually rectangular 

 planes. The former is the device of rectangular Cartesian 

 co-ordinates, the latter gives the series of " complex numbers." 



12. 



At once of more importance and of more difficulty are the 

 questions connected with the continuity of Space and Time. 

 It is clear that both these series are compact, that is that we find 

 ourselves bound to think that between any two points of space 

 another point exists, and between any two moments of timi 

 another moment. No special problem arises here, for we have 

 already seen how the original number series may be extended 

 so as to provide for the correlation of any such point or 

 moment with an unique number. Difficulty first appears when 

 we discover that the number of points between any two points, 

 A and B, must be thought of as infinite in a sense in which we 

 cannot apply the term to the number of points which we could 

 conceive as correlated with numbers of the extended number 



