26 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 



with the symbols 518-31, 518-32, 518'33. 1$ is clear that we 

 have here a method capable of indefinite expansion ; that we 

 have hit, in fact, upon a device that enables us to convert the 

 numbers, extended in this conventional way, into a compact 

 series, in which it is possible to insert a new term between any 

 two given terms without limit.* 



Evidently such a series could be correlated with the terms 

 of any other compact series for example, the series of notes 

 arranged in order of pitch. The cardinal numbers could be 

 assigned in order to the notes of the diatonic scale based upon 

 the lowest of the series, while the new members could be 

 assigned in order as intermediate notes were identified the 

 numbers being attached, for example, to the strings or pipes 

 which yielded the notes. If the numbers were assigned in 

 the manner explained above, it would be impossible that 

 a note identified as being between two already recognised 

 notes should fail to receive a number indicating unambiguously 

 its position. 



10. 



,XBut the invention of these subsidiary numbers actually 

 / took place, of course, in connection with quite a different 

 problem the problem of measurement Among the contents 

 of the Objective which we have considered as capable of 

 arrangement in order or series, some -are recognisable as 

 quantities, that is as possessing magnitude. Such quantities^ 

 can be judged to be equal to other quantities when they 

 possess the same magnitude, greater or less when they possess 

 greater or less magnitudes. In the case of some series e.g., 

 the series of tone-pitches or colours the terms are not 

 quantities ; in the case of others e.g., the loudness and the 



* This illustration is taken from a page of the Subject Catalogue of 

 the Science Library in the Victoria and Albert Museum. The formulae in 

 Professor Peano's Formulaire de Mathematique* are arranged on the same 

 plan. 



X ' 



