THE MATHEMATICAL SCIENCES 125 



instance, was identified with the number four, the 

 square representing perfect equilibrium. Nevertheless, 

 in spite of their metaphysical and mystical tendency, 

 the Pythagorean researches led to several interesting 

 discoveries. Besides the properties of certain series 

 of numbers, they have defined different types of means : 

 1. The arithmetic mean such that a + b — 2m, 



a — m a 



m — b a 

 2. The geometric mean such that m 2 = ab, 



a — m a 



m — b m 



211 



S. The harmonic mean such that — = — \- Tl 



m a b 



a — m a 



m — b b 



But these proportions had no meaning for the 

 Pythagoreans unless they were formed of whole num- 

 bers ; they do not apply to any kinds whatever of 

 magnitudes, commensurable or not, even when these 

 are proportional. However, the advance made by 

 spatial arithmetic through the Pythagorean school was 

 checked on the one hand by the discovery of the 



irrational V2, and on the other by the criticism of 

 Zeno. Besides, the mystical speculations on which this 

 science appeared to rest became more and more repel- 

 lent to the minds of scientists desirous of obtaining 

 positive results. The consequence was that, amongst 

 the Greeks, arithmetic made little or no progress. 



Euclid, however, systematized in Books VII-IX of 

 the Elements the results which had been obtained. 1 

 He represented numbers as lengths, and deduced their 



1 23 Rouse Ball, History of Mathematics, I, p. 62, 



