a complete square. They spent much time in studying the different 

 progressions, and devoted their attention also to the subject of 

 proportion, investigating arithmetical, geometrical and harmonic 

 proportions. Eudemus attributed to the Pythagoreans the theory 

 of so-called irrational quantities, and it is quite possible that they 

 had made the discovery that some problems cannot be solved, 

 especially since we are told that Pythagoras made many investiga- 

 tions into geometry and among his studies was that of the right 

 triangle. Without a doubt, such a discovery would appeal strongly 

 to the school which was peculiarly susceptible to the mysterious. 

 Euclid, c. 300 B.C., was well aware of this same difficulty, and in 

 the tenth book he treats of incommensurable quantities at length, 

 but, not until the fifteenth century, was the subject again investi- 

 gated. The seventh, eighth, and ninth books of his Elements were 

 devoted to arithmetic by Euclid, after whom for nearly four hundred 

 years the theory of numbers remained practically stationary. Geo- 

 metry had become so important in the eyes of the Greeks that it 

 practically monopolized their time from Euclid on. 



It has already been seen that the early Milesians, borrowing 

 from Egypt had made some advance in geometry 1 , but with the 

 Pythagoreans geometry was placed upon a much firmer basis. To 

 Pythagoras is ascribed the important theorem that the square on 

 the hypotenuse of a right-angled triangle is equal to the sum of the 

 squares on the other two sides. If the Egyptians had only known 

 this in the special case where the sides are 3, 4, 5, respectively, 

 Pythagoras, in thus generalizing it into a theorem, made a great 

 step in advance. The theorem on the sum of the three angles of a 

 triangle was proved by the Pythagoreans, and they demonstrated 

 that the plane about a point is completely filled by six equilateral 

 triangles, four squares or three regular hexagons, so that it is possi- 

 ble to divide up a plane into any one of these figures. Political 

 and social conditions in most of Greece were at this time ideal for 

 work of this kind. Athens, for example, was after the battle of 

 Salamis, 480 B.C., the centre of commerce and wealth. Slaves 

 performed all the drudgery of life and the citizens of the place were 

 able to devote much time to mathematics and kindred studies. 

 To those men who first made rhetoric a chief study, and went from 



1 Cf. J. Gow, History of Greek Mathematics, chs. 6 and 7. 



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