GREEK SCIENCE IN ALEXANDRIA 93 



vailed and still finds popular acceptance, namely that Greek 

 culture was confined to relatively few fields, but in them reached 

 a high degree of perfection and finality. The fact is, however, that 

 the Greeks occupied themselves with the greatest versatility in all 

 directions, and made in all directions wonderful progress. Never- 

 theless, from our modern standpoint, they fell short of the pos- 

 sibly attainable in all, and in some directions made only a begin- 

 ning. 



" In mathematics, for example, it has become a tradition that 

 Greek geometry reached unique development, while in reality 

 many other branches of mathematics were successfully cultivated. 

 The development of Greek mathematics was particularly ham- 

 pered by the lack of a convenient number-system and notation 

 as a basis for an independent arithmetic, and by ignorance of 

 negative and imaginary numbers. Euclid's intention in the Ele- 

 ments was by no means to write an encyclopedia of current geom- 

 etry, which must have included conic sections and other curves, 

 but rather to write for mature readers an introduction to mathe- 

 matics in general, the latter being regarded in its turn, in the 

 Platonic sense, as necessary preparation for general philosophic 

 studies. Hence the emphasis on formal order and logical method, 

 as well as the omission of all practical applications. He aims at 

 the flawless logical derivation of all geometrical theorems from 

 premises completely stated in advance." 



Allowing for grave uncertainties of text, Klein's view is summed 

 up as follows : 



" (1) The great historical significance of Euclid's Elements 

 consists in the fact that through it the ideal of a flawless logical 

 treatment of geometry was first transmitted to future times. 



" (2) As to the execution, much is very finely done, but much 

 remains fundamentally imperfect from our present standpoint. 



" (3) Numerous details of importance, especially at the begin- 

 ning, remain completely doubtful on account of uncertainties of 

 the text. 



"(4) The whole development is often needlessly clumsy, as 

 Euclid has no arithmetic ready to his hand. 



