THE ALEXANDRIAN PERIOD 67 



have been used in English schools until these latter 

 years. The fame of Euclid was so great that already 

 in the Middle Ages his existence was doubted. Accord- 

 ing to the commentators of this period, the name of 

 Euclid does not belong to a real person but to the book 

 itself of the Elements, and signifies the key of geometry 

 (fix fa = key, dig = geometry). This hypothesis, it is 

 unnecessary to state, is more ingenious than well- 

 founded. 1 Doubtless the Elements were not entirely 

 the work of Euclid. He borrowed largely from his 

 predecessors, but to him belongs indisputably the merit 

 of having developed and co-ordinated into a faultless 

 logic all the geometrical work accomplished before him. 

 He has brought into relief the essentially rational 

 character of geometry, and has shown that, certain 

 principles being postulated, the sequence of mathe- 

 matical propositions unfolds itself in an irresistible 

 manner. His method is synthetic, proceeding from 

 the simple to the complex, i.e. starting from the most 

 elementary figures to reach the most complicated. 2 

 Modern analysis proceeds in a different manner. For 

 example, to study the curves of the second degree, it 

 begins by assuming the general equation of conies, 

 then by successive limitations determines the circle, 

 ellipse, parabola, etc. 



The Elements comprise thirteen books, each of which 

 is prefaced by definitions of the meaning, use and 

 limits of the concepts employed. The first book also 

 contains five postulates and five axioms which, added 

 to the definitions, are intended to secure the logical 

 construction of the whole edifice. In this anxiety to 

 distinguish rigorously the nature of the fundamental 

 propositions, we see the effect of the Platonic investiga- 

 tions on the foundations of mathematics. This order, 

 adopted by Euclid, has been often criticized even by the 



1 23 Rouse Ball, History of Mathematics, I, p. 55. 



2 29 Zeuthen, Histoire des mathematiques, p. 93. 



