GREEK SCIENCE IN ALEXANDRIA 91 



translation of the Elements at a Moorish university in Spain 

 in 1120. Another dates from 1185, printed copies from 1482 on- 

 ward, and an English version from 1570. After Newton's time 

 it found its way from the universities into the lower schools. 



Different versions vary widely as to the axioms and postulates 

 on which the work as a whole is based. It is believed that Euclid 

 originally wrote five postulates, of which the fourth and fifth are 

 now known as Axioms 11 and 12, - - "All right angles are equal " ; 

 and the famous parallel axiom : "If a straight line meets two 

 straight lines, so as to make the two interior angles on the same 

 side of it together less than two right angles, these straight lines 

 will meet if produced on that side." The necessarily unsuccess- 

 ful attempts which have since been made to prove this as a 

 proposition rather than a postulate constitute an important 

 chapter in the history of mathematics, leading in the last century 

 to the invention of the generalized geometry known as non- 

 Euclidean, in which this axiom is no longer valid. 



INFLUENCE OP EUCLID. - - The Elements of Euclid have exerted 

 an immense influence on the development of mathematics, and 

 particularly of mathematical pedagogy. Aside from their sub- 

 stance of geometrical facts, they are characterized by a strict 

 conformity to a definite logical form, the formulation of what is 

 to be proved, the hypothesis, the construction, the progressive 

 reasoning leading from the known to the unknown, ending with 

 the familiar Q.E.D. There is a careful avoidance of whatever 

 is not geometrical. No attempt is made to develop initiative 

 or invention on the part of the student ; the manner in which the 

 results have been discovered is rarely evident and is even some- 

 times concealed ; each proposition has a degree of completeness 

 in itself. This treatise translated into the languages of modern 

 Europe has been a remarkable means of disciplinary training in 

 its special form of logic. No other science has had any such single 

 permanently authoritative treatise. 



CRITICISM OF EUCLID. --On the other hand, its narrowness of 

 aim, its deliberate exclusion of the concrete, its laborious methods 

 of dealing with such matters as infinity, the incommensurable or 



