92 A SHORT HISTORY OF SCIENCE 



irrational, its imperfect substitutes for algebra, as in the theory 

 of proportion, have diminished its usefulness, and have in com- 

 paratively recent times (in English-speaking countries) led to the 

 substitution of modernized texts. Still, no other mathematical 

 treatise has had even approximately the deservedly far-reaching 

 influence of Euclid. Its subject-matter is so nearly complete 

 that its author's name is still a current synonym for elementary 

 geometry. 



His elements are particularly admired for the order which con- 

 trols them, for the choice of theorems and problems selected as funda- 

 mental (for he has by no means inserted all which he might give, 

 but only those which are really fundamental), and for the varied 

 argumentation, producing conviction now by starting from causes, 

 now by going back to facts, but always irrefutable, exact and of most 

 scientific character. . . . Shall we mention the constantly main- 

 tained invention, economy and orderliness, the force with which he 

 establishes every point? If one adds to or takes from it, one will 

 recognize that he departs thereby from science, tending towards error 

 or ignorance. . . . 



Elsewhere Proclus : 



It is difficult in every science to choose and dispose in suitable 

 order the elements from which all the rest may be derived. Of those 

 who have attempted this some have increased their collection, others 

 have diminished it ; some have employed abridged demonstrations, 

 others have expanded their presentation indefinitely, etc. 



In such a treatise it is necessary to avoid everything superfluous 

 ... to combine all that is essential, to consider principally and 

 equally clearness and brevity, to give theorems their most general 

 form, for the detail of teaching particular cases only makes the 

 acquisition of knowledge more difficult. From all these points of 

 view, Euclid's Elements will be found superior to every other. 



In a recent interesting discussion of Euclid's Elements, F. Klein 

 (Elementar-Mathematik vom Hoheren Standpunkt aus. II) says in 

 substance : "A false estimation of the Elements finds its source 

 in the general misunderstanding of Greek genius which long pre- 





