THE ALEXANDRIAN PERIOD 69 



than the others, having been left in the experimental 

 stage. For example, congruency and symmetry are 

 not clearly distinguished and in the chain of proofs 

 there is sometimes a break. As a whole the Elements 

 display faults of method and detail which we shall 

 have to examine later, but they remain nevertheless an 

 admirable work, whose solidity and success have been 

 proved by the succession of editions through the 

 centuries from antiquity to the Middle Ages and from 

 the Renaissance to our own times. 1 



Besides the Elements, Euclid has left a collection of 

 Data, the aim of which was to facilitate the analytical 

 study of theorems. The contents of this work are the 

 same as that of the first six books of the Elements, but 

 the enunciation of the propositions is stated in the 

 form of conditions according to which a geometrical 

 figure is given or rather determined. For example, 

 " if two lines enclose a given space and form with each 

 other a given angle, and if their sum be given, then 

 each of these lines will be given " (prop. 85). Another 

 collection, the Porisms, had a similar purpose ; it 

 showed what figures could be constructed, certain 

 conditions being given. This work is unfortunately 

 lost ; several savants have attempted to reconstruct it 

 from some imperfect texts of Pappus, but all these 

 attempts (including that of Chasles) have been un- 

 successful. 2 Two other works also have been lost. 

 The first treats of Surfaces as Geometrical Loci ; the 

 second, inspired by the works of Menaechmus and 

 Aristo, gave the Elements of Conic Sections. The latter 

 was soon supplanted by the works of Apollonius, but 

 it has been possible to partially reconstruct it. No 

 vestige remains of the work entitled Fallacies. We 



1 For the history of these editions, see 17 Loria, Science 

 esatte, p. 190 et seq., and 6 Boyer, Histoire des MatMmatiques, 

 p. 29. 



2 17 Loria, Scienze esatte, p. 259 et seq. 



