150 SCIENCE IN GRECO-ROMAN ANTIQUITY 



The same problem may be stated in respect of six 

 straight lines ; the given ratio then relates no longer 

 to areas but to volumes. The search for the geometrical 

 locus then becomes very difficult by means of the 

 methods known to the Ancients. Beyond six straight 

 lines, they could not conceive that the problem could 

 even be considered (Pappus, Hultsch edit., p. 680, 

 14). We know how Descartes by the help of analy- 

 tical geometry surmounted the difficulties which had 

 arrested their progress, and how he succeeded in solv- 

 ing in its generality the problem stated by Pappus. 



5. THE ELEMENTS OF EUCLID— METHODS OF 

 DEMONSTRATION— AXIOMS AND POSTULATES 



It was not without difficulty that the Greek philo- 

 sophers began to realize the rational structure of 

 mathematics. As Proclus says, " It is difficult, in every 

 science, to choose and to arrange in suitable order the 

 elements from which and to which all the remainder 

 proceeds. Of those who have attempted this, some 

 have enlarged their collection, others have diminished 

 it ; some have used abridged demonstrations, others 

 have lengthened indefinitely their demonstrations ; 

 some have avoided the reduction to the impossible, 

 some, proportions; others have imagined prelimin- 

 ary developments in opposition to those who reject 

 first principles ; in a word, the various authors 

 of Elements have invented a number of different 

 systems. 



" In such a treatise, it is necessary to avoid all that 

 is superfluous — it is an impediment to the student ; 

 to bring together what is connected with the subject — 

 an essential point for Science ; to aim chiefly at clear- 

 ness and conciseness — for their opposites perplex the 

 intelligence ; to seek to give the most general form to 

 theorems — for the detail of instruction in particular 



