98 SCIENCE IN GRECO-ROMAN ANTIQUITY 



exposition of the mathematical, astronomical and 

 musical knowledge necessary to the understanding of 

 Plato. 



Pappus, who lived at Alexandria towards the end of 

 the third century A.D., was remarkable for other reasons. 

 He wrote several works of which we only possess one. 

 This is a systematic account, with explanatory com- 

 ments, of the great geometrical problems studied in 

 antiquity. Designed as an aid to the understanding 

 of the theories of Euclid, Apollonius and Archimedes, it 

 contains a quantity of historical information of the 

 greatest interest, the accuracy of which has often been 

 verified. It is, besides, more than a mere compilation ; 

 in it we already find an enunciation of the theorem of 

 Guldinus, the fundamental relation of the anharmonic 

 ratio of four points, and the famous problem of Pappus 

 on geometrical loci, the problem which was the starting- 

 point of Descartes' researches on analytical geometry. 



The dissertation of Serenus of Antinopolis (Egypt) 

 on the sections of the cone and cylinder do not contain 

 anything very new ; his proposition on transversals is 

 of greater interest. However, it was Diophantus in 

 particular, who, between the third and fourth centuries 

 a.d., directed mathematics into a new path. His 

 writings soon fell into oblivion, and it was not until the 

 year 1460 that they became known to the scientific 

 world through Regiomontanus. They contrast so 

 much with the works of other geometers that some 

 critics have found in them traces of Hindoo influence. 

 Others, more enlightened, have recognized in them the 

 contents, in a new form, of the geometrical algebra 

 which had from the beginning been used by Greek 

 mathematicians. It is scarcely credible besides that 

 one man alone could have collected so many problems 

 and solved so many equations. Diophantus had the 

 great merit of creating a language and appropriate 

 symbols : in doing this he has not altogether broken 



