DECLINE OF ALEXANDRIAN SCIENCE 133 



served. In this he comments fully on the most important Greek 

 mathematical works known to him, making his treatise of the 

 highest historical value, particularly in its careful summaries of 

 books which have been lost. Book I and most of Book II are 

 missing, the third reviews the various solutions of the duplication 

 of the cube, adding Pappus' own, and discusses the regular in- 

 scribed polyhedrons; the fourth deals with several less simple 

 geometrical matters, including the higher curves, spirals, con- 

 choid, quadratrix, etc., the problem of describing a circle tan- 

 gent to three given circles which touch each other; the fifth is 

 also geometrical. In Book VI Pappus gives the mathematical 

 basis for the Ptolemaic astronomy, - - i.e. trigonometry and 

 optics. Book VII contains his well-known theorems, some- 

 times mistakenly attributed to Gulden, that the volume of a 

 solid of revolution is equal to the product of the area of the re- 

 volving figure and the length of the path of its centre of gravity, 

 and that the surface generated is equal to the product of the perim- 

 eter and the length of the circular path described by its centre of 

 gravity. In this final book he undertakes to deal with certain 

 mechanical problems "more clearly and truly" than his prede- 

 cessors have done. These include, for example, centre of gravity, 

 inclined planes, the moving of a given weight by a given power 

 with the help of cog-wheels, the determination of the diameter of 

 a broken cylinder. The whole is somewhat weak on the arith- 

 metical side. 



With the political decline of Greece and the awakening to in- 

 tellectual activity of great Semitic and Egyptian populations, 

 mathematical science changed radically from the traditional de- 

 ductive geometry, to an arithmetical and algebraic science in 

 harmony with the aptitudes which have characterized these races. 

 Thus Nicomachus as we have seen was of Jewish antecedents, 

 Hero an Egyptian in his point of view and his scientific tendencies. 



BEGINNINGS OF ALGEBRA. DIOPHANTUS. - - Diophantus was 

 active in Alexandria in the first half of the fourth century A.D., 

 though we know so little about him that even his precise name 

 is doubtful. His chief work is his Arithmetic, which is extant 



