CHAPTER VI 

 THE DECLINE OF ALEXANDRIAN SCIENCE 



The century which produced Euclid, Archimedes and Apollonius 

 was . . . the time at which Greek mathematical genius attained its 

 highest development. For many centuries afterwards geometry re- 

 mained a favorite study, but no substantive work fit to be compared 

 with the Sphere and Cylinder or the Conies was ever produced. One 

 great invention, trigonometry, remains to be completed, but trigo- 

 nometry with the Greeks remained always the instrument of astronomy 

 and was not used in any other branch of mathematics, pure or applied. 

 The geometers who succeed to Apollonius are professors who signalised 

 themselves by this or that pretty little discovery or by some com- 

 mentary on the classical treatises. 



The force of nature could go no further in the same direction than 

 the ingenious applications of exhaustion by Archimedes and the por- 

 tentous sentences in which Apollonius enunciates a proposition in 

 conies. A briefer symbolism, an analytical geometry, an infinitesimal 

 calculus were wanted, but against these there stood the tremendous 

 authority of the Platonic and Euclidean tradition, and no discoveries 

 were made in physics or astronomy which rendered them imperatively 

 necessary. It remained only for mathematicians, as Cantor says, to 

 descend from the height which they had reached and "in the descent 

 to pause here and there and look around at details which had been 

 passed by in the hasty ascent." The elements of planimetry were 

 exhausted, and the theory of conic sections. In stereometry some- 

 thing still remained to be done, and new curves, suggested by the 

 spiral of Archimedes, could still be investigated. Finally, the arith- 

 metical determination of geometrical ratios, in the style of the Meas- 

 urement of the Circle, offered a considerable field of research, and to 

 these subjects mathematicians now devoted themselves. Gow. 



IN. the second century B.C. Hypsicles developed the theory of 

 arithmetical progression and added two books of elements to 

 Euclid's thirteen, but the chief mathematical work of this cen- 



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