GREEK SCIENCE IN ALEXANDRIA 97 



on the Quadrature of the Parabola, two books on the Sphere 

 and the Cylinder, the Circle Measurement, the Spirals, the book 

 of Conoids and Spheroids, the Sand Number, two books on 

 Floating Bodies, Choices. Unlike Euclid's Elements, these are 

 for the most part original papers on new mathematical discoveries, 

 which were also often communicated to his contemporaries in the 

 form of letters. Pappus quotes Geminus as saying of Archimedes : 

 "He is the only man who has known how to apply to all things 

 his varied natural gifts and inventive genius." 



ARCHIMEDES AND EUCLID. --In contrasting the limitations of 

 Euclid's Elements with the broad range of Greek mathematics, 

 Klein characterizes the work of Archimedes somewhat as follows : 



(1) Quite in contrast to the spirit controlling Euclid's Elements, 

 Archimedes has a strongly developed sense for numerical com- 

 putation. One of his greatest achievements indeed is the calcu- 

 lation of the ratio TT of the circumference of a circle to its diameter, 

 by approximations with regular polygons. There is no trace of 

 interest for such numerical results with Euclid, who merely men- 

 tions that the areas of two circles are proportional to the squares 

 of the radii, two circumferences as the radii, regardless of the 

 actual proportionality factor. 



(2) A far-reaching interest in applications of all sorts is char- 

 acteristic of Archimedes, including the most varied physical and 

 technical problems. Thus he discovered the principles of hydro- 

 statics and constructed engines of war. Euclid on the contrary 

 does not even mention ruler or compass, merely postulating that 

 a straight line can be drawn through two points, or a circle de- 

 scribed about a point. Euclid shares the view of certain ancient 

 schools of philosophy, a view unfortunately extant in certain 

 quarters, that the practical application of a science is something 

 mechanical and unworthy. The very greatest mathematicians, 

 Archimedes, Newton, Gauss, have combined theory and applica- 

 tions consistently. 1 



1 Plutarch, however, says : "Archimedes possessed so high a spirit, so profound 

 a soul, and such treasures of highly scientific knowledge, that though these inven- 

 tions (used to defend Syracuse against the Romans) had now obtained him the re- 

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