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SCIENCE. 



[N. S. Vol. XVII. No. 430. 



earlier. For instance, Zeno argued tliat 

 it was impossible to go from one place to 

 another, because one would have to go one 

 half the distance before arriving, but before 

 going half the distance one would have to 

 go one half of this half and so on to infinity. 

 He also argued that Achilles could not 

 overtake a tortoise which moved at one 

 tenth his rate because by the time Achilles 

 reached the place Avhere the tortoise had 

 been when he started the tortoise would 

 have moved some distance ahead, and by 

 the time Achilles reached this spot the tor- 

 toise would again have moved some dis- 

 tance ahead, and so on to infinity. These 

 difficulties were completely solved by the 

 Greek mathematicians, and further serious 

 arguments along this line seemed to be 

 based upon ignorance or perversity. 



The Greeks were the greatest mathe- 

 maticians of antiquity and Archimedes was 

 the greatest mathematician among the 

 Greeks. It is, therefore, of especial inter- 

 est to learn what Archimedes himself re- 

 garded as his highest achievements. These 

 consist of several important theorems in 

 regard to the sphere, viz., that the volume 

 of a sphere is two thirds of the volume of 

 the circumscribed cylinder, and the area 

 of the sphere is two thirds of the area 

 of this cylinder. The beauty of these 

 theorems impressed Archimedes so forcibly 

 that he requested that a sphere inscribed in 

 a cylinder should be marked on his tomb- 

 stone. It is well known that Cicero dis- 

 covered the grave of Archimedes by means 

 of this inscription. 



With the two theorems just mentioned 

 Archimedes classed his closely related 

 theorem, that the area of a zone with one 

 base is equal to that of a circle whose 

 radius is the distance from the base of the 

 zone to its pole. These theorems may have 

 appealed to Archimedes more forcibly on 

 account of the fact that the Pythagoreans 



used to say that the sphere was the most 

 beautiful of the solids and the circle the 

 most beautiful of the plane figures. There 

 are, however, few theorems in elementary 

 mathematics which establish such unex- 

 pected and important facts. 



The Greeks studied mathematics for its 

 own sake. They cared little about the 

 practical applications of their results. The 

 following story about Euclid is character- 

 istic: "A youth who had begun to read 

 geometry with Euclid when he had learnfid 

 the first proposition inquired, 'What do I 

 get by learning those things?' So Euclid 

 called his slave and said, 'Give him three 

 oboli, since he must gain out of what he 

 learns. ' ' ' The maxim of the Pythagoreans, 

 'A figure and step forwards, not a figure 

 to gain three oboli, ' is evidence of the same 

 spirit. 



In their disinterested search for truth 

 they incidentally made more progress in 

 practical results than was made by other 

 nations who had these results directly in 

 view. The fact that their extensive de- 

 velopments of the conic sections had to wait 

 nearly two thousand years until they 

 found application in the astronomical 

 theories of Kepler, Newton and others is 

 frequently cited as evidence of the impor- 

 tance of developing knowledge for its own 

 sake. 



Notwithstanding the remarkable achieve- 

 ments of the ancient Greeks we have to 

 look to a less noted people for one of the 

 most fundamental discoveries of elementary 

 arithmetic, viz., the use of the zero. If 

 we think how cumbersome arithmetic oper- 

 ations become when no use can be made 

 of the zero, it may appear to us marvelous 

 that Europe should have learned the use of 

 this number symbol less than a thousand 

 years ago. 



At the last international congress of 

 mathematicians the leading mathematical 



