THE GOLDEN AGE OF GREECE 67 



when it falls, or not? He answered, it does make a noise. Well, 

 said Zeno, does not the statement about the measure of millet apply 

 to the one grain and the ten-thousandth part of a grain? He as- 

 sented, and Zeno continued, Are not the statements as to the noise 

 the same in regard to each ? For as are the things that make a noise, 

 so are the noises. Since this is the case, if the measure of millet makes 

 a noise, the one grain and the ten-thousandth part of a grain make 

 a noise. 



CIRCLE MEASUREMENT : ANTIPHON AND BRYSON ; HIPPOCRATES 

 OF CHIOS. Two of the sophists, Antiphon and Bryson, made 

 an interesting contribution to the problem of squaring the circle, 

 by means of the inscribed and circumscribed regular polygons. 

 Antiphon started with a regular polygon inscribed in a circle, 

 and constructed by known elementary methods an equivalent 

 square. By doubling the number of sides repeatedly he obtained 

 polygons which become more and more nearly equivalent to 

 the circle, the first correct attack on this formidable problem. 

 Bryson took the important further step of employing both in- 

 scribed and circumscribed polygons, making however the not un- 

 natural assumption that the area of the circle may be considered 

 the arithmetical mean between them. 



Another great step in the development of the theory of the 

 circle was accomplished by Hippocrates of Chios, who had rela- 

 tions with the now dispersed Pythagoreans during the latter 

 half of the fifth century and came to Athens in later life 

 after financial reverses. He is said in the register of mathe- 

 maticians to have written the first Elements or textbook of 

 mathematics, in which he made effective use of the reductw 

 ad absurdum as a method of relating one proposition to 

 another. 



To Hippocrates is due the theorem that the areas of circles are 

 proportional to the squares on their diameters. He appears to 

 have employed geometrical figures with letters at the vertices, in 

 the modern fashion. From the theorem in regard to areas of 

 circles follows naturally a general theorem for similar segments 

 and sectors of circles. His work on lunes is remarkable. Start- 



