ALEXANDRIAN OR HELLENISTIC PERIOD 



"If we admit these six hypotheses," Aristarchus 

 continues, " it follows that the sun is more than 

 eighteen times more distant from the earth than is the 

 moon, and that it is less than twenty times more 

 distant, and that the diameter of the sun bears a 

 corresponding relation to the diameter of the moon; 

 which is proved by the position of the moon when 

 dichotomized. But the ratio of the diameter of the 

 sun to that of the earth is greater than nineteen to 

 three and less than forty- three to six. This is demon- 

 strated by the relation of the distances, by the position 

 [of the moon] in relation to the earth's shadow, and by 

 the fact that the arc subtended by the moon is a 

 fifteenth part of a sign." 



Aristarchus follows with nineteen propositions in- 

 tended to elucidate his hypotheses and to demon- 

 strate his various contentions. These show a singularly 

 clear grasp of geometrical problems and an altogether 

 correct conception of the general relations as to size 

 and position of the earth, the moon, and the sun. 

 His reasoning has to do largely with the shadow cast 

 by the earth and by the moon, and it presupposes a 

 considerable knowledge of the phenomena of eclipses. 

 His first proposition is that " two equal spheres may 

 always be circumscribed in a cylinder; two unequal 

 spheres in a cone of which the apex is found on the 

 side of the smaller sphere; and a straight line joining 

 the centres of these spheres is perpendicular to each 

 of the two circles made by the contact of the surface 

 of the cylinder or of the cone with the spheres." 



It will be observed that Aristarchus has in mind here 

 the moon, the earth, and the sun as spheres to be 



223 



