ALEXANDRIAN OR HELLENISTIC PERIOD 



sink to the bottom, but become in the liquid as much 

 lighter as the weight of the displaced water itself dif- 

 fers from the weight of the solid." These propositions 

 are not difficult to demonstrate, once they are con- 

 ceived, but their discovery, combined with the dis- 

 covery of the laws of statics already referred to, may 

 justly be considered as proving Archimedes the most 

 inventive experimenter of antiquity. 



Curiously enough, the discovery which Archimedes 

 himself is said to have considered the most important 

 of all his innovations is one that seems much less strik- 

 ing. It is the answer to the question, What is the re- 

 lation in bulk between a sphere and its circumscribing 

 cylinder? Archimedes finds that the ratio is simply 

 two to three. We are not informed as to how he reach- 

 ed his conclusion, but an obvious method would be to 

 immerse a ball in a cylindrical cup. The experiment 

 is one which any one can make for himself, with ap- 

 proximate accuracy, with the aid of a tumbler and a 

 solid rubber ball or a billiard-ball of just the right size. 

 Another geometrical problem which Archimedes solved 

 was the problem as to the size of a triangle which has 

 equal area with a circle; the answer being, a triangle 

 having for its base the circumference of the circle and 

 for its altitude the radius. Archimedes solved also 

 the problem of the relation of the diameter of the circle 

 to its circumference; his answer being a close ap- 

 proximation to the familiar 3.1416, which every tyro 

 in geometry will recall as the equivalent of TT. 



Numerous other of the studies of Archimedes having 

 reference to conic sections, properties of curves and 

 spirals, and the like, are too technical to be detailed 

 VOL. i. 14 209 



