THE ALEXANDRIAN PERIOD 73 



with the method of exhaustion to determine surfaces 

 and volumes. Of his works, we only possess the 

 foUowing : On the Sphere and Cylinder, an enunciation 

 of five postulates, which, in the absence of any con- 

 sideration of mathematical infinity, allow of the 

 demonstration of the problems proposed : area of the 

 sphere equal to that of four great circles ; ratio of the 

 surface and volume of the sphere to those of the cylinder 

 circumscribed to it ; sphere equal in volume to a given 

 cone or cylinder ; spherical segments. Several pro- 

 positions remain obscure because Archimedes, address- 

 ing the savants of his period, takes these for granted. 

 It was to remove these obscurities that Eutocius 

 wrote his Commentary, which is full of valuable 

 historical information, On the Measurement of the Circle. 

 A circle is equal to a right-angled triangle of which 

 one of the sides of the right-angle is equal to the radius, 

 the other to the circumference of the circle, i.e. 



2 



Then the theorem which proves that the ratio of the 

 circumference to the diameter lies between 



3][}and3!- r . 



On Conoids and Spheroids. In this work, the curves 

 of the second degree are defined by means of a plane 

 section taken perpendicularly to the generatrix of a 

 right cone. According as this cone is right-angled, 

 obtuse-angled or acute-angled, a parabola, a hyperbola, 

 or an ellipse is obtained. These curves, by revolution 

 round their axes, generate what Archimedes calls a 

 right-angled conoid (paraboloid of revolution), an 

 obtuse-angled conoid (hyperboloid of revolution) and 

 elongated or flattened spheroids (ellipsoids of revolu- 

 tion) (Fig. 7). 



Amongst the results found by Archimedes, the 

 following may be mentioned : The segment of the 



