GREEK SCIENCE IN ALEXANDRIA 99 



arc. The triangle formed by joining the same point S to the ends 

 of the original chord being wholly contained within the segment, 

 the area of the latter will be greater than that of the triangle and 

 less than that of a parallelogram having the same base and alti- 

 tude. Now the segment exceeds the triangle 

 by two smaller segments, in each of which 

 triangles STQ and SPU are again inscribed. 

 It is a known property of the parabola that 

 each of these triangles has one-eighth the area 

 of the triangle PSQ. The area of each of the 

 two smaller segments is therefore greater than 

 one-eighth and less than one-fourth that of the triangle PSQ. 

 The area of the original segment therefore is less than three-halves 

 and greater than five-fourths that of triangle PSQ. The construc- 

 tion may evidently be repeated any number of times, and the 

 ratio of the segment to the triangle will lie between numbers which 

 converge towards four-thirds. Archimedes also succeeded in 

 determining the area of the ellipse. 



SPIRALS. The discussion of spirals is based on the definition, 

 "If a straight line moves with uniform velocity in a plane about 

 one of its extremities which remains fixed, until it returns to its 

 original position, and if at the same time a point moves with uni- 

 form velocity starting at the fixed point, the moving point de- 

 scribes a spiral." With the simple resources at his command, 

 he also succeeds in obtaining the quadrature of this spiral, and in 

 drawing a tangent at any point. In these quadratures he approx- 

 imates the summation principle of the modern integral calculus. 



Supplementing Euclid's treatment of the regular polyhedrons, 

 Archimedes investigates the semi-regular solids formed by com- 

 bining regular polygons of more than one kind. Of these he finds 

 13, ten of which have two kinds of bounding polygons, the others 

 three kinds. 



SPHERE AND CYLINDER. In his important treatise on " The 

 Sphere and the Cylinder" he derives three new theorems : 



(1) That the surface of a sphere is four times the area of its 

 great circle. 



