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 ( \ 



i i^ r- \ 



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. 



