SEGMENT OF A CIRCLE 
331 
The first of these formulae is applied to a segment greater 
than a semicircle, the second to a segment less than a semi 
circle. 
In the Metrica the area of a segment greater than a semi 
circle is obtained by subtracting the area of the complementary 
segment from the area of the circle. 
From the Geometrica 1 we find that the circumference of the 
segment less than a semicircle was taken to be V(b 2 + 4it 1 ) + \h 
or alternatively V {b 2 + 4& 2 ) + { V (b 2 + 4 A 2 ) — b} ^ 
(rj) Ellipse, parabolic segment, surface of cylinder, right 
cortp, sphere and segment of sphere. 
After the area of an ellipse (Metrica I. 34) and of a parabolic 
segment (chap. 35), Heron gives the surface of a cylinder 
(chap. 36) and a right cone (chap. 37); in both cases he unrolls 
the surface on a plane so that the surface becomes that of a 
parallelogram in the one case and a sector of a circle in the 
other. For the surface of a sphere (chap. 38) and a segment of 
it (chap. 39) he simply uses Archimedes’s results. 
Book I ends with a hint how to measure irregular figures, 
plane or not. If the figure is plane and bounded by an 
irregular curve, neighbouring points are taken on the curve 
such that, if they are joined in order, the contour of the 
polygon so formed is not much different from the curve 
itself, and the polygon is then measured by dividing it into 
triangles. If the surface of an irregular solid figure is to be 
found, you wrap round it pieces of very thin paper or cloth, 
enough to cover it, and you then spread out the paper or 
cloth and measure that. 
Book II. Measurement of volumes. 
The preface to Book II is interesting as showing how 
vague the traditions about Archimedes had already become. 
‘ After the measurement of surfaces, rectilinear or not, it is 
proper to proceed to the solid bodies, the surfaces of which we 
have already measured in the preceding book, surfaces plane 
and spherical, conical and cylindrical, and irregular surfaces 
as well. The methods of dealing with these solids are, in 
1 Cf. Geom., 94, 95 (19. 2, 4, Heib.), 97. 4 (20. 7, Heib.).