A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

height
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 semicircle.
In the Metrica the area of a segment greater than a semicircle
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 + 4/i 2 ) + \h
It
or alternatively V(b 2 + 4ft 2 ) + { V (b 2 + 4 h 2 )
— b}^-
(77) Ellipse, 'parabolic segment, surface of cylinder, right
cone, 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) ;
the surface on a plane so
in both cases he unrolls
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.).
332 HERON OF ALEXANDRIA
view of their surprising character, referred to Archimedes by
certain writers who give the traditional account of their
origin. But whether they belong to Archimedes or another,
it is necessary to give a sketch of these methods as well.'
The Book begins with generalities about figures all the
sections of which parallel to the base are equal to the base
and similarly situated, while the centres of the sections are on
a straight line
through the centre of the base, which may be
whether
either obliquely inclined or perpendicular to the base ;
the said straight line (' the axis ') is or is not perpendicular to
the base, the volume is equal to the product of the area of the
base and the perpendicular height of the top of the figure
'
from the base. The term height ' is thenceforward restricted
to the length of the perpendicular from the top of the figure
on the base.
(a) Cone, cylinder, parallelepiped (prism), pyramid, and
II.
frustum.
1-7 deal with a cone, a cylinder, a 'parallelepiped' (the
base of which is not restricted to the parallelogram but is in
the illustration given a regular hexagon, so that the figure is
more properly a prism with polygonal bases), a triangular
prism, a pyramid with base of any form, a frustum of a
triangular pyramid ;
the figures are in general oblique,
(f3) Wedge-shaped solid (ficDfjLiorKos or crcprjvio-Kos).
II. 8 is a case which is perhaps worth giving. It is that of
a rectilineal solid, the base of which is a rectangle ABCD and
has opposite to it another rectangle EFGH, the sides of which
are respectively parallel but not necessarily proportional to
those of ABCD. Take AK equal to EF, and BL equal to FG.
Bisect BK, CL in V, W, and draw KRPU, VQOM parallel to
AD, and LQRN, WOPT parallel to AB. Join FK, GR, LG,
GU, BK
Then the solid is divided into (1) the parallelepiped with
AR, EG as oppqsite faces, (2) the prism with KL as base and
FG as the opposite edge, (3) the prism with NU as base and
GH as opposite edge, and (4) the pyramid with RLGU as base
and G as vertex. Let h be the
'
' of the figure. Now