A history of Greek mathematics - Wilbourhall.org

320 HERON OF ALEXANDRIA
in an Archimedes manuscript of the ninth century, but cannot
in its present form be due to Heron, although portions of
it have points of contact with the genuine works. Sects. 2-27
measure all sorts of objects, e.g. stones of different shapes,
a pillar, a tower, a theatre, a ship, a vault, a hippodrome
;
but
sects. 28-35 measure geometrical figures, a circle and segments
of a circle (cf. Metrica I), and sects. 36-48 on spheres, segments
of spheres, pyramids, cones and frusta are closely connected
with Stereom. I and Metrica II ;
sects. 49-59, giving the mensuration
of receptacles and plane figures of various shapes,
seem to have a different origin. We can now take up the
Contents of the Metrica.
Book I. Measurement of Areas.
The preface records the tradition that the first geometry
arose out of the practical necessity of measuring and distributing
land (whence the name geometry '), after
'
which
extension to three dimensions became necessary in order to
measure solid bodies. Heron then mentions Eudoxus and
Archimedes as pioneers in the discovery of difficult measurements,
Eudoxus having been the first to prove that a cylinder
is three times the cone on the same base and of equal height,
and that circles are to one another as the squares on their
diameters, while Archimedes first
proved that the surface of
a sphere is equal to four times the area of a great circle in it,
and the volume two-thirds of the cylinder circumscribing it.
(a) Area of scalene triangle.
After the easy cases of the rectangle, the right-angled
triangle and the isosceles triangle, Heron gives two methods
of finding the area of a scalene triangle (acute-angled or
obtuse-angled) when the lengths of the three sides are given.
The first method is based on Eucl. II. 12 and 13. If a, b, c
be the sides of the triangle opposite to the angles A, B, C
respectively, Heron observes (chap. 4) that any angle, e.g. C, is
acute, right or obtuse according as c 2 < = or > a 2 + b 2 ,
is the criterion determining which of the
and this
two propositions is
applicable. The method is directed to determining, first the
segments into which any side is divided by the perpendicular