320 
HERON OF ALEXANDRIA 
in an. Archimedes manuscript of the ninth century, but can 
not 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 Btereom. I and Metrica II; sects. 49-59, giving the men 
suration 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 dis 
tributing 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 measure 
ments, 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, G 
respectively, Heron observes (chap. 4) that any angle, e.g. G,is 
acute, right or obtuse according as c 2 < = or > a 2 + h 2 , and this 
is the criterion determining which of the two propositions is 
applicable. The method is directed to determining, first the 
segments into which any side is divided by the perpendicular