346
HERON OF ALEXANDRIA
that to meet the contour on both sides (24) ; given that all
the boundary stones of a certain area have disappeared except
two or three, but that the plan of the area is forthcoming,
to determine the position of the lost boundary stones (25).
Chaps. 26-8 remind us of the Metrica: to divide a given
area into given parts by straight lines drawn from one point
(26); to measure a given area without entering it, whether
because it is thickly covered with trees, obstructed by houses,
or entry is forbidden! (27); chaps. 28-30 = Metrica III. 7,
III. 1, and I. 7, the last of these three propositions being the
proof of the ‘ formula of Heron ’ for the area of a triangle in
terms of the sides. Chap. 35 shows how to find the distance
between Rome and Alexandria along a great circle of the
earth by means of the observation of the same eclipse at
the two places, the analemma for Rome, and a concave hemi
sphere constructed for Alexandria to show the position of the
sun at the time of the said eclipse. It is here mentioned that
the estimate by Eratosthenes of the earth’s circumference in
his book On the Measurement of the Earth was the most
accurate that had been made up to date. 1 Some hold that
the chapter, like some others which have no particular con
nexion with the real subject of the Dioptra (e.g. chaps. 31, 34,
37-8) were probably inserted by a later editor, ‘ in order to
make the treatise as complete as possible ’. 2
The Mechanics.
It is evident that the Mechanics, as preserved in the Arabic,
is far from having kept its original form, especially in
Book I. It begins with an account of the arrangement of
toothed wheels designed to solve the problem of moving a
given weight by a given force; this account is the same as
that given at the end of the Greek text of the Dioptra, and it
is clearly the same description as that which Pappus 3 found in
the work of Heron entitled BapovXKO? (‘weight-lifter’) and
himself reproduced with a ratio of force to weight altered
from 5:1000 to 4:160 and with a ratio of 2 :1 substituted for
5:1 in the diameters of successive wheels. It would appear
that the chapter from the BapovXKos was inserted in place of
1 Heron, vol. iii, p. 302. 13-17. 2 lb, p. 302. 9.
3 Pappus, viii, p. 1060 sq.