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.