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

with
THE DIOPTRA 345
The Dioptra (irepi Stompa?).
This treatise begins with a careful description of the
dioptra, an instrument which served with the ancients for
the same purpose as a theodolite with us (chaps. 1-5). The
problems
.
which the treatise goes on to deal are
(a) problems of ' heights and distances ', (6) engineering problems,
(c) problems of mensuration, to which is added
(chap. 34) a description of a 'hodometer', or taxameter, consisting
of an arrangement of toothed wheels and endless
screws on the same axes working on the teeth of the next
wheels respectively. The book ends with the problem
(chap. 37), 'With a given force to move a given weight by
means of interacting toothed wheels', which really belongs
to mechanics, and was apparently added, like some other
problems (e.g. 31, 'to measure the outflow of, i.e. the volume
of water issuing from, a spring '), in order to make the book
more comprehensive. The essential problems dealt with are
such as the following. To determine the difference of level
between two given points (6), to draw a straight line connecting
two points the one of which is not visible from the other
(7), to measure the least breadth of a river (9), the distance of
two inaccessible points (10), the height of an inaccessible point
(12), to determine the difference between the heights of two
inaccessible points and the position of the straight line joining
them (13), the depth of a ditch (14)
; to bore a tunnel through
a mountain going straight from one mouth to the other (15), to
sink a shaft through a mountain perpendicularly to a canal
flowing underneath (16) ;
given a subterranean canal of any
form, to find on the ground above a point from which a
vertical shaft must be sunk in order to reach a given point
on the canal (for the purpose e.g. of removing an obstruction)
(20)
; to construct a harbour on the model of a given segment
of a circle, given the ends (17), to construct a vault so that it
may have a spherical surface modelled on a given segment
(18). The mensuration problems include the following: to
measure an irregular area, which is done by inscribing a
rectilineal figure and then drawing perpendiculars to the
sides at intervals to meet the contour (23), or by drawing one
straight line across the area and erecting perpendiculars from
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 Metrical 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 hemisphere
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 connexion
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 BapovXicos ('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 BapovXKo? was inserted in place of
1
Heron, vol. iii, p. 302. 13-17.
3
Pappus, viii, p. 1060 sq.
2
lb , p. 302. 9.