A history of Greek mathematics - Wilbourhall.org

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
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with 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