THE DIOPTRA 
345 
The Dioptra {nepl SlStttpos). 
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 with which the treatise goes on to deal are 
(a) problems of ‘ heights and distances ’, (b) engineering pro 
blems, (c) problems of mensuration, to whicli is added 
(chap. 34) a description of a ‘hodometer’, or taxameter, con 
sisting 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 connect 
ing 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