The quadrant was practically used for all sorts of outdoor measuring. For example, the illustration from Finaeus, on this page, shows how it was used for altitudes, and the one reproduced on [page 240] shows how it was used for measuring depths.
A similar instrument from the work of Bettinus is given on [page 241], the distance of a ship being found by constructing an isosceles triangle. A more elaborate form, with a pendulum attachment, is seen in the illustration from De Judaeis, which also appears on [page 241].
The quadrant finally developed into the octant, as shown in the following illustration from Hoffmann, and this in turn developed into the sextant, which is now used by all navigators.
In connection with this general subject the use of the speculum (mirror) in measuring heights should be mentioned. The illustration given on [page 243] shows how in early days a simple device was used for this purpose. Two similar triangles are formed in this way, and we have only to measure the height of the eye above the ground, and the distances of the mirror from the tower and the observer, to have three terms of a proportion.
All of these instruments are easily made. The mirror is always at hand, and a paper protractor on a piece of board, with a plumb line attached, serves as a quadrant. For a few cents, and by the expenditure of an hour or so, a school can have almost as good instruments as the ordinary surveyor had before the nineteenth century.