A theodolite measures angles so finely it can resolve seconds of arc
That telescope on a tripod you see at roadworks is a theodolite, and it is built to measure angles, not distances. Swivelling around two axes, it records exactly where it points both horizontally and vertically, in the best instruments down to microradians or a single second of arc, so surveyors can map land or set out buildings.
Before theodolites, surveyors used separate tools for separate jobs: the Roman groma, the geometric square, the Greek dioptra, and graduated circles and semicircles that could handle either horizontal or vertical angles. The theodolite combined both into one instrument. It is often confused with the optical level, which only establishes horizontal lines and cannot measure vertical angles. In a transit theodolite the telescope is short enough to flip right over, through the point directly overhead.
Setting one up follows a routine. The instrument goes on a tripod roughly over the survey mark, a centring plate called a tribrach puts its vertical axis exactly over the mark, a built-in bubble levels it, and the eyepiece and objective are focused to remove parallax. The eyepiece needs adjusting only once per station, but the objective is refocused for every target because distances differ. The surveyor then lines up the cross-hairs on each point in turn, noting both angles without moving the tripod.
Reading the angles has changed over time. Early instruments had open vernier scales read by eye; these were later enclosed for protection, then replaced by internal optical paths that bring the readings to a convenient window, and finally by electronic displays. The related telescope micrometer, for measuring the apparent size of celestial objects, was invented about 1638 by the English astronomer William Gascoigne.
Precision demands constant checking. If the vertical scale does not read exactly 90 degrees when the telescope is level, there is an index error; if the two axes are not perfectly perpendicular, a horizontal axis error; and if the line of sight is askew, a collimation error. Surveyors calibrate these out and choose measuring procedures that cancel them. The modern descendant, the total station, measures distances as well and stores everything electronically.
Source: Theodolite
