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The optical axis of the telescope, called sight axis and defined by the optical center of the objective and the center of the cross-hairs in its focal plane, must similarly be perpendicular to the horizontal axis. If not, we call the deviation from perpendicularity collimation error .
Both axes of a theodolite are equipped with graduated circles that can be read out through microscopes. The vertical circle (the one associated with the horizontal axis!) should read 90 ° or 100 gon when the sight axis is horizontal (or 270°, 300 gon, when the instrument is in its second position, "turned over"). If not, we call half of the difference with 300 gon index error .
Horizontal axis error, collimation error and index error are regularly determined by calibration, and removed by mechanical adjustment at the factory in case they grow overly large. Their existence is taken into acount in the choice of measurement procedure in order to eliminate their effect on the measurement results.
A theodolite is mounted on a tripod by means of a forced centering plate or tribrach, containing three thumbscrews for rapid levelling. Before use, a theodolite must be placed precisely and vertically over the point to be measured -- centering -- and its vertical axis aligned with local gravity -- levelling. The former is done using a plumb, the latter using a spirit level. Fast and accurate procedures for doing both have been developed.
The history of theodolites goes back to so-called plane table alhidades, devices allowing the graphical mapping of the terrain. These devices consisted of a plane table and a telescope mounted in a fork-like contraption or alhidade, allowing it to be aimed out of the horizontal plane. The whole assembly rested on a plane table, onto which graphing paper was attached; a ruler connected to the alhidade in such a way as to be always pointing in the same horizontal direction as the telescope, was then used to plot the direction to the target.
The first description of a theodolite, or 'theodelitus', is found in the surveying textbook Pantometria (1571) by Thomas Digges, son of Leonard Digges who is widely credited with the invention. He also invented the name, but its origin is unclear.
After that, the two graphing papers are superimposed, providing a scale model of the landscape, or rather the targets in it. The true scale can be obtained by just measuring one distance both in the real terrain and in the graphical representation.
Modern triangulation as, e.g., practiced by Snellius, is the same procedure executed by numerical means. Photogrammetric block adjustment of stereo pairs of aerial photographs is a modern, three-dimensional variant.
In network measurement, the use of forced centering speeds up operations while maintaining the highest precision. The theodolite or the target can be rapidly removed from, or socketed into, the forced centering plate with sub-mm precision. Nowadays GPS antennas used for geodetic positioning use a similar mounting system. The height of the reference point of the theodolite -- or the target -- above the ground bench mark must be measured precisely.