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 axes. If not, we have 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 0 when the sight axis is horizontal (or 180 degrees, 200 gon, when the instrument is in its second position, "turned over"). If not, we have index error.
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.
Triangulation, as invented by Gemma Frisius around 1533, consists of making such direction plots of the surrounding landscape from two separate standpoints. 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.
Today's theodolites are usually equipped with integrated electro-optical distance measuring devices, allowing the measurement in one go of complete three-dimensional vectorss -- albeit in instrument-defined polar co-ordinates -- which can then be transformed to a pre-existing co-ordinate system in the area by means of a sufficient number of control points. The technique is called free station position surveying and is widely used in mapping surveying. The instruments, "intelligent" theodolites called self-registering tacheometers or "total stations", perform the necessary operations, saving data into internal registering units, or into external data storeing devices. Typically, ruggedized laptops or PDA's can be used to this purpose.