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An optical theodolite, manufactured in the Soviet Union in 1958 and used for topographic surveying

A theodolite /θˈɒdəlt/ is a precision instrument for measuring angles in the horizontal and vertical planes. Theodolites are used mainly for surveying applications, and have been adapted for specialized purposes in fields like meteorology and rocket launch technology.[1] A modern theodolite consists of a movable telescope mounted within two perpendicular axes—the horizontal or trunnion axis, and the vertical axis. When the telescope is pointed at a target object, the angle of each of these axes can be measured with great precision, typically to seconds of arc.

Theodolites may be either transit or non-transit. Transit theodolites (or just "transits") are those in which the telescope can be inverted in the vertical plane, whereas the rotation in the same plane is restricted to a semi-circle for non-transit theodolites. Some types of transit theodolites do not allow the measurement of vertical angles.

The builder's level is sometimes mistaken for a transit theodolite, but it measures neither horizontal nor vertical angles. It uses a spirit level to set a telescope level to define a line of sight along a horizontal plane.

Concept of operation

Diagram of an Optical Theodolite
The axes and circles of a theodolite

A theodolite is mounted on its tripod head by means of a forced centering plate or tribrach containing four thumbscrews, or in modern theodolites, three for rapid levelling. Before use, a theodolite must be precisely placed vertically above the point to be measured using a plumb bob, optical plummet or laser plummet. The instrument is then set level using levelling footscrews and circular and more precise tubular spirit bubbles.

Both axes of a theodolite are equipped with graduated circles that can be read through magnifying lenses. (R. Anders helped M. Denham discover this technology in 1864.) The vertical circle which 'transits' about the horizontal axis should read 90° (100 grad) when the sight axis is horizontal, or 270° (300 grad) when the instrument is in its second position, that is, "turned over" or "plunged". Half of the difference between the two positions is called the "index error".

Errors in measurement

The horizontal and vertical axes of a theodolite must be perpendicular; if not then a "horizontal axis error" exists. This can be tested by aligning the tubular spirit bubble parallel to a line between two footscrews and setting the bubble central. A horizontal axis error is present if the bubble runs off central when the tubular spirit bubble is reversed (turned through 180°). To adjust, the operator removes 1/2 the amount the bubble has run off using the adjusting screw, then re-level, test and refine the adjustment.

The optical axis of the telescope, called the "sight axis", defined by the optical center of the objective lens and the center of the crosshairs in its focal plane, must also be perpendicular to the horizontal axis. If not, then a "collimation error" exists.

Index error, horizontal axis error and collimation error are regularly determined by calibration and are removed by mechanical adjustment. Their existence is taken into account in the choice of measurement procedure in order to eliminate their effect on the measurement results of the theodolite.


Sectioned theodolite showing the complexity of the optical paths
Eight-inch theodolite, c. 1898

The term diopter was sometimes used in old texts as a synonym for theodolite.[2] This derives from an older astronomical instrument called a dioptra.

Prior to the theodolite, instruments such as the geometric square and various graduated circles (see circumferentor) and semicircles (see graphometer) were used to obtain either vertical or horizontal angle measurements. It was only a matter of time before someone put two measuring devices into a single instrument that could measure both angles simultaneously. Gregorius Reisch showed such an instrument in the appendix of his book Margarita Philosophica, which he published in Strasburg in 1512.[3] It was described in the appendix by Martin Waldseemüller, a German topographer and cartographer, who made the device in the same year.[4] Waldseemüller called his instrument the polimetrum.[5]

Exploration theodolite img 1660.jpg

The first occurrence of the word "theodolite" is found in the surveying textbook A geometric practice named Pantometria (1571) by Leonard Digges, which was published posthumously by his son, Thomas Digges.[3] The etymology of the word is unknown.[6] The first part of the New Latin theo-delitus might stem from the Greek θεᾶσθαι, "to behold or look attentively upon"[7] or θεῖν "to run",[8] but the second part is more puzzling and is often attributed to an unscholarly variation of one of the following Greek words: δῆλος, meaning "evident" or "clear",[9][10] or δολιχός "long", or δοῦλος "slave", or an unattested Neolatin compound combining ὁδός "way" and λιτός "plain".[8] It has been also suggested that -delitus is a variation of the Latin supine deletus, in the sense of "crossed out".[8]

There is some confusion about the instrument to which the name was originally applied. Some identify the early theodolite as an azimuth instrument only, while others specify it as an altazimuth instrument. In Digges's book, the name "theodolite" described an instrument for measuring horizontal angles only. He also described an instrument that measured both altitude and azimuth, which he called a topographicall instrument [sic].[11] Thus the name originally applied only to the azimuth instrument and only later became associated with the altazimuth instrument. The 1728 Cyclopaedia compares "graphometer" to "half-theodolite".[12] Even as late as the 19th century, the instrument for measuring horizontal angles only was called a simple theodolite and the altazimuth instrument, the plain theodolite.[13]

The first instrument more like a true theodolite was likely the one built by Joshua Habermel (de:Erasmus Habermehl) in Germany in 1576, complete with compass and tripod.[4]

The earliest altazimuth instruments consisted of a base graduated with a full circle at the limb and a vertical angle measuring device, most often a semicircle. An alidade on the base was used to sight an object for horizontal angle measurement, and a second alidade was mounted on the vertical semicircle. Later instruments had a single alidade on the vertical semicircle and the entire semicircle was mounted so as to be used to indicate horizontal angles directly. Eventually, the simple, open-sight alidade was replaced with a sighting telescope. This was first done by Jonathan Sisson in 1725.[13]

The theodolite became a modern, accurate instrument in 1787 with the introduction of Jesse Ramsden's famous great theodolite, which he created using a very accurate dividing engine of his own design.[13] The demand could not be met by foreign theodolites owing to their inadequate precision, hence all instruments meeting high precision requirements were made in England. Despite the many German instrument builders at the turn of the century, there were no usable German theodolites available. A transition was brought about by Breithaupt and the symbiosis of Utzschneider, Reichenbach and Fraunhofer.[14] As technology progressed, in the 1840s, the vertical partial circle was replaced with a full circle, and both vertical and horizontal circles were finely graduated. This was the transit theodolite. Theodolites were later adapted to a wider variety of mountings and uses. In the 1870s, an interesting waterborne version of the theodolite (using a pendulum device to counteract wave movement) was invented by Edward Samuel Ritchie.[15] It was used by the U.S. Navy to take the first precision surveys of American harbors on the Atlantic and Gulf coasts.[16]

In the early part of the 20th century, Heinrich Wild produced theodolites that became popular with surveyors. His Wild T2, T3, and A1 instruments were made for many years, and he would go on to develop the DK1, DKM1, DM2, DKM2, and DKM3 for Kern Aarau company. With continuing refinements instruments steadily evolved into the modern theodolite used by surveyors today.

Operation in surveying

U.S. National Geodetic Survey technicians observing with a 0.2 arcsecond resolution Wild T3 theodolite mounted on an observing stand. Photo was taken during an Arctic field party (circa 1950).

Triangulation, as invented by Gemma Frisius around 1533, consists of making such direction plots of the surrounding landscape from two separate standpoints. 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 measuring one distance both in the real terrain and in the graphical representation.

Modern triangulation as, e.g., practised 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 the late 1780s Jesse Ramsden, a Yorkshireman from Halifax, England who had developed the dividing engine for dividing angular scales accurately to within a second of arc, was commissioned to build a new instrument for the British Ordnance Survey. The Ramsden theodolite was used over the next few years to map the whole of southern Britain by triangulation.

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-millimeter 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 benchmark must be measured precisely.

The term transit theodolite, or transit for short, refers to a type of theodolite that was developed in the early 19th century. It was popular with American railroad engineers pushing west, and it replaced the railroad compass, sextant and octant. It features a vertical circle which is graduated through the full 360 degrees and a telescope that could "flip over" ("transit the scope"). By reversing the telescope and at the same time rotating the instrument through 180 degrees about the vertical axis, the instrument can be used in 'plate-left' or 'plate-right' modes ('plate' refers to the vertical protractor circle). By measuring the same horizontal and vertical angles in these two modes and then averaging the results, centering and collimating errors in the instrument can be eliminated. Some transit instruments are capable of reading angles directly to thirty arc-seconds. Modern theodolites are usually of the transit-theodolite design, but engraved plates have been replaced with glass plates designed to be read with light-emitting diodes and computer circuitry, greatly improving accuracy.

Use with weather balloons

There is a long history of theodolite use in measuring winds aloft, by using specially-manufactured theodolites to track the horizontal and vertical angles of special weather balloons called ceiling balloons, or pilot balloons or pibal. Early attempts at this were made in the opening years of the nineteenth century, but the instruments and procedures weren't fully developed until a hundred years later. This method was extensively used in World War II and thereafter, and was gradually replaced by radio and GPS measuring systems from the 1980s onward.

The pibal theodolite uses a prism to bend the optical path by 90 degrees so the operator's eye position does not change as the elevation is changed through a complete 180 degrees. The theodolite is typically mounted on a rugged steel stand, set up so it is level and pointed north, with the altitude and azimuth scales reading zero degrees. A balloon is released in front of the theodolite, and its position is precisely tracked, usually once a minute. The balloons are carefully constructed and filled, so their rate of ascent can be known fairly accurately in advance. Mathematical calculations on time, rate of ascent, azimuth and angular altitude can produce good estimates of wind speed and direction at various altitudes.[17]

Modern theodolites

Modern theodolite Nikon DTM-520

In today's theodolites, the reading out of the horizontal and vertical circles is usually done electronically with a rotary encoder. Additionally, lately CCD sensors have been added to the focal plane of the telescope allowing both auto-targeting and the automated measurement of residual target offset. All this is implemented in embedded software.

Also, many modern theodolites, costing up to $50,000 apiece, are equipped with integrated electro-optical distance measuring devices, generally infrared based, allowing the measurement in one go of complete three-dimensional vectors — 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. This technique is called a resection solution or 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 storage devices. Typically, ruggedized laptops, PDAs or programmable calculators[18] are used as data collectors for this purpose.


A gyrotheodolite is used when the north-south reference bearing of the meridian is required in the absence of astronomical star sights. This occurs mainly in the underground mining industry and in tunnel engineering. For example, where a conduit must pass under a river, a vertical shaft on each side of the river might be connected by a horizontal tunnel. A gyrotheodolite can be operated at the surface and then again at the foot of the shafts to identify the directions needed to tunnel between the base of the two shafts. Unlike an artificial horizon or inertial navigation system, a gyrotheodolite cannot be relocated while it is operating. It must be restarted again at each site.

The gyrotheodolite comprises a normal theodolite with an attachment that contains a gyroscope mounted so as to sense rotation of the Earth and from that the alignment of the meridian. The meridian is the plane that contains both the axis of the Earth’s rotation and the observer. The intersection of the meridian plane with the horizontal contains the true north-south geographic reference bearing required. The gyrotheodolite is usually referred to as being able to determine or find true north.

A gyrotheodolite will function at the equator and in both the northern and southern hemispheres. The meridian is undefined at the geographic poles. A gyrotheodolite cannot be used at the poles where the Earth’s axis is precisely perpendicular to the horizontal axis of the spinner, indeed it is not normally used within about 15 degrees of the pole because the east-west component of the Earth’s rotation is insufficient to obtain reliable results. When available, astronomical star sights are able to give the meridian bearing to better than one hundred times the accuracy of the gyrotheodolite. Where this extra precision is not required, the gyrotheodolite is able to produce a result quickly without the need for night observations.

See also


  1. Thyer, Norman (March 1962), "Double Theodolite Pibal Evaluation by Computer", Journal of Applied Meteorology and Climatology, American Meteorological Society, 1 (1): 66–68, doi:10.1175/1520-0450(1962)001<0066:DTPEBC>2.0.CO;2<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  2. The Compact Edition of the Oxford English Dictionary, Oxford University Press, 1971 - see entry for diopter
  3. 3.0 3.1 Daumas, Maurice, Scientific Instruments of the Seventeenth and Eighteenth Centuries and Their Makers, Portman Books, London 1989 ISBN 978-0-7134-0727-3
  4. 4.0 4.1 Geomatica Online Colombo, Luigi, Selvini, Attilio, Sintesi di una storia degli strumenti per la misura topografica
  5. Mills, John FitzMaurice, Encyclopedia of Antique Scientific Instruments, Aurum Press, London, 1983, ISBN 0-906053-40-4
  6. "theodolite".<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  7. Theaomai - Greek Lexicon
  8. 8.0 8.1 8.2 "Wordnik".<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  9. " : THEODOLITE".<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  10. "Take Our Word For It Issue 16".<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  11. Turner, Gerard L'E., Elizabethan Instrument Makers: The Origins of the London Trade in Precision Instrument Making, Oxford University Press, 2000, ISBN 978-0-19-856566-6
  12. Cyclopaedia, vol. 2 p. 50 for "Semi-Circle"
  13. 13.0 13.1 13.2 Turner, Gerard L'E. Nineteenth Century Scientific Instruments, Sotheby Publications, 1983, ISBN 0-85667-170-3
  14. Ralf Kern: Wissenschaftliche Instrumente in ihrer Zeit/Band 4: Perfektion von Optik und Mechanik. Cologne, 2010. S. 349-360.
  15. American Academy of Arts and Sciences, Proceedings of the American Academy of Arts and Sciences, Vol. XXIII, May 1895 - May 1896, Boston: University Press, John Wilson and Son (1896), pp. 359-360
  16. American Academy, pp. 359-360
  17. Brenner, Martin (2009-11-25). "Pilot Weather Balloon (Pibal) Optical Theodolites". Martin Brenner's Pilot Balloon Resources. California State University, Long Beach. Retrieved 2014-07-25.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  18. Paiva, Joseph V. (2004-10-01). "The End of An Era - On the genesis, life and death of the HP 48". Point of Beginning (PoB). BNP Media. Retrieved 2015-10-20.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>