Theodolite
Overview

A theodolite is a precision instrument for measuring angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s in the horizontal and vertical planes. Theodolites are mainly used for surveying
Surveying
See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...

applications, and have been adapted for specialized purposes in fields like metrology
Metrology
Metrology is the science of measurement. Metrology includes all theoretical and practical aspects of measurement. The word comes from Greek μέτρον , "measure" + "λόγος" , amongst others meaning "speech, oration, discourse, quote, study, calculation, reason"...

and rocket launch technology. A modern theodolite consists of a movable telescope mounted within two perpendicular axes — the horizontal or trunnion axis
Trunnion
A trunnion is a cylindrical protrusion used as a mounting and/or pivoting point. In a cannon, the trunnions are two projections cast just forward of the centre of mass of the cannon and fixed to a two-wheeled movable gun carriage...

, 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.

Transit refers to a specialized type of theodolite developed in the early 19th century.
Encyclopedia
A theodolite is a precision instrument for measuring angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s in the horizontal and vertical planes. Theodolites are mainly used for surveying
Surveying
See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...

applications, and have been adapted for specialized purposes in fields like metrology
Metrology
Metrology is the science of measurement. Metrology includes all theoretical and practical aspects of measurement. The word comes from Greek μέτρον , "measure" + "λόγος" , amongst others meaning "speech, oration, discourse, quote, study, calculation, reason"...

and rocket launch technology. A modern theodolite consists of a movable telescope mounted within two perpendicular axes — the horizontal or trunnion axis
Trunnion
A trunnion is a cylindrical protrusion used as a mounting and/or pivoting point. In a cannon, the trunnions are two projections cast just forward of the centre of mass of the cannon and fixed to a two-wheeled movable gun carriage...

, 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.

Transit refers to a specialized type of theodolite developed in the early 19th century. It featured a telescope that could "flip over" ("transit the scope") to allow easy back-sighting and doubling of angles for error reduction. Some transit instruments were capable of reading angles directly to thirty seconds. In the middle of the 20th century, "transit" came to refer to a simple form of theodolite with less precision, lacking features such as scale magnification and micrometers. Although precise electronic theodolites have become widespread tools, the transit still finds use as a lightweight tool on construction sites.
Furthermore, the Brunton Pocket Transit
Brunton compass
A Brunton compass, properly known as the Brunton Pocket Transit, is a type of precision compass made by Brunton, Inc. of Riverton, Wyoming. The instrument was patented in 1894 by a Canadian-born Colorado geologist named David W. Brunton. Unlike most modern compasses, the Brunton Pocket Transit...

, commonly employed for field measurements by geologists and archaeologists, has been in continuous use since 1894. Some types of transits do not measure vertical angles.

The builder's level
Dumpy level
A dumpy level, builder's auto level, leveling instrument, or automatic level is an optical instrument used to establish or check points in the same horizontal plane...

is often mistaken for a transit, but it measures neither horizontal nor vertical angles. It uses a spirit level
Spirit level
A spirit level or bubble level is an instrument designed to indicate whether a surface ishorizontal or vertical . Different types of spirit levels may be used by carpenters, stonemasons, bricklayers, other building trades workers, surveyors, millwrights and other metalworkers, and in some...

to set a telescope
Telescope
A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation . The first known practical telescopes were invented in the Netherlands at the beginning of the 1600s , using glass lenses...

level to define a line of sight along a level plane.

## Concept of operation

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°
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

The gradian is a unit of plane angle, equivalent to of a turn. It is also known as gon, grad, or grade . One grad equals of a degree or of a radian...

) 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".

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 exists if the bubble runs off central when the tubular spirit bubble is reversed (turned through 180°). To adjust, remove half the amount the bubble has run off using the adjusting screw, then relevel, 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 crosshair
Crosshair
A reticle is a net of fine lines or fibers in the eyepiece of a sighting device, such as a telescope, a telescopic sight, a microscope, or the screen of an oscilloscope. The word reticle comes from the Latin "reticulum," meaning "net." Today, engraved lines or embedded fibers may be replaced by a...

s 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
Calibration
Calibration is a comparison between measurements – one of known magnitude or correctness made or set with one device and another measurement made in as similar a way as possible with a second device....

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.

A theodolite is mounted on its tripod
Tripod (surveying)
A surveyor's tripod is a device used to support any one of a number of surveying instruments, such as theodolites, total stations, levels or transits.- History :...

head by means of a forced centering plate or tribrach
Tribrach (instrument)
right|thumb|A tribrach with an optical plummet .In surveying science, a tribrach means an instrument attachment plate containing three thumbscrews . The device consists of two triangular metal plates, which are connected at their corners by thumbscrews...

containing four thumbscrews, or in modern theodolites, three for rapid levelling. Before use, a theodolite must be precisely placed vertical 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.

## History

The term diopter was sometimes used in old texts as a synonym for theodolite. This derives from an older astronomical instrument called a dioptra
Dioptra
A dioptra is a classical astronomical and surveying instrument, dating from the 3rd century BCE. The dioptra was a sighting tube or, alternatively, a rod with a sight at both ends, attached to a stand...

.

Prior to the theodolite, instruments such as the geometric square and various graduated circles (see circumferentor
Circumferentor
A circumferentor, or surveyor's compass, is an instrument used in surveying to measure horizontal angles, now superseded by the theodolite. See also alidade....

) and semicircles (see graphometer
Graphometer
The graphometer or semicircle is a surveying instrument used for angle measurements. It consists of a semicircular limb divided into 180 degrees and sometimes subdivided into minutes. The limb is subtended by the diameter with two sights at its ends. In the middle of the diameter a "box and needle"...

) 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
Strasbourg
Strasbourg is the capital and principal city of the Alsace region in eastern France and is the official seat of the European Parliament. Located close to the border with Germany, it is the capital of the Bas-Rhin département. The city and the region of Alsace are historically German-speaking,...

in 1512. It was described in the appendix by Martin Waldseemüller
Martin Waldseemüller
Martin Waldseemüller was a German cartographer...

, a Rhineland
Rhineland
Historically, the Rhinelands refers to a loosely-defined region embracing the land on either bank of the River Rhine in central Europe....

topographer and cartographer, who made the device in the same year. Waldseemüller called his instrument the polimetrum.
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. The etymology
Etymology
Etymology is the study of the history of words, their origins, and how their form and meaning have changed over time.For languages with a long written history, etymologists make use of texts in these languages and texts about the languages to gather knowledge about how words were used during...

of the word is unknown. The first part of the New Latin
New Latin
The term New Latin, or Neo-Latin, is used to describe the Latin language used in original works created between c. 1500 and c. 1900. Among other uses, Latin during this period was employed in scholarly and scientific publications...

theo-delitus might stem from the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

θεᾶσθαι, "to behold or look attentively upon" or θεῖν "to run", 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", or δολιχός "long", or δοῦλος "slave", or an unattested Neolatin compound combining ὁδός "way" and λιτός "plain". It has been also suggested that -delitus is a variation of the Latin supine
Supine
In grammar a supine is a form of verbal noun used in some languages.-In Latin:In Latin there are two supines, I and II . They are originally the accusative and dative or ablative forms of a verbal noun in the fourth declension, respectively. The first supine ends in -um. It has two uses. The first...

deletus, in the sense of "crossed out".

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]. Thus the name originally applied only to the azimuth instrument and only later became associated with the altazimuth instrument. The 1728 Cyclopaedia compares "graphometer
Graphometer
The graphometer or semicircle is a surveying instrument used for angle measurements. It consists of a semicircular limb divided into 180 degrees and sometimes subdivided into minutes. The limb is subtended by the diameter with two sights at its ends. In the middle of the diameter a "box and needle"...

" to "half-theodolite". 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.

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.

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
Telescope
A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation . The first known practical telescopes were invented in the Netherlands at the beginning of the 1600s , using glass lenses...

. This was first done by Jonathan Sisson in 1725.

The theodolite became a modern, accurate instrument in 1787 with the introduction of Jesse Ramsden
Jesse Ramsden
Jesse Ramsden FRSE was an English astronomical and scientific instrument maker.Ramsden was born at Salterhebble, Halifax, West Riding of Yorkshire, England. After serving his apprenticeship with a cloth-worker in Halifax, he went in 1755 to London, where in 1758 he was apprenticed to a...

's famous great theodolite, which he created using a very accurate dividing engine
Dividing engine
A dividing engine is a device specifically employed to mark graduations on measuring instruments.-History:There has always been a need for accurate measuring instruments...

of his own design. The demand could not be met by foreign theodolites due 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.
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
Edward Samuel Ritchie
Edward Samuel Ritchie , an American inventor and physicist, is considered to be the most innovative instrument maker in nineteenth-century America, making important contributions to both science and navigation.-Early life and career:...

. It was used by the U.S. Navy to take the first precision surveys of American harbors on the Atlantic and Gulf coasts. With continuing refinements, the instrument steadily evolved into the modern theodolite used by surveyors today.

## Operation in surveying

Triangulation
Triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly...

, as invented by Gemma Frisius
Gemma Frisius
Gemma Frisius , was a physician, mathematician, cartographer, philosopher, and instrument maker...

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
Jesse Ramsden
Jesse Ramsden FRSE was an English astronomical and scientific instrument maker.Ramsden was born at Salterhebble, Halifax, West Riding of Yorkshire, England. After serving his apprenticeship with a cloth-worker in Halifax, he went in 1755 to London, where in 1758 he was apprenticed to a...

, a Yorkshireman from Halifax
Halifax, West Yorkshire
Halifax is a minster town, within the Metropolitan Borough of Calderdale in West Yorkshire, England. It has an urban area population of 82,056 in the 2001 Census. It is well-known as a centre of England's woollen manufacture from the 15th century onward, originally dealing through the Halifax Piece...

, England who had developed the dividing engine
Dividing engine
A dividing engine is a device specifically employed to mark graduations on measuring instruments.-History:There has always been a need for accurate measuring instruments...

for dividing angular scales accurately to within a second of arc, was commissioned to build a new instrument for the British Ordnance Survey
Ordnance Survey
Ordnance Survey , an executive agency and non-ministerial government department of the Government of the United Kingdom, is the national mapping agency for Great Britain, producing maps of Great Britain , and one of the world's largest producers of maps.The name reflects its creation together with...

. The Ramsden theodolite
Ramsden theodolite
The Ramsden theodolite is a large theodolite that was specially constructed for use in the first Ordnance Survey of Southern Britain. It was also known as the Great or 36 inch theodolite....

was used over the next few years to map the whole of southern Britain
Kingdom of Great Britain
The former Kingdom of Great Britain, sometimes described as the 'United Kingdom of Great Britain', That the Two Kingdoms of Scotland and England, shall upon the 1st May next ensuing the date hereof, and forever after, be United into One Kingdom by the Name of GREAT BRITAIN. was a sovereign...

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-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 benchmark
Benchmark (surveying)
The term bench mark, or benchmark, originates from the chiseled horizontal marks that surveyors made in stone structures, into which an angle-iron could be placed to form a "bench" for a leveling rod, thus ensuring that a leveling rod could be accurately repositioned in the same place in the future...

must be measured precisely.

The American transit gained popularity during the 19th century with American railroad engineers pushing west. The transit replaced the railroad compass, sextant
Sextant
A sextant is an instrument used to measure the angle between any two visible objects. Its primary use is to determine the angle between a celestial object and the horizon which is known as the altitude. Making this measurement is known as sighting the object, shooting the object, or taking a sight...

and octant
Octant (instrument)
The octant, also called reflecting quadrant, is a measuring instrument used primarily in navigation. It is a type of reflecting instrument.-Etymology:...

and was distinguished by having a telescope shorter than the base arms, allowing the telescope to be vertically rotated past straight down. The transit had the ability to 'flip' over on its vertical circle and easily show the exact 180 degree sight to the user. This facilitated the viewing of long straight lines, such as when surveying the American West. Previously the user rotated the telescope on its horizontal circle to 180 and had to carefully check the angle when turning 180 degree turns.

## Modern theodolites

In today's theodolites, the reading out of the horizontal and vertical circles is usually done electronically. The readout is done by a rotary encoder
Rotary encoder
A rotary encoder, also called a shaft encoder, is an electro-mechanical device that converts the angular position or motion of a shaft or axle to an analog or digital code. The output of incremental encoders provides information about the motion of the shaft which is typically further processed...

, which can be absolute, e.g. using Gray code
Gray code
The reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system where two successive values differ in only one bit. It is a non-weighted code....

s, or incremental, using equidistant light and dark radial bands. In the latter case the circles spin rapidly, reducing angle measurement to electronic measurement of time differences. Additionally, lately CCD
Charge-coupled device
A charge-coupled device is a device for the movement of electrical charge, usually from within the device to an area where the charge can be manipulated, for example conversion into a digital value. This is achieved by "shifting" the signals between stages within the device one at a time...

sensors have been added to the focal plane of the telescope
Telescope
A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation . The first known practical telescopes were invented in the Netherlands at the beginning of the 1600s , using glass lenses...

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 \$10,000 apiece, are equipped with integrated electro-optical distance measuring devices, generally infrared
Infrared
Infrared light is electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 micrometres , and extending conventionally to 300 µm...

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
Resection (orientation)
Resection is a method for determining a position using a compass and topographic map .-Resection versus intersection:...

solution or free station position surveying and is widely used in mapping surveying. The instruments, "intelligent" theodolites called self-registering tacheometers or "total station
Total station
A total station is an electronic/optical instrument used in modern surveying. The total station is an electronic theodolite integrated with an electronic distance meter to read slope distances from the instrument to a particular point....

s", perform the necessary operations, saving data into internal registering units, or into external data storage devices. Typically, ruggedized laptops or PDA
Personal digital assistant
A personal digital assistant , also known as a palmtop computer, or personal data assistant, is a mobile device that functions as a personal information manager. Current PDAs often have the ability to connect to the Internet...

s are used as data collectors for this purpose.

## Gyrotheodolites

A gyrotheodolite is used when the north-south reference bearing of the meridian is required in the absence of astronomical star sights. This mainly occurs 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.

• Clinometer
• Dumpy level
Dumpy level
A dumpy level, builder's auto level, leveling instrument, or automatic level is an optical instrument used to establish or check points in the same horizontal plane...

• Leica Geosystems
Leica Geosystems
Leica Geosystems based in eastern Switzerland produces products and systems for surveying and geographical measurement...

• LIDAR
LIDAR
LIDAR is an optical remote sensing technology that can measure the distance to, or other properties of a target by illuminating the target with light, often using pulses from a laser...

• Macrometer
Macrometer
A macrometer is an instrument for measuring the size and distance of distant objects. Distant in this sense means a length that can not be readily measured by a calibrated length. The optical version of this instrument used two mirrors on a common sextant...

• Plane table
Plane table
A plane table is a device used in surveying and related disciplines to provide a solid and level surface on which to make field drawings, charts and maps...

• Rankine's method
Rankine's method
Rankine's method is a technique for laying out circular curves by a combination of chaining and angles at circumference, fully exploiting the theodolite and making a substantial improvement in accuracy and productivity over existing methods....

• Surveying
Surveying
See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...

• Tacheometry
Tacheometry
Tacheometry , is a system of rapid surveying, by which the positions, both horizontal and vertical, of points on the earth surface relatively to one another are determined without using a chain or tape or a separate levelling instrument....

• Total station
Total station
A total station is an electronic/optical instrument used in modern surveying. The total station is an electronic theodolite integrated with an electronic distance meter to read slope distances from the instrument to a particular point....

• Tribrach
Tribrach (instrument)
right|thumb|A tribrach with an optical plummet .In surveying science, a tribrach means an instrument attachment plate containing three thumbscrews . The device consists of two triangular metal plates, which are connected at their corners by thumbscrews...

• Tripod
Tripod (surveying)
A surveyor's tripod is a device used to support any one of a number of surveying instruments, such as theodolites, total stations, levels or transits.- History :...