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A graticule on the Earth as a sphere or an ellipsoid. The lines from pole to pole are lines of constant longitude, or meridians. The circles parallel to the Equator are circles of constant latitude, or parallels. The graticule shows the latitude and longitude of points on the surface. In this example, meridians are spaced at 6deg intervals and parallels at 4deg intervals. Division of the Earth into Gauss-Krueger zones - Globe.svg
A graticule on the Earth as a sphere or an ellipsoid. The lines from pole to pole are lines of constant longitude, or meridians. The circles parallel to the Equator are circles of constant latitude, or parallels. The graticule shows the latitude and longitude of points on the surface. In this example, meridians are spaced at 6° intervals and parallels at 4° intervals.

Longitude ( /ˈlɒnɪtjd/ , AU and UK also /ˈlɒŋɡɪ-/ ) [1] [2] is a geographic coordinate that specifies the eastwest position of a point on the Earth's surface, or the surface of a celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter lambda (λ). Meridians (lines running from pole to pole) connect points with the same longitude. The prime meridian, which passes near the Royal Observatory, Greenwich, England, is defined as 0° longitude by convention. Positive longitudes are east of the prime meridian, and negative ones are west.


Because of the earth's rotation, there is a close connection between longitude and time. Local time (for example from the position of the sun) varies with longitude, a difference of 15° longitude corresponding to a one-hour difference in local time. Comparing local time to an absolute measure of time allows longitude to be determined. Depending on the era, the absolute time might be obtained from a celestial event visible from both locations, such as a lunar eclipse, or from a time signal transmitted by telegraph or wireless. The principle is straightforward, but in practice finding a reliable method of determining longitude took centuries and required the effort of some of the greatest scientific minds.

A location's northsouth position along a meridian is given by its latitude, which is approximately the angle between the local vertical and the equatorial plane.

Longitude is generally given using the geometrical or astronomical vertical. This can differ slightly from the gravitational vertical because of small variations in Earth's gravitational field.


The concept of longitude was first developed by ancient Greek astronomers. Hipparchus (2nd century BCE) used a coordinate system that assumed a spherical earth, and divided it into 360° as we still do today. His prime meridian passed through Alexandria. [3] :31 He also proposed a method of determining longitude by comparing the local time of a lunar eclipse at two different places, thus demonstrating an understanding of the relationship between longitude and time. [3] :11. [4] Claudius Ptolemy (2nd century CE) developed a mapping system using curved parallels that reduced distortion. He also collected data for many locations, from Britain to the Middle East. He used a prime meridian through the Canary Islands, so that all longitude values would be positive. While Ptolemy's system was sound, the data he used were often poor, leading to a gross over-estimate (by about 70%) of the length of the Mediterranean. [5] [6] :551–553 [7]

After the fall of the Roman Empire, interest in geography greatly declined in Europe. [8] :65 Hindu and Muslim astronomers continued to develop these ideas, adding many new locations and often improving on Ptolemy's data. [9] [10] For example al-Battānī used simultaneous observations of two lunar eclipses to determine the difference in longitude between Antakya and Raqqa with an error of less than 1°. This is considered to be the best that can be achieved with the methods then available - observation of the eclipse with the naked eye, and determination of local time using an astrolabe to measure the altitude of a suitable "clock star". [11] [12]

In the later Middle Ages, interest in geography revived in the west, as travel increased, and Arab scholarship began to be known through contact with Spain and North Africa. In the 12th Century, astronomical tables were prepared for a number of European cities, based on the work of al-Zarqālī in Toledo. The lunar eclipse of September 12, 1178 was used to establish the longitude differences between Toledo, Marseilles, and Hereford. [13] :85

Christopher Columbus made two attempts to use lunar eclipses to discover his longitude, the first in Saona Island, on 14 September 1494 (second voyage), and the second in Jamaica on 29 February 1504 (fourth voyage). It is assumed that he used astronomical tables for reference. His determinations of longitude showed large errors of 13 and 38° W respectively. [14] Randles (1985) documents longitude measurement by the Portuguese and Spanish between 1514 and 1627 both in the Americas and Asia. Errors ranged from 2-25°. [15]

The telescope was invented in the early 17th-century. Initially an observation device, developments over the next half century transformed it into an accurate measurement tool. [16] [17] The pendulum clock was patented by Christiaan Huygens in 1657 [18] and gave an increase in accuracy of about 30 fold over previous mechanical clocks. [19] These two inventions would revolutionise observational astronomy and cartography. [20]

On land, the period from the development of telescopes and pendulum clocks until the mid 18th-Century saw a steady increase in the number of places whose longitude had been determined with reasonable accuracy, often with errors of less than a degree, and nearly always within 2-3°. By the 1720s errors were consistently less than 1°. [21] At sea during the same period, the situation was very different. Two problems proved intractable. The first was the need of a navigator for immediate results. The second was the marine environment. Making accurate observations in an ocean swell is much harder than on land, and pendulum clocks do not work well in these conditions.

In response to the problems of navigation, a number of European maritime powers offered prizes for a method to determine longitude at sea. The best-known of these is the Longitude Act passed by the British parliament in 1714. [22] :8 It offered two levels of rewards, for solutions within 1° and 0.5°. Rewards were given for two solutions: lunar distances, made practicable by the tables of Tobias Mayer [23] developed into an nautical almanac by the Astronomer Royal Nevil Maskelyne; and for the chronometers developed by the Yorkshire carpenter and clock-maker John Harrison. Harrison built five chronometers over more than three decades. This work was supported and rewarded with thousands of pounds from the Board of Longitude, [24] but he fought to receive money up to the top reward of £20,000, finally receiving an additional payment in 1773 after the intervention of parliament [22] :26. It was some while before either method became widely used in navigation. In the early years, chronometers were very expensive, and the calculations required for lunar distances were still complex and time-consuming. Lunar distances came into general use after 1790. [25] Chronometers had the advantages that both the observations and the calculations were simpler, and as they became cheaper in the early 19th-Century they started to replace lunars, which were seldom used after 1850. [26]

The first working telegraphs were established in Britain by Wheatstone and Cooke in 1839, and in the US by Morse in 1844. It was quickly realised that the telegraph could be used to transmit a time signal for longitude determination. [27] The method was soon in practical use for longitude determination, especially in North America, and over longer and longer distances as the telegraph network expanded, including western Europe with the completion of transatlantic cables. The US Coast Survey was particularly active in this development, and not just in the United States. The Survey established chains of mapped locations through Central and South America, and the West Indies, and as far as Japan and China in the years 1874–90. This contributed greatly to the accurate mapping of these areas. [28] [29]

While mariners benefited from the accurate charts, they could not receive telegraph signals while under way, and so could not use the method for navigation. This changed when wireless telegraphy became available in the early 20th-Century. [30] Wireless time signals for the use of ships were transmitted from Halifax, Nova Scotia, starting in 1907 [31] and from the Eiffel Tower in Paris from 1910. [32] These signals allowed navigators to check and adjust their chronometers on a frequent basis. [33]

Radio navigation systems came into general use after World War II. The systems all depended on transmissions from fixed navigational beacons. A ship-board receiver calculated the vessel's position from these transmissions. [34] They allowed accurate navigation when poor visibility prevented astronomical observations, and became the established method for commercial shipping until replaced by GPS in the early 1990s.


The main methods for determining longitude are listed below. With one exception (magnetic declination) they all depend on a common principle, which was to determine an absolute time from an event or measurement and to compare the corresponding local time at two different locations.

With the exception of magnetic declination, all proved practicable methods. Developments on land and sea, however, were very different.

There is no other physical principle determining longitude directly but with time. Longitude at a point may be determined by calculating the time difference between that at its location and Coordinated Universal Time (UTC). Since there are 24 hours in a day and 360 degrees in a circle, the sun moves across the sky at a rate of 15 degrees per hour (360° ÷ 24 hours = 15° per hour). So if the time zone a person is in is three hours ahead of UTC then that person is near 45° longitude (3 hours × 15° per hour = 45°). The word near is used because the point might not be at the centre of the time zone; also the time zones are defined politically, so their centres and boundaries often do not lie on meridians at multiples of 15°. In order to perform this calculation, however, a person needs to have a chronometer (watch) set to UTC and needs to determine local time by solar or astronomical observation. The details are more complex than described here: see the articles on Universal Time and on the equation of time for more details.


Longitude is given as an angular measurement ranging from 0° at the Prime Meridian to +180° eastward and −180° westward. The Greek letter λ (lambda), [35] [36] is used to denote the location of a place on Earth east or west of the Prime Meridian.

Each degree of longitude is sub-divided into 60 minutes, each of which is divided into 60 seconds. A longitude is thus specified in sexagesimal notation as 23° 27′ 30″ E. For higher precision, the seconds are specified with a decimal fraction. An alternative representation uses degrees and minutes, where parts of a minute are expressed in decimal notation with a fraction, thus: 23° 27.5′ E. Degrees may also be expressed as a decimal fraction: 23.45833° E. For calculations, the angular measure may be converted to radians, so longitude may also be expressed in this manner as a signed fraction of π (pi), or an unsigned fraction of 2π.

For calculations, the West/East suffix is replaced by a negative sign in the western hemisphere. The international standard convention (ISO 6709)—that East is positive—is consistent with a right-handed Cartesian coordinate system, with the North Pole up. A specific longitude may then be combined with a specific latitude (positive in the northern hemisphere) to give a precise position on the Earth's surface. Confusingly, the convention of negative for East is also sometimes seen, most commonly in the United States; the Earth System Research Laboratory used it on an older version of one of their pages, in order "to make coordinate entry less awkward" for applications confined to the Western Hemisphere. They have since shifted to the standard approach. [37]

Note that the longitude is singular at the Poles and calculations that are sufficiently accurate for other positions may be inaccurate at or near the Poles. Also the discontinuity at the ±180° meridian must be handled with care in calculations. An example is a calculation of east displacement by subtracting two longitudes, which gives the wrong answer if the two positions are on either side of this meridian. To avoid these complexities, consider replacing latitude and longitude with another horizontal position representation in calculation.

Length of a degree of longitude

The length of a degree of longitude (east–west distance) depends only on the radius of a circle of latitude. For a sphere of radius a that radius at latitude φ is a cos φ, and the length of a one-degree (or π/180 radian) arc along a circle of latitude is

110.574 km111.320 km
15°110.649 km107.551 km
30°110.852 km96.486 km
45°111.133 km78.847 km
60°111.412 km55.800 km
75°111.618 km28.902 km
90°111.694 km0.000 km
Length of one degree (black), minute (blue) and second (red) of latitude and longitude in metric (upper half) and imperial units (lower half) at a given latitude (vertical axis) in WGS84. For example, the green arrows show that Donetsk (green circle) at 48degN has a Dlong of 74.63 km/deg (1.244 km/min, 20.73 m/sec etc) and a Dlat of 111.2 km/deg (1.853 km/min, 30.89 m/sec etc). WGS84 angle to distance conversion.svg
Length of one degree (black), minute (blue) and second (red) of latitude and longitude in metric (upper half) and imperial units (lower half) at a given latitude (vertical axis) in WGS84. For example, the green arrows show that Donetsk (green circle) at 48°N has a Δlong of 74.63 km/° (1.244 km/min, 20.73 m/sec etc) and a Δlat of 111.2 km/° (1.853 km/min, 30.89 m/sec etc).

When the Earth is modelled by an ellipsoid this arc length becomes [38] [39]

where e, the eccentricity of the ellipsoid, is related to the major and minor axes (the equatorial and polar radii respectively) by

An alternative formula is

; here is the so-called parametric or reduced latitude.

Cos φ decreases from 1 at the equator to 0 at the poles, which measures how circles of latitude shrink from the equator to a point at the pole, so the length of a degree of longitude decreases likewise. This contrasts with the small (1%) increase in the length of a degree of latitude (north–south distance), equator to pole. The table shows both for the WGS84 ellipsoid with a = 6378137.0 m and b = 6356752.3142 m. Note that the distance between two points 1 degree apart on the same circle of latitude, measured along that circle of latitude, is slightly more than the shortest (geodesic) distance between those points (unless on the equator, where these are equal); the difference is less than 0.6 m (2 ft).

A geographical mile is defined to be the length of one minute of arc along the equator (one equatorial minute of longitude), therefore a degree of longitude along the equator is exactly 60 geographical miles or 111.3 kilometers, as there are 60 minutes in a degree. The length of 1 minute of longitude along the equator is 1 geographical mile or 1.855 km or 1.153 miles, while the length of 1 second of it is 0.016 geographical mile or 30.916 m or 101.43 feet.

See also

Related Research Articles

Latitude Geographic coordinate specifying north–south position

In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes that are used in special applications.

Mercator projection Map projection for navigational use that distorts areas far from the equator

The Mercator projection is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and south as down everywhere while preserving local directions and shapes. The map is thereby conformal. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator but accelerates with increasing latitude to become infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

Geographic coordinate system Coordinate system to specify locations on Earth

A geographic coordinate system (GCS) is a coordinate system associated with positions on Earth. A GCS can give positions:

Prime meridian A line of longitude, at which longitude is defined to be 0°

A prime meridian is the meridian in a geographic coordinate system at which longitude is defined to be 0°. Together, a prime meridian and its anti-meridian form a great circle. This great circle divides a spheroid into two hemispheres. If one uses directions of East and West from a defined prime meridian, then they can be called the Eastern Hemisphere and the Western Hemisphere.

Celestial navigation Navigation using astronomical objects to determine position

Celestial navigation, also known as astronavigation, is the ancient and modern practice of position fixing that enables a navigator to transition through a space without having to rely on estimated calculations, or dead reckoning, to know their position. Celestial navigation uses "sights", or angular measurements taken between a celestial body and the visible horizon. The Sun is most commonly used, but navigators can also use the Moon, a planet, Polaris, or one of 57 other navigational stars whose coordinates are tabulated in the nautical almanac and air almanacs.

Rhumb line Arc crossing all meridians of longitude at the same angle

In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north.

Great-circle distance

The great-circle distance, orthodromic distance, or spherical distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere. The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called great circles.

Reference ellipsoid Ellipsoid that approximates the figure of the Earth

In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, which is the truer, imperfect figure of the Earth, or other planetary body, as opposed to a perfect, smooth, and unaltered sphere, which factors in the undulations of the bodies' gravity due to variations in the composition and density of the interior, as well as the subsequent flattening caused by the centrifugal force from the rotation of these massive objects . Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which geodetic network computations are performed and point coordinates such as latitude, longitude, and elevation are defined.

Transverse Mercator projection

The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the Universal Transverse Mercator. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

Scale (map) Ratio of distance on a map to the corresponding distance on the ground

The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.

Equirectangular projection

The equirectangular projection (also called the equidistant cylindrical projection or la carte parallélogrammatique projection, and which includes the special case of the plate carrée projection, is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. The projection maps meridians to vertical straight lines of constant spacing, and circles of latitude to horizontal straight lines of constant spacing. The projection is neither equal area nor conformal. Because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. In particular, the plate carrée has become a standard for global raster datasets, such as Celestia, NASA World Wind, and Natural Earth, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth. In addition it is frequently used in panoramic photography to represent a spherical panoramic image.

Universal Transverse Mercator coordinate system

The Universal Transverse Mercator (UTM) is a system for assigning coordinates to locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, which means it ignores altitude and treats the earth as a perfect ellipsoid. However, it differs from global latitude/longitude in that it divides earth into 60 zones and projects each to the plane as a basis for its coordinates. Specifying a location means specifying the zone and the x, y coordinate in that plane. The projection from spheroid to a UTM zone is some parameterization of the transverse Mercator projection. The parameters vary by nation or region or mapping system.

Longitude by chronometer is a method, in navigation, of determining longitude using a marine chronometer, which was developed by John Harrison during the first half of the eighteenth century. It is an astronomical method of calculating the longitude at which a position line, drawn from a sight by sextant of any celestial body, crosses the observer's assumed latitude. In order to calculate the position line, the time of the sight must be known so that the celestial position i.e. the Greenwich Hour Angle and Declination, of the observed celestial body is known. All that can be derived from a single sight is a single position line, which can be achieved at any time during daylight when both the sea horizon and the sun are visible. To achieve a fix, more than one celestial body and the sea horizon must be visible. This is usually only possible at dawn and dusk.

Lunar distance (navigation) Angular distance between the Moon and another celestial body

In celestial navigation, lunar distance is the angular distance between the Moon and another celestial body. The lunar distances method uses this angle, also called a lunar, and a nautical almanac to calculate Greenwich time if so desired, or by extension any other time. That calculated time can be used in solving a spherical triangle. The theory was first published by Johannes Werner in 1524, before the necessary almanacs had been published. A fuller method was published in 1763 and used until about 1850 when it was superseded by the marine chronometer. A similar method uses the positions of the Galilean moons of Jupiter.

History of longitude

The history of longitude is a record of the effort, by astronomers, cartographers and navigators over the centuries, to discover a means of determining longitude.

Great ellipse

A great ellipse is an ellipse passing through two points on a spheroid and having the same center as that of the spheroid. Equivalently, it is an ellipse on the surface of a spheroid and centered on the origin, or the curve formed by intersecting the spheroid by a plane through its center. For points that are separated by less than about a quarter of the circumference of the earth, about , the length of the great ellipse connecting the points is close to the geodesic distance. The great ellipse therefore is sometimes proposed as a suitable route for marine navigation. The great ellipse is special case of an earth section path.

In geodesy, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length. The purpose of measuring meridian arcs is to determine a figure of the Earth.

<i>Longitude</i> (book)

Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time is a best-selling book by Dava Sobel about John Harrison, an 18th-century clockmaker who created the first clock (chronometer) sufficiently accurate to be used to determine longitude at sea—an important development in navigation. The book was made into a television series entitled Longitude. In 1998, The Illustrated Longitude was published, supplementing the earlier text with 180 images of characters, events, instruments, maps and publications.

Geographical distance Distance measured along the surface of the earth

Geographical distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic problem.

Geodesics on an ellipsoid

The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points on a curved surface, analogous to a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry.


  1. "Definition of LONGITUDE". www.merriam-webster.com. Merriam-Webster . Retrieved 14 March 2018.
  2. Oxford English Dictionary
  3. 1 2 Dicks, D.R. (1953). Hipparchus : a critical edition of the extant material for his life and works (PhD). Birkbeck College, University of London.
  4. Hoffman, Susanne M. (2016). "How time served to measure the geographical position since Hellenism". In Arias, Elisa Felicitas; Combrinck, Ludwig; Gabor, Pavel; Hohenkerk, Catherine; Seidelmann, P.Kenneth (eds.). The Science of Time. Astrophysics and Space Science Proceedings. 50. Springer International. pp. 25–36. doi:10.1007/978-3-319-59909-0_4. ISBN   978-3-319-59908-3.
  5. Mittenhuber, Florian (2010). "The Tradition of Texts and Maps in Ptolemy's Geography". In Jones, Alexander (ed.). Ptolemy in Perspective: Use and Criticism of his Work from Antiquity to the Nineteenth Century . Archimedes. 23. Dordrecht: Springer. pp.  95-119. doi:10.1007/978-90-481-2788-7_4. ISBN   978-90-481-2787-0.
  6. Bunbury, E.H. (1879). A History of Ancient Geography. 2. London: John Murray.
  7. Shcheglov, Dmitry A. (2016). "The Error in Longitude in Ptolemy's Geography Revisited". The Cartographic Journal. 53 (1): 3–14. doi:10.1179/1743277414Y.0000000098. S2CID   129864284.
  8. Wright, John Kirtland (1925). The geographical lore of the time of the Crusades: A study in the history of medieval science and tradition in Western Europe. New York: American geographical society.
  9. Ragep, F.Jamil (2010). "Islamic reactions to Ptolemy's imprecisions". In Jones, A. (ed.). Ptolemy in Perspective. Archimedes. 23. Dordrecht: Springer. doi:10.1007/978-90-481-2788-7. ISBN   978-90-481-2788-7.
  10. Tibbetts, Gerald R. (1992). "The Beginnings of a Cartographic Tradition" (PDF). In Harley, J.B.; Woodward, David (eds.). The History of Cartography Vol. 2 Cartography in the Traditional Islamic and South Asian Societies. University of Chicago Press.
  11. Said, S.S.; Stevenson, F.R. (1997). "Solar and Lunar Eclipse Measurements by Medieval Muslim Astronomers, II: Observations". Journal for the History of Astronomy. 28 (1): 29–48. Bibcode:1997JHA....28...29S. doi:10.1177/002182869702800103. S2CID   117100760.
  12. Steele, John Michael (1998). Observations and predictions of eclipse times by astronomers in the pre-telescopic period (PhD). University of Durham (United Kingdom).
  13. Wright, John Kirtland (1923). "Notes on the Knowledge of Latitudes and Longitudes in the Middle Ages". Isis. 5 (1). Bibcode:1922nkll.book.....W.
  14. Pickering, Keith (1996). "Columbus's Method of Determining Longitude: An Analytical View". The Journal of Navigation. 49 (1): 96–111. Bibcode:1996JNav...49...95P. doi:10.1017/S037346330001314X.
  15. Randles, W.G.L. (1985). "Portuguese and Spanish attempts to measure longitude in the 16th century". Vistas in Astronomy. 28 (1): 235–241. Bibcode:1985VA.....28..235R. doi:10.1016/0083-6656(85)90031-5.
  16. Pannekoek, Anton (1989). A history of astronomy. Courier Corporation. pp. 259–276.
  17. Van Helden, Albert (1974). "The Telescope in the Seventeenth Century". Isis. 65 (1): 38–58. doi:10.1086/351216. JSTOR   228880.
  18. Grimbergen, Kees (2004). Fletcher, Karen (ed.). Huygens and the advancement of time measurements. Titan - From Discovery to Encounter. Titan - from Discovery to Encounter. 1278. ESTEC, Noordwijk, Netherlands: ESA Publications Division. pp. 91–102. Bibcode:2004ESASP1278...91G. ISBN   92-9092-997-9.
  19. Blumenthal, Aaron S.; Nosonovsky, Michael (2020). "Friction and Dynamics of Verge and Foliot: How the Invention of the Pendulum Made Clocks Much More Accurate". Applied Mechanics. 1 (2): 111–122. doi: 10.3390/applmech1020008 .
  20. Olmsted, J.W. (1960). "The Voyage of Jean Richer to Acadia in 1670: A Study in the Relations of Science and Navigation under Colbert". Proceedings of the American Philosophical Society. 104 (6): 612–634. JSTOR   985537.
  21. See, for example, Port Royal, Jamaica: Halley, Edmond (1722). "Observations on the Eclipse of the Moon, June 18, 1722. and the Longitude of Port Royal in Jamaica". Philosophical Transactions. 32 (370–380): 235–236.; Buenos Aires: Halley, Edm. (1722). "The Longitude of Buenos Aires, Determin'd from an Observation Made There by Père Feuillée". Philosophical Transactions. 32 (370–380): 2–4.Santa Catarina, Brazil: Legge, Edward; Atwell, Joseph (1743). "Extract of a letter from the Honble Edward Legge, Esq; F. R. S. Captain of his Majesty's ship the Severn, containing an observation of the eclipse of the moon, Dec. 21. 1740. at the Island of St. Catharine on the Coast of Brasil". Philosophical Transactions. 42 (462): 18–19.
  22. 1 2 Siegel, Jonathan R. (2009). "Law and Longitude". Tulane Law Review. 84: 1–66.
  23. Forbes, Eric Gray (2006). "Tobias Mayer's lunar tables". Annals of Science. 22 (2): 105–116. doi:10.1080/00033796600203075. ISSN   0003-3790.
  24. "There was no such thing as the Longitude Prize". Royal Museums Greenwich. 2012-03-07. Retrieved 2021-01-27.
  25. Wess, Jane (2015). "Navigation and Mathematics: A Match Made in the Heavens?". In Dunn, Richard; Higgitt, Rebekah (eds.). Navigational Enterprises in Europe and its Empires, 1730-1850. London: Palgrave Macmillan UK. pp. 201–222. doi:10.1057/9781137520647_11. ISBN   978-1-349-56744-7.
  26. Littlehales, G.W. (1909). "The Decline of the Lunar Distance for the Determination of the Time and Longitude at". Bulletin of the American Geographical Society. 41 (2): 83–86. doi:10.2307/200792. JSTOR   200792.
  27. Walker, Sears C (1850). "Report on the experience of the Coast Survey in regard to telegraph operations, for determination of longitude &c". American Journal of Science and Arts. 10 (28): 151–160.
  28. Knox, Robert W. (1957). "Precise Determination of Longitude in the United States". Geographical Review. 47: 555–563. JSTOR   211865.
  29. Green, Francis Mathews; Davis, Charles Henry; Norris, John Alexander (1883). Telegraphic Determination of Longitudes in Japan, China, and the East Indies: Embracing the Meridians of Yokohama, Nagasaki, Wladiwostok, Shanghai, Amoy, Hong-Kong, Manila, Cape St. James, Singapore, Batavia, and Madras, with the Latitude of the Several Stations. Washington: US Hydrographic Office.
  30. Munro, John (1902). "Time-Signals by Wireless Telegraphy". Nature. 66 (1713): 416. Bibcode:1902Natur..66..416M. doi:10.1038/066416d0. ISSN   0028-0836. S2CID   4021629.
  31. Hutchinson, D.L. (1908). "Wireless Time Signals from the St. John Observatory of the Canadian Meteorological Service". Proceedings and Transactions of the Royal Society of Canada. Ser. 3 Vol. 2: 153–154.
  32. Lockyer, William J. S. (1913). "International Time and Weather Radio-Telegraphic Signals". Nature. 91 (2263): 33–36. Bibcode:1913Natur..91...33L. doi: 10.1038/091033b0 . ISSN   0028-0836. S2CID   3977506.
  33. Zimmerman, Arthur E. "The first wireless time signals to ships at sea" (PDF). antiquewireless.org. Antique Wireless Association. Retrieved 9 July 2020.
  34. Pierce, J.A. (1946). "An introduction to Loran". Proceedings of the IRE. 34 (5): 216–234. doi:10.1109/JRPROC.1946.234564. S2CID   20739091.
  35. "Coordinate Conversion". colorado.edu. Archived from the original on 29 September 2009. Retrieved 14 March 2018.
  36. "λ = Longitude east of Greenwich (for longitude west of Greenwich, use a minus sign)."
    John P. Snyder, Map Projections, A Working Manual , USGS Professional Paper 1395, page ix
  37. NOAA ESRL Sunrise/Sunset Calculator (deprecated). Earth System Research Laboratory . Retrieved October 18, 2019.
  38. Osborne, Peter (2013). "Chapter 5: The geometry of the ellipsoid". The Mercator Projections: The Normal and Transverse Mercator Projections on the Sphere and the Ellipsoid with Full Derivations of all Formulae (PDF). Edinburgh. doi:10.5281/zenodo.35392. Archived from the original (PDF) on 2016-05-09. Retrieved 2016-01-24.
  39. Rapp, Richard H. (April 1991). "Chapter 3: Properties of the Ellipsoid". Geometric Geodesy Part I. Columbus, Ohio.: Department of Geodetic Science and Surveying, Ohio State University. hdl:1811/24333.