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'{{Short description|Geographic coordinate that specifies the east-west position of a point on the Earth's surface}} {{about||Dava Sobel's book about John Harrison|Longitude (book){{!}}''Longitude'' (book)|the adaptation of Sobel's book| Longitude (TV series){{!}}''Longitude'' (TV series)|other uses|Longitude (disambiguation)}} {{Use British English|date=May 2021}} [[File:Division of the Earth into Gauss-Krueger zones - Globe.svg|thumb|upright=0.9|right|A [[Geographic coordinate system|graticule]] on the [[Earth]] as a [[sphere]] or an [[ellipsoid]]. The lines from pole to pole are lines of constant longitude, or [[meridian (geography)|meridians]]. The circles parallel to the [[Equator]] are circles of constant [[latitude]], or [[circle of latitude|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.]] {{longlat}}{{Geodesy|notshowfig=1}} Longitude is the measurement east or west of the prime meridian. Longitude is measured by imaginary lines that run around the Earth vertically (up and down) and meet at the North and South Poles. These lines are known as meridians. Each meridian measures one arcdegree of longitude. The distance around the Earth measures 360 degrees. '''Longitude''' ({{IPAc-en|ˈ|l|ɒ|n|dʒ|ᵻ|tj|uː|d}}, {{small|AU and UK also}} {{IPAc-en|ˈ|l|ɒ|ŋ|ɡ|ᵻ|-}})<ref>{{cite web |url=http://www.merriam-webster.com/dictionary/longitude |title=Definition of LONGITUDE |website=www.merriam-webster.com |publisher=[[Merriam-Webster]] |access-date=14 March 2018}}</ref><ref>[[Oxford English Dictionary]]</ref> is a [[geographic coordinate system|geographic coordinate]] that specifies the [[east]]–[[west]] position of a point on the [[Earth]]'s surface, or the surface of a celestial body. It is an angular measurement, usually expressed in [[degree (angle)|degrees]] and denoted by the [[Greek alphabet|Greek letter]] [[lambda]] (λ). [[meridian (geography)|Meridians]] (lines running from [[geographical pole|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 varies with longitude: a difference of 15° longitude corresponds to a one-hour difference in local time, due to the differing position in relation to the Sun. 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 radio. 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 north–south position along a meridian is given by its [[latitude]], which is approximately the angle between the normal from the ground at the location and the equatorial plane. Longitude is generally given using the [[geodetic normal]] or the [[gravity direction]]. The '''astronomical longitude''' can differ slightly from the ordinary longitude because of ''[[vertical deflection]]'', small variations in Earth's gravitational field (see also: [[astronomical latitude]]). == History == {{Main|History of longitude}} 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]].<ref name="Dicks">{{cite thesis |type=PhD|last1=Dicks |first1=D.R. |title=Hipparchus : a critical edition of the extant material for his life and works |date=1953 |publisher=Birkbeck College, University of London |url=https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.720566}}</ref>{{rp|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.{{r|Dicks|p=11}}.<ref>{{cite book |last1=Hoffman |first1=Susanne M. |title=The Science of Time |chapter=How time served to measure the geographical position since Hellenism|date=2016 |editor-last1=Arias |editor-first1=Elisa Felicitas |editor-last2=Combrinck |editor-first2=Ludwig |editor-last3=Gabor |editor-first3=Pavel |editor-last4=Hohenkerk |editor-first4=Catherine |editor-last5=Seidelmann |editor-first5=P.Kenneth |publisher=Springer International |series=Astrophysics and Space Science Proceedings|volume=50 |doi=10.1007/978-3-319-59909-0_4|pages=25–36|isbn=978-3-319-59908-3 }}</ref> [[Ptolemy|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.<ref>{{cite book |last1=Mittenhuber |first1=Florian |title=Ptolemy in Perspective: Use and Criticism of his Work from Antiquity to the Nineteenth Century|url=https://archive.org/details/ptolemyperspecti00jone |url-access=limited |chapter=The Tradition of Texts and Maps in Ptolemy's Geography |series=Archimedes |date=2010 |volume=23 |editor-last1=Jones | editor-first1=Alexander |publisher=Springer |location=Dordrecht |pages=[https://archive.org/details/ptolemyperspecti00jone/page/n106 95]-119|doi=10.1007/978-90-481-2788-7_4|isbn=978-90-481-2787-0 }}</ref><ref name="Bunbury">{{cite book |last1=Bunbury |first1=E.H. |title=A History of Ancient Geography |volume=2|date=1879 |publisher=John Murray |location=London|url=https://archive.org/details/historyofancientgeographybunburye.h.vol21879_648_H}}</ref>{{rp|551–553}}<ref>{{cite journal |last1=Shcheglov |first1=Dmitry A. |s2cid=129864284 |title=The Error in Longitude in Ptolemy's Geography Revisited |journal=The Cartographic Journal |date=2016 |volume=53 |issue=1 |pages=3–14 |doi=10.1179/1743277414Y.0000000098}}</ref> After the fall of the Roman Empire, interest in geography greatly declined in Europe.<ref name="Wright1925">{{cite book |last1=Wright |first1=John Kirtland |title=The geographical lore of the time of the Crusades: A study in the history of medieval science and tradition in Western Europe |date=1925 |publisher=American geographical society |location=New York |url=https://archive.org/details/geographicallore00wrig}}</ref>{{rp|65}} Hindu and Muslim astronomers continued to develop these ideas, adding many new locations and often improving on Ptolemy's data.<ref name="Ragep">{{cite book |last1=Ragep |first1=F.Jamil|editor-last=Jones|editor-first=A.|title=Ptolemy in Perspective |publisher=Springer |place=Dordrecht |date=2010 |chapter=Islamic reactions to Ptolemy's imprecisions |series=Archimedes|volume=23|isbn=978-90-481-2788-7 |doi=10.1007/978-90-481-2788-7|url=https://authors.library.caltech.edu/21360/}}</ref><ref name="Tibbett">{{cite book |last1=Tibbetts|first1=Gerald R.|editor1-last=Harley|editor1-first=J.B.|editor2-last=Woodward |editor2-first=David |title=The History of Cartography Vol. 2 Cartography in the Traditional Islamic and South Asian Societies |publisher=University of Chicago Press |date=1992 |chapter=The Beginnings of a Cartographic Tradition |chapter-url=https://press.uchicago.edu/books/HOC/HOC_V2_B1/HOC_VOLUME2_Book1_chapter4.pdf }}</ref> For example [[Al-Battani|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".<ref name="Said2">{{cite journal |last1=Said |first1=S.S. |last2=Stevenson |first2=F.R. |s2cid=117100760 |title=Solar and Lunar Eclipse Measurements by Medieval Muslim Astronomers, II: Observations |journal=Journal for the History of Astronomy |date=1997 |volume=28 |issue=1 |pages=29–48 |doi=10.1177/002182869702800103|bibcode=1997JHA....28...29S }}</ref><ref name="Steele">{{cite thesis |type=PhD |last=Steele|first=John Michael |date=1998 |title=Observations and predictions of eclipse times by astronomers in the pre-telescopic period |publisher=University of Durham (United Kingdom)}}</ref> 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 [[Abū Ishāq Ibrāhīm al-Zarqālī|al-Zarqālī]] in [[Toledo, Spain|Toledo]]. The lunar eclipse of September 12, 1178 was used to establish the longitude differences between Toledo, [[Marseille]]s, and [[Hereford]].<ref name="Wright1923">{{cite journal |last1=Wright |first1=John Kirtland |title=Notes on the Knowledge of Latitudes and Longitudes in the Middle Ages |journal=Isis |date=1923 |volume=5 |issue=1 |bibcode=1922nkll.book.....W |url=https://archive.org/details/wright-1923-isisacad-05acaduoft}}</ref>{{rp|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.<ref name="Pickering">{{cite journal |last1=Pickering |first1=Keith |title=Columbus's Method of Determining Longitude: An Analytical View |journal=The Journal of Navigation |date=1996 |volume=49 |issue=1 |pages=96–111 |doi=10.1017/S037346330001314X|bibcode=1996JNav...49...95P }}</ref> Randles (1985) documents longitude measurement by the Portuguese and Spanish between 1514 and 1627 both in the Americas and Asia. Errors ranged from 2° to 25°.<ref name="Randles">{{cite journal |last1=Randles |first1=W.G.L. |title=Portuguese and Spanish attempts to measure longitude in the 16th century |journal=Vistas in Astronomy |date=1985 |volume=28 |issue=1 |pages=235–241|doi=10.1016/0083-6656(85)90031-5 |bibcode=1985VA.....28..235R }}</ref> 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.<ref name="Pannekoek">{{cite book |last1=Pannekoek |first1=Anton |title=A history of astronomy |date=1989 |publisher=Courier Corporation |pages=259–276 |url=https://archive.org/details/historyofastrono0000pann}}</ref><ref name="Van Helden">{{cite journal |last1=Van Helden |first1=Albert |title=The Telescope in the Seventeenth Century |journal=Isis |date=1974 |volume=65 |issue=1 |pages=38–58 |doi=10.1086/351216 |jstor=228880 |s2cid=224838258 }}</ref> The [[pendulum clock]] was patented by [[Christiaan Huygens]] in 1657<ref name="Grimbergen">{{cite conference |last1=Grimbergen |first1=Kees |title=Huygens and the advancement of time measurements |journal=Titan - from Discovery to Encounter |conference=Titan - From Discovery to Encounter |editor-last=Fletcher | editor-first=Karen |location=ESTEC, Noordwijk, Netherlands |date=2004 |volume=1278 |pages=91–102 |publisher=ESA Publications Division |bibcode=2004ESASP1278...91G |isbn=92-9092-997-9 }}</ref> and gave an increase in accuracy of about 30 fold over previous mechanical clocks.<ref>{{cite journal |last1=Blumenthal |first1=Aaron S. |last2=Nosonovsky |first2=Michael |title=Friction and Dynamics of Verge and Foliot: How the Invention of the Pendulum Made Clocks Much More Accurate |journal= Applied Mechanics|date=2020 |volume=1 |issue=2 |pages=111–122 |doi=10.3390/applmech1020008|doi-access=free }}</ref> These two inventions would revolutionise observational astronomy and cartography.<ref name="Olmsted">{{cite journal |last1=Olmsted |first1=J.W. |title=The Voyage of Jean Richer to Acadia in 1670: A Study in the Relations of Science and Navigation under Colbert |journal=Proceedings of the American Philosophical Society |date=1960 |volume=104 |issue=6 |pages=612–634 |jstor=985537 }}</ref> 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° to 3°. By the 1720s errors were consistently less than 1°.<ref>See, for example, Port Royal, Jamaica: {{cite journal |last1=Halley |first1=Edmond |title=Observations on the Eclipse of the Moon, June 18, 1722. and the Longitude of Port Royal in Jamaica |journal=Philosophical Transactions |date=1722 |volume=32 |issue=370–380 |pages=235–236 |url=https://archive.org/details/jstor-103607}}; Buenos Aires: {{cite journal |last1=Halley |first1=Edm. |title=The Longitude of Buenos Aires, Determin'd from an Observation Made There by Père Feuillée |journal=Philosophical Transactions |date=1722 |volume=32 |issue=370–380 |pages=2–4 |url=https://archive.org/details/jstor-103565}}Santa Catarina, Brazil: {{cite journal |last1=Legge |first1=Edward |last2=Atwell |first2=Joseph |title=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 |journal=Philosophical Transactions |date=1743 |volume=42 |issue=462 |pages=18–19 |url=https://archive.org/details/jstor-104132}}</ref> 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. ===The chronometer=== 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.<ref name="Siegel">{{cite journal |last1=Siegel |first1=Jonathan R. |title=Law and Longitude |journal=Tulane Law Review |date=2009 |volume=84 |pages=1–66}}</ref>{{rp|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]]<ref name="Forbes2006">{{cite journal |last1=Forbes |first1=Eric Gray |title=Tobias Mayer's lunar tables|journal=Annals of Science |volume=22 |issue=2 |year=2006 |pages=105–116 |issn=0003-3790|doi=10.1080/00033796600203075}}</ref> 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,<ref>{{Cite web|date=2012-03-07|title=There was no such thing as the Longitude Prize|url=https://www.rmg.co.uk/discover/behind-the-scenes/blog/there-was-no-such-thing-longitude-prize|access-date=2021-01-27|website=Royal Museums Greenwich|language=en}}</ref> 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{{r|"Siegel"|p=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.<ref name="Wess2015">{{cite book|title=Navigational Enterprises in Europe and its Empires, 1730-1850 |editor1-last=Dunn |editor1-first=Richard |editor2-last=Higgitt |editor2-first=Rebekah |last1=Wess|first1=Jane|chapter=Navigation and Mathematics: A Match Made in the Heavens?|year=2015|pages=201–222|publisher=Palgrave Macmillan UK |location=London |doi=10.1057/9781137520647_11|isbn=978-1-349-56744-7 }}</ref> 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.<ref name="Littlehales">{{cite journal |last1=Littlehales |first1=G.W. |title=The Decline of the Lunar Distance for the Determination of the Time and Longitude at |journal=Bulletin of the American Geographical Society |date=1909 |volume=41 |issue=2 |pages=83–86 |doi=10.2307/200792 |jstor=200792 |url=https://archive.org/details/jstor-200792}}</ref> The first working [[Telegraphy|telegraphs]] were established in Britain by [[Charles Wheatstone|Wheatstone]] and [[William Fothergill Cooke|Cooke]] in 1839, and in the US by [[Samuel Morse|Morse]] in 1844. It was quickly realised that the telegraph could be used to transmit a time signal for longitude determination.<ref name="Walker 1850">{{cite journal |last1=Walker |first1=Sears C |title=Report on the experience of the Coast Survey in regard to telegraph operations, for determination of longitude &c. |journal=American Journal of Science and Arts |date=1850 |volume=10 |issue=28 |pages=151–160 |url=https://archive.org/details/appendix-telegraphic-longitude-the-american-journal-of-science-and-arts }}</ref> 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 [[U.S. National Geodetic Survey|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.<ref name="Knox">{{cite journal |last1=Knox |first1=Robert W. |title=Precise Determination of Longitude in the United States |journal=Geographical Review |date=1957 |volume=47 |issue=4 |pages=555–563 |doi=10.2307/211865 |jstor=211865}}</ref><ref name="Green1883">{{cite book |last1=Green |first1=Francis Mathews |last2=Davis |first2=Charles Henry |last3=Norris |first3=John Alexander |title=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 |date=1883 |publisher=US Hydrographic Office |location=Washington |url=https://archive.org/details/in.ernet.dli.2015.177254}}</ref> 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 (radio) became available in the early 20th century.<ref name="Munro1902">{{cite journal|last1=Munro|first1=John|s2cid=4021629|title=Time-Signals by Wireless Telegraphy |journal=Nature |volume=66 |issue=1713 |year=1902 |pages=416 |issn=0028-0836 |doi=10.1038/066416d0 |bibcode=1902Natur..66..416M |url=https://zenodo.org/record/2080631}}</ref> Wireless time signals for the use of ships were transmitted from [[Halifax, Nova Scotia]], starting in 1907<ref name="Hutchnson">{{cite journal |last1=Hutchinson |first1=D.L. |title=Wireless Time Signals from the St. John Observatory of the Canadian Meteorological Service. |journal=Proceedings and Transactions of the Royal Society of Canada |date=1908 |volume=Ser. 3 Vol. 2 |pages=153–154 |url=https://archive.org/details/hutchinson-1908-proceedingstrans-32roya}}</ref> and from the [[Eiffel Tower]] in Paris from 1910.<ref name="Lockyer1913">{{cite journal|last1=Lockyer|first1=William J. S.|s2cid=3977506|title=International Time and Weather Radio-Telegraphic Signals |journal=Nature |volume=91 |issue=2263 |year=1913 |pages=33–36 |issn=0028-0836 |doi=10.1038/091033b0 |bibcode=1913Natur..91...33L |doi-access=free}}</ref> These signals allowed navigators to check and adjust their chronometers frequently.<ref name="Zimmerman">{{cite web |last1=Zimmerman |first1=Arthur E. |title=The first wireless time signals to ships at sea |url=https://www.antiquewireless.org/wp-content/uploads/50-the_first_wireless_time_signals_to_ships_at_sea.pdf |website=antiquewireless.org |publisher=Antique Wireless Association |access-date=9 July 2020}}</ref> [[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.<ref name="Pierce">{{cite journal |last1=Pierce |first1=J.A. |s2cid=20739091 |title=An introduction to Loran |journal=Proceedings of the IRE |date=1946 |volume=34 |issue=5 |pages=216–234 |doi=10.1109/JRPROC.1946.234564}}</ref> They allowed accurate navigation when poor visibility prevented astronomical observations, and became the established method for commercial shipping until replaced by [[Global Positioning System|GPS]] in the early 1990s. == Determination == {{further|Celestial navigation#Longitude}} {{see also|Latitude determination}} 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. * [[Lunar distance (navigation)|Lunar distances]]. In its orbit around the Earth, the Moon moves relative to the stars at a rate of just over 0.5°/hour. The angle between the Moon and a suitable star is measured with a [[sextant]], and (after consulting tables and lengthy calculations) gives a value for absolute time. *Satellites of Jupiter. [[Galileo Galilei|Galileo]] proposed that with sufficiently accurate knowledge of the orbits of the satellites, their positions could provide a measure of absolute time. The method requires a telescope, as the moons are not visible to the naked eye. *Appulses, occultations, and eclipses. An [[appulse]] is the least apparent distance between two objects (the Moon, a star or a planet); an [[occultation]] occurs when a star or planet passes behind the Moon — essentially a type of eclipse. Lunar eclipses continued to be used. The times of any of these events can be used as the measure of absolute time. *[[Marine chronometer|Chronometers]]. A clock is set to the local time of a starting point whose longitude is known, and the longitude of any other place can be determined by comparing its local time with the clock time. *Magnetic declination. A compass needle does not in general point exactly north. The [[Magnetic declination|variation]] from true north varies with location, and it was suggested that this could provide a basis for determination of longitude. 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.{{clarify|date=November 2021}} 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 a location's [[time zone]] is three hours ahead of UTC then that location 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, one needs a [[marine chronometer|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. ==Values== Longitude is given as an [[angle|angular measurement]] ranging from 0° at the Prime Meridian to +180° eastward and −180° westward. The Greek letter λ (lambda)<ref>{{cite web|url=http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|title=Coordinate Conversion|website=colorado.edu|access-date=14 March 2018|archive-url=https://web.archive.org/web/20090929121405/http://www.colorado.edu/geography/gcraft/notes/datum/gif/llhxyz.gif|archive-date=29 September 2009|url-status=dead}}</ref><ref>"λ = Longitude east of Greenwich (for longitude west of Greenwich, use a minus sign)."<br>John P. Snyder, ''[https://pubs.er.usgs.gov/usgspubs/pp/pp1395 Map Projections, A Working Manual]'', [[USGS]] Professional Paper 1395, page ix</ref> 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 [[minute of arc|minutes]], each of which is divided into 60 [[arcsecond|seconds]]. A longitude is thus specified in [[sexagesimal]] notation as, for example, 23°&nbsp;27′&nbsp;30″&nbsp;E. For higher precision, the seconds are specified with a [[Decimal#Decimal fractions|decimal fraction]]. An alternative representation uses degrees and minutes, and parts of a minute are expressed in decimal notation, thus: 23°&nbsp;27.5′&nbsp;E. Degrees may also be expressed as a decimal fraction: 23.45833°&nbsp;E. For calculations, the angular measure may be converted to [[radian]]s, so longitude may also be expressed in this manner as a signed fraction of {{pi}} ([[pi]]), or an unsigned fraction of 2{{pi}}. 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.<ref>[https://www.esrl.noaa.gov/gmd/grad/solcalc/sunrise.html NOAA ESRL Sunrise/Sunset Calculator] (deprecated). ''[[Earth System Research Laboratory]]''. Retrieved October 18, 2019.</ref> Note that the longitude is [[mathematical singularity|singular]] at the [[Geographical pole|Poles]] and calculations that are sufficiently accurate for other positions may be inaccurate at or near the Poles. Also the [[Discontinuity (mathematics)|discontinuity]] at the ±[[180th meridian|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== {{see also|Length of a degree of latitude}} 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 {{mvar|a}} that radius at latitude {{mvar|φ}} is {{math|''a'' [[cosine|cos]] ''φ''}}, and the length of a one-degree (or {{sfrac|{{pi}}|180}} [[radian]]) arc along a circle of latitude is :<math>\Delta^1_{\rm long}= \frac{\pi}{180}a \cos \phi </math> {| class="wikitable" style="float: right; margin-left:1em; text-align:right;" !{{mvar|φ}}||{{math|Δ{{su|p=1|b=lat}}}}||{{math|Δ{{su|p=1|b=long}}}} |- | 0° || 110.574&nbsp;km || 111.320&nbsp;km |- | 15° || 110.649&nbsp;km || 107.551&nbsp;km |- | 30° || 110.852&nbsp;km || 96.486&nbsp;km |- | 45° || 111.133&nbsp;km || 78.847&nbsp;km |- | 60° || 111.412&nbsp;km || 55.800&nbsp;km |- | 75° || 111.618&nbsp;km || 28.902&nbsp;km |- | 90° || 111.694&nbsp;km || 0.000&nbsp;km |} {{WGS84_angle_to_distance_conversion.svg}} When the Earth is modelled by an [[ellipsoid]] this arc length becomes<ref name=osborne>{{Cite book |last=Osborne |first=Peter |year=2013 |url=http://www.mercator99.webspace.virginmedia.com/mercator.pdf |doi=10.5281/zenodo.35392 |title=The Mercator Projections: The Normal and Transverse Mercator Projections on the Sphere and the Ellipsoid with Full Derivations of all Formulae |chapter=Chapter 5: The geometry of the ellipsoid |location=Edinburgh |access-date=2016-01-24 |archive-url=https://web.archive.org/web/20160509180529/http://www.mercator99.webspace.virginmedia.com/mercator.pdf |archive-date=2016-05-09 |url-status=dead }}</ref><ref name=rapp>{{cite book |last=Rapp |first=Richard H. |date=April 1991 |title=Geometric Geodesy Part I |chapter=Chapter 3: Properties of the Ellipsoid |publisher=Department of Geodetic Science and Surveying, Ohio State University |location=Columbus, Ohio. |hdl=1811/24333 }}</ref> :<math>\Delta^1_{\rm long}=\frac{\pi a\cos\phi}{180 \sqrt{1 - e^2 \sin^2 \phi}}</math> where {{mvar|e}}, the eccentricity of the ellipsoid, is related to the major and minor axes (the equatorial and polar radii respectively) by :<math>e^2=\frac{a^2-b^2}{a^2}</math> An alternative formula is :<math>\Delta^1_{\rm long}= \frac{\pi}{180}a \cos \beta \quad \mbox{where }\tan \beta = \frac{b}{a} \tan \phi</math>; here <math>\beta</math> is the so-called [[Latitude#Parametric (or reduced) latitude|'''parametric''' or '''reduced''' latitude]]. cos {{mvar|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 {{mvar|a}} = {{val|6378137.0|u=m}} and {{mvar|b}} = {{val|6356752.3142|u=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 {{convert|0.6|m|ft|0|abbr=on}}. 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 {{convert|1.855|km|mi|disp=or|abbr=in}}, while the length of 1 second of it is 0.016 geographical mile or {{convert|30.916|m|ft|disp=or|abbr=in}}. ==See also== {{div col|colwidth=20em}} * [[American Practical Navigator]] * [[Cardinal direction]] * [[Dead reckoning]] * [[Ecliptic longitude]] * [[Geodesy]] * [[Geodetic system]] * [[Geographic coordinate system]] * [[Geographical distance]] * [[Geotagging]] * [[Great-circle distance]] * [[History of longitude]] * ''[[The Island of the Day Before]]'' * [[Latitude]] * [[Meridian arc]] * [[Natural Area Code]] * [[Navigation]] * [[Orders of magnitude (length)|Orders of magnitude]] * [[Planetary coordinate system#Longitude]] * [[Right ascension]] on [[celestial sphere]] * [[World Geodetic System]] {{div col end}} ==References== {{reflist}} ==Further reading== * {{Cite book|last=Andrews|first=William J. H.|title=The Quest for Longitude|location=[[Cambridge, Massachusetts]]|publisher=[[Harvard University Press]]|year=1996|isbn=978-0-9644329-0-1|oclc=59617314 |ref=none}} * {{Cite book |title=Greenwich Time and the Discovery of the Longitude |last=Howse |first=Derek |year=1980 |publisher=Philip Wilson Publishers, Ltd |isbn=978-0-19-215948-9 |url=https://archive.org/details/greenwichtimedis0000hows |ref=none}} ==External links== {{sisterlinks}} * [http://jan.ucc.nau.edu/~cvm/latlon_find_location.html Resources for determining your latitude and longitude] * [http://www.hnsky.org/iau-iag.htm IAU/IAG Working Group On Cartographic Coordinates and Rotational Elements of the Planets and Satellites] * [http://entertainment.timesonline.co.uk/tol/arts_and_entertainment/the_tls/article5136819.ece "Longitude forged"]: an essay exposing a hoax solution to the problem of calculating longitude, undetected in Dava Sobel's Longitude, from [http://www.the-tls.co.uk TLS], November 12, 2008. * [http://cudl.lib.cam.ac.uk/collections/longitude Board of Longitude Collection, Cambridge Digital Library] – complete digital version of the Board's archive * [http://www.thegpscoordinates.com Longitude And Latitude Of Points of Interest] * [http://www.csgnetwork.com/degreelenllavcalc.html Length Of A Degree Of Latitude And Longitude Calculator] * [http://articles.adsabs.harvard.edu/full/seri/RA.../0002//0000185.000.html Esame critico intorno alla scoperta di Vespucci ...] * [https://exhibits.museogalileo.it/waldseemuller/ewal.php?c%5B%5D=54117 A land beyond the stars - Museo Galileo] {{Map projection}} {{TimeSig}} {{Time Topics}} {{Time measurement and standards}} {{Authority control}} [[Category:Meridians (geography)|*]] [[Category:Navigation]] [[Category:Geodesy]]'