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62-476: A painted stick dating from 2300 BC that was excavated at the archeological site of Taosi is the oldest gnomon known in China. The gnomon was widely used in ancient China from the second millennium BC onward in order to determine the changes in seasons, orientation, and geographical latitude. The ancient Chinese used shadow measurements for creating calendars that are mentioned in several ancient texts. According to

124-484: A geographic coordinate system as defined in the specification of the ISO 19111 standard. Since there are many different reference ellipsoids , the precise latitude of a feature on the surface is not unique: this is stressed in the ISO standard which states that "without the full specification of the coordinate reference system, coordinates (that is latitude and longitude) are ambiguous at best and meaningless at worst". This

186-465: A jaw harp . A single bronze bell was also found at a Taosi grave. Several Chinese archaeologists postulate that Taosi was the site where the state of Tang ( 有唐 ) was conquered by Emperor Yao (traditionally c. 2356–2255 BC), who later instituted Taosi as the capital. In Chinese classic documents Yao Dian ( Document of Yao ) in Shang Shu ( Book of Ancient Time ), and Wudibenji ( Records for

248-560: A 300-by-300-pixel sphere, so illustrations usually exaggerate the flattening. The graticule on the ellipsoid is constructed in exactly the same way as on the sphere. The normal at a point on the surface of an ellipsoid does not pass through the centre, except for points on the equator or at the poles, but the definition of latitude remains unchanged as the angle between the normal and the equatorial plane. The terminology for latitude must be made more precise by distinguishing: Geographic latitude must be used with care, as some authors use it as

310-559: A function similar to the Thirteen Towers of the Chankillo Observatory , having been intentionally constructed for calendrical observation of the sunrise on particular given days, in order to follow the local solar calendar, which would have been crucial for rituals and also for the practice of agriculture at that time. A painted pole discovered in a tomb at the prehistoric site dating from perhaps 2000 or 2300 BCE

372-404: A larger parallelogram. Indeed, the gnomon is the increment between two successive figurate numbers , including square and triangular numbers. The ancient Greek mathematician and engineer Hero of Alexandria defined a gnomon as that which, when added or subtracted to an entity (number or shape), makes a new entity similar to the starting entity. In this sense Theon of Smyrna used it to describe

434-545: A number which added to a polygonal number produces the next one of the same type. The most common use in this sense is an odd integer especially when seen as a figurate number between square numbers . Vitruvius mentions the gnomon as " gnonomice " in the first sentence of chapter 3 in volume 1 of his book De Architectura . That Latin term " gnonomice " leaves room for interpretation. Despite its similarity to " γνωμονικός " (or its feminine form " γνωμονική "), it appears unlikely that Vitruvius refers to judgement on

496-528: A rebellion against the ruling class. Latitude In geography , latitude is a coordinate that specifies the north – south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from −90° at the south pole to 90° at the north pole, with 0° at the Equator . Lines of constant latitude , or parallels , run east–west as circles parallel to

558-443: A survey but, with the advent of GPS , it has become natural to use reference ellipsoids (such as WGS84 ) with centre at the centre of mass of the Earth and minor axis aligned to the rotation axis of the Earth. These geocentric ellipsoids are usually within 100 m (330 ft) of the geoid. Since latitude is defined with respect to an ellipsoid, the position of a given point is different on each ellipsoid: one cannot exactly specify

620-555: A synonym for geodetic latitude whilst others use it as an alternative to the astronomical latitude . "Latitude" (unqualified) should normally refer to the geodetic latitude. The importance of specifying the reference datum may be illustrated by a simple example. On the reference ellipsoid for WGS84, the centre of the Eiffel Tower has a geodetic latitude of 48° 51′ 29″ N, or 48.8583° N and longitude of 2° 17′ 40″ E or 2.2944°E. The same coordinates on

682-433: Is 42m in diameter and over 1000 sq m in area, and can be reconstructed as a three-level altar. The most important construction preserved was a semi-cirular structure of rammed earth, facing East. Depending on the interpretation, this was (a) a tall wall pierced with a number of irregularly spaced and separated slots, or (b) a series of pillars, separated by small somewhat irregular vertical spaces. This wall or line of pillars

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744-411: Is also used in the current literature. The parametric latitude is related to the geodetic latitude by: The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section. In terms of Cartesian coordinates p , the distance from the minor axis, and z , the distance above the equatorial plane, the equation of the ellipse is: The Cartesian coordinates of

806-665: Is an archaeological site in Xiangfen County , Shanxi , China . Taosi is considered to be part of the late phase of the Longshan culture in southern Shanxi, also known as the Taosi phase (2300 BC to 1900 BC). Taosi was surrounded by a gigantic rammed-clay enclosure. This was discovered from 1999 to 2001 by the archaeologists from the Institute of Archaeology , Chinese Academy of Social Sciences ; they attributed this wall to

868-597: Is associated with the 1475 placement of a bronze plate with a round hole in the dome of the Cathedral of Santa Maria del Fiore in Florence to project an image of the Sun on the cathedral's floor. With markings on the floor it tells the exact time of each midday (reportedly to within half a second) as well as the date of the summer solstice. Italian mathematician, engineer, astronomer and geographer Leonardo Ximenes reconstructed

930-590: Is commonly used in CAD and computer graphics as an aid to positioning objects in the virtual world . By convention, the x -axis direction is colored red, the y -axis green and the z -axis blue. The Gnomon of Saint-Sulpice inside the Parisian church, Église Saint-Sulpice , built to assist in determining the date of Easter , was fictionalized as a " Rose Line " in the novel The Da Vinci Code . Taosi Taosi ( Chinese : 陶寺 ; pinyin : Táosì )

992-484: Is determined by the shape of the ellipse which is rotated about its minor (shorter) axis. Two parameters are required. One is invariably the equatorial radius, which is the semi-major axis , a . The other parameter is usually (1) the polar radius or semi-minor axis , b ; or (2) the (first) flattening , f ; or (3) the eccentricity , e . These parameters are not independent: they are related by Many other parameters (see ellipse , ellipsoid ) appear in

1054-453: Is determined with the meridian altitude method. More precise measurement of latitude requires an understanding of the gravitational field of the Earth, either to set up theodolites or to determine GPS satellite orbits. The study of the figure of the Earth together with its gravitational field is the science of geodesy . The graticule is formed by the lines of constant latitude and constant longitude, which are constructed with reference to

1116-466: Is of great importance in accurate applications, such as a Global Positioning System (GPS), but in common usage, where high accuracy is not required, the reference ellipsoid is not usually stated. In English texts, the latitude angle, defined below, is usually denoted by the Greek lower-case letter phi ( ϕ or φ ). It is measured in degrees , minutes and seconds or decimal degrees , north or south of

1178-451: Is the angle between the equatorial plane and the normal to the surface at that point: the normal to the surface of the sphere is along the radial vector. The latitude, as defined in this way for the sphere, is often termed the spherical latitude, to avoid ambiguity with the geodetic latitude and the auxiliary latitudes defined in subsequent sections of this article. Besides the equator, four other parallels are of significance: The plane of

1240-421: Is the meridional radius of curvature . The quarter meridian distance from the equator to the pole is For WGS84 this distance is 10 001 .965 729  km . The evaluation of the meridian distance integral is central to many studies in geodesy and map projection. It can be evaluated by expanding the integral by the binomial series and integrating term by term: see Meridian arc for details. The length of

1302-406: Is the probably the oldest gnomon known in China. From ancient texts, we know that the gnomon was widely used in ancient China from the second century BC onward in order determine the changes in seasons,and to determine positional orientation, including geographical latitudes. The ancient Chinese used shadow measurements for creating calendars that are mentioned in several ancient texts. According to

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1364-464: Is within 1° of the north celestial pole . On some sundials, the gnomon is vertical. These were usually used in former times for observing the altitude of the Sun , especially when on the meridian . The style is the part of the gnomon that casts the shadow. This can change as the Sun moves. For example, the upper west edge of the gnomon might be the style in the morning and the upper east edge might be

1426-522: The Philosophiæ Naturalis Principia Mathematica , in which he proved that a rotating self-gravitating fluid body in equilibrium takes the form of an oblate ellipsoid. (This article uses the term ellipsoid in preference to the older term spheroid .) Newton's result was confirmed by geodetic measurements in the 18th century. (See Meridian arc .) An oblate ellipsoid is the three-dimensional surface generated by

1488-580: The zenith ). On map projections there is no universal rule as to how meridians and parallels should appear. The examples below show the named parallels (as red lines) on the commonly used Mercator projection and the Transverse Mercator projection . On the former the parallels are horizontal and the meridians are vertical, whereas on the latter there is no exact relationship of parallels and meridians with horizontal and vertical: both are complicated curves. \ In 1687 Isaac Newton published

1550-515: The 29 days of some lunar months. Most lunar months have 30 days, and thus the 29 day lunar months would have been exceptional, requiring special treatment. One could therefore link this to the observatory as well, assuming that it was also a calendrical device. The cemetery of Taosi covered an area of 30,000 square meters (3ha) at its height. The cemetery contained over 1,500 burials. The burials at Taosi were highly stratified (the most stratified of Longshan sites), with burial wealth concentrated in

1612-401: The Earth's orbit about the Sun is called the ecliptic , and the plane perpendicular to the rotation axis of the Earth is the equatorial plane. The angle between the ecliptic and the equatorial plane is called variously the axial tilt, the obliquity, or the inclination of the ecliptic, and it is conventionally denoted by i . The latitude of the tropical circles is equal to i and the latitude of

1674-509: The Five Kings ) in Shiji ( Historic Records ), King Yao assigned astronomic officers to observe celestial phenomena, including time and position of sunrise, sunset, and stars in culmination, in order to systematically establish a lunisolar calendar with 366 days a year with leap month. The observatory found at Taosi coincides with these records. It is theorized that the city collapsed with

1736-749: The Linfen basin of the Yellow River, and is possibly a regional center. The settlement represents the most politically organized system on the Central Plains at the time. The polities in the Taosi site are considered an advanced chiefdom, but may not have developed into a higher political organization. It was not the Taosi polities but the less socially complex Central Plains Longshan sites, the scattered, multi-system competing systems that gave rise to early states in this region. An astronomical observatory

1798-460: The Middle Taosi period (4,100 to 4,000 BP). Rectangular in form with an inner area of 280 ha. An internal rammed-earth wall separated the residential and ceremonial areas of the elite from the areas inhabited by commoners, signifying the development of a stratified society. The Huaxia settlement outgrew the perimeter of the wall. The settlement is the largest Longshan site discovered in

1860-691: The Sun is overhead at some point of the Tropic of Capricorn . The south polar latitudes below the Antarctic Circle are in daylight, whilst the north polar latitudes above the Arctic Circle are in night. The situation is reversed at the June solstice, when the Sun is overhead at the Tropic of Cancer. Only at latitudes in between the two tropics is it possible for the Sun to be directly overhead (at

1922-714: The Sun whose location can be measured to tell the time of day and year were described in the Chinese Zhoubi Suanjing , possibly dating as early as the early Zhou (11th century BC) but surviving only in forms dating to the Eastern Han (3rd century). In the Middle East and Europe, it was separately credited to the Egyptian astronomer and mathematician Ibn Yunus around AD 1000. The Italian astronomer, mathematician and cosmographer Paolo Toscanelli

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1984-571: The WGS84 spheroid is The variation of this distance with latitude (on WGS84 ) is shown in the table along with the length of a degree of longitude (east–west distance): A calculator for any latitude is provided by the U.S. Government's National Geospatial-Intelligence Agency (NGA). The following graph illustrates the variation of both a degree of latitude and a degree of longitude with latitude. There are six auxiliary latitudes that have applications to special problems in geodesy, geophysics and

2046-425: The angle subtended at the centre by the meridian arc from the equator to the point concerned. If the meridian distance is denoted by m ( ϕ ) then where R denotes the mean radius of the Earth. R is equal to 6,371 km or 3,959 miles. No higher accuracy is appropriate for R since higher-precision results necessitate an ellipsoid model. With this value for R the meridian length of 1 degree of latitude on

2108-584: The centre of the Earth and perpendicular to the rotation axis intersects the surface at a great circle called the Equator . Planes parallel to the equatorial plane intersect the surface in circles of constant latitude; these are the parallels. The Equator has a latitude of 0°, the North Pole has a latitude of 90° North (written 90° N or +90°), and the South Pole has a latitude of 90° South (written 90° S or −90°). The latitude of an arbitrary point

2170-519: The collection of Zhou Chinese poetic anthologies Classic of Poetry , one of the distant ancestors of King Wen of the Zhou dynasty used to measure gnomon shadow lengths to determine orientation around the 14th-century BC. In a tomb at the site, a copper object resembling a gear was also discovered. All lunar months always have an integer-dependent number of days, since the half-days of lunar months do not exist in practice. The 29 open spaces might match

2232-499: The collection of Zhou Chinese poetic anthologies Classic of Poetry , one of the distant ancestors of King Wen of the Zhou dynasty used to measure gnomon shadow lengths to determine the orientation around the 14th century BC. The ancient Greek philosopher Anaximander (610–546 BC) is credited with introducing this Babylonian instrument to the Ancient Greeks. The ancient Greek mathematician and astronomer Oenopides used

2294-408: The datum ED50 define a point on the ground which is 140 metres (460 feet) distant from the tower. A web search may produce several different values for the latitude of the tower; the reference ellipsoid is rarely specified. The length of a degree of latitude depends on the figure of the Earth assumed. On the sphere the normal passes through the centre and the latitude ( ϕ ) is therefore equal to

2356-529: The definitions of latitude and longitude. In the first step the physical surface is modeled by the geoid , a surface which approximates the mean sea level over the oceans and its continuation under the land masses. The second step is to approximate the geoid by a mathematically simpler reference surface. The simplest choice for the reference surface is a sphere , but the geoid is more accurately modeled by an ellipsoid of revolution . The definitions of latitude and longitude on such reference surfaces are detailed in

2418-402: The ellipsoid to that point Q on the surrounding sphere (of radius a ) which is the projection parallel to the Earth's axis of a point P on the ellipsoid at latitude ϕ . It was introduced by Legendre and Bessel who solved problems for geodesics on the ellipsoid by transforming them to an equivalent problem for spherical geodesics by using this smaller latitude. Bessel's notation, u ( ϕ ) ,

2480-517: The equator. For navigational purposes positions are given in degrees and decimal minutes. For instance, The Needles lighthouse is at 50°39.734′ N 001°35.500′ W. This article relates to coordinate systems for the Earth: it may be adapted to cover the Moon, planets and other celestial objects ( planetographic latitude ). For a brief history, see History of latitude . In celestial navigation , latitude

2542-452: The equator. Latitude and longitude are used together as a coordinate pair to specify a location on the surface of the Earth. On its own, the term "latitude" normally refers to the geodetic latitude as defined below. Briefly, the geodetic latitude of a point is the angle formed between the vector perpendicular (or normal ) to the ellipsoidal surface from the point, and the plane of the equator . Two levels of abstraction are employed in

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2604-438: The following sections. Lines of constant latitude and longitude together constitute a graticule on the reference surface. The latitude of a point on the actual surface is that of the corresponding point on the reference surface, the correspondence being along the normal to the reference surface, which passes through the point on the physical surface. Latitude and longitude together with some specification of height constitute

2666-399: The geocentric latitude ( θ ) and the geodetic latitude ( ϕ ) is: For points not on the surface of the ellipsoid, the relationship involves additionally the ellipsoidal height h : where N is the prime vertical radius of curvature. The geodetic and geocentric latitudes are equal at the equator and at the poles but at other latitudes they differ by a few minutes of arc. Taking the value of

2728-508: The gnomon according to his new measurements in 1756. In the Northern Hemisphere , the shadow-casting edge of a sundial gnomon is normally oriented so that it points due northward and is parallel to the rotational axis of Earth . That is, it is inclined to the northern horizon at an angle that equals the latitude of the sundial's location. At present, such a gnomon should thus point almost precisely at Polaris , as this

2790-455: The graves of a few males (nine large graves). The largest graves were placed in separated rooms with murals, had a large cache of grave goods (some with over 200 objects, including jades, copper bells, wooden and crocodile skin musical instruments); middle-size graves featured painted wooden coffins and luxury objects; most of the small graves did not have grave goods. Musical instruments have been unearthed at Taosi, including drums, chimes, and

2852-451: The latitude and longitude of a geographical feature without specifying the ellipsoid used. Many maps maintained by national agencies are based on older ellipsoids, so one must know how the latitude and longitude values are transformed from one ellipsoid to another. GPS handsets include software to carry out datum transformations which link WGS84 to the local reference ellipsoid with its associated grid. The shape of an ellipsoid of revolution

2914-538: The meridian arc between two given latitudes is given by replacing the limits of the integral by the latitudes concerned. The length of a small meridian arc is given by When the latitude difference is 1 degree, corresponding to ⁠ π / 180 ⁠ radians, the arc distance is about The distance in metres (correct to 0.01 metre) between latitudes ϕ {\displaystyle \phi }  − 0.5 degrees and ϕ {\displaystyle \phi }  + 0.5 degrees on

2976-403: The one hand or to the design of sundials on the other. It appears to be more appropriate to assume that he refers to geometry, a science upon which gnomons rely heavily. In those days, calculations were carried out geometrically, in contrast to the algebraic methods in use today. Thus, it seems that he indirectly refers to mathematics and geodesy . Perforated gnomons projecting a pinhole image of

3038-400: The phrase drawn gnomon-wise to describe a line drawn perpendicular to another. Later, the term was used for an L -shaped instrument like a steel square used to draw right angles. This shape may explain its use to describe a shape formed by cutting a smaller square from a larger one. Euclid extended the term to the plane figure formed by removing a similar parallelogram from a corner of

3100-463: The polar circles is its complement (90° - i ). The axis of rotation varies slowly over time and the values given here are those for the current epoch . The time variation is discussed more fully in the article on axial tilt . The figure shows the geometry of a cross-section of the plane perpendicular to the ecliptic and through the centres of the Earth and the Sun at the December solstice when

3162-509: The reference ellipsoid to the plane or in calculations of geodesics on the ellipsoid. Their numerical values are not of interest. For example, no one would need to calculate the authalic latitude of the Eiffel Tower. The expressions below give the auxiliary latitudes in terms of the geodetic latitude, the semi-major axis, a , and the eccentricity, e . (For inverses see below .) The forms given are, apart from notational variants, those in

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3224-473: The rotation axis of the Earth. The primary reference points are the poles where the axis of rotation of the Earth intersects the reference surface. Planes which contain the rotation axis intersect the surface at the meridians ; and the angle between any one meridian plane and that through Greenwich (the Prime Meridian ) defines the longitude: meridians are lines of constant longitude. The plane through

3286-417: The rotation of an ellipse about its shorter axis (minor axis). "Oblate ellipsoid of revolution" is abbreviated to 'ellipsoid' in the remainder of this article. (Ellipsoids which do not have an axis of symmetry are termed triaxial .) Many different reference ellipsoids have been used in the history of geodesy . In pre-satellite days they were devised to give a good fit to the geoid over the limited area of

3348-519: The semi-major axis and the inverse flattening, ⁠ 1 / f ⁠ . For example, the defining values for the WGS84 ellipsoid, used by all GPS devices, are from which are derived The difference between the semi-major and semi-minor axes is about 21 km (13 miles) and as fraction of the semi-major axis it equals the flattening; on a computer monitor the ellipsoid could be sized as 300 by 299 pixels. This would barely be distinguishable from

3410-420: The sphere is 111.2 km (69.1 statute miles) (60.0 nautical miles). The length of one minute of latitude is 1.853 km (1.151 statute miles) (1.00 nautical miles), while the length of 1 second of latitude is 30.8 m or 101 feet (see nautical mile ). In Meridian arc and standard texts it is shown that the distance along a meridian from latitude ϕ to the equator is given by ( ϕ in radians) where M ( ϕ )

3472-403: The squared eccentricity as 0.0067 (it depends on the choice of ellipsoid) the maximum difference of ϕ − θ {\displaystyle \phi {-}\theta } may be shown to be about 11.5 minutes of arc at a geodetic latitude of approximately 45° 6′. The parametric latitude or reduced latitude , β , is defined by the radius drawn from the centre of

3534-458: The standard reference for map projections, namely "Map projections: a working manual" by J. P. Snyder. Derivations of these expressions may be found in Adams and online publications by Osborne and Rapp. The geocentric latitude is the angle between the equatorial plane and the radius from the centre to a point of interest. When the point is on the surface of the ellipsoid, the relation between

3596-472: The study of geodesy, geophysics and map projections but they can all be expressed in terms of one or two members of the set a , b , f and e . Both f and e are small and often appear in series expansions in calculations; they are of the order ⁠ 1 / 298 ⁠ and 0.0818 respectively. Values for a number of ellipsoids are given in Figure of the Earth . Reference ellipsoids are usually defined by

3658-520: The style in the afternoon. Gnomons have been used in space missions to the Moon and Mars. The gnomon used by the Apollo astronauts was a gimballed stadia rod mounted on a tripod. While the rod's shadow indicated the direction of the Sun, the grayscale paints of varying reflectivity and the red, green and blue patches facilitated proper photography on the surface on the Moon. MarsDials have been used on Mars Exploration Rovers . A three-dimensional gnomon

3720-407: The theory of map projections: The definitions given in this section all relate to locations on the reference ellipsoid but the first two auxiliary latitudes, like the geodetic latitude, can be extended to define a three-dimensional geographic coordinate system as discussed below . The remaining latitudes are not used in this way; they are used only as intermediate constructs in map projections of

3782-491: Was also partially preserved at Taosi, the oldest in East Asia. This was discovered in 2003-2004. Archaeologists unearthed a Middle Taosi period semi-circular foundation just beside the southern wall of the Middle Taosi enclosure, which could have been used for calendrical observations. The structure consists of an outer semi-ring-shaped path, and a semi-circular rammed-earth platform with a diameter of about 60 m. The platform

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3844-541: Was linked to a central position from which observations could be made by peering through the empty spaces. Standing in the center of the altar and looking out, one finds that most of slots are oriented toward a given point on the Chongfen Mountain to the East. In ancient times, sunrises related to the winter and summer solstices might have been visible through different slots. This means these slots might share

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