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rigonometry edit the great circle model is applied to calculate the qibla using spherical trigonometry a branch of geometry that deals with the mathematical relations between the sides and angles of triangles formed by three great circles of a sphere as opposed to the conventional trigonometry which deals with those of a two dimensional triangle for example the qibla from the city of yogyakarta indonesia can be calculated as follows the city s coordinates ϕ displaystyle phi are 7 801389 s 110 364444 e while the kaaba s coordinates ϕ q displaystyle phi _ q are 21 422478 n 39 825183 e the longitude difference δ l displaystyle delta l is 110 364444 minus 39 825183 70 539261 substituting the values into the 3 obtains an answer of approximately 295 or 25 north of west 28 if a location o displaystyle o the kaaba q displaystyle q and the north pole n displaystyle n form a triangle on the sphere of the earth then the qibla is indicated by o q displaystyle oq which is the direction of the great circle passing through both o displaystyle o and q displaystyle q the qibla can also be expressed as an angle n o q displaystyle angle noq or q displaystyle angle q of the qibla with respect to the north also called the inhiraf al qibla this angle can be calculated as a mathematical function of the local latitude ϕ displaystyle phi the latitude of the kaaba ϕ q displaystyle phi _ q and the longitude difference between the locality and the kaaba δ l displaystyle delta l 29 this function is derived from the cotangent rule which applies to any spherical triangle with angles a displaystyle a b displaystyle b c displaystyle c and sides a displaystyle a b displaystyle b c displaystyle c cos a cos c cot b sin a cot b sin c displaystyle cos a cos c cot b sin a cot b sin c 30 1 applying this formula in the spherical triangle n o q displaystyle triangle noq substituting b q n o q displaystyle b angle q angle noq 30 and applying trigonometric identities obtain tan q sin ϕ cos δ l cos ϕ tan ϕ q sin δ l displaystyle tan q frac sin phi cos delta l cos phi tan phi _ q sin delta l or 2 q arctan sin ϕ cos δ l cos ϕ tan ϕ q sin δ l displaystyle q arctan left frac sin phi cos delta l cos phi tan phi _ q sin delta l right 29 3 this formula was derived by modern scholars but equivalent methods have been known to muslim astronomers since the 9th century 3rd century ah developed by various scholars including habash al hasib active in damascus and baghdad c 850 al nayrizi baghdad c 900 31 ibn yunus 10th 11th century 32 ibn al haytham 11th century 32 and al biruni 11th century 33 today spherical trigonometry also underlies nearly all applications or websites which calculate the qibla 23 when the qibla angle with respect to the north q displaystyle angle q is known true north needs to be known to find the qibla in practice common practical methods to find it include the observation of the shadow at the culmination of the sun when the sun crosses exactly the local meridian at this point any vertical object would cast a shadow oriented in the north south direction the result of this observation is very accurate but it requires an accurate determination of the local time of culmination as well as making the correct observation at that exact moment 34 another common method is using the compass which is more practical because it can be done at any time the disadvantage is that the north indicated by a magnetic compass differs from true north 16 35 this magnetic declination can measure up to 20 which can vary in different places on earth and changes over time 16 shadow observation edit main article qibla observation by shadows twice a year the sun passes directly above the kaaba allowing the observation of its direction from the shadow of a vertical object as observed from earth the sun appears to shift between the northern and southern tropics seasonally additionally it appears to move from east to west daily as a consequence of the earth s rotation the combination of these two apparent motions means that every day the sun crosses the meridian once usually not precisely overhead but to the north or to the south of the observer in locations between the two tropics latitudes lower than 23 5 north or south at certain moments of the year usually twice a year the sun passes almost directly overhead this happens when the sun crosses the meridian while being at the local latitude at the same time 36 the city of mecca is among the places where this occurs due to its location at 21 25 n it occurs twice a year firstly on 27 28 may at about 12 18 saudi arabia standard time sast or 09 18 utc and secondly on 15 16 july at 12 27 sast 09 27 utc 36 37 b as the sun reaches the zenith of the kaaba any vertical object on earth that receives sunlight cast a shadow that indicates the qibla see picture 36 this method of finding the qibla is called rasd al qiblat observing the qibla 38 28 since night falls on the hemisphere opposite of the kaaba half the locations on earth including australia as well as most of the americas and the pacific ocean cannot observe this directly 39 instead such places observe the opposite phenomenon when the sun passes above the antipodal point of the kaaba in other words the sun passes directly underneath the kaaba causing shadows in the opposite direction from those observed during rasd al qiblat 36 40 this occurs twice a year on 14 january 00 30 sast 21 30 utc the previous day and 29 november 00 09 sast 21 09 utc the previous day 41 observations made within five minutes of the rasd al qiblat moments or its antipodal counterparts or at the same time of the day two days before or after each event still show accurate directions with negligible difference 36 37 on the world map edit a map generated using the craig retroazimuthal projection centered on mecca unlike most map projections it preserves the direction from any other point on the map to the centre spherical trigonometry provides the shortest path from any point on earth to the kaaba even though the indicated direction might seem counterintuitive when imagined on a flat world map for example the qibla from alaska obtained through spherical trigonometry is almost due north 23 this apparent counter intuitiveness is caused by projections used by world maps which by necessity distort the surface of the earth a straight line shown by the world map in using the mercator projection is called the rhumb line or the loxodrome which is used to indicate the qibla by a minority of muslims 42 c it can result in a dramatic difference in some places for example in some parts of north america the flat map shows mecca in the southeast while the great circle calculation shows it to the northeast 23 in japan the map shows it to the southwest while the great circle shows it to the northwest 43 the majority of muslims however follow the great circle method 23 a retroazimuthal projection is any map projection which preserves the angular direction the azimuth of the great circle path from any point of the map to a point selected as the centre of the map the initial purpose of its development was to help finding the qibla by choosing the kaaba as the centre point 44 the earliest surviving works using this projection were two astrolabe shaped brass instruments created in 18th century iran 45 they contain grids covering locations between spain and china label the locations of major cities along with their names but do not show any coastline 45 46 the first of the two was discovered in 1989 its diameter is 22 5 centimetres 8 9 in and it has a ruler with which one can read the direction of mecca from the markings on the instrument s circumference and the distance to mecca from the markings on the ruler 46 d only the second one is signed by its creator muhammad husayn 47 the first formal design of a retroazimuthal projection in the western literature is the craig projection or the mecca projection created by the scottish mathematician james ireland craig who worked at the survey department of egypt in 1910 48 his map is centered in mecca and its range is limited to show the predominantly muslim lands 48 extending the map further than 90 in longitude from the centre will result in crowding and overlaps 49 50 traditional methods edit historical records and surviving old mosques show that throughout history the qibla was often determined by simple methods based on tradition or folk science not based on mathematical astronomy some early muslims used due south everywhere as the qibla literally following muhammad s instruction to face south while he was in medina mecca is due south of medina some mosques as far away as al andalus to the west and central asia to the east face south even though mecca is nowhere near that direction 51 in various places there are also the qiblas of the companions qiblat al sahaba those which were used there by the companions of the prophet the first generation of muslims who are considered role models in islam such directions continued to be used by some muslims in the following centuries side by side with other directions even after muslim astronomers began to use calculations to determine more accurate directions to mecca among the directions described as the qiblas of the companions are due south in syria and palestine 52 the direction of the winter sunrise in egypt and the direction of the winter sunset in iraq 53 the direction of the winter sunrise and sunset are also traditionally favoured because they are parallel to the walls of the kaaba 54 development of methods edit pre astronomy edit the determination of qibla has been an important problem for muslim communities throughout history muslims are required to know the qibla to perform their daily prayers and it is also needed to determine the orientation of mosques 55 when muhammad lived among the muslims in medina which like mecca is also in the hejaz region he prayed due south according to the known direction of mecca within the few generations following muhammad s death in 632 muslims had reached places far away from mecca presenting the new challenge of determining the qibla in locations far removed from those in which the earlier methods were employed 56 mathematical methods based on astronomy would be developed only around the end of the 8th century and were not initially popular therefore early muslims relied on non astronomical methods 57 there was a wide range of traditional methods in determining the qibla during this early islamic period resulting in different directions used even in the same place in addition to due south and the qiblas of the companions the arabs also knew a form of folk astronomy called so by the historian of astronomy david a king to distinguish it from the more precise mathematical astronomy originating from pre islamic traditions 53 it used natural phenomena including the observation of the sun the moon the stars and wind without any basis in mathematics these methods yield specific directions in individual localities often using the fixed setting and rising points of a specific star the sunrise or sunset at the equinoxes or at the summer or the winter solstices 58 historical sources record several such qiblas for example sunrise at the equinoxes due east in the maghreb sunset at the equinoxes due west in india the origin of the north wind or the fixed location of the north star in yemen the rising point of the star suhayl canopus in syria and the midwinter sunset in iraq 58 such directions appear in texts of fiqh islamic jurisprudence and texts of folk astronomy astronomers aside from folk astronomers typically do not comment on these methods but they were not opposed by islamic legal scholars 59 the traditional directions were still in use when methods were developed to calculate the qibla more accurately and they still appear in some surviving medieval mosques today 52 with astronomy edit a portion of the qibla table compiled by astronomer and muwaqqit shams al din al khalili of damascus in the 14th century the qibla directions are listed in the arabic sexagesimal notation the study of astronomy known as ilm al falak science of the celestial orbs in the islamic intellectual tradition began to appear in the islamic world in the second half of the 8th century centred in baghdad the principal city of the abbasid caliphate initially the science was introduced through the works of indian authors but after the 9th century the works of greek astronomers such as ptolemy were translated into arabic and became the main references in the field 60 muslim astronomers preferred greek astronomy because they considered it to be better supported by theoretical explanations and therefore it could be developed as an exact science however the influence of indian astronomy survives especially in the compilation of astronomical tables 61 this new science was applied to develop new methods of determining the qibla making use of the concept of latitude and longitude taken from ptolemy s geography as well as trigonometric formulae developed by muslim scholars 62 most textbooks of astronomy written in the medieval islamic world contain a chapter on the determination of the qibla considered one of the many things connecting astronomy with islamic law sharia 29 63 according to david a king various medieval solutions for the determination of the qibla bear witness to the development of mathematical methods from the 3rd 9th to the 8th 14th centuries and to the level of sophistication in trigonometry and computational techniques attained by these scholars 29 the first mathematical methods developed in the early 9th century were approximate solutions to the mathematical problem usually using a flat map or two dimensional geometry since in reality the earth is spherical the directions found were inexact but they were sufficient for locations relatively close to mecca including as far away as egypt and iran because the errors were less than 2 64 exact solutions based on three dimensional geometry and spherical trigonometry began to appear in the mid 9th century habash al hasib wrote an early example using an orthographic projection 65 e another group of solutions uses trigonometric formulas for example al nayrizi s four step application of menelaus s theorem 66 f subsequent scholars including ibn yunus abu al wafa ibn al haitham and al biruni proposed other methods which are confirmed to be accurate from the viewpoint of modern astronomy 67 muslim astronomers subsequently used these methods to compile tables showing the qibla from a list of locations grouped by their latitude and longitude differences from mecca the oldest known example from c 9th century baghdad contained entries for each degree and arc minute up to 20 68 in the 14th century shams al din al khalili an astronomer who served as a muwaqqit timekeeper in the umayyad mosque of damascus compiled a qibla table for 2 880 coordinates with longitude differences of up to 60 from mecca a...