Muslim Contributions to Mathematics
Astronomical Motion • Systems & Numbers • Accuracy & Precision • Trigonometry (Functions & Tables • Identities) • Algebra & Algebraic Theory (Differential & Integral Calculus) • Geometry (Spherical Triangles • Number Theory & Summation Series) • Navigational, Celestial & Planetary Calculators • List of Mathematicians • See also • Sources • (Footnotes • References • Acknowledgements) • External Links • Total Inventions & Discoveries Listed: 52 
The Islamic Golden Age was brought about by three dynasties; the Abbasids (750—1258^{[1]}), the Fatimids (909—1171^{[2]}) and the Umayyads of Cordoba (929—1031^{[3]}).^{[4]} The Abbasids were directly descended from Muhammad's (571—632^{[5]}) bloodline through his uncle,^{[6]}^{[7]} the Fatimids were descended from the daughter of Muhammad, Fatimah (605—633^{[8]}),^{[4]} and the Umayyad's claimed no direct ancestry.^{[9]} The longest of these periods was that of the Abbasids, who were pivotal in leading two crucial revolutions, one military and the other intellectual (particularly following the Battle of Talas (751) which was hugely influential in spreading the technology of papermaking throughout the world—which until then, had remained a tightly guarded secret; although this is somewhat disputed as some evidence suggests that it was already known to the Muslims). The rise of the Islamic Golden Age thus begins in 750 and ends in 1258, when all the dynasties had ceased to exist. The end of the Abbasid dynasty was particularly traumatic, as the Mongols (who were largely illiterate^{[10]}) devastated the capital of Baghdad,^{[11]} destroying much of it's intellectual and historical heritage, and incorporated it into the Mongol Empire (the only people spared were Nestorian Christians, who at the request of Hulagu Khan's (1218—1265^{[12]}) Christian wife, Dokuz Khatun (d. 1265^{[13]}), asked only for them not to be massacred^{[14]}). Many Armenians (12,000 cavalrymen and 40,000 infantrymen^{[15]})^{[16]} and Georgians also participated in the massacre—of genocidal proportion—murdering between 800,000—2,000,000 people.^{[17]} Khan died several years after the siege and was buried on Shahi Island, Iran, along with his wealth that is yet to be found.^{[18]} The Armenian Kingdom of Cilicia (1198—1375^{[19]}) itself ceased to exist in 1375,^{[20]} when the Mamluk Empire (1250—1517^{[21]}) conquered it.^{[22]}

Working Title:List of Inventions and Discoveries in Mathematics During the Islamic Golden Age  Original Publisher: Materia Islamica  Publication Date: February 14th, 2020  Written by: Canadian786  Article N^{o.} 95. 
Article Methodology
 In accordance with the principle of peerreview and the hierarchy of evidence and relevance of research papers, this list has been compiled using the best evidence available and has been taken from a range of scientific and historical research databases. The majority of the sources hence can be easily found using the DOI numbers (similar to an ISBN number for books, except which these unique identifiers identify research papers which have undergone thorough checks and rechecks through academic scholarship). This article consists of a List of Inventions and Discoveries in Mathematics During the Islamic Golden Age. This is by no means an exhaustive list, and thus should be considered incomplete.
Historical Context
The historiography of Muslim science can be summarised by the Uni. of St. Andrews; it is important to remember this when reading through this article;
 Quote: "Recent research paints a new picture of the debt that we owe to Arabic/Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects the mathematics studied today is far closer in style to that of the Arabic/Islamic contribution than to that of the Greeks".^{[23]}^{[24]} This is especially stark when comparing the two.
It is important to also mention that there has been a deliberate historical narrative to eliminate^{[25]} any mention of Muslims in Europe's scientific history;
 Quote: "There is a widely held view that, after a brilliant period for mathematics when the Greeks laid the foundations for modern mathematics, there was a period of stagnation before the Europeans took over where the Greeks left off at the beginning of the sixteenth century. The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century. That such views should be generally held is of no surprise [since it is a widely held view]".^{[23]}^{[24]}
Indeed, discrimination against Muslims by Western chroniclers and historians has been so rampant throughout European historiography that;
 Quote: "Many leading historians of mathematics have contributed to the perception by either omitting any mention of Arabic/Islamic mathematics in the historical development of the subject or with statements such as that made by Duhem" who stated "Arabic science only reproduced the teachings received from Greek science", when this isn't true.^{[23]}^{[24]} Only recently have scholars begun to challenge this longheld Eurocentric viewpoint.
Systems & Numbers (2)
 This is by no means an exhaustive list, and thus should be considered incomplete.
 Ghubar Numerals—The modern numerical system which consists of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, also known as "ghubar numerals" ("ghubar" meaning "abacus" and not simply "dust" as is erroneously believed^{[28]}), "Western Arabic numerals", "Arabic numerals" or "Islamic numerals" (not to be confused with the "Eastern Arabic numerals"—which are also confusingly known as "Hindu numerals"^{[28]}—which are represented as ١,٢,٣,٤,٥,٦,٧,٨,٩;^{[29]} despite the fact that these newer glyphs were invented by the Eastern Arabs and not Hindus; indeed the last true "Hindu" numerals date to the Gwailor period of India in the 9th century—see righthand image^{[26]}) were first invented in the 9th century in Muslim Spain.^{[28]} The ghubar numerals evolved from the primitive numbers developed on the Indian subcontinent (much like how English branched off from earlier IndoEuropean languages).
 The fact that this number system is sometimes also called "HinduArabic" is erroneous. The ghubar numerals appear totally different to those from India, and shouldn't really be called "Hindu" at all. According to the "National Institute of Sciences of India" itself, the "Ghubar numerals resemble modern European numerals much more closely than do the Hindu numerals".^{[30]} This is actually quite obvious when tracing their history; see right handimage for example. This otherwise would be akin to erroneously calling the English language "German" just because it is directly related in the vast tree of the 448 IndoEuropean languages.^{[31]}
 The earliest documents in history which depict ghubar numerals are from two surviving manuscripts; one dated 874, and the other 888; and curiously the oldest European documents to depict these numerals is a Latin manuscript written by Gerbert (of Aurillac; 930/946—1003^{[28]}^{[32]}) in 976 (however this latter source described them without reference to the number zero).^{[28]}
 Regarding the etymological origins of the name "ghubar", historians have noted "[t]he ghubar, originated in Spain, is not an original Arabic term alluding to the Hindu custom of writing on the sand, but it is an attempt to arabicise the Roman term abacus. When the Arabs arrived in Spain, they learned to know the abacus and the abacus numerals; thus they rendered the word abacus by ghubar, and the numerals found on the abacus were called huruifalghubdr, the abacus numerals".^{[28]}
 Interestingly, Gerbert later became Pope, and was Christened with the name Pope Sylvester II^{[32]} (reigning as head of the Catholic church from 999 to 1003^{[33]}). As a student, prior to becoming pope, he was taught the numbers by his Arab teachers, when he was residing in Spain for his education.^{[32]} The Muslim Arabs had originally conquered Spain in 711 and flourished until 1492.^{[34]} Interestingly, Pope Sylvester II was a curious character himself; his death was said to have been particularly horrendous; caused as a result of him having "made a pact with the devil", for which he was punished by dismemberment; but oddly at his own request as payment for his repentance to God, which itself was said to have been "lastminute".^{[35]}
 A Note on the History and Invention of 0 (Zero)—The worlds oldest representation of the number zero (i.e. the symbol for "0") is to be found in ancient Pakistan.^{[36]}^{[37]} Evidence for this lies in an ancient manuscript called the "Bakhshali",^{[36]} which was discovered by chance by a farmer tending to his fields in 1881 during the reign of the British Raj (1858—1947^{[38]}), and subsequently brought to the attention of the British Indologist A. F. R. Hoernle (1841—1918^{[39]}).^{[36]} The name of the manuscript itself, "Bakhshali", is named after the village from where it was found, 80 km^{[40]} from Peshawar.^{[36]} It was then transferred over to the United Kingdom in 1902 for safekeeping, and has since lain dormant at the University of Oxford (or more specifically at it's prestigious Bodleian Library).^{[36]}^{[37]}
 Interestingly, the famous South Asian mathematician, Brahmagupta (598—668^{[41]}), was born in (or near^{[42]}^{[43]}) Multan, Punjab (modernday Pakistan),^{[44]}^{[45]}^{[46]}^{[47]}^{[48]} or Bhillamala, Sind (modernday Sindh, Pakistan).^{[49]}^{[50]}^{[51]} Others claim Abu, Rajasthan (modernday India).^{[48]}^{[52]}
 In the history of mathematics however he is known as "Brahmagupta of Multan" or "Brahmagupta of Sind", who is known to have popularised the use of zero hundreds of years after its conception.^{[44]}^{[53]} However, at some point he moved to Ujjain (modernday Madhya Pradesh) where he lived and worked.^{[52]} To the Chinese he was known as "Pilomolo".^{[49]} His contributions themselves were popularised by alBiruni (973—1048).^{[54]}
 Prior to the discovery of the manuscript, it was thought the earliest depiction of zero as a symbol was first developed in modernday India, as the earliest use was found in a 9th century inscription on the wall of a temple in Gwalior, Madhya Pradesh.^{[36]} According to the University of Oxford, "[t]he findings are highly significant for the study of the early history of mathematics" as the newer evidence predates the use of zero by at least five centuries. ^{[36]}
 One of the reasons why this wasn't known until only recently is perhaps because the manuscripts used to depict zero were made out of birch bark—a natural biodegradable perishable material that was extensively used in the medieval age (especially in ancient Buddhist Pakistan) prior to the invention of paper in China and its subsequent spread by the Sogdians (Iranian nomads) and Arabs (Abbasids) in the 8th century, after the Battle of Talas (751).
 The concept of zero however may be even older; the Babylonians "used a character for the absence of [a] number" and "they made use of a primitive kind of place value" but "they did not create a system of numeration in which the zero played any such part as it does in the one which we now use".^{[55]}
 Interestingly, the famous South Asian mathematician, Brahmagupta (598—668^{[41]}), was born in (or near^{[42]}^{[43]}) Multan, Punjab (modernday Pakistan),^{[44]}^{[45]}^{[46]}^{[47]}^{[48]} or Bhillamala, Sind (modernday Sindh, Pakistan).^{[49]}^{[50]}^{[51]} Others claim Abu, Rajasthan (modernday India).^{[48]}^{[52]}
 A Note on the History and Invention of 0 (Zero)—The worlds oldest representation of the number zero (i.e. the symbol for "0") is to be found in ancient Pakistan.^{[36]}^{[37]} Evidence for this lies in an ancient manuscript called the "Bakhshali",^{[36]} which was discovered by chance by a farmer tending to his fields in 1881 during the reign of the British Raj (1858—1947^{[38]}), and subsequently brought to the attention of the British Indologist A. F. R. Hoernle (1841—1918^{[39]}).^{[36]} The name of the manuscript itself, "Bakhshali", is named after the village from where it was found, 80 km^{[40]} from Peshawar.^{[36]} It was then transferred over to the United Kingdom in 1902 for safekeeping, and has since lain dormant at the University of Oxford (or more specifically at it's prestigious Bodleian Library).^{[36]}^{[37]}
 Arithmetic, Pen and Paper Algorithms—When the Indian numerals gradually evolved into the newly invented Arabic numerals, a new method of calculation was also invented to make them even easier to use. It is important here then to distinguish between the dustboard algorithms that the old Indian system used versus the newly devised penandpaper algorithms developed by the Arabs. The dust board algorithms were "gradually replaced by algorithms performed with pen and paper. The earliest text to describe these new methods was al Uqlidisi's Arithmetic, written in Baghdad in 945 C.E.; in it the author argues for penandpaper techniques so that arithmetics could be distinguished from the dust boardwielding astrologers".^{[56]}
 The dust board algorithms from India were written down by alKhwarizmi, but there was significant resistance to the adoption of the new numerical system largely because of how difficult people found it.^{[57]} Indeed, "[c]ertainly the fact that the Indian system required a dust board had been one of the main obstacles to its acceptance".^{[57]} Others at the time testified that "[o]fficial scribes nevertheless avoid using [the Indian system] because it requires equipment [like a dust board] and they consider that a system that requires nothing but the members of the body is more secure and more fitting to the dignity of a leader".^{[57]} The crucial step forward was then developed by Abu'l Hasan Ahmad ibn Ibrahim alUqlidisi (920—980^{[58]}) when he invented the penandpaper algorithms (indeed, it was he "who gave algorithms for use with pen and paper, as opposed to those of alKhwarizmi which were for dust board").^{[59]} AlUqlidisi was therefore the first to invent penandpaper algorithms which are still used today.^{[56]}^{[n. 1]}
 It should be noted that the dust board's physical appearance itself is not known (a modern statue of the Indian mathematician Bhaskara II (1114—1185^{[60]}) depicts him holding a board on his knees but historians have noted it "would have been difficult to perform computations in this way" with other historians speculating the dust board could have been a "sort of laptop blackboard"); but what is known is that it's computation technique (it's algorithms) were not the same as those created by alUqlidisi since he clearly distinguishes the old techniques from the new in his book, "Kitab alfusul fi alhisab alHindi", where he directly states he presented the "airthmatic of the Indians that has been done on the takht [dust board], but here with no takht and no erasing; we carry it out on a sheet of paper, thus dispensing with the dust and the board".^{[61]}
 In India itself, the first computations without rubbing out appeared in the 15th century, several centuries after alUqlidisi.^{[61]}
 It should be noted that the dust board's physical appearance itself is not known (a modern statue of the Indian mathematician Bhaskara II (1114—1185^{[60]}) depicts him holding a board on his knees but historians have noted it "would have been difficult to perform computations in this way" with other historians speculating the dust board could have been a "sort of laptop blackboard"); but what is known is that it's computation technique (it's algorithms) were not the same as those created by alUqlidisi since he clearly distinguishes the old techniques from the new in his book, "Kitab alfusul fi alhisab alHindi", where he directly states he presented the "airthmatic of the Indians that has been done on the takht [dust board], but here with no takht and no erasing; we carry it out on a sheet of paper, thus dispensing with the dust and the board".^{[61]}
 The dust board algorithms from India were written down by alKhwarizmi, but there was significant resistance to the adoption of the new numerical system largely because of how difficult people found it.^{[57]} Indeed, "[c]ertainly the fact that the Indian system required a dust board had been one of the main obstacles to its acceptance".^{[57]} Others at the time testified that "[o]fficial scribes nevertheless avoid using [the Indian system] because it requires equipment [like a dust board] and they consider that a system that requires nothing but the members of the body is more secure and more fitting to the dignity of a leader".^{[57]} The crucial step forward was then developed by Abu'l Hasan Ahmad ibn Ibrahim alUqlidisi (920—980^{[58]}) when he invented the penandpaper algorithms (indeed, it was he "who gave algorithms for use with pen and paper, as opposed to those of alKhwarizmi which were for dust board").^{[59]} AlUqlidisi was therefore the first to invent penandpaper algorithms which are still used today.^{[56]}^{[n. 1]}
Accuracy & Precision (4)
 This is by no means an exhaustive list, and thus should be considered incomplete.
 Pi (π) to 16 Decimal Places (^{[62]})—Ghiyath alDin Jamshid Masʿud alKashi (AlKhashi/AlKashani;^{[63]} c. 1380—c. 1429^{[64]}) was the first in history to compute pi to 16 decimal places (other sources claim it was 14;^{[65]}^{[66]} but these sources are notably prior to the advent of the popularity of the internet and sources such as Google Scholar; being written in 2000 and 1996 respectively when not enough information was available online).^{[67]}^{[68]} He was the assistant to Ulugh Beg (1394—1449^{[69]}), the Royal Astronomer of Persia.^{[70]} He did this by initially computing the value of 2π (leading to the result ), which was the best up until 1700.^{[71]}
 Axial Tilt—Abu Mahmud Hamid ibn Khidr Khojandi (Khujundi; 900—1000^{[73]}) was the first to discover that the axial tilt of the earth is always in a state of change (during his time found it was on the decline), and not a constant value as was previously believed.^{[73]} The earth's axial tilt varies between 21.5° and 24.5°.^{[74]} This is important because the tilt influences the amount of sunlight which reaches the poles in the summer.^{[74]}
 As of 2016, the value is 23.5°.^{[74]} In a "staggering" feat, he was the first to calculate it accurately to modern standards (23.3219° vs. 23.34°).^{[73]}
 Axial Tilt—Abu Mahmud Hamid ibn Khidr Khojandi (Khujundi; 900—1000^{[73]}) was the first to discover that the axial tilt of the earth is always in a state of change (during his time found it was on the decline), and not a constant value as was previously believed.^{[73]} The earth's axial tilt varies between 21.5° and 24.5°.^{[74]} This is important because the tilt influences the amount of sunlight which reaches the poles in the summer.^{[74]}
 Mediterranean Length—Abu Ishaq Ibrahim ibnYahya al Naqqash ibn alZarqal(a/i; Arzachel; 1029—1087^{[75]}/1100^{[76]}) was the first to discover the true length of the Mediterranean sea. Although he was not the first to attempt calculating it, earlier attempts were wildly inaccurate.^{[77]} For centuries, most had accepted the value Ptolemy (fl. 127—145^{[78]}) had calculated which was 62°, however this was very wrong.^{[77]}^{[79]} Muhammad ibn Musa alKhwarizmi (alKhwarizni; 780—850^{[80]}) attempted to correct this, arriving at a value of 52°.^{[77]}^{[79]} The length was actually 42° and was first found by Arzachel.^{[77]}^{[79]}
 Solar Apogee—Abu Ishaq Ibrahim ibnYahya al Naqqash ibn alZarqal(a/i; Arzachel; 1029—1087^{[75]}/1100^{[76]}) was the first to calculate the motion of the solar apogee with reference to a star.^{[81]} The apogee is "an endpoint of the line that passe[s] through the center of the sun's orb and the center of the earth".^{[82]} The Greeks had a primitive way of calculating it but this relied only on solstices and equinoxes (not stars);^{[n. 3]} which was timeconsuming.^{[82]} Muslim scientists found a way which did not depend on either.^{[82]} They instead found that four observations of the sun's position separated by 90° was enough to calculate it.^{[82]}
Trigonometric Functions (7)
 This is by no means an exhaustive list, and thus should be considered incomplete.
 Secant ()—Abu ʿAbd Allah Muḥammad ibn Jabir ibn Sinan alRaqqi alḤarrani asSabiʾ alBattani (Albategnius; 858—929^{[85]}) was the first to discover the secant trigonometric function.^{[86]}^{[87]}^{[88]}^{[89]}^{[90]}^{[91]} However, it is particularly interesting that "The Cambridge History of Islam" says the Abu alWafa' Buzjani (known as "alWafa" or "l'wafa"; 940—998^{[92]}) was the first to discover it.^{[93]} It is one of the six trigonometric functions used in math (the others being sine [sin], cosine [cos], tangent [tan], cotangent [cot], and cosecant [csc]). He was able to discover this when he was studying shadow triangles cast by sun dials.^{[94]}
 Indeed this claim is back up by Owen J. Gingerich (1930—Present^{[95]}), professor emeritus of astronomy and of the history of science at Harvard University and a senior astronomer emeritus at the Smithsonian Astrophysical Observatory, the secant function, one of the main trigonometric functions in mathematics, was first discovered by Muslim mathematicians.^{[96]}^{[n. 5]} This is especially relevant since Harvard is the amongst the best research universities in the world.^{[97]}
 Cosecant ()—Abu ʿAbd Allah Muhammad ibn Jabir ibn Sinan alRaqqi alHarrani asSabiʾ alBattani (Albategnius; 858—929^{[85]}) was the first to discover the cosecant trigonometric function.^{[98]}^{[99]}^{[100]}^{[101]}^{[102]}^{[103]} However, it is particularly interesting that "The Cambridge History of Islam" says the Abu alWafa' Buzjani (known as "alWafa" or "l'wafa"; 940—998^{[92]}) was the first to discover it.^{[104]} It is one of the six trigonometric functions used in math (the others being sine [sin], cosine [cos], tangent [tan], cotangent [cot], and secant [sec]). He was able to discover this when he was studying shadow triangles cast by sun dials.^{[105]}
 Indeed this is backed up by Owen J. Gingerich (1930—Present^{[95]}), professor emeritus of astronomy and of the history of science at Harvard University and a senior astronomer emeritus at the Smithsonian Astrophysical Observatory, the cosecant function, one of the main trigonometric functions in mathematics, was first discovered by Muslim mathematicians.^{[96]}^{[n. 6]} This is especially relevant since Harvard is the amongst the best research universities in the world.^{[97]}
 Tangent () and Cotangent ()—The tangent and cotangent originated from Islamic mathematicians.^{[107]} They first appeared as shadows of the gnomon, or "miqyas".^{[107]} Previously, vague ideas did exist through recognition of shadows amongst the ancient Egyptians and Greeks but who "however made no use of these functions of an angle"; even though Western authors have keen to stress that the Greek Thales "measured the heights of pyramids by means of shadows", but that which wasn't a "real use" of it as a function (since this relied on ratios rather than angles^{[108]}).^{[106]}
 However the Arabs were clearly the "first to have made any real use of [the tangent] as a function",^{[106]} and hence have been credited with what was hiding in plain sight by numerous historians. Indeed this claim is backed up by Owen J. Gingerich (1930—Present^{[95]}), professor emeritus of astronomy and of the history of science at Harvard University and a senior astronomer emeritus at the Smithsonian Astrophysical Observatory, who notes that the tangent and cotangent trigonometric functions in mathematics was first discovered by medieval Muslim mathematicians.^{[96]}^{[n. 7]}
 Tangent () and Cotangent ()—The tangent and cotangent originated from Islamic mathematicians.^{[107]} They first appeared as shadows of the gnomon, or "miqyas".^{[107]} Previously, vague ideas did exist through recognition of shadows amongst the ancient Egyptians and Greeks but who "however made no use of these functions of an angle"; even though Western authors have keen to stress that the Greek Thales "measured the heights of pyramids by means of shadows", but that which wasn't a "real use" of it as a function (since this relied on ratios rather than angles^{[108]}).^{[106]}
 Regarding Thales of Miletus^{[108]} / Miletos^{[109]} (c. 625—547 BC^{[108]}), he calculated the height of objects using the ratio method mentioned above (and not the tan function of the angle); "he argued that the ratio of the height h of the column to the height h' of his staff was equal to the ratio of the length s of the column's shadow to the length s' of the staff's shadow...[s]ince three of these quantities are known, [he] was able to calculate the height of the column...[and] used a similar method to find the height of the Great Pyramid in Egypt".^{[108]}
 Thales did not invent this method. In fact "Thales has been given credit for discoveries that were not really his",^{[110]} with the height calculation method being one of them; "Thales is believed to have visited Egypt, where he learned of Egyptian methods for calculating heights and distances by measuring the lengths of shadows".^{[111]} He was one of many Greeks that brought Egyptian science to Greece where they also learned many other methods such as "ropeandstake" (this was also where the Greeks first learned geometry).^{[110]}
 Like many Greeks in Egypt, Thales was an immigrant. Specifically "[w]hat Thales brought back...was the Egyptian system of measurements and calculations by which they built pyramids and palaces, and through which they measured heights and distances. Thus, it is traditionally attributed to Thales that he learned from the Egyptians to calculate the measurement of the height of the pyramids by the length of their shadow...[a]nd...then applied this knowledge to calculate the distance of ships at sea".^{[109]}
 Thales did not invent this method. In fact "Thales has been given credit for discoveries that were not really his",^{[110]} with the height calculation method being one of them; "Thales is believed to have visited Egypt, where he learned of Egyptian methods for calculating heights and distances by measuring the lengths of shadows".^{[111]} He was one of many Greeks that brought Egyptian science to Greece where they also learned many other methods such as "ropeandstake" (this was also where the Greeks first learned geometry).^{[110]}
 Regarding Thales of Miletus^{[108]} / Miletos^{[109]} (c. 625—547 BC^{[108]}), he calculated the height of objects using the ratio method mentioned above (and not the tan function of the angle); "he argued that the ratio of the height h of the column to the height h' of his staff was equal to the ratio of the length s of the column's shadow to the length s' of the staff's shadow...[s]ince three of these quantities are known, [he] was able to calculate the height of the column...[and] used a similar method to find the height of the Great Pyramid in Egypt".^{[108]}
 Tangents Table—A tangent ("alzill alma kus"^{[112]}) table (also; "tangent table of shadows" or the "umbra recta" as it was known in Latin translations of Islamic texts) was first developed in 860 by Muslim mathematicians,^{[113]} specifically by Ahmad ibn Abdallah alMarwazi alBaghdadi (c. 770—870^{[114]}).^{[115]} Both the tangent and cotangent are superior to using the chord and sine functions, largely because "the more practical measurements of heights and distances first required the tangent and cotangent—the gnomon and shadow respectively".^{[116]}
 Cotangents Table—A cotangent table (also known as the "cotangent table of shadows" or the "umbra versa" as it was known in Latin translations of Islamic mathematics) was first developed in 860 by Arabic Islamic mathematicians.^{[117]} Both the tangent and cotangent is superior to using the chord and sine functions, largely because "the more practical measurements of heights and distances first required the tangent and cotangent—the gnomon and shadow respectively".^{[118]} The cotangent is also known as the "alzill almustawi" in Arabic.^{[119]}
 Trigonometric Unification—Abu alWafa' Buzjani (known as "alWafa" or "l'wafa"; 940—998^{[92]}) was the first to unite all six trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant); indeed according to the University of Cambridge, he "brought them together and established the relations between the six fundamental trigonometric functions for the first time".^{[120]} From this he was able to determine a number of notable trigonometric identities.^{[121]} A few of these are (alt.; ) and (alt.; ).^{[121]}
 Interestingly, scholars have noted that the "advantages of including these functions in the trigonometric stable were not appreciated by all of Abu'lWafa's colleagues. Ibn Yunus [Abu alHasan 'Ali ibn 'Abd alRahman ibn Ahmad ibn Yunus alSadafi alMisri; 950—1009], for example, either did not hear of the innovations or chose to continue under the old regime. But with alBiruni's adoption, the continued life and acceptance of the new functions were assured. Their advantages in spherical astronomy, as Abu'lWafa demonstrates...were too great to be ignored".^{[121]}
 Trigonometric Unification—Abu alWafa' Buzjani (known as "alWafa" or "l'wafa"; 940—998^{[92]}) was the first to unite all six trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant); indeed according to the University of Cambridge, he "brought them together and established the relations between the six fundamental trigonometric functions for the first time".^{[120]} From this he was able to determine a number of notable trigonometric identities.^{[121]} A few of these are (alt.; ) and (alt.; ).^{[121]}
Trigonometric Identities (10)
 This is by no means an exhaustive list, and thus should be considered incomplete.
 Doublesine trigonometric Identity—Abu alWafa' Buzjani (known as "alWafa" or "l'wafa"; 940—998^{[92]}) was the first to discover the trigonometric identity in 980.^{[123]}^{[124]} This was an important discovery in the history of mathematics, as it was "essential to calculate the tables that were used for astronomy and engineering".^{[123]} The equation can be found in the "Kitab alKhamil".^{[123]} Interestingly, scientists immortalised his name by naming one of the craters on the moon after him as a mark of respect. An image of the crater can be seen here (or found on the "WayBackMachine"; here).
 Doublecosine trigonometric Identity—Abu alWafa' Buzjani (known as "alWafa" or "l'wafa"; 940—998^{[92]}) was the first to discover the trigonometric identity .^{[125]} The discovery of this equation, amongst several new trigonometric identities, "brought real progress to trigonometry".^{[125]} Interestingly, scientists in the West in the 20th century later immortalised his name by naming one of the craters on the moon after him as a mark of respect. An image of the crater can be seen here (or found on the "WayBackMachine" at Archive.org; here).
 Triplesine trigonometric Identity—Ghiyath alDin Jamshid Mas'ud alKashi (better known as alKashi; c. 1380—1429^{[58]}) was the first to discover the trigonometric formula .^{[126]}^{[n. 11]} It is sometimes also expressed as .^{[126]} It was found when Muslim scientists wanted to find a far more quicker and far more efficient method of computing values of Sin1°. The previous method was too "cumbersome" to use.^{[126]} The new formula was mistakenly attributed to be a discovery of the Frenchman, François Viète (1540—1603).^{[126]}
 Multiple cosine trigonometric Identity—Abu alHasan 'Ali ibn 'Abd alRahman ibn Ahmad ibn Yunus alSadafi alMisri (Ibn Yunus; 950—1009) was the first to construct a trigonometric identity labelled here as the multiple cosine trigonometric identity (top).^{[127]} This was an important discovery as it permitted the "passage from a sum to a product, an operation which was to be of importance in the logarithmic system of calculation invented later".^{[127]} This system proved useful for the industrial revolution.
 Sine Arc of the First Quadrant Trigonometric Identity—Abū ʿAbd Allah Muḥammad ibn Jabir ibn Sinān alRaqqi alḤarrani aṣSabiʾ alBattani (Albategnius; 858—929^{[85]}) was the first to solve the equation which lead him to the discovery of the Sine Arc of the First Quadrant trigonometric identity (below).^{[128]} This trigonometric identity is shown on the right.^{[128]} Keeping in mind that this was solved in the 9th century, it represents an important step in mathematical history.

 Secant trigonometric Identity—Abu alWafa' Buzjani (known as "alWafa" or "l'wafa"; 940—998^{[92]}) was the first to discover the trigonometric identity .^{[133]}^{[134]} This equation in particular was important in the development modernday satellite geolocation (GPS) systems.^{[134]} It is also sometimes written as .^{[134]}^{[135]} Interestingly, scientists in the West in the 20th century later immortalised his name by naming one of the craters on the moon after him as a mark of respect. An image of the crater can be seen here (or found at Archive.org; here).
 Cosecant trigonometric Identity—Abu alWafa' Buzjani (known as "alWafa" or "l'wafa"; 940—998^{[92]}) was the first to discover the trigonometric identity .^{[136]}^{[137]} This equation in particular was important in the development modernday satellite geolocation (GPS) systems.^{[137]} It is also sometimes written as .^{[137]}^{[138]} Interestingly, scientists in the West in the 20th century later immortalised his name by naming one of the craters on the moon after him as a mark of respect. An image of the crater can be seen here (or found at Archive.org; here).
Algebra & Algebraic Theory (3)
 This is by no means an exhaustive list, and thus should be considered incomplete.
 Khwarizmi Identity—Muḥammad ibn Musa alKhwarizmi (780—850^{[80]}) was the first to formulate what is now known as the "Khwarizmi Identity" (shown on the right). This is a crucial identity which is actually a solution of the quadratic equation (itself long known since the Babylonian Empire; 1,895 BC—539 BC^{[139]}), which lead to the discovery of the now wellknown quadratic formula.^{[140]}
 The quadratic formula itself is a crucial tool used to factorise quadratic polynomials;it took thousands of years to solve. When used practically on a factorised quadratic equation, the resulting solution pinpoints the exact locations of parabolic coordinates on a cartesian diagram. According to the University of New South Wales, Khwarizmis identity is "one of the really great derivations" in math history.^{[140]}
 Quadratic Formula—In 1145, Abraham bar Ḥiyya haNasi (also known as Savasorda; 1070—1136^{[141]}), a Jewish translator of the works of Islamic treatises, is notable for having written a book (the "Hibbur hameshihah vehatishboret"; sometimes also spelled as "Hibbur hameshihah wehatishboret";^{[142]} or the "Treatise on Measurement and Calculation"^{[143]}) which contained the first known complete solution of the quadratic formula.^{[143]} However he was the not the one to develop the formula; as it was first developed by Khwarizmi since it was already apparent that it appeared in a translation of his work in 1145; the same year that it appeared in Savasorda's book.^{[n. 13]}^{[144]} Of note, Ḥiyya's book was more of an encyclopedia than an original work of mathematics.^{[142]}
 Savasorda, along with Plato of Tivoli (fl. 1134—1145^{[145]}) and Rudolph of Bruges (fl. 1144^{[146]}), worked as a translator of Muslim scientific manuscripts (and indeed this was how Europeans became acquainted with the science of the Islamic Golden Age).^{[142]} He is also known to have "turned out original works on astronomy, chronology, philosophy and mathematics".^{[142]} However, crucially these "original works" weren't really original at all, but instead "[t]he originality of the best of these works consisted, rather, in their methods of attack, the range of their problems, and in the depth of their integration of Muslim, ancient, and Greek science".^{[142]} Indeed "one of these original masterpieces is exemplified by the Hibbur hameshihah wehatishboret".^{[142]}
 Furthermore; "[t]his work on geometry gathered into its text the geometric theory of the Greeks and geometric algebra and trigonometry of the Muslims. Further, it elevated the completely practical approach so characteristic of Babylonian and Egyptian mathematics to a new level of vitality, an approach which a later mathematician, Leonardo Fibonacci, used as a stepping stone to help lay the basis of the new European mathematical science".^{[142]}
 Leonardo Fibonacci (1170—1240^{[147]}) was also a translator of Muslim science into Latin, who helped disseminate this knowledge across the Europe.
 Savasorda, along with Plato of Tivoli (fl. 1134—1145^{[145]}) and Rudolph of Bruges (fl. 1144^{[146]}), worked as a translator of Muslim scientific manuscripts (and indeed this was how Europeans became acquainted with the science of the Islamic Golden Age).^{[142]} He is also known to have "turned out original works on astronomy, chronology, philosophy and mathematics".^{[142]} However, crucially these "original works" weren't really original at all, but instead "[t]he originality of the best of these works consisted, rather, in their methods of attack, the range of their problems, and in the depth of their integration of Muslim, ancient, and Greek science".^{[142]} Indeed "one of these original masterpieces is exemplified by the Hibbur hameshihah wehatishboret".^{[142]}
 Quadratic Formula—In 1145, Abraham bar Ḥiyya haNasi (also known as Savasorda; 1070—1136^{[141]}), a Jewish translator of the works of Islamic treatises, is notable for having written a book (the "Hibbur hameshihah vehatishboret"; sometimes also spelled as "Hibbur hameshihah wehatishboret";^{[142]} or the "Treatise on Measurement and Calculation"^{[143]}) which contained the first known complete solution of the quadratic formula.^{[143]} However he was the not the one to develop the formula; as it was first developed by Khwarizmi since it was already apparent that it appeared in a translation of his work in 1145; the same year that it appeared in Savasorda's book.^{[n. 13]}^{[144]} Of note, Ḥiyya's book was more of an encyclopedia than an original work of mathematics.^{[142]}
 Newton's Method—In the 12th century, Sharaf alDin alMuzaffar ibn Muhammad ibn alMuẓaffar alTusi (Tusi; 1201—1274^{[129]}) was the first to develop what later became known as "Newton's Method" (named after Isaac Newton; 1642—1727^{[148]}).^{[149]} Indeed, "[a] method algebraically equivalent to Newton’s method was known to the 12th century algebraist Sharaf alDin alTusi".^{[149]} It was also known to Ghiyath alDin Jamshīd Masʿud alKashi in the 15th century who used it compute , finding the roots of N.^{[149]} It first appeared in Europe in the "Trigonometria Britannica" in 1633 by Henry Briggs (1561—1630^{[150]}); though it is said that Newton did not know of this latter particular mathematician's work.^{[149]} Newton himself first published the method in 1669.^{[149]} It is also known as the "NewtonRaphson Method" (with the second named after Joseph Raphson; 1648—1715^{[151]}).^{[149]}
Differential & Integral Calculus (5)
 This is by no means an exhaustive list, and thus should be considered incomplete.
 Derivative / Differentiation—Sharaf alDin alMuzaffar ibn Muḥammad ibn alMuzaffar alTusi (Tusi; c. 1135—c. 1213^{[152]}) was the first to develop the notion of a derivative in 1209.^{[153]}^{[154]}^{[155]} He achieved this by calculating "the value of the variable x for which the derivative of the above function is equal to zero. AlTusi did not use the Arabic equivalent for the word derivative, but he clearly introduced...the notion of a derivativeall of which were crucial concepts for the development of algebraic geometry".^{[153]} In essence, "in its analytic approach, alTusi's work on equations marks the beginning of the discipline of algebraic geometry: the study of curves by means of equations".^{[156]} He has thus also been credited with the discovery of local analysis of plotted graphical functions, and for being the first to study [its] maxima.^{[153]}
 Despite the mathematical precision and accuracy in his result, he is not credited by Western historians as he "did not extend this result to more general functions".^{[157]}^{[154]} They instead give credit to Isaac Newton (1642—1727^{[148]}) and Gottfried Wilhelm Leibniz (1646—1716^{[158]}).^{[154]} However, Tusi needn't have to, since he clearly came up with the first derivative in mathematical history,^{[153]} and how this was an entirely "new way" of thinking.^{[159]} Indeed, historians who are not Western (however, this appears to be changing^{[153]}) have stated clearly that "he was using the idea of derivatives already in the 12th century".^{[160]}
 Derivative / Differentiation—Sharaf alDin alMuzaffar ibn Muḥammad ibn alMuzaffar alTusi (Tusi; c. 1135—c. 1213^{[152]}) was the first to develop the notion of a derivative in 1209.^{[153]}^{[154]}^{[155]} He achieved this by calculating "the value of the variable x for which the derivative of the above function is equal to zero. AlTusi did not use the Arabic equivalent for the word derivative, but he clearly introduced...the notion of a derivativeall of which were crucial concepts for the development of algebraic geometry".^{[153]} In essence, "in its analytic approach, alTusi's work on equations marks the beginning of the discipline of algebraic geometry: the study of curves by means of equations".^{[156]} He has thus also been credited with the discovery of local analysis of plotted graphical functions, and for being the first to study [its] maxima.^{[153]}
 Paraboloid Volume (Using a Rotational Parabola)—Ḥasan Ibn alHaytham (Alhazan; 965—1040^{[161]}) was the first to calculate the volume of a paraboloid obtained from the rotation of a parabola around its ordinate.^{[162]} It is important to stress that the volume of a revolving paraboloid itself alone had been studied by Thabit ibn Qurra (826—901^{[163]}) and Abu Sahl Wayjan ibn Rustam alQuhi (940—1000^{[164]}) but the difference was in the method of calculation.^{[162]}^{[165]}
 He came up with the following equation shown right (his own invention^{[162]}); his calculation was the equivalent of the integral shown (what this equation means is that the volume is 8/15th the size of a circumscribed cylinder.^{[162]}
 Integration—Ḥasan Ibn alHaytham (Alhazan; 965—1040^{[161]}) was the first to use integration to calculate the volume of an object. This would otherwise have been impossible had he not discovered the sums of integral law (see Figure 7; Fig. 7), which was directly useful in calculating the paraboloid volume. He also additionally used the sums of integrals law to also calculate the area of a sphere.^{[166]}
 An understanding of maths is required to appreciate how Haytham managed to formulate the first integral. This was 650 years before Isaac Newton (1642—1727^{[148]}) and Gottfried Leibniz (1646—1716^{[158]}) who are traditionally given credit for inventing it.^{[167]} Curiously, Haytham is only just just beginning to be recognised as the true father of integration after centuries of neglect.
 The best explanation is illustrated by the historian Victor J. Katz (1942—Present^{[168]}), in "A History of Mathematics" (1998),^{[165]} and by Marlow Anderson (1950—Present^{[169]}), Victor J. Katz and Robin J. Wilson (1943—Present^{[170]}) in "Sherlock Holmes in Babylon: And Other Tales of Mathematical History" (2004).^{[171]} Consider a parabola, cut through the middle with xaxis value of , with a horizontal bar of height and , and length . Haytham used his result for the sums of integral powers (see "summation of the fourth power and beyond" in the "summation series") to "perform what we would call an integration".^{[171]}
 He applied this to find the "volume of the solid formed by rotating the parabola around the line perpendicular to the axis of the parabola, and showed that this volume is 8/15 of the volume of the cylinder of radius and height ". Next, he used the Greekstyle exhaustion argument using "a double reductio ad absurdum".^{[171]} However, this is where things get interesting.
 Now, Haytham followed steps from Figures 1 (Fig. 1), 2 (Fig. 2), 3 (Fig. 3) to determine 4 (Fig. 4);^{[171]} uniquely going further, stating as n approaches infinity, the area of the paraboloid gives Figure 5 (indeed Haytham takes it "one step further and notes that we can find a better approximation by making n sufficiently large...[i]n essence he found" the following equation (Fig. 4) and thus for the first time in history "used a method similar to integration to correctly find the volume of a solid of revolution"; Fig. 5)^{[167]}
 Integration—Ḥasan Ibn alHaytham (Alhazan; 965—1040^{[161]}) was the first to use integration to calculate the volume of an object. This would otherwise have been impossible had he not discovered the sums of integral law (see Figure 7; Fig. 7), which was directly useful in calculating the paraboloid volume. He also additionally used the sums of integrals law to also calculate the area of a sphere.^{[166]}
 Katz et al. (2004) explains after Step 4 (i.e. Fig. 4); "[n]ow the volume of a typical slice of the circumscribing cylinder is , and therefore the total volume of the cylinder is while the volume of the cylinder less its "top slice" is ...[t]he inequality then shows that the volume of the paraboloid is bounded between 8/15 of the cylinder less its top slice and 8/15 of the entire cylinder. Because the top slice can be made as small as desired by taking n sufficiently large, it follows that the volume of the paraboloid is exactly 8/15 of the volume of the cylinder as asserted".^{[171]}
 Thus, the integration is complete and in it's modern form, can be expressed in the following way (Figure 6^{[162]});
 Katz et al. (2004) explains after Step 4 (i.e. Fig. 4); "[n]ow the volume of a typical slice of the circumscribing cylinder is , and therefore the total volume of the cylinder is while the volume of the cylinder less its "top slice" is ...[t]he inequality then shows that the volume of the paraboloid is bounded between 8/15 of the cylinder less its top slice and 8/15 of the entire cylinder. Because the top slice can be made as small as desired by taking n sufficiently large, it follows that the volume of the paraboloid is exactly 8/15 of the volume of the cylinder as asserted".^{[171]}
^{(Fig. 6.)} ^{(Fig. 7.)} ^{(Fig. 8.)}
 If this has been difficult to follow; consider more simply; by the end of classical Greek mathematics, the Greeks had left many problems unsolved; "including the problem of finding the areas of even such simple geometric figures as a domain bounded by a hyperbola ".^{[172]} The most that they had done was develop the "Greekstyle exhaustion method"; a primitive way to solve this, but which was not general.^{[173]}^{[174]} By contrast, Haytham "found the formulas for the area of a domain bounded by the curve (in modern terms [this is Fig. 8]) for an arbitrary nonnegative integer k. This discovery enabled Alhazen to compute the areas and volumes of the curves and surfaces bounded by polynomial equations ".^{[172]}
 Exhaustion was pioneered by an ancient Greek called Eudoxus (c. 390 BC—c. 340 BC^{[175]}) which was extensively used by Archimedes (c. 287 BC—c. 212 BC^{[176]}).^{[165]} The latter wrote a book called "On Conoids and Spheroids" where he had shown how to calculate the volume of the solid formed by a revolving parabola about its axis which used Greek exhaustion.^{[165]} Crucially, Islamic mathematicians did not have access to this and invented their own methods of determining this volume.^{[165]} Several different scientists developed solutions for this but they were long and complicated.^{[165]} Haytham himself tried his own, and used Greekexhaustion, but crucially his was different. This is most evident when looking at an example of what Greekexhaustion is; the following illustrates an example. The method only involves putting known geometric shapes inside another shape and determining it's area or volume;
 The Greeks—including Archimedes—did not use the same method as Haytham's (even though they approximated^{[n. 15]} the area or volume of a parabola^{[n. 16]}).^{[173]} When using Greek exhaustion, they took "some geometrical figure and fill[ed] it with some large, but finite number of polygons...[t]he sum of the areas of these polygons would be the area of the figure...[b]ut they did not consider this as a series...[t]he math boils down to a convergent series, usually an easy to work with geometric series".^{[173]}^{[n. 17]} Because of this finite nature of area calculation, they did not consider what would happen if it was infinitely extended using a series.^{[174]} This is precisely what Haytham achieved,^{[167]} even though he also incorporated Greek exhaustion into his calculation.^{[171]} The Greeks never considered infinity precisely because they did not really understand it; in fact, "the framework of Greek mathematics had no ability to deal with the infinite, infinite sums, or limit processes...[b]ecause they could not be considered in any sense that would allow for mathematical rigour, they were taboo in...Ancient Greece".^{[174]}^{[n. 18]}
 Historians in the West have finally started to truly recognise Haytham's contributions, with one historian remarking; "[i]t was he, as we shall see, who radically overhauled the study of lunes, making him a worthy successor to Hippocrates of Chios, but one who was infinitely closer to Euler. It was he who developed the ancient methods of integration, taking them further in the infinitesimal direction, and achieving results that were later rediscovered by Kepler and Cavalieri".^{[178]} The BBC have even honoured him by calling him the first true scientist.^{[179]}^{[180]}
Geometry (3)
 This is by no means an exhaustive list, and thus should be considered incomplete.
 Geometric Motion—Ḥasan Ibn alHaytham (Alhazan; 965—1040^{[161]}) was the first to introduce the concept of motion in geometry.^{[183]} He "wrote that if a straight line moves so that one end always lies on a second straight line and so that it always remains perpendicular to that line, then the other end of the moving line will trace out a straight line parallel to the second line...this...characterized parallel lines as lines always equidistant from one another and also introduced the concept of motion into geometry".^{[183]} The crucial defined step was; "[i]f two straight lines are drawn at right angles to the two endpoints of a fixed straight line, then every perpendicular line dropped from the one line to the other is equal to the fixed line".^{[183]}
 Mathematicians initially found this difficult to accept, most notably Omar Khayyam (1048—1131^{[184]}), but it was eventually accepted.^{[183]}
 HaythamLambert Quadrilateral—Ḥasan Ibn alHaytham (Alhazan; 965—1040^{[161]}) was the first to discover the Lambert Quadrilateral (named after the FrancoGerman—or Swiss^{[185]}—scientist Johann H. Lambert; 1728—1777^{[186]}).^{[187]} The shape is a foursided polygon that has three rightangles at the vertices.^{[188]}^{[189]} In hyperbolic geometry rectangles do not exist, however, the HaythamLambert Quadrilateral is one of two geometric shapes (the other being the KhayyamSaccheri Quadrilateral;^{[190]} discussed below) that is the closest to the shape of rectangle.^{[190]}^{[188]} This quadrilateral has played an important role in the history of nonEuclidean geometry.^{[187]} They are also used in pangeometry.^{[191]}
 KhayyamSaccheri Quadrilateral—Omar Khayyam (Khayyam; 1048—1131^{[184]}) was the first to discover the Saccheri quadrilateral (named after the Italian mathematician Giovanni G. Saccheri; 1667—1733^{[194]}).^{[195]} This quadrilateral has played an important role in the history of nonEuclidean geometry.^{[195]} It is defined as "an isosceles biright quadrilateral, that is, a quadrilateral in which the opposite sides adjacent to the right angles are congruent", where the "[t]he midline of a KSquadrilateral is the segment joining the midpoint of the base with the midpoint of the summit".^{[192]} It is also defined as "a quadrilateral having two opposite edges of the same length and making right angles with a third common edge".^{[196]} Albert Einstein (1879—1955^{[197]}) used this to develop the theory of relativity.^{[198]}
 There is a strong possibility that Saccheri plagiarised the work of Khayyam since the way that Saccheri's figures are shown and lettered are done in the exact same way as Khayyams.^{[199]} This isn't surprising since Saccheri lived in the era of European colonialism (c. 15th Century—20th century^{[200]}^{[201]}). A lot of historical scientific research from colonized countries was directly taken and used by these colonists (for example one notable invention that was borrowed was the Flush Deck which had never been seen before elsewhere). Europeans also extensively borrowed knowledge from the Islamic world through looting during the Crusades (1095—1492^{[202]}).
 KhayyamSaccheri Quadrilateral—Omar Khayyam (Khayyam; 1048—1131^{[184]}) was the first to discover the Saccheri quadrilateral (named after the Italian mathematician Giovanni G. Saccheri; 1667—1733^{[194]}).^{[195]} This quadrilateral has played an important role in the history of nonEuclidean geometry.^{[195]} It is defined as "an isosceles biright quadrilateral, that is, a quadrilateral in which the opposite sides adjacent to the right angles are congruent", where the "[t]he midline of a KSquadrilateral is the segment joining the midpoint of the base with the midpoint of the summit".^{[192]} It is also defined as "a quadrilateral having two opposite edges of the same length and making right angles with a third common edge".^{[196]} Albert Einstein (1879—1955^{[197]}) used this to develop the theory of relativity.^{[198]}
Spherical Triangle Trigonometric Laws (3)
 This is by no means an exhaustive list, and thus should be considered incomplete.
 Law of Sines—Abu alWafa' Buzjani (known as "alWafa" or "l'wafa"; 940—998^{[92]}) was the first to discover the law of sines for spherical triangles.^{[203]}^{[204]}^{[205]} According to St. Andrews University, Scotland, a spherical triangle is described as being "made up of three arcs of great circles, all less than 180°. The sum of the angles is not fixed, but will always be greater than 180°. If any side of the triangle is exactly 90°, the triangle is called quadrantal. There are many formulae relating the sides and angles of a spherical triangle [two of which we discuss in the article]".^{[206]}
 It is also important to remember; "[t]he cosine rule will solve almost any triangle if it is applied often enough...[t]he sine rule is simpler to remember but not always applicable". Additionally, "both formulae can suffer from ambiguity" since two answers can be obtained but this can resolved by "check[ing] to see if the answer is sensible".^{[206]}
 Law of Sines—Abu alWafa' Buzjani (known as "alWafa" or "l'wafa"; 940—998^{[92]}) was the first to discover the law of sines for spherical triangles.^{[203]}^{[204]}^{[205]} According to St. Andrews University, Scotland, a spherical triangle is described as being "made up of three arcs of great circles, all less than 180°. The sum of the angles is not fixed, but will always be greater than 180°. If any side of the triangle is exactly 90°, the triangle is called quadrantal. There are many formulae relating the sides and angles of a spherical triangle [two of which we discuss in the article]".^{[206]}
 ; also known as the Law of Cosines—Ghiyath alDin Jamshid Masʿud alKashi (AlKhashi/AlKashani;^{[207]} c. 1380—c. 1429^{[64]}) was the first to discover the law of cosines.^{[208]}^{[209]}^{[n. 23]} Euclid's Elements (c. 300 BC^{[210]}) details that the Greek came close to discovering it himself but ultimately failed; hence historians give his work credit at the very most containing the "seeds" of the concept.^{[208]} It is also known as the theorem of alKashi (and in France; "théorème d'AlKashi".^{[211]} Sometimes the law is expressed with different symbols, as .^{[208]}
 The law was transmitted to Europe—specifically to Johannes Kepler (1571—1630^{[212]})—through the work of Abu ʿAbd Allah Muhammad ibn Jabir ibn Sinan alRaqqi alḤarrani asSabiʾ alBattani ("alBattani"; 858—929).^{[213]} Interestingly, alBattani also greatly influenced the work of Nicolaus Copernicus (1473—1543^{[214]}), Tycho Brahe (1546—1601^{[215]}) and Galileo Galilei (1564—1642^{[216]}).^{[213]} This is notable because it is already known Western mathematicians such as had François Viète (1540—1603^{[217]}) plagiarised the work of Muslim mathematicians without having given them any credit at all (one example which has already been discussed and illustrated above; ).
 Law of Tangents—Muhammad ibn Muhammad ibn alHasan alTusi (1201—1274^{[129]}) was the first to discover the law of tangents for spherical triangles.^{[218]}^{[219]}^{[220]} Others say it was in fact Abu alWafa' Buzjani (known as "alWafa" or "l'wafa"; 940—998^{[92]}), who preceded the discovery by Tusi by centuries according to French translations of his work.^{[221]} It is certainly possible that Tusi might have independently discovered it as well.
 Similarly, Ghiyath alDin Jamshid Masʿud alKashi (AlKhashi/AlKashani;^{[207]} c. 1380—c. 1429^{[64]}) thought himself as having independently discovered decimal fractions, when in fact he himself was preceded centuries earlier by Abu'lHasan alUqlidisi (920—980) in Baghdad indicating a coincidence.^{[222]} The particular copy of work that demonstrates this historical record is located in Istanbul (at the Yeni Gami Library^{[223]}), and also represents the earliest extant mathematical book on Arabic arithmetic.^{[224]} This is the only copy in existence, first published in Damascus in 952.^{[223]}
 Law of Tangents—Muhammad ibn Muhammad ibn alHasan alTusi (1201—1274^{[129]}) was the first to discover the law of tangents for spherical triangles.^{[218]}^{[219]}^{[220]} Others say it was in fact Abu alWafa' Buzjani (known as "alWafa" or "l'wafa"; 940—998^{[92]}), who preceded the discovery by Tusi by centuries according to French translations of his work.^{[221]} It is certainly possible that Tusi might have independently discovered it as well.
Number Theory & Summation Series (4)
 This is by no means an exhaustive list, and thus should be considered incomplete.
 Wilson's Theorem (if p is prime then is divisible by p)—A theorem known as "Wilson's Theorem" was first discovered by Ḥasan Ibn alHaytham (Alhazen; 965—1040).^{[225]} He developed it when he was researching how to solve the "Chinese Remainder Problem" (or "Theorem").^{[226]}
 The theorem was selfnamed, eponymously, after John Wilson (1741—1793^{[227]}) in 1770, who allegedly claimed that he had discovered it when he made a remark about it to another English mathematician, Edward Waring (1736—1798^{[228]}).^{[225]} Another mathematician, this time the FrancoItalian JosephLouis Lagrange (1736—1813^{[229]}), provided the first known European proof for the theorem in 1772.^{[225]} However, they were all preceded by 750 years by Haytham.^{[225]} Another source claims Waring himself attributed the discovery to Wilson in his book, "Meditationes Algebraicae" (1782).^{[230]}
 Wilson himself was greatly inspired by Haytham, and curiously, the latter has also been "nicknamed Ptolemy the Second" in Western history.^{[231]}
 It is important to stress that neither Wilson nor Waring was able to prove the theorem (just Lagrange).^{[230]} It is however also important to stress that Haytham was not only the first to discover the theorem, but was also the first to provide proof for it.^{[225]} Indeed, it is known that he solved equations using this theorem; specifically he "solved problems involving congruences using what is now called Wilson's theorem".^{[232]}
 Despite this, some Western authors still refuse to give him credit for being the first to prove it, and instead give credit to Lagrange for providing the first known proof.^{[233]} Crucially, there has been resistance to this claim as there appears ample evidence that Haytham did know it's proof; enough such that, historically, mathematicians on the Indian subcontinent also used it, and more importantly, cited texts "directly dependent on Arabic mathematics".^{[234]}
 Graham Everest and Thomas Ward in their book, "An Introduction to Number Theory" (2007), explain the proof in the following way;^{[225]}
 "We prove that the congruence is satisfied when n is prime and leave the converse as an exercise. Assume that is an odd prime. (The congruence is clear for .)...Each of the integers has a unique multiplicative inverse distinct from a modulo p (see Corollary 1.25). Uniqueness is obvious; for distinctness, note that modulo p implies , forcing modulo p by primality. Thus in the product all the terms cancel out modulo p except the first and the last. Their product is clearly 1 modulo p".^{[225]}
 Graham Everest and Thomas Ward in their book, "An Introduction to Number Theory" (2007), explain the proof in the following way;^{[225]}
 The theorem was selfnamed, eponymously, after John Wilson (1741—1793^{[227]}) in 1770, who allegedly claimed that he had discovered it when he made a remark about it to another English mathematician, Edward Waring (1736—1798^{[228]}).^{[225]} Another mathematician, this time the FrancoItalian JosephLouis Lagrange (1736—1813^{[229]}), provided the first known European proof for the theorem in 1772.^{[225]} However, they were all preceded by 750 years by Haytham.^{[225]} Another source claims Waring himself attributed the discovery to Wilson in his book, "Meditationes Algebraicae" (1782).^{[230]}
 Wilson's Theorem (if p is prime then is divisible by p)—A theorem known as "Wilson's Theorem" was first discovered by Ḥasan Ibn alHaytham (Alhazen; 965—1040).^{[225]} He developed it when he was researching how to solve the "Chinese Remainder Problem" (or "Theorem").^{[226]}
 Summation of a Series of Cubes ()—The formula for a summation of a series of cubes was first discovered by Aryabhatta (476—550^{[235]}) in 499, however, he was not able to prove it.^{[237]} The first to prove it was Abu Bakr alKaraji (c. 953—c. 1029).^{[238]} However, according to some Western historians, the summation of a series of cubes was known to the Greeks (specifically by Nicomachus; c. 60—c. 120) and the Romans before them (through the agrimensores—who were Roman landsurveyors).^{[239]}^{[240]} However, both these claims are based on assumptions, as no explicit evidence exists (what little does is only implicitly assumed^{[237]}).^{[238]}
 Worse still, such historians acknowledge the proof for sum of a series of cubes appears first in the works of Abu Bakr alKaraji (c. 953—c. 1029) but, in highly Eurocentric opines, claim that it belongs to the Greeks because it can "be inferred from the fact that alKarkhi, the Arabian algebraist, who mainly followed Greek models, gives in his algebra entitled alFakhri a proof by means of a figure with gnomons drawn about squares in the traditional Greek fashion".^{[239]} Inferral, however, does not mean explicit. The explicit proof can be seen online at the Mathematics Association of America (MAA) website, written by prof. J. Beery.^{[236]}
 Importantly, Karkhi did not cite a single Greek or Roman source from where he got the proof from; the author even later admits as much in another book; postulating that"two alternatives are possible, alkarkhi may have devised the proof himself in the Greek manner, following the hint supplied by Nicomachus's theorem. Or he may have found the whole proof set out in some Greek treatise now lost and reproduced it".^{[241]} Most reliable historians however do not believe this, and realise that it first appeared elsewhere, from nonGreek, nonRoman sources, notably from the Indian subcontinent in the 5th century.^{[237]}
 Another Western mathematical historian has clarified the issue, noting, "[i]t seems likely, from the work of the neoPythagorean Nicomachus of Gerasa in the first century c.e., that the mathematicians of ancient Greece knew this too; while it is not explicit in extant work, it is implicit in a fact about sums of odd numbers and cubic numbers found in Nicomachus’s Introductio Arithmetica", but later admits that it was actually Aryabhatta who discovered it first in 499, but adding this was "[w]ithout any proof or justification", and that it was actually alKaraji who provided the "earliest proof we have of the sum of cubes formula".^{[237]}
 Additionally, the agrimensores having known about this formula seems even more dubious when historians have already stated that the agrimensores obtained all their mathematics from the Greeks (and since there isn't any explicit proof in Greek mathematics that they had discovered the summation of a series of cubes in their entire corpus, it is quite obvious that they did not discover or prove it).^{[242]} The credit therefore both deservedly goes to Aryabhatta and alKarkhi for it's existence.
 Worse still, such historians acknowledge the proof for sum of a series of cubes appears first in the works of Abu Bakr alKaraji (c. 953—c. 1029) but, in highly Eurocentric opines, claim that it belongs to the Greeks because it can "be inferred from the fact that alKarkhi, the Arabian algebraist, who mainly followed Greek models, gives in his algebra entitled alFakhri a proof by means of a figure with gnomons drawn about squares in the traditional Greek fashion".^{[239]} Inferral, however, does not mean explicit. The explicit proof can be seen online at the Mathematics Association of America (MAA) website, written by prof. J. Beery.^{[236]}
 Summation of a Series of Fourth Powers (Sums of Integral Law; )—Historians of mathematics have noted that the summation series for a square (i.e. ) is "not difficult to discover", and the summation of a cube (i.e. ) is "virtually obvious, given some experimentation"; however the summation of the fourth power by contrast is "not obvious" and therefore more difficult to obtain.^{[243]}
 The formula for the summation series of the fourth power was first^{[244]} discovered by Ḥasan Ibn alHaytham (965—1040) who also used it further to find the summation series of the powers of 1, 2 and 3—which, using his method—proved these previously known summation series using the same algorithm (whereas before, individual algorithms were used for finding the summation of the powers of 1, 2 and 3^{[245]}^{[n. 25]}—clearly disadvantageous because no obvious pattern linked all these summations, making finding algorithms for higher powers progressively more and more difficult.^{[243]}^{[245]} Haytham however managed to discover the pattern.
 Crucially, Haytham's discovery was so pivotal that it also went beyond the fourth power, and could be used to calculate any summation series of any power.^{[243]}^{[246]}^{[247]} Indeed, "if one can discover a method for determining this formula, one can discover a method for determining the formula for the sum of any integral powers".^{[243]} As a result, Haytham's "proofs were similar in nature and easily generalizable to the discovery and proof of formulas for the sum of any given powers of the integers".^{[243]}
 Prior to Haytham's discovery "no general pattern" emerged "for the details of the formula for various values of k" i.e. "k" being the power of a number series, i.e. , "and worse, all the formulas we obtained emerged from ad hoc methods, each demanding separate verification".^{[245]} However, Haytham "gives us the first steps along a path toward understanding these formulas in general".^{[245]} Although Haytham "did not state a completely general result" he did generalise for 1, 2, 3 and 4 (as he used the same method for finding the 4th summation power as he did with the summation series for 1, 2 and 3).^{[245]}
 He however did not go beyond the fourth power—however, he needn't have to; as he was only interested in calculating the volume of a paraboloid who's power only goes up to four.^{[246]} Indeed, Haytham's motivation for developing the fourth power formula was in finding it's volume.^{[243]} He "used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him" to arrive at the answer.^{[243]}^{[248]} He did this "by cutting it into everthinner slices" which "predates the official discovery of calculus in the 1600s".^{[249]}
 The formula for the summation series of the fourth power was first^{[244]} discovered by Ḥasan Ibn alHaytham (965—1040) who also used it further to find the summation series of the powers of 1, 2 and 3—which, using his method—proved these previously known summation series using the same algorithm (whereas before, individual algorithms were used for finding the summation of the powers of 1, 2 and 3^{[245]}^{[n. 25]}—clearly disadvantageous because no obvious pattern linked all these summations, making finding algorithms for higher powers progressively more and more difficult.^{[243]}^{[245]} Haytham however managed to discover the pattern.
 Summation of a Series of Fourth Powers (Sums of Integral Law; )—Historians of mathematics have noted that the summation series for a square (i.e. ) is "not difficult to discover", and the summation of a cube (i.e. ) is "virtually obvious, given some experimentation"; however the summation of the fourth power by contrast is "not obvious" and therefore more difficult to obtain.^{[243]}
 This is by no means an exhaustive list, and thus should be considered incomplete. See also List of Inventions and Discoveries in Mechanics During the Islamic Golden Age
 Sine Quadrant—Muhammad ibn Musa alKhwarizmi (Khwarizmi; 780—850^{[80]}) was the first person to invent the sine quadrant (the "alrub almujayyab"/"aljayb") in the 9th century.^{[250]} It was originally used as a graphical device for finding time as a function of altitude using the universal approximate formula.^{[250]} It contained an altitude scale along the rim, and a set of parallel lines, horizontal or vertical, drawn with equal divisions of the altitude scale to one of the radii, parallel to the other radius.^{[250]} Many variants were later made.^{[250]} Abu Ja'far alKhazin (c. 900—971) designed a trigonometric grid (known in Europe as the "sexagenarium").^{[250]}
 Sextant—Abu Mahmud Hamid ibn Khidr Khojandi (Khujundi; 900—1000^{[73]}) was the first person in history to invent the sextant, which he called the "sudsifakhri" (or "Fakhri's Sextant"), naming it after Fakhr alDawla (d. 997), the ruler of Rayy, Iran.^{[251]}^{[252]} Prior to it's invention visual observations were only made by the naked eye to measure the distance between two objects.^{[251]} A star's rays would pass through a dioptric lens placed in the upper part of a darkened chamber and the star itself would be caught as a reflection on the scale of the sextants arc.^{[251]} It is also described by Abu Rayḥan Muḥammad ibn Aḥmad AlBiruni (Biruni; 973–1050).^{[251]}
 Trigonometric Quadrant—The was first invented in 9th century Baghdad (and was used to solve trigonometric problems without the need for manual calculation).^{[253]}^{[254]} The tan function itself was discovered by Ahmad ibn Abdallah Habash Hasib Marwazi (d. 864/874^{[255]} or 869^{[256]}) in 830.^{[257]} By the 10th century, more sophisticated designs were available, with complicated mathematical markings resembling graph paper.^{[254]} These quadrants would be constructed on the back of astrolabes or astrolabic quadrants.^{[254]} It was so popular than during the Ottoman period (1299—1922) it replaced the astrolabe.^{[254]}
 Plate Conjunction—Jamshid ibn Masʿud ibn Maḥmud Ghiyath alDin alKashi (Kashani; 1380—1429) was the first to invent the plate conjunction (also known as the "Plate of Conjunctions").^{[258]} It functions as a computer which mathematically illustrates when during the time of day a planetary conjunction will happen (defined as "the instance of two or more events occurring at the same point in time or space").^{[258]} Data could be programmed directly into the instrument in order to obtain the timings.^{[258]} The manuscript is held in the Garrett Collection of Persian, Turkish, and Indic Manuscripts at Princeton University Library.^{[258]}
 Plate Zone—Jamshid ibn Masʿud ibn Maḥmud Ghiyath alDin alKashi (Kashani; 1380—1429) was the first to invent the plate zone (also known as the "Plate of Zones").^{[259]} The instrument was used in "its application to the problem of finding the true longitude of the sun and moon at a given time".^{[259]} However some believe that the instrument was created even earlier, specifically by two Arabs living in Spain, ibn asSamh (c. 1020) and Azaquiel (azZarqali; c. 1060), but this possibly may only be conjecture.^{[259]} The historian of science George A. L. Sarton (1884—1956) attributes the invention to these earlier scientists.^{[259]}
 Saphaea Arzachelis—Abu Ishaq Ibrahim b. Yahya lNaqqash alTujibi Ibn alZarqalluh (Arzachel/Azarquiel; 1029—1087^{[75]}/1100^{[76]}) was the first to invent the Saphaea Arzachelis (also known as the "Safiha Flatus").^{[81]} It was a flat sphere astrolabe (also known as the "Arzachel's Sphere").^{[81]} The invention was significant since it was the first universal astrolabe (it could be used from any position anywhere on earth), and did not depend on the latitude of the observer, unlike previous astrolabes, which were limited to one position.^{[260]} A crater on the moon was named^{[261]} after him by Giovanni Riccioli (1598—1671).^{[81]}
 Reversed Astrolabe—Abu alḤasan Ala al‐Din Ali ibn Ibrahim alAnsari (Ibn alShatir; 1305—1375^{[263]}) was the first to invent the reverse astrolabe, which contains a set of horizons rotated over a fixed stereographic projection of the stars.^{[264]} Although it is not known when the first normal astrolabe was first constructed,^{[265]} it is an ancient ancestor of the computer, used to determine the position of the zodiac and various stars at a specific time of day.^{[266]} It is made up of several pieces of metal.^{[266]} Shatir worked elsewhere besides in astronomy, namely as the chief muwaqqit of the Umayyad Mosque in Damascus.^{[267]}
 Astrolabe Clock—Abu alḤasan Ala al‐Din Ali ibn Ibrahim alAnsari (Ibn alShatir; 1305—1375^{[263]}) was the first to invent the astrolabe clock.^{[264]} Although it is not known when the first normal astrolabe was first constructed,^{[268]} it is an ancient ancestor of the computer, used to determine the position of the zodiac and various stars at a specific time of day.^{[266]} It is made up of several pieces of metal.^{[266]} The historian Khalil bin Aybak alṢafadi (1296—1363) confirms this that it was hung from his personal home.^{[264]} Shatir worked elsewhere besides in astronomy, namely as the chief muwaqqit of the Umayyad Mosque in Damascus.^{[267]}
 Linear Astrolabe—Sharaf alDin alMuẓaffar ibn Muḥammad ibn alMuẓaffar alTusi (Sharif alDin; 1135—1213^{[152]}) was the first to invent the linear astrolabe (also known as the "rod of alTuusi").^{[269]}^{[270]} It consists of a single rod with markings on the sides. The rod "represents the meridian of the planispheric astrolabe, and two threads attached to it, with movable beads on them, can be positioned at various points along the rod to serve in place of the rete (the top plate in the usual planispheric astrolabe, whose pointers indicate the position of certain prominent stars)".^{[271]}^{[n. 27]} His student, Kamāl alDīin Ibn Yūnus, later improved it.^{[270]}
Astronomical Motion (2)
 This is by no means an exhaustive list, and thus should be considered incomplete.
 Tusi Couple—Muhammad ibn Muhammad ibn alHasan alTusi (Tusi; 1201—1274^{[129]}) was the first to discover the Tusi Couple;^{[273]} an ingenious model which explained planetary motion which avoided the use of the equant; in other words he discovered a model to represent the motion of the inferior and superior planets in relation to one another.^{[274]} Scientists such as Ptolemy (fl. 127—145^{[78]}) and alHaytham (965—1040^{[161]}) had long been seeking such a model in their own time but ultimately failed.^{[272]} Interestingly Nicolaus Copernicus (1473—1543)—who could read Arabic^{[275]}—copied Tusi's ideas;^{[273]}^{[276]} indeed these can be found in his book "De revolutionibus" (1543).^{[277]} Tusi's theories made their way to the Polish scientist through the work of a SpanishJew, Abner of Burgos (1270—1340).^{[273]}
 Craig G. Fraser, in "The Cosmos: A Historical Perspective" (2006), explains the model in the following way; "[a]ssume that a circle of radius r rolls on the interior perimeter of a larger circle of radius 2r...[d]uring this motion a point on the perimeter of the smaller circle will trace out a straight line...the point...will move in a reciprocating motion back and forth on the line...[b]y means of this construction, two circular motions are able to generate a straightline motion, a fact in itself that seemed to challenge conceptually the Aristotelian opposition between rectilinear motion (terrestrial) and circular motion (celestial)...[a]lTusi was able to use the couple device in a somewhat complicated way to represent the motion of the inferior and superior planets with suitable accuracy, without using the equant".^{[274]}
 Additionally, "[t]he theorem is very easily proved: provided the lengths along the circumferences of the two circles that have been in contact have to be equal, and each is the product of a radius and an angle. For the fixed circle, the angle is only half as great, but the radius is twice as great, for the rolling circle".^{[272]}
 Of further interest is that Tusi's model is the first theory about circles rolling inside other circles, which was later to prove pivotal in explaining its applications in mechanics and engineering, particularly in the use of crankslider mechanism (first discovered^{[278]} by AlShaykh Rais AlAmal Badii AlZaman Abu AlIzz Ibn Ismail Ibn AlRazzaz AlJazari (Jazari; 1136–1206; see Mechanics During the Islamic Golden Age for further information) and in the design of a rotary engine.^{[277]}
 Craig G. Fraser, in "The Cosmos: A Historical Perspective" (2006), explains the model in the following way; "[a]ssume that a circle of radius r rolls on the interior perimeter of a larger circle of radius 2r...[d]uring this motion a point on the perimeter of the smaller circle will trace out a straight line...the point...will move in a reciprocating motion back and forth on the line...[b]y means of this construction, two circular motions are able to generate a straightline motion, a fact in itself that seemed to challenge conceptually the Aristotelian opposition between rectilinear motion (terrestrial) and circular motion (celestial)...[a]lTusi was able to use the couple device in a somewhat complicated way to represent the motion of the inferior and superior planets with suitable accuracy, without using the equant".^{[274]}
 Tusi Couple—Muhammad ibn Muhammad ibn alHasan alTusi (Tusi; 1201—1274^{[129]}) was the first to discover the Tusi Couple;^{[273]} an ingenious model which explained planetary motion which avoided the use of the equant; in other words he discovered a model to represent the motion of the inferior and superior planets in relation to one another.^{[274]} Scientists such as Ptolemy (fl. 127—145^{[78]}) and alHaytham (965—1040^{[161]}) had long been seeking such a model in their own time but ultimately failed.^{[272]} Interestingly Nicolaus Copernicus (1473—1543)—who could read Arabic^{[275]}—copied Tusi's ideas;^{[273]}^{[276]} indeed these can be found in his book "De revolutionibus" (1543).^{[277]} Tusi's theories made their way to the Polish scientist through the work of a SpanishJew, Abner of Burgos (1270—1340).^{[273]}
 A basic, simplified Tusi couple (click to see animation)
(click here to see larger version).  An ellipses astroid type Tusi couple (click to see animation)
(click here to see larger version).  Hypotrochoid ellipse type Tusi couple (click to see animation)
(click here to see larger version).  A Rohn ellipsograph Tusi couple (click to see animation)
(click here to see larger version).  An antiparallelogram ellipse Tusi couple (click to see animation)
(click here to see larger version).  A slidercrank ellipses Tusi couple (click to see animation)
(click here to see larger version).
List of Muslim Scientists Mentioned Within This Article
 Muhammad ibn Musa alKhwarizmi^{[279]} (Khwarizmi; 780—850^{[80]})—A mathematician who coined the words "algebra" and "algorithm".^{[280]} He is also known as the father of algebra.^{[281]} He wrote the "Hisab alJabr w'almuqabala" (or the "Science of Reduction and Confrontation", or "Science of Equations").^{[280]} His was born in Khwarizm, Uzbekistan.^{[282]} He is mentioned in the "History of Envoys" by the Islamic historian alTabari.^{[282]}
 Ahmad ibn Abdallah Habash Hasib Marwazi (d. 864/874^{[255]} or 869^{[256]})—Born in Marw, Turkestan (now Mary, Turkmenistan), then a part of the Abbasid Empire,^{[255]} he was an astronomer by profession, and was notably the first astronomer to calculate when the new crescent moon appeared.^{[255]} Not much of his work survives, but fortunately some does; most notably the "Kitab alajram walab 'ad" (or "The Book of Bodies and Distances").^{[255]}
 Abu ʿAbd Allah Muḥammad ibn Jabir ibn Sinan alRaqqi alḤarrani asSabiʾ alBattani (Albategnius; 858—929^{[85]})—was an an astronomer and mathematician, most well known for his accurate and correct determination of the length of a year at 365.24 days; which was directly used by the Catholic Papacy to create the "Gregorian" calendar (replacing the "Julian" calendar) in 1582.^{[283]} He died in Qar alJiss (where modern Iraq now stands).^{[283]}
 Abu Mahmud Hamid ibn Khidr Khojandi (Khujundi; 900—1000^{[73]})—was an astronomer and mathematician, who is notable for having constructed an observatory in Tehran and building a colossal mural sextent in 994 in order to measure Earth's axial tilt.^{[284]} He was also the first astronomer capable of measuring to an accuracy of arcseconds.^{[284]} He was born in Central Asia and is said to have been Central Asian himself.^{[284]}
 Abu'lHasan alUqlidisi (920—980)—was a mathematician who lived in Damascus.^{[285]} He is most well known for his book, "Kitab alFusul fi' lhisab alhindi" (or the "Book of the Sections on Indian Arithmetic").^{[285]} It was written in 945 and published in 952.^{[285]} He was also one of many scholars who took the work of alKhwarizmi and furthered it by improving it.^{[286]} His work entered Europe through first entering Muslim Spain.^{[286]}
 Abu alHasan 'Ali ibn 'Abd alRahman ibn Ahmad ibn Yunus alSadafi alMisri (Ibn Yunus; 950—1009)—M. Ismail, M. Z. Nashed, A. I. Zayed and A. F. Ghaleb in "Mathematical Analysis, Wavelets, and Signal Processing" (1994) label him the "greatest astronomer of medieval Islam".^{[287]} He tabulated the Sine function to nine decimal places, produced the "Very Useful Tables" for "reckoning time by the sun", and work on the Prosthaphaeresis.^{[287]}
 Abu alWafa' Buzjani (940—998^{[288]})—was born in Khorasan, Persia, and died in Baghdad.^{[288]} A particular caliph named Adud adDawlah ruled Western Persia and Iraq at the time, and who supported scientific research, especially in that of the astronomy and mathematics.^{[288]} The dynasty that he belonged to was called the Buyid Dynasty which lasted from 945 to 1055.^{[260]} In 959 Buzjani joined the caliph's court as a researcher.^{[288]}
 Hasan Ibn alHaytham (Alhazan; 965—1040^{[161]})—has been described as the "first ever true scientist".^{[289]} He was commissioned by the CaliphaImam alHakim bi Amr Allah (r. 996—1021) of the Fatimid Caliphate, to construct a dam to control the "ebb and flow of the Nile" c. 1008—1010.^{[290]} Notably this was one of his few failures^{[287]} given his extensive list of contributions towards science.^{[290]} He was also a contemporary of ibn Yunis.^{[290]}
 Abu Ishaq Ibrahim b. Yahya lNaqqash alTujibi Ibn alZarqalluh^{[291]} (Arzachel^{[292]}/Azarquiel^{[291]}; 1029—1087^{[75]}/1100^{[76]})—An astronomer in Islamic Spain (711—1492) notable for having invented the equatorium which plotted the positions of the sun, moon and planets.^{[293]} One of the craters on the moon is named after him.^{[294]} He is also famously constructed the "Toledan Tables" which were used until the beginning of the 14th century.^{[295]}
 Omar Khayyam (Khayyam; 1048—1131^{[184]})—known also for his famed story, the "Rubáiyát of Omar Khayyám", and chiefly known in the West as a poet, was also an astronomer, mathematician and philosopher.^{[296]} He extensively travelled throughout his eighty or so years of his life, living in Samarkand, Bokhara, Merv, Mecca and Isfahan.^{[296]} For part of his life he worked for the Seljuks, living 100 years before the Mongol Hordes.^{[296]}
 Sharaf alDin alMuzaffar ibn Muḥammad ibn alMuzaffar alTusi (Tusi; c. 1135—c. 1213^{[152]})—was a mathematician born in Tus.^{[297]} Little is known about him; but it is known he existed through other scientists, notably ibn alQifti, Ibn alHalikan and Tas Kubra Zad.^{[297]} Around 1165 he settled in Damascus and taught classical mathematics.^{[297]} He also settled in Aleppo where he taught distinguished Jews mathematics and astronomy.^{[297]}
 Muhammad ibn Muhammad ibn alHasan alTusi (Nasir alDin Tusi; 1201—1274^{[129]})—lived during the times of the Mongol Hordes.^{[298]} He was famed as a mathematician even amongst these foreign invaders.^{[298]} Hulagu Khan commissioned him to build an observatory the very same year he had destroyed Baghdad.^{[298]} Hulagu's brother, Mangu Khan, even asked for him to be sent to his own capital in China to construct an observatory.^{[298]}
 Abu alḤasan Ala al‐Din Ali ibn Ibrahim alAnsari (Ibn alShatir; 1305—1375^{[263]})—was a distinguished astronomer, who came to prominence in the West in the 1950s when it was discovered that Copernicus had copied his planetary and mathematical models.^{[263]} He is known for his books, the astronomical table handbook the "Zij", the "Ta'liq alarsad", the "Nihayat al su'l fi tashih alusul" and the "alZij alJadid".^{[263]}
 Ghiyath alDin Jamshid Masʿud alKashi (Alkhashi/Alkashani;^{[207]} c. 1380—c. 1429^{[64]})—Born in Persia, he is best known for being a prolific science writer with several well known books to his name; most notably "Sulam alSama" (or "The Stairway to Heaven"; 1407), "Mukhtasar dar ilmi hayat" (or "Compendium of the Science of Astronomy") and the "Khaqani Zij" (a set of astronomical tables; 1413—1414) used up until 1917.^{[299]}
Notable Libraries
 This is by no means an exhaustive list, and thus should be considered incomplete.
 The Library of the House of Wisdom—In the 12th century, this library contained at least 700,000 handwritten volumes.^{[300]} There were 63 libraries littered across ancient Baghdad,^{[301]} with one of the smaller libraries alone containing 150,000 volumes.^{[300]} For context, the University of Oxford in England around this time hardly had a few chestsworth of volumes.^{[300]}
 The Library of AlAzhar University—The library of the University of AlAzhar was founded in 969 in Egypt. It is still in operation, over 1,050 years since it's founding. In 1977, it's collection contains 80,000 volumes and 20,000 historical manuscripts.^{[301]} By 2016, the ancient manuscript collection grew to some 40,000, but many of them still require urgent digitization.^{[302]}
 The Library of Cordova—Established in 976 in Cordova, Spain, it contained 400,000—600,000 hand written volumes.^{[303]} Approximately 500 people were employed to maintain it. It was one of 70 libraries established by the Muslims in Spain. After the end of the Cordoba Caliphate in 1031, the books were scattered amongst various Moorish Muslim kingdoms.^{[303]}
 The Library of the House of Science—The library was established in 1004, in Cairo, by the Fatimad Caliphate, and contained 100,000—600,000 hand written volumes of bound books, and 2,400 copies of the Qu'ran written in gold and silver. The library contained volumes on "jurisprudence, grammar, rhetoric, history, biography, astronomy, and chemistry".^{[303]}
 The Library of Tripoli, Syria—This library contained 3,000,000 hand written volumes. It contained 50,000 copies of the Qu'ran, and 80,000 commentaries. The rest were all on science. It was deliberately burned down by a European Christian priest during the First Crusade, his reason being that the library contained copies of the Qu'ran. Foreigners used to study in it.^{[303]}
See also
Sources
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 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} Salim Ayduz (2014). The Oxford Encyclopedia of Philosophy, Science, and Technology in Islam. Oxford University Press. p. 390. ISBN 9780199812578.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} Norriss S. Hetherington (2006). Planetary Motions: A Historical Perspective. Greenwood Publishing Group. p. 78. ISBN 9780313332418.
 ^ ^{a} ^{b} Hunt Janin (2005). The Pursuit of Learning in the Islamic World, 6102003. McFarland. p. 108. ISBN 9780786419548.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} C.G. Weeramantry (19 September 1988). Islamic Jurisprudence: An International Perspective. Palgrave Macmillan UK. p. 16. ISBN 9781349194568.
 ^ ^{a} ^{b} ^{c} ^{d} Ibrahim AlKindilchie, Amer (2014). Libraries in Iraq and Egypt: A comparative study. International Library Review. 9 (1): 113–123. doi:10.1016/00207837(77)900541. ISSN 00207837.
 ^ ^{a} ^{b} Walid Ghali Nasr (2016). The State of Manuscript Digitisation Projects in Egypt. Aga Khan University. Institute for the Study of Muslim Civilisations, London. Library and Information Science in the Middle East and North Africa, Vol. 3. p. 302318. WayBackMachine Link. Retrieved January 27th, 2019.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} Elayyan, Ribhi Mustafa (2014). The history of the ArabicIslamic Libraries: 7th to 14th Centuries. International Library Review. 22 (2): 119–135. doi:10.1016/00207837(90)900147. ISSN 00207837.
Footnotes
 ^
In 952, Abu'l Hasan Ahmad ibn Ibrahim alUqlidisi became the first person in the world to solve the Indian numerical system without having to use the dust board as the Indians had been doing for centuries (the Indians had never used penandpaper algorithms to solve their mathematical problems); this is proven by the following peerreviewed sources on the history of the numerals;
 Quote: "A step forward in the development of the decimalplace value system was due to Abu'l Hasan Ahmad ibn Ibrahim alUqlidisi (920980)...who gave algorithms for use with pen and paper [in 1952], as opposed to those of alKhwarizmi which were for dust board, and, more importantly introduced decimal fractions."
 Ethan D. Bloch (14 May 2011). The Real Numbers and Real Analysis. Springer Science & Business Media. p. 54. ISBN 9780387721774.
 The Indian numerical system never used penandpaper style algorithmic arithmetic:
 Historians note that "[f]rom their origins in the late eighth and early ninth centuries, the numerals spread throughout the Islamic world, though not without resistance or confusion. Many conservative scribes and bookkeepers resisted the new numerals in favour of older calculation on the fingers and with numerical words".
 Historians say "[t]he Arabs borrowed not only the Indian numerals, but also a host of computational techniques and devices, including the dustboard, a flat tablet strewn with sand into which figures could be written for undertaking computations...Other techniques available included complex Greekderived system of finger reckoning and the use of shells; accordingly, the use of written penandpaperarithmetic was apparently not part of the initial practice of Indian derived numeration".
 After the work of alUqlidisi, only then did penandpaperarithmetic began to be widely used; well after his algorithms on penandpaper arithmetic been published and disseminated across the world; "[y]et once the [treatise] had been established by the eleventh and twelfth centuries, Arabic mathematical texts began to advocate doing computations with paper and ink instead of the dustboard".
 Stephen Chrisomalis (2010). Numerical Notation: A Comparative History. Cambridge University Press. p. 215. ISBN 9780521878180.
 ^
In 952, Abu'l Hasan Ahmad ibn Ibrahim alUqlidisi became the first person in the world to solve the Indian numerical system without having to use the dust board as the Indians had been doing for centuries (the Indians had never used penandpaper algorithms to solve their mathematical problems); this is proven by the following peerreviewed sources on the history of the numerals;
 Quote: "A step forward in the development of the decimalplace value system was due to Abu'l Hasan Ahmad ibn Ibrahim alUqlidisi (920980)...who gave algorithms for use with pen and paper [in 1952], as opposed to those of alKhwarizmi which were for dust board, and, more importantly introduced decimal fractions."
 Ethan D. Bloch (14 May 2011). The Real Numbers and Real Analysis. Springer Science & Business Media. p. 54. ISBN 9780387721774.
 The Indian numerical system never used penandpaper style algorithmic arithmetic:
 Historians note that "[f]rom their origins in the late eighth and early ninth centuries, the numerals spread throughout the Islamic world, though not without resistance or confusion. Many conservative scribes and bookkeepers resisted the new numerals in favour of older calculation on the fingers and with numerical words".
 Historians say "[t]he Arabs borrowed not only the Indian numerals, but also a host of computational techniques and devices, including the dustboard, a flat tablet strewn with sand into which figures could be written for undertaking computations...Other techniques available included complex Greekderived system of finger reckoning and the use of shells; accordingly, the use of written penandpaperarithmetic was apparently not part of the initial practice of Indian derived numeration".
 After the work of alUqlidisi, only then did penandpaperarithmetic began to be widely used; well after his algorithms on penandpaper arithmetic been published and disseminated across the world; "[y]et once the [treatise] had been established by the eleventh and twelfth centuries, Arabic mathematical texts began to advocate doing computations with paper and ink instead of the dustboard".
 Stephen Chrisomalis (2010). Numerical Notation: A Comparative History. Cambridge University Press. p. 215. ISBN 9780521878180.
 ^ Quote: "To determine the location of the apogee, one needed to find, by measuring the times between either the solstices and the equinoxes or the midpoints of the seasons, the direction in which the earth was removed from the center of the sun's path".
 Robert Morrison (10 September 2007). Islam and Science: The Intellectual Career of Nizam AlDin AlNisaburi. Routledge. p. 168. ISBN 9781135981143.
 ^ Quote: "To determine the location of the apogee, one needed to find, by measuring the times between either the solstices and the equinoxes or the midpoints of the seasons, the direction in which the earth was removed from the center of the sun's path".
 Robert Morrison (10 September 2007). Islam and Science: The Intellectual Career of Nizam AlDin AlNisaburi. Routledge. p. 168. ISBN 9781135981143.
 ^ Quote: "By the ninth century the six modern trigonometric functionssine and cosine, tangent and cotangent, secant and cosecanthad been identified, whereas Ptolemy knew only a single chord function. Of the six, five seem to be essentially Arabic in origin; only the sine function was introduced into Islam from India".
 Gingerich, Owen (1986). "Islamic Astronomy". Scientific American. Vol. 254. Issue No. 4. pp. 74–83. WayBackMachine Link. Retrieved December 15th, 2019.
 ^ Quote: "By the ninth century the six modern trigonometric functionssine and cosine, tangent and cotangent, secant and cosecanthad been identified, whereas Ptolemy knew only a single chord function. Of the six, five seem to be essentially Arabic in origin; only the sine function was introduced into Islam from India".
 Gingerich, Owen (1986). "Islamic Astronomy". Scientific American. Vol. 254. Issue No. 4. pp. 74–83. WayBackMachine Link. Retrieved December 15th, 2019.
 ^ Quote: "By the ninth century the six modern trigonometric functionssine and cosine, tangent and cotangent, secant and cosecanthad been identified, whereas Ptolemy knew only a single chord function. Of the six, five seem to be essentially Arabic in origin; only the sine function was introduced into Islam from India".
 Gingerich, Owen (1986). "Islamic Astronomy". Scientific American. Vol. 254. Issue No. 4. pp. 74–83. WayBackMachine Link. Retrieved December 15th, 2019.
 ^ Quote: "By the ninth century the six modern trigonometric functionssine and cosine, tangent and cotangent, secant and cosecanthad been identified, whereas Ptolemy knew only a single chord function. Of the six, five seem to be essentially Arabic in origin; only the sine function was introduced into Islam from India".
 Gingerich, Owen (1986). "Islamic Astronomy". Scientific American. Vol. 254. Issue No. 4. pp. 74–83. WayBackMachine Link. Retrieved December 15th, 2019.
 ^ Quote: "By the ninth century the six modern trigonometric functionssine and cosine, tangent and cotangent, secant and cosecanthad been identified, whereas Ptolemy knew only a single chord function. Of the six, five seem to be essentially Arabic in origin; only the sine function was introduced into Islam from India".
 Gingerich, Owen (1986). "Islamic Astronomy". Scientific American. Vol. 254. Issue No. 4. pp. 74–83. WayBackMachine Link. Retrieved December 15th, 2019.
 ^ Quote: "By the ninth century the six modern trigonometric functionssine and cosine, tangent and cotangent, secant and cosecanthad been identified, whereas Ptolemy knew only a single chord function. Of the six, five seem to be essentially Arabic in origin; only the sine function was introduced into Islam from India".
 Gingerich, Owen (1986). "Islamic Astronomy". Scientific American. Vol. 254. Issue No. 4. pp. 74–83. WayBackMachine Link. Retrieved December 15th, 2019.
 ^ Quote: "In this massive labor of computing trigonometric tables, knowing the precise value of sin1° is of fundamental importance. From the value of sin1° and the values of a few other basic sines, the trigonometric formulas generate sinp° for all integere values of p. The halfangle formulas can then be used to compute the sines in intervals of 1/2 ° and 1/4 °. Finally, the interpolation algorithms yield values for finer subdivisions. Although Ptolemy's interpolation method [1] for the approximate calculation of chords, in particular the chord of 1°, had been refined and used by Muslim mathematicians, they knew well the inherent limitations of this procedure, which quickly grows cumbersome and whose accuracy is restricted by the very inequalities from which it proceeds. To find a simpler and rapidly converging method for the evaluation of sin1° was highly desirable, and this became alKashi's goal. (See Figure 1 for the relation between crd 2α, the chord of angle 2α, and sinα.) He begins by setting up an equation expressing sin1° in terms of sin3°. His method is purely geometrical and general, so that he can be justly considered the first to derive the well known formula otherwise attributed to Viete in the late sixteenth century".
 Marlow Anderson; Victor Katz; Robin Wilson (14 October 2004). Sherlock Holmes in Babylon: And Other Tales of Mathematical History. MAA. p. 139. ISBN 9780883855461.
 ^ Quote: "In this massive labor of computing trigonometric tables, knowing the precise value of sin1° is of fundamental importance. From the value of sin1° and the values of a few other basic sines, the trigonometric formulas generate sinp° for all integere values of p. The halfangle formulas can then be used to compute the sines in intervals of 1/2 ° and 1/4 °. Finally, the interpolation algorithms yield values for finer subdivisions. Although Ptolemy's interpolation method [1] for the approximate calculation of chords, in particular the chord of 1°, had been refined and used by Muslim mathematicians, they knew well the inherent limitations of this procedure, which quickly grows cumbersome and whose accuracy is restricted by the very inequalities from which it proceeds. To find a simpler and rapidly converging method for the evaluation of sin1° was highly desirable, and this became alKashi's goal. (See Figure 1 for the relation between crd 2α, the chord of angle 2α, and sinα.) He begins by setting up an equation expressing sin1° in terms of sin3°. His method is purely geometrical and general, so that he can be justly considered the first to derive the well known formula otherwise attributed to Viete in the late sixteenth century".
 Marlow Anderson; Victor Katz; Robin Wilson (14 October 2004). Sherlock Holmes in Babylon: And Other Tales of Mathematical History. MAA. p. 139. ISBN 9780883855461.
 ^
Quote: "Abraham barHiyya HaNasi (1060 – 1136 A.D.) was a Spanish mathematician and astronomer who wrote the earliest book in Europe that expanded on the algebra of the Islamic world. His work, Treatise on Measurement and Calculation, was the first text in Europe, which contained a complete solution of the general quadratic equation. It was translated into Latin as Liber Embadorum and published in 1145. In a coincidence of history, alKhwarizmi’s algebra text was translated into Latin in this year as well and also included a complete solution of the quadratic".
 Raymond F. Tennant (February 2003). Using History in the Mathematics Classroom: Pythagoras, Quadratics and More. Teachers, Learners, and Curriculum Journal. Vol. 1. WayBackMachine Link. Retrieved January 2nd, 2020.
 ^
Quote: "Abraham barHiyya HaNasi (1060 – 1136 A.D.) was a Spanish mathematician and astronomer who wrote the earliest book in Europe that expanded on the algebra of the Islamic world. His work, Treatise on Measurement and Calculation, was the first text in Europe, which contained a complete solution of the general quadratic equation. It was translated into Latin as Liber Embadorum and published in 1145. In a coincidence of history, alKhwarizmi’s algebra text was translated into Latin in this year as well and also included a complete solution of the quadratic".
 Raymond F. Tennant (February 2003). Using History in the Mathematics Classroom: Pythagoras, Quadratics and More. Teachers, Learners, and Curriculum Journal. Vol. 1. WayBackMachine Link. Retrieved January 2nd, 2020.
 ^ Quote: "Among other 'integrations' by Archimedes were the volume and surface area of a sphere, the volume and area of a cone, the surface area of an ellipse, the volume of any segment of a paraboloid of revolution and a segment of an hyperboloid of revolution".
 J. J. O'Connor; E. F. Robertson (February 1996). A history of the calculus. Math History. MacTutor. WayBackMachine Link. Retrieved February 1st, 2020.
 ^ Haytham's method of calculating the area and volume of a parabola was different from the Greeks. This becomes evident when comparing Haytham's method to Archimedes':
 Quote: "Archimedes applied the Method to several curves, including solid figures such as spheres and cones, as well as plane curves such as parabolas. Here is a summary of Archimedes’ method for finding the area bounded by a parabola. Archimedes begins with a parabola and constructs several triangles from it. By using various properties of triangles and segments, Archimedes is able to find that the area under a parabolic arc is 4/3 of the area of a triangle inscribed into that arc. Here is where the method of exhaustion comes into play. Archimedes considered more and more triangles to fill the area under the parabola. What he finds is equivalent to a modern day geometric series. What is quite interesting is that Archimedes uses Proof by Contradiction to show that this series cannot differ from the true value by any finite amount. This would imply that the series converges to that value. However, Archimedes did not use concepts such as limits and convergence. As is usual of the method of exhaustion, Archimedes stops after some finite number of terms and is satisfied that he can make the disagreement between the approximation and the correct value as small as possible (while it of course remains finite)".
 Saul Foresta; Lawrence Goldman (????). Principia Mathematica Historallis Integratus. p. 3. Cite Seer X. Rutgers School of Art & Sciences. [WayBackMachine Link]. Retrieved February 1st, 2020.
 ^ Quote: "For all this work, how much did the Greeks actually accomplish? True, the method of exhaustion was a work of creative genius, but it did have two major flaws. First, it was not general. For each different problem, a different ingenious way of drawing triangles or some other polygon needed to be devised. The analytic approach of the modern era is completely general to the point that advanced math courses do not actually use numbers. The second, and larger, flaw was that the method of exhaustion was not at all rigorous by modern standards. Quite simply, there was no inclusion of a limit concept. The geometers all used the same argument. Take some geometrical figure and fill it with some large, but finite number of polygons. The sum of the areas of these polygons would be the area of the figure. But they did not consider this as a series. The math boils down to a convergent series, usually an easy to work with geometric series. Without a concept of infinity, which the Greeks lacked, it would have been impossible for them to rigorize the method of exhaustion".
 Saul Foresta; Lawrence Goldman (????). Principia Mathematica Historallis Integratus. p. 3. Cite Seer X. Rutgers School of Art & Sciences. [WayBackMachine Link]. Retrieved February 1st, 2020.
 ^
Quote: "While the method of exhaustion was used to great effect in Ancient Greece, it, and mathematics at the time, had its limitation; for one, the method needed to be adapted to each new shape  there was no general procedure that could be applied with minimal modifications to any problem. Furthermore, the framework of Greek mathematics had no ability to deal with the infinite, infinite sums, or limit processes. Because they could not be considered in any sense that would allow for mathematical rigour, they were taboo in the mathematical world of Ancient Greece [4]. Even though Archimedes and others came close to these concepts in their work with the method of exhaustion, they were still limited in their ability to progress the subject and approach the modern concept of an integral".
 Olle Hammarström (2016). Origins of Integration. Uppsala University. [WayBackMachine Link]. p. 45. Retrieved February 1st, 2020.
 ^ Quote: "Among other 'integrations' by Archimedes were the volume and surface area of a sphere, the volume and area of a cone, the surface area of an ellipse, the volume of any segment of a paraboloid of revolution and a segment of an hyperboloid of revolution".
 J. J. O'Connor; E. F. Robertson (February 1996). A history of the calculus. Math History. MacTutor. WayBackMachine Link. Retrieved February 1st, 2020.
 ^ Haytham's method of calculating the area and volume of a parabola was different from the Greeks. This becomes evident when comparing Haytham's method to Archimedes':
 Quote: "Archimedes applied the Method to several curves, including solid figures such as spheres and cones, as well as plane curves such as parabolas. Here is a summary of Archimedes’ method for finding the area bounded by a parabola. Archimedes begins with a parabola and constructs several triangles from it. By using various properties of triangles and segments, Archimedes is able to find that the area under a parabolic arc is 4/3 of the area of a triangle inscribed into that arc. Here is where the method of exhaustion comes into play. Archimedes considered more and more triangles to fill the area under the parabola. What he finds is equivalent to a modern day geometric series. What is quite interesting is that Archimedes uses Proof by Contradiction to show that this series cannot differ from the true value by any finite amount. This would imply that the series converges to that value. However, Archimedes did not use concepts such as limits and convergence. As is usual of the method of exhaustion, Archimedes stops after some finite number of terms and is satisfied that he can make the disagreement between the approximation and the correct value as small as possible (while it of course remains finite)".
 Saul Foresta; Lawrence Goldman (????). Principia Mathematica Historallis Integratus. p. 3. Cite Seer X. Rutgers School of Art & Sciences. [WayBackMachine Link]. Retrieved February 1st, 2020.
 ^ Quote: "For all this work, how much did the Greeks actually accomplish? True, the method of exhaustion was a work of creative genius, but it did have two major flaws. First, it was not general. For each different problem, a different ingenious way of drawing triangles or some other polygon needed to be devised. The analytic approach of the modern era is completely general to the point that advanced math courses do not actually use numbers. The second, and larger, flaw was that the method of exhaustion was not at all rigorous by modern standards. Quite simply, there was no inclusion of a limit concept. The geometers all used the same argument. Take some geometrical figure and fill it with some large, but finite number of polygons. The sum of the areas of these polygons would be the area of the figure. But they did not consider this as a series. The math boils down to a convergent series, usually an easy to work with geometric series. Without a concept of infinity, which the Greeks lacked, it would have been impossible for them to rigorize the method of exhaustion".
 Saul Foresta; Lawrence Goldman (????). Principia Mathematica Historallis Integratus. p. 3. Cite Seer X. Rutgers School of Art & Sciences. [WayBackMachine Link]. Retrieved February 1st, 2020.
 ^
Quote: "While the method of exhaustion was used to great effect in Ancient Greece, it, and mathematics at the time, had its limitation; for one, the method needed to be adapted to each new shape  there was no general procedure that could be applied with minimal modifications to any problem. Furthermore, the framework of Greek mathematics had no ability to deal with the infinite, infinite sums, or limit processes. Because they could not be considered in any sense that would allow for mathematical rigour, they were taboo in the mathematical world of Ancient Greece [4]. Even though Archimedes and others came close to these concepts in their work with the method of exhaustion, they were still limited in their ability to progress the subject and approach the modern concept of an integral".
 Olle Hammarström (2016). Origins of Integration. Uppsala University. [WayBackMachine Link]. p. 45. Retrieved February 1st, 2020.
 ^ Quote: "It is also known that alKashi was the first person who provided a clear statement of the law of cosine (i.e. where a, b, and c are side lengths of a triangle and γ is the angle between the sides with lengths a and b opposed to the side with length c)".
 Yoshihide Igarashi; Tom Altman; Mariko Funada (27 May 2014). Computing: A Historical and Technical Perspective. CRC Press. p. 78. ISBN 9781482227420.
 ^ Quote: "It is also known that alKashi was the first person who provided a clear statement of the law of cosine (i.e. where a, b, and c are side lengths of a triangle and γ is the angle between the sides with lengths a and b opposed to the side with length c)".
 Yoshihide Igarashi; Tom Altman; Mariko Funada (27 May 2014). Computing: A Historical and Technical Perspective. CRC Press. p. 78. ISBN 9781482227420.
 ^ Quote: "all the formulas we obtained emerged from ad hoc methods, each demanding separate verification".
 Knoebel, Arthur; Lodder, Jerry; Laubenbacher, Reinhard; Pengelley, David (2007). "The Bridge Between Continuous and Discrete: 1–82. doi:10.1007/9780387330624_1. WayBackMachine Link. Retrieved January 12th, 2019.
 ^ Quote: "all the formulas we obtained emerged from ad hoc methods, each demanding separate verification".
 Knoebel, Arthur; Lodder, Jerry; Laubenbacher, Reinhard; Pengelley, David (2007). "The Bridge Between Continuous and Discrete: 1–82. doi:10.1007/9780387330624_1. WayBackMachine Link. Retrieved January 12th, 2019.
 ^ Quote: "AlTūsī Linear Astrolabe. Sharaf alDin was also the inventor of a linear astrolabe, a single rod with markings on it (sometimes called “the rod of alTūusī”). The rod represents the meridian of the planispheric astrolabe, and two threads attached to it, with movable beads on them, can be positioned at various points along the rod to serve in place of the rete (the top plate in the usual planispheric astrolabe, whose pointers indicate the position of certain prominent stars). The rod has a number of scales, one of which represents the intersections of the altitude circles with the meridian. Another represents the intersections with the meridian of concentric circles that are the stereographic projections of the circles containing the beginnings of the zodiacal signs. (See Figure 1.)".
 Berggren, J. Lennart (2008). AlTūsī, Sharaf AlDīn AlMuzaffar Ibn Muhammad Ibn AlMuzaffar. Complete Dictionary of Scientific Biography. Charles Scribner & Sons. Online Copy Found Here (WayBackMachine Link). ISBN 9780684315591. Retrieved February 13th, 2019.
 ^ Quote: "AlTūsī Linear Astrolabe. Sharaf alDin was also the inventor of a linear astrolabe, a single rod with markings on it (sometimes called “the rod of alTūusī”). The rod represents the meridian of the planispheric astrolabe, and two threads attached to it, with movable beads on them, can be positioned at various points along the rod to serve in place of the rete (the top plate in the usual planispheric astrolabe, whose pointers indicate the position of certain prominent stars). The rod has a number of scales, one of which represents the intersections of the altitude circles with the meridian. Another represents the intersections with the meridian of concentric circles that are the stereographic projections of the circles containing the beginnings of the zodiacal signs. (See Figure 1.)".
 Berggren, J. Lennart (2008). AlTūsī, Sharaf AlDīn AlMuzaffar Ibn Muhammad Ibn AlMuzaffar. Complete Dictionary of Scientific Biography. Charles Scribner & Sons. Online Copy Found Here (WayBackMachine Link). ISBN 9780684315591. Retrieved February 13th, 2019.
Acknowledgements
I dedicate this article to a special someone who I love. My little sister Farheen. You're intelligent, you're funny, you're beautiful. And you're very understanding! As your big brother I just wanted to say thank you for being my sister! And a good sister at that! I love you and always will! —February 14th, 2020.