Mathematics in the medieval Islamic world

<h2 id="origin-and-spread-of-arab-islamic-mathematics">Origin and spread of Arab-Islamic mathematics</h2>
Arabic mathematics, particularly algebra, developed significantly during the <a href="/facts/Islamic_Golden_Age/yAeG2rBA">medieval period</a>. <a href="/facts/Al-Khwarizmi/24KdxyEQ">Muhammad ibn Musa al-Khwārizmī</a>'s (<a href="/facts/Arabic_language/A1e66hVV">Arabic</a>: محمد بن موسى الخوارزمي; c. 780 – c. 850) work between AD 813 and 833 in Baghdad was a turning point. He introduced the term "algebra" in the title of his book, "<a href="/facts/Al-Jabr/lxWsG6WD">Kitab al-jabr wa al-muqabala</a>," marking it as a distinct discipline. He regarded his work as "a short work on Calculation by (the rules of) Completion and Reduction, confining it to what is easiest and most useful in arithmetic".<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> Later, people commented his work was not just a theoretical treatise but also practical, aimed at solving problems in areas like commerce and land measurement.
<a href="/facts/Al-Khwarizmi/24KdxyEQ">Al-Khwārizmī</a>'s approach was groundbreaking in that it did not arise from any previous "arithmetical" tradition, including that of <a href="/facts/Diophantus/XTWRgRWP">Diophantus</a>. He developed a new vocabulary for algebra, distinguishing between purely algebraic terms and those shared with arithmetic. Al-Khwārizmī noticed that the representation of numbers is crucial in daily life. Thus, he wanted to find or summarize a way to simplify the mathematical operation, so-called later, the algebra.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> His algebra was initially focused on linear and quadratic equations and the elementary arithmetic of binomials and trinomials. This approach, which involved solving equations using radicals and related algebraic calculations, influenced mathematical thinking long after his death.
Al-Khwārizmī's proof of the rule for solving <a href="/facts/Quadratic_equation/9IMnkJAT">quadratic equations</a> of the form (ax^2 + bx = c), commonly referred to as "squares plus roots equal numbers," was a monumental achievement in the history of algebra. This breakthrough laid the groundwork for the systematic approach to solving quadratic equations, which became a fundamental aspect of algebra as it developed in the Western world.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> Al-Khwārizmī's method, which involved completing the square, not only provided a practical solution for equations of this type but also introduced an abstract and generalized approach to mathematical problems. His work, encapsulated in his seminal text "Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala" (The Compendious Book on Calculation by Completion and Balancing), was translated into <a href="/facts/Latin/pM3Kge85">Latin</a> in the 12th century. This translation played a pivotal role in the transmission of algebraic knowledge to Europe, significantly influencing mathematicians during the Renaissance and shaping the evolution of modern mathematics.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> Al-Khwārizmī's contributions, especially his proof for quadratic equations, are a testament to the rich mathematical heritage of the Islamic world and its enduring impact on Western mathematics.
The spread of Arabic mathematics to the West was facilitated by several factors. The practicality and general applicability of al-Khwārizmī's methods were significant. They were designed to convert numerical or geometrical problems into equations in normal form, leading to canonical solution formulae. His work and that of his successors like <a href="/facts/Al-Karaji/HM6UBsc0">al-Karaji</a> laid the foundation for advances in various mathematical fields, including <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a>, and <a href="/facts/Diophantine_equation/mYaHp7oX">rational Diophantine analysis</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
Al-Khwārizmī's algebra was an autonomous discipline with its historical perspective, eventually leading to the "arithmetization of algebra". His successors expanded on his work, adapting it to new theoretical and technical challenges and reorienting it towards a more arithmetical direction for abstract algebraic calculation.
Arabic mathematics, epitomized by al-Khwārizmī's work, was crucial in shaping the mathematical landscape. Its spread to the West was driven by its practical applications, the expansion of mathematical concepts by his successors, and the translation and adaptation of these ideas into the Western context. This spread was a complex process involving economics, politics, and cultural exchange, greatly influencing Western mathematics.
The period known as the <a href="/facts/Islamic_Golden_Age/yAeG2rBA">Islamic Golden Age</a> (8th to 14th century) was characterized by significant advancements in various fields, including <a href="/facts/Mathematics/pxTouaz4">mathematics</a>. Scholars in the Islamic world made substantial contributions to <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, <a href="/facts/Astronomy/0ArJVM1p">astronomy</a>, <a href="/facts/Medicine/2fnp90Je">medicine</a>, and other <a href="/facts/Science/IJoaD7rs">sciences</a>. As a result, the intellectual achievements of Islamic scholars attracted the attention of scholars in medieval Europe who sought to access this wealth of knowledge. Trade routes, such as the <a href="/facts/Silk_Road/TLBryY39">Silk Road</a>, facilitated the movement of goods, ideas, and knowledge between the East and West. Cities like <a href="/facts/Baghdad/RbtLzzHe">Baghdad</a>, <a href="/facts/Cairo/dXjj60ar">Cairo</a>, and <a href="/facts/C%C3%B3rdoba,_Spain/t4c3oIc0">Cordoba</a> became centers of learning and attracted scholars from different cultural backgrounds. Therefore, mathematical knowledge from the Islamic world found its way to Europe through various channels. Meanwhile, the <a href="/facts/Crusades/mblv9nwI">Crusades</a> connected Western Europeans with the Islamic world. While the primary purpose of the Crusades was military, there was also cultural exchange and exposure to Islamic knowledge, including mathematics. European scholars who traveled to the Holy Land and other parts of the Islamic world gained access to Arabic manuscripts and mathematical treatises. During the 14th to 17th century, the translation of Arabic mathematical texts, along with <a href="/facts/Greek_language/pzDd6eIz">Greek</a> and <a href="/facts/Romance_languages/TMK4KHI3">Roman</a> ones, played a crucial role in shaping the intellectual landscape of the Renaissance. Figures like <a href="/facts/Fibonacci/d5rA6p8c">Fibonacci</a>, who studied in North Africa and the Middle East, helped introduce and popularize Arabic numerals and mathematical concepts in Europe.

<h2 id="concepts">Concepts</h2>

<h3>Algebra</h3>
Further information: <a href="/facts/History_of_algebra/duLGFrgf">History of algebra</a>
The study of <a href="/facts/Algebra/nMdqczjo">algebra</a>, the name of which is derived from the <a href="/facts/Arabic_language/A1e66hVV">Arabic</a> word meaning completion or "reunion of broken parts",<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> flourished during the <a href="/facts/Islamic_golden_age/yAeG2rBA">Islamic golden age</a>. <a href="/facts/Muhammad_ibn_Musa_al-Khwarizmi/24KdxyEQ">Muhammad ibn Musa al-Khwarizmi</a>, a Persian scholar in the <a href="/facts/House_of_Wisdom/Iyo9lg8a">House of Wisdom</a> in <a href="/facts/Baghdad/RbtLzzHe">Baghdad</a> was the founder of algebra, is along with the <a href="/facts/Greek_people/wXwd2xlb">Greek</a> mathematician <a href="/facts/Diophantus/XTWRgRWP">Diophantus</a>, known as the father of algebra. In his book <a href="/facts/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing/lxWsG6WD">The Compendious Book on Calculation by Completion and Balancing</a>, Al-Khwarizmi deals with ways to solve for the <a href="/facts/Positive_number/z0xollg1">positive</a> <a href="/facts/Properties_of_polynomial_roots/YKNYVKcj">roots</a> of first and second-degree (linear and quadratic) <a href="/facts/Polynomial_equations/1QauIGxs">polynomial equations</a>. He introduces the method of <a href="/facts/Reduction_(mathematics)/tTVTXTKN">reduction</a>, and unlike Diophantus, also gives general solutions for the equations he deals with.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
Al-Khwarizmi's algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the algebraic work of Diophantus, which was syncopated, meaning that some symbolism is used. The transition to symbolic algebra, where only symbols are used, can be seen in the work of <a href="/facts/Ibn_al-Banna%27_al-Marrakushi/aWqx81bS">Ibn al-Banna' al-Marrakushi</a> and <a href="/facts/Ab%C5%AB_al-%E1%B8%A4asan_ibn_%CA%BFAl%C4%AB_al-Qala%E1%B9%A3%C4%81d%C4%AB/mulBlHCs">Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>
On the work done by Al-Khwarizmi, J. J. O'Connor and <a href="/facts/Edmund_F._Robertson/jQMcnY79">Edmund F. Robertson</a> said:<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

"Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed <a href="/facts/Rational_numbers/VJQmyY05">rational numbers</a>, <a href="/facts/Irrational_number/Psv19TET">irrational numbers</a>, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for the future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before." — <a href="/facts/MacTutor_History_of_Mathematics_archive/xtwLmN5q">MacTutor History of Mathematics archive</a>
Several other mathematicians during this time period expanded on the algebra of Al-Khwarizmi. <a href="/facts/Abu_Kamil_Shuja%27/rL52YrHL">Abu Kamil Shuja'</a> wrote a book of algebra accompanied with geometrical illustrations and proofs. He also enumerated all the possible solutions to some of his problems. <a href="/facts/Abu_al-Jud/5Cf2E1W7">Abu al-Jud</a>, <a href="/facts/Omar_Khayyam/M9ARtmAe">Omar Khayyam</a>, along with <a href="/facts/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%AB/uYR64wm4">Sharaf al-Dīn al-Tūsī</a>, found several solutions of the <a href="/facts/Cubic_equation/Eqm3HCMx">cubic equation</a>. Omar Khayyam found the general geometric solution of a cubic equation.

<h3>Cubic equations</h3>

Further information: <a href="/facts/Cubic_equation/Eqm3HCMx">Cubic equation</a>
<a href="/facts/Omar_Khayyam/M9ARtmAe">Omar Khayyam</a> (c. 1038/48 in <a href="/facts/Iran/24yukge8">Iran</a> – 1123/24)<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> wrote the Treatise on Demonstration of Problems of Algebra containing the systematic solution of <a href="/facts/Cubic_equation/Eqm3HCMx">cubic or third-order equations</a>, going beyond the Algebra of al-Khwārizmī.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> Khayyám obtained the solutions of these equations by finding the intersection points of two <a href="/facts/Conic_section/teqgLptX">conic sections</a>. This method had been used by the Greeks,<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> but they did not generalize the method to cover all equations with positive <a href="/facts/Zero_of_a_function/mlK3dhoT">roots</a>.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>
<a href="/facts/Sharaf_al-D%C4%ABn_al-%E1%B9%AC%C5%ABs%C4%AB/uYR64wm4">Sharaf al-Dīn al-Ṭūsī</a> (? in <a href="/facts/Tus,_Iran/vKSN0N1J">Tus, Iran</a> – 1213/4) developed a novel approach to the investigation of cubic equations—an approach which entailed finding the point at which a cubic polynomial obtains its maximum value. For example, to solve the equation 
 
 
 
  
 
 x
 
 3
 
 
 +
 a
 =
 b
 x
 
 
 {\displaystyle \ x^{3}+a=bx}
 
, with a and b positive, he would note that the maximum point of the curve 
 
 
 
  
 y
 =
 b
 x
 −
 
 x
 
 3
 
 
 
 
 {\displaystyle \ y=bx-x^{3}}
 
 occurs at 
 
 
 
 x
 =
 
 
 
 
 b
 3
 
 
 
 
 
 
 {\displaystyle x=\textstyle {\sqrt {\frac {b}{3}}}}
 
, and that the equation would have no solutions, one solution or two solutions, depending on whether the height of the curve at that point was less than, equal to, or greater than a. His surviving works give no indication of how he discovered his formulae for the maxima of these curves. Various conjectures have been proposed to account for his discovery of them.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>

<h3>Induction</h3>
See also: <a href="/facts/Mathematical_induction/KxAPqNrH">Mathematical induction § History</a>
The earliest implicit traces of mathematical induction can be found in <a href="/facts/Euclid/dHpVyL4R">Euclid</a>'s <a href="/facts/Euclid%27s_theorem/Nl4SM0P0">proof that the number of primes is infinite</a> (c. 300 BCE). The first explicit formulation of the principle of induction was given by <a href="/facts/Blaise_Pascal/Huy5LenL">Pascal</a> in his Traité du triangle arithmétique (1665).
In between, implicit <a href="/facts/Mathematical_proof/7ECDrU80">proof</a> by induction for <a href="/facts/Arithmetic_progression/c4ew9Xoq">arithmetic sequences</a> was introduced by <a href="/facts/Al-Karaji/HM6UBsc0">al-Karaji</a> (c. 1000) and continued by <a href="/facts/Ibn_Yahy%C4%81_al-Maghrib%C4%AB_al-Samaw%27al/XCXM56qJ">al-Samaw'al</a>, who used it for special cases of the <a href="/facts/Binomial_theorem/hd4SmkUd">binomial theorem</a> and properties of <a href="/facts/Pascal%27s_triangle/tb6BKXbc">Pascal's triangle</a>.

<h3>Irrational numbers</h3>
The Greeks had discovered <a href="/facts/Irrational_number/Psv19TET">irrational numbers</a>, but were not happy with them and only able to cope by drawing a distinction between magnitude and number. In the Greek view, magnitudes varied continuously and could be used for entities such as line segments, whereas numbers were discrete. Hence, irrationals could only be handled geometrically; and indeed Greek mathematics was mainly geometrical. Islamic mathematicians including <a href="/facts/Ab%C5%AB_K%C4%81mil_Shuj%C4%81%CA%BF_ibn_Aslam/rL52YrHL">Abū Kāmil Shujāʿ ibn Aslam</a> and <a href="/facts/Ibn_Tahir_al-Baghdadi/e1PmYTd9">Ibn Tahir al-Baghdadi</a> slowly removed the distinction between magnitude and number, allowing irrational quantities to appear as coefficients in equations and to be solutions of algebraic equations.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a><a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> They worked freely with irrationals as mathematical objects, but they did not examine closely their nature.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a>
In the twelfth century, <a href="/facts/Latin/pM3Kge85">Latin</a> translations of <a href="/facts/Al-Khwarizmi/24KdxyEQ">Al-Khwarizmi</a>'s <a href="/facts/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB/24KdxyEQ">Arithmetic</a> on the <a href="/facts/Hindu%E2%80%93Arabic_numeral_system/z73KvotW">Indian numerals</a> introduced the <a href="/facts/Decimal/7omfVNIM">decimal</a> <a href="/facts/Positional_notation/31DECoN4">positional number system</a> to the <a href="/facts/Western_world/UQl4e7du">Western world</a>.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a> His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of <a href="/facts/Linear_equation/JfCgALwf">linear</a> and <a href="/facts/Quadratic_equation/9IMnkJAT">quadratic equations</a>. In <a href="/facts/Renaissance/4D9Bc6KU">Renaissance</a> Europe, he was considered the original inventor of algebra, although it is now known that his work is based on older Indian or Greek sources.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a><a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> He revised <a href="/facts/Ptolemy/htudGQTQ">Ptolemy</a>'s <a href="/facts/Geography_(Ptolemy)/gbCXgJR7">Geography</a> and wrote on astronomy and astrology. However, <a href="/facts/C.A._Nallino/jdE8kguY">C.A. Nallino</a> suggests that al-Khwarizmi's original work was not based on Ptolemy but on a derivative world map,<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> presumably in <a href="/facts/Syriac_language/mLhEtdSW">Syriac</a> or <a href="/facts/Arabic_language/A1e66hVV">Arabic</a>.

<h3>Spherical trigonometry</h3>
Further information: <a href="/facts/Law_of_sines/PXNDVjlN">Law of sines</a> and <a href="/facts/History_of_trigonometry/75R19TDD">History of trigonometry</a>
The spherical <a href="/facts/Law_of_sines/PXNDVjlN">law of sines</a> was discovered in the 10th century: it has been attributed variously to <a href="/facts/Abu-Mahmud_Khojandi/Jv2sIskb">Abu-Mahmud Khojandi</a>, <a href="/facts/Nasir_al-Din_al-Tusi/BFScoqbf">Nasir al-Din al-Tusi</a> and <a href="/facts/Abu_Nasr_Mansur/bLDY9TEg">Abu Nasr Mansur</a>, with <a href="/facts/Abu_al-Wafa%27_Buzjani/ynklB2fP">Abu al-Wafa' Buzjani</a> as a contributor.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> <a href="/facts/Ibn_Mu%CA%BF%C4%81dh_al-Jayy%C4%81n%C4%AB/KUwkQ9po">Ibn Muʿādh al-Jayyānī</a>'s The book of unknown arcs of a sphere in the 11th century introduced the general law of sines.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a> The plane law of sines was described in the 13th century by <a href="/facts/Nas%C4%ABr_al-D%C4%ABn_al-T%C5%ABs%C4%AB/BFScoqbf">Nasīr al-Dīn al-Tūsī</a>. In his On the Sector Figure, he stated the law of sines for plane and spherical triangles and provided proofs for this law.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a>

<h3>Negative numbers</h3>
Further information: <a href="/facts/Negative_numbers/LzErzCmI">Negative numbers</a>
In the 9th century, Islamic mathematicians were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a> <a href="/facts/Al-Khwarizmi/24KdxyEQ">Al-Khwarizmi</a> did not use negative numbers or negative coefficients.<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a> But within fifty years, <a href="/facts/Abu_Kamil/rL52YrHL">Abu Kamil</a> illustrated the rules of signs for expanding the multiplication 
 
 
 
 (
 a
 ±
 b
 )
 (
 c
 ±
 d
 )
 
 
 {\displaystyle (a\pm b)(c\pm d)}
 
.<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a> <a href="/facts/Al-Karaji/HM6UBsc0">Al-Karaji</a> wrote in his book al-Fakhrī that "negative quantities must be counted as terms".<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a> In the 10th century, <a href="/facts/Ab%C5%AB_al-Waf%C4%81%27_al-B%C5%ABzj%C4%81n%C4%AB/ynklB2fP">Abū al-Wafā' al-Būzjānī</a> considered debts as negative numbers in A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen.<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a>
By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve <a href="/facts/Polynomial_division/MgJlJ2G7">polynomial divisions</a>.<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a> As <a href="/facts/Al-Samaw%27al/XCXM56qJ">al-Samaw'al</a> writes:

the product of a negative number—al-nāqiṣ—by a positive number—al-zāʾid—is negative, and by a negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. If we subtract a positive number from an empty power (martaba khāliyya), the remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number.<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a>
<h3>Double false position</h3>
Further information: <a href="/facts/False_position_method/bMuhM6pl">False position method</a>
Between the 9th and 10th centuries, the <a href="/facts/Egyptians/kBkt6yV8">Egyptian</a> mathematician <a href="/facts/Abu_Kamil/rL52YrHL">Abu Kamil</a> wrote a now-lost treatise on the use of double false position, known as the Book of the Two Errors (Kitāb al-khaṭāʾayn). The oldest surviving writing on double false position from the <a href="/facts/Middle_East/8CD6pLWv">Middle East</a> is that of <a href="/facts/Qusta_ibn_Luqa/FWdnx4Jq">Qusta ibn Luqa</a> (10th century), an <a href="/facts/Arab/E77duPsg">Arab</a> mathematician from <a href="/facts/Baalbek/5pHygt5K">Baalbek</a>, <a href="/facts/Lebanon/2dReTXyp">Lebanon</a>. He justified the technique by a formal, <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean-style geometric proof</a>. Within the tradition of Golden Age Muslim mathematics, double false position was known as hisāb al-khaṭāʾayn ("reckoning by two errors"). It was used for centuries to solve practical problems such as commercial and juridical questions (estate partitions according to rules of <a href="/facts/Islamic_inheritance_jurisprudence/cNlxHxRI">Quranic inheritance</a>), as well as purely recreational problems. The algorithm was often memorized with the aid of <a href="/facts/Mnemonics/6HfDn0ks">mnemonics</a>, such as a verse attributed to <a href="/facts/Ibn_al-Yasamin/NbLDNf54">Ibn al-Yasamin</a> and balance-scale diagrams explained by <a href="/facts/Al-Hassar/5qyGNHKq">al-Hassar</a> and <a href="/facts/Ibn_al-Banna/aWqx81bS">Ibn al-Banna</a>, who were each mathematicians of <a href="/facts/Moroccan_people/jy9XcnHN">Moroccan</a> origin.<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a>

<h2 id="influences">Influences</h2>
The influence of medieval Arab-Islamic mathematics to the rest of the world is wide and profound, in both the realm of science and mathematics. The knowledge of the Arabs went into the western world through <a href="/facts/Spain/bIyrB8EY">Spain</a> and <a href="/facts/Sicily/E6fug03z">Sicily</a> during the translation movement. "The Moors (western Mohammedans from that part of North Africa once known as Mauritania) crossed over into Spain early in the seventh century, bringing with them the cultural resources of the Arab world".<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a> In the 13th century, King <a href="/facts/Alfonso_X_of_Castile/xE7fc27D">Alfonso X of Castile</a> established the <a href="/facts/Toledo_School_of_Translators/lmpuoynI">Toledo School of Translators</a>, in the <a href="/facts/Kingdom_of_Castile/eA28464s">Kingdom of Castile</a>, where scholars translated numerous scientific and philosophical works from <a href="/facts/Arabic/A1e66hVV">Arabic</a> into <a href="/facts/Latin/pM3Kge85">Latin</a>. The translations included Islamic contributions to <a href="/facts/Trigonometry/K4hfaC6F">trigonometry</a>, which helps European mathematicians and astronomers in their studies. European scholars such as <a href="/facts/Gerard_of_Cremona/dToQQnXQ">Gerard of Cremona</a> (1114–1187) played a key role in translating and disseminating these works, thus making them accessible to a wider audience. Cremona is said to have translated into Latin "no fewer than 90 complete Arabic texts."<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a> European mathematicians, building on the foundations laid by Islamic scholars, further developed practical trigonometry for applications in navigation, cartography, and celestial navigation, thus pushing forward the age of discovery and scientific revolution. The practical applications of trigonometry for navigation and astronomy became increasingly important during the Age of Exploration.
<a href="/facts/Al-Battani/mzsBlt1T">Al-Battānī</a> is one of the islamic mathematicians who made great contributions to the development of trigonometry. He "innovated new trigonometric functions, created a table of cotangents, and made some formulas in spherical trigonometry."<a class="footnote-ref" id="fnref:40" href="#fn:40">40</a> These discoveries, together with his astronomical works which are praised for their accuracy, greatly advanced astronomical calculations and instruments. 
<a href="/facts/Omar_Khayyam/M9ARtmAe">Al-Khayyām</a> (1048–1131) was a Persian mathematician, astronomer, and poet, known for his work on algebra and geometry, particularly his investigations into the solutions of cubic equations. He was "the first in history to elaborate a geometrical theory of equations with degrees ≤ 3",<a class="footnote-ref" id="fnref:41" href="#fn:41">41</a> and has great influence on the work of Descartes, a French mathematician who is often regarded as the founder of analytical geometry. Indeed, "to read <a href="/facts/Ren%C3%A9_Descartes/ohqj6DMf">Descartes</a>' Géométrie is to look upstream towards al-Khayyām and al-Ṭūsī; and downstream towards Newton, Leibniz, Cramer, Bézout and the Bernoulli brothers".<a class="footnote-ref" id="fnref:42" href="#fn:42">42</a> Numerous problems that appear in "La Géométrie" (Geometry) have foundations that date back to al-Khayyām.
<a href="/facts/Abu_Kamil/rL52YrHL">Abū Kāmil</a> (Arabic: أبو كامل شجاع بن أسلم بن محمد بن شجاع, also known as Al-ḥāsib al-miṣrī—lit. "The Egyptian Calculator") (c. 850 – c. 930), was studied algebra following the author of Algebra, al-Khwārizmī. His Book of Algebra (Kitāb fī al-jabr wa al-muqābala) is "essentially a commentary on and elaboration of al-Khwārizmī's work; in part for that reason and in part for its own merit, the book enjoyed widespread popularity in the Muslim world".<a class="footnote-ref" id="fnref:43" href="#fn:43">43</a> It contains 69 problems, which is more than al-Khwārizmī who had 40 in his book.<a class="footnote-ref" id="fnref:44" href="#fn:44">44</a> Abū Kāmil's Algebra plays a significant role in shaping the trajectory of Western mathematics, particularly in its impact on the works of the Italian mathematician Leonardo of Pisa, widely recognized as Fibonacci. In his Liber Abaci (1202), Fibonacci extensively incorporated ideas from Arabic mathematicians, using approximately 29 problems from Book of Algebra with scarce modification.<a class="footnote-ref" id="fnref:45" href="#fn:45">45</a>

<h2 id="western-historians-perception-of-the-contribution-of-arab-mathematicians">Western historians' perception of the contribution of Arab mathematicians</h2>
Despite the fundamental works Arabic mathematicians have done on the development of Algebra and algebraic geometry, Western historians in the 18th and early 19th century still regarded it as a fact that <a href="/facts/Classical_science/pdBLPwP0">Classical science</a> and math were unique phenomena of the West. Even though some math contributions from Arab mathematicians are occasionally acknowledged, they are considered to be "outside history or only integrated in so far as it contributed to science, which is essentially European",<a class="footnote-ref" id="fnref:46" href="#fn:46">46</a> and just some technical innovations to the <a href="/facts/Greeks/wXwd2xlb">Greek</a> heritage rather than open up a completely new branch of mathematics. In the French philosopher <a href="/facts/Ernest_Renan/feDzxarQ">Ernest Renan</a>'s work, Arabic math is merely "a reflection of <a href="/facts/Greece/lqHw8bTB">Greece</a>, combined with <a href="/facts/Persians/4GhM3jl2">Persian</a> and Indian influences". And according to <a href="/facts/Pierre_Duhem/coxgQp9u">Duhem</a>, "Arabic science only reproduced the teachings received from Greek science". Besides being considered as merely some insignificant additions or reflections to the great tradition of Greek classical science, math works from Arabic mathematicians are also blamed for lacking rigor and too focused on practical applications and calculations, and this is why Western historians argued they could never reach the level of Greek mathematicians.<a class="footnote-ref" id="fnref:47" href="#fn:47">47</a> As <a href="/facts/Paul_Tannery/4l9UBtYj">Tannery</a> wrote, Arabic math "in no way superseded the level attained by Diophantus". On the other hand, they perceived that Western mathematicians went into a very different way both in its method employed and ultimate purpose, "the hallmark of Western science in its Greek origins as well as in its modern renaissance, is its conformity to rigorous standards".<a class="footnote-ref" id="fnref:48" href="#fn:48">48</a> Thus, the perceived non-rigorous proof in Arabic mathematicians' book authorizes <a href="/facts/Bourbaki_Panorama/FChgxJwv">Bourbaki</a> to exclude the Arabic period when he retraced the evolution of algebra.<a class="footnote-ref" id="fnref:49" href="#fn:49">49</a> And instead, the history of classical algebra is written as the work of the <a href="/facts/Renaissance/4D9Bc6KU">Renaissance</a> and the origin of algebraic geometry is traced back to <a href="/facts/Ren%C3%A9_Descartes/ohqj6DMf">Descartes</a>, while Arabic mathematicians' contributions are ignored deliberately. In Rashed's words: "To justify the exclusion of science written in Arabic from the history of science, one invokes its absence of rigor, its calculatory appearance and its practical aims. Furthermore, strictly dependent on Greek science and, lastly, incapable of introducing experimental norms, scientists of that time were relegated to the role of conscientious guardians of the Hellenistic museum."<a class="footnote-ref" id="fnref:50" href="#fn:50">50</a>
In 18th century <a href="/facts/Germany/paluuWBX">Germany</a> and <a href="/facts/France/CxIeIKM3">France</a>, the prevailing <a href="/facts/Orientalist_scholars_of_Islam/kNWshAlX">Orientalist</a> view was "East and West oppose each other not as geographical but as historical positivities",<a class="footnote-ref" id="fnref:51" href="#fn:51">51</a> which labeled "<a href="/facts/Rationalism/MNk171TG">Rationalism</a>" as the essence of the West, while the "Call of the <a href="/facts/Orient/uPK9os02">Orient</a>" movement emerged in the 19th century was interpreted as "against Rationalism"<a class="footnote-ref" id="fnref:52" href="#fn:52">52</a> and a return to a more "spiritual and harmonious" lifestyle. Thus, the prevailing <a href="/facts/Orientalism/56Rfn244">Orientalism</a> in that period was one of the main reasons why Arabic mathematicians were often ignored for their contributions, as people outside the West were considered to be lacking the necessary rationality and scientific spirit to made significant contributions to math and science.

<h2 id="conclusion">Conclusion</h2>
The medieval Arab-Islamic world played a crucial role in shaping the trajectory of mathematics, with <a href="/facts/Al-Khwarizmi/24KdxyEQ">al-Khwārizmī</a>'s algebraic innovations serving as a cornerstone. The dissemination of Greek mathematics to the West during the <a href="/facts/Islamic_Golden_Age/yAeG2rBA">Islamic Golden Age</a>, facilitated by cultural exchanges and translations, left a lasting impact on Western mathematical thought. Mathematicians like <a href="/facts/Al-Batt%C4%81n%C4%AB/mzsBlt1T">Al-Battānī</a>, <a href="/facts/Omar_Khayyam/M9ARtmAe">Al-Khayyām</a>, and <a href="/facts/Abu_Kamil/rL52YrHL">Abū Kāmil</a>, with their contributions to <a href="/facts/Trigonometry/K4hfaC6F">trigonometry</a>, <a href="/facts/Algebra/nMdqczjo">algebra</a>, and <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, extended their influence beyond their time. Despite the foundational contributions of Arab mathematicians, Western historians in the 18th and early 19th centuries, influenced by <a href="/facts/Orientalism/56Rfn244">Orientalist</a> views, sometimes marginalized these achievements. The East lacking rationality and scientific spirit perpetuated a biased perspective, hindering the recognition of the significant role played by Arabic mathematics in the development of algebra and other mathematical disciplines. Reevaluating the history of mathematics necessitates acknowledging the interconnectedness of diverse mathematical traditions and dispelling the notion of a uniquely European mathematical heritage. The contributions of Arab mathematicians, marked by practical applications and theoretical innovations, form an integral part of the rich tapestry of mathematical history, and deserves recognition.

<h2 id="other-major-figures">Other major figures</h2>
<ul><li><a href="/facts/%27Abd_al-Ham%C4%ABd_ibn_Turk/g9uHTKN3">'Abd al-Hamīd ibn Turk</a> (fl. 830) (quadratics)</li>
<li><a href="/facts/Sind_ibn_Ali/hmvmo9S1">Sind ibn Ali</a> (d. after 864)</li>
<li><a href="/facts/Thabit_ibn_Qurra/zuHkTsqK">Thabit ibn Qurra</a> (826–901)</li>
<li><a href="/facts/Al-Battani/mzsBlt1T">Al-Battānī</a> (before 858 – 929)</li>
<li><a href="/facts/Abu_Kamil/rL52YrHL">Abū Kāmil</a> (c. 850 – c. 930)</li>
<li><a href="/facts/Abu%27l-Hasan_al-Uqlidisi/xK4pXxW8">Abu'l-Hasan al-Uqlidisi</a> (fl. 952) (arithmetic)</li>
<li><a href="/facts/Abu_al-Saqr_al-Qabisi_%27Abd_al-%27Aziz_ibn_Uthman/8RvqJxoQ">'Abd al-'Aziz al-Qabisi</a> (d. 967)</li>
<li><a href="/facts/Ab%C5%AB_Sahl_al-Q%C5%ABh%C4%AB/E15RVsNl">Abū Sahl al-Qūhī</a> (c. 940–1000) (centres of gravity)</li>
<li><a href="/facts/Ibn_al-Haytham/Q5H9hm4J">Ibn al-Haytham</a> (c. 965–1040)</li>
<li><a href="/facts/Ab%C5%AB_al-Ray%E1%B8%A5%C4%81n_al-B%C4%ABr%C5%ABn%C4%AB/4XbTzrYl">Abū al-Rayḥān al-Bīrūnī</a> (973–1048) (trigonometry)</li>
<li><a href="/facts/Omar_Khayyam/M9ARtmAe">Al-Khayyām</a> (1048–1131)</li>
<li><a href="/facts/Ibn_Ma%E1%B8%8D%C4%81%CA%BE/aqVFDMB1">Ibn Maḍāʾ</a> (c. 1116–1196)</li>
<li><a href="/facts/Ismail_al-Jazari/ycMzl2tU">Ismail al-Jazari</a> (1136–1206)</li>
<li><a href="/facts/Jamsh%C4%ABd_al-K%C4%81sh%C4%AB/BHJWSgKK">Jamshīd al-Kāshī</a> (c. 1380–1429) (decimals and estimation of the circle constant)</li></ul>
<h2 id="gallery">Gallery</h2>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Arabic_numerals/LryNhh26">Arabic numerals</a></li>
<li><a href="/facts/Indian_influence_on_Islamic_science/A8UfVooL">Indian influence on Islamic mathematics in medieval Islam</a></li>
<li><a href="/facts/History_of_calculus/kJbq5owp">History of calculus</a></li>
<li><a href="/facts/History_of_geometry/2eShd8kg">History of geometry</a></li>
<li><a href="/facts/Science_in_the_medieval_Islamic_world/ajn0zRSc">Science in the medieval Islamic world</a></li>
<li><a href="/facts/Timeline_of_science_and_engineering_in_the_Muslim_world/8RbsW9gp">Timeline of science and engineering in the Muslim world</a></li></ul>

<h2 id="sources">Sources</h2>

<ul><li>Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Victor J. Katz (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (2nd ed.). Princeton, New Jersey: <a href="/facts/Princeton_University_Press/kwvFlKEj">Princeton University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-11485-9.</li>
<li><a href="/facts/Carl_Benjamin_Boyer/H58yXLK2">Boyer, Carl B.</a> (1991), "Greek Trigonometry and Mensuration, and The Arabic Hegemony", <a href="https://archive.org/details/historyofmathema00boye">A History of Mathematics</a> (2nd ed.), New York City: John Wiley & Sons, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-54397-7</li>
<li>Katz, Victor J. (1993). A History of Mathematics: An Introduction. HarperCollins college publishers. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-673-38039-4.</li>
<li><a href="/facts/C.A._Nallino/jdE8kguY">Nallino, C.A.</a> (1939), "Al-Ḥuwārismī e il suo rifacimento della Geografia di Tolomeo", Raccolta di scritti editi e inediti (in Italian), vol. V, Rome: Istituto per l'Oriente, pp. 458–532</li>
<li>Rosen, Fredrick (1831). <a href="https://archive.org/details/algebraofmohamme00khuwrich">The Algebra of Mohammed Ben Musa</a>. Kessinger Publishing. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-4179-4914-7.</li>
<li><a href="/facts/David_Eugene_Smith/0YqkAX82">Smith, David E.</a> (1958). History of Mathematics. Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-20429-4.</li>
<li><a href="/facts/Dirk_Jan_Struik/D7cVVb9r">Struik, Dirk J.</a> (1987), <a href="https://archive.org/details/concisehistoryof0000stru_m6j1">A Concise History of Mathematics</a> (4th rev. ed.), Dover Publications, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-60255-9</li></ul>

<h2 id="further-reading">Further reading</h2>

Books on Islamic mathematics
<ul><li>Berggren, J. Lennart (1986). Episodes in the Mathematics of Medieval Islam. New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-96318-9.
<ul><li>Review: <a href="/facts/Gerald_J._Toomer/awP7X8xD">Toomer, Gerald J.</a>; Berggren, J. L. (1988). "Episodes in the Mathematics of Medieval Islam". <a href="/facts/American_Mathematical_Monthly/Te6CHvkg">American Mathematical Monthly</a>. 95 (6). Mathematical Association of America: 567. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2322777">10.2307/2322777</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2322777">2322777</a>.</li>
<li>Review: Hogendijk, Jan P.; Berggren, J. L. (1989). "Episodes in the Mathematics of Medieval Islam by J. Lennart Berggren". Journal of the American Oriental Society. 109 (4). American Oriental Society: 697–698. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F604119">10.2307/604119</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/604119">604119</a>.</li></ul></li>
<li><a href="/facts/Ali_Abdullah_Al-Daffa/GFa6lSvs">Daffa', Ali Abdullah al-</a> (1977). The Muslim contribution to mathematics. London: Croom Helm. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-85664-464-1.</li>
<li><a href="/facts/Colin_Ronan/UmySwBh3">Ronan, Colin A.</a> (1983). The Cambridge Illustrated History of the World's Science. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-25844-8.</li>
<li><a href="/facts/Roshdi_Rashed/z5hX3dUa">Rashed, Roshdi</a> (2001). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Translated by A. F. W. Armstrong. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7923-2565-6.</li>
<li><a href="/facts/Adolph_Pavlovich_Yushkevich/l01o9FcC">Youschkevitch, Adolf P.</a>; Rozenfeld, Boris A. (1960). Die Mathematik der Länder des Ostens im Mittelalter. Berlin.{{cite book}}: CS1 maint: location missing publisher (link) Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62–160.</li>
<li>Youschkevitch, Adolf P. (1976). Les mathématiques arabes: VIIIe–XVe siècles. translated by M. Cazenave and K. Jaouiche. Paris: Vrin. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-2-7116-0734-1.</li></ul>
Book chapters on Islamic mathematics
<ul><li>Lindberg, D.C., and M. H. Shank, eds. The Cambridge History of Science. Volume 2: Medieval Science (Cambridge UP, 2013), chapters 2 and 3 mathematics in Islam.</li></ul>
<ul><li>Cooke, Roger (1997). "Islamic Mathematics". <a href="https://archive.org/details/historyofmathema0000cook">The History of Mathematics: A Brief Course</a>. Wiley-Interscience. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-18082-3.</li></ul>
Books on Islamic science
<ul><li>Daffa, Ali Abdullah al-; Stroyls, J.J. (1984). Studies in the exact sciences in medieval Islam. New York: Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-90320-5.</li>
<li><a href="/facts/Edward_Stewart_Kennedy/SSXvUb3A">Kennedy, E. S.</a> (1984). Studies in the Islamic Exact Sciences. Syracuse Univ Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8156-6067-7.</li></ul>
Books on the history of mathematics
<ul><li>Joseph, George Gheverghese (2000). <a href="https://archive.org/details/crestofpeacockno00jose">The Crest of the Peacock: Non-European Roots of Mathematics</a> (2nd ed.). Princeton University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-691-00659-8. (Reviewed: Katz, Victor J.; Joseph, George Gheverghese (1992). "<a href="/facts/The_Crest_of_the_Peacock:_Non-European_Roots_of_Mathematics/kxQphUSM">The Crest of the Peacock: Non-European Roots of Mathematics</a> by George Gheverghese Joseph". The College Mathematics Journal. 23 (1). Mathematical Association of America: 82–84. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2686206">10.2307/2686206</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2686206">2686206</a>.)</li>
<li>Youschkevitch, Adolf P. (1964). Gesichte der Mathematik im Mittelalter. Leipzig: BG Teubner Verlagsgesellschaft.</li></ul>
Journal articles on Islamic mathematics
<ul><li>Høyrup, Jens. <a href="http://akira.ruc.dk/~jensh/Publications/1987_Formation%20of%20Islamic%20mathematics.PDF">“The Formation of «Islamic Mathematics»: Sources and Conditions”</a>. Filosofi og Videnskabsteori på Roskilde Universitetscenter. 3. Række: Preprints og Reprints 1987 Nr. 1.</li></ul>
Bibliographies and biographies
<ul><li><a href="/facts/Carl_Brockelmann/D78VeHGF">Brockelmann, Carl</a>. Geschichte der Arabischen Litteratur. 1.–2. Band, 1.–3. Supplementband. Berlin: Emil Fischer, 1898, 1902; Leiden: Brill, 1937, 1938, 1942.</li>
<li><a href="/facts/Jos%C3%A9_Augusto_S%C3%A1nchez_P%C3%A9rez/xxArIfFV">Sánchez Pérez, José A.</a> (1921). Biografías de Matemáticos Árabes que florecieron en España. Madrid: Estanislao Maestre.</li>
<li><a href="/facts/Fuat_Sezgin/qlbAvEtM">Sezgin, Fuat</a> (1997). Geschichte Des Arabischen Schrifttums (in German). Brill Academic Publishers. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 90-04-02007-1.</li>
<li><a href="/facts/Heinrich_Suter/Egtg3RQt">Suter, Heinrich</a> (1900). Die Mathematiker und Astronomen der Araber und ihre Werke. Abhandlungen zur Geschichte der Mathematischen Wissenschaften Mit Einschluss Ihrer Anwendungen, X Heft. Leipzig.{{cite book}}: CS1 maint: location missing publisher (link)</li></ul>
Television documentaries
<ul><li><a href="/facts/Marcus_du_Sautoy/7LabtSsz">Marcus du Sautoy</a> (presenter) (2008). "The Genius of the East". <a href="/facts/The_Story_of_Maths/NYWNGCpI">The Story of Maths</a>. <a href="/facts/BBC/qFateIz8">BBC</a>.</li>
<li><a href="/facts/Jim_Al-Khalili/G9QzgXaT">Jim Al-Khalili</a> (presenter) (2010). <a href="/facts/Science_and_Islam_(documentary)/3kHjSDbe">Science and Islam</a>. <a href="/facts/BBC/qFateIz8">BBC</a>.</li></ul>

<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Mathematics of the Islamic Golden Age.

<ul><li><a href="/facts/Jan_Hogendijk/eh4J4lx0">Hogendijk, Jan P.</a> (January 1999). <a href="http://www.jphogendijk.nl/publ/Islamath.html">"Bibliography of Mathematics in Medieval Islamic Civilization"</a>.</li>
<li>O'Connor, John J.; <a href="/facts/Edmund_F._Robertson/jQMcnY79">Robertson, Edmund F.</a> (1999), <a href="https://mathshistory.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html">"Arabic mathematics: forgotten brilliance?"</a>, <a href="/facts/MacTutor_History_of_Mathematics_Archive/xtwLmN5q">MacTutor History of Mathematics Archive</a>, <a href="/facts/University_of_St_Andrews/IwXjmljw">University of St Andrews</a></li>
<li><a href="http://www.saudiaramcoworld.com/issue/200703/rediscovering.arabic.science.htm">Richard Covington, Rediscovering Arabic Science, 2007, Saudi Aramco World</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Katz (1993): "A complete history of mathematics of medieval Islam cannot yet be written, since so many of these Arabic manuscripts lie unstudied... Still, the general outline... is known. In particular, Islamic mathematicians fully developed the decimal place-value number system to include decimal fractions, systematised the study of algebra and began to consider the relationship between algebra and geometry, studied and made advances on the major Greek geometrical treatises of Euclid, Archimedes, and Apollonius, and made significant improvements in plane and spherical geometry."
^ Smith (1958), Vol. 1, Chapter VII.4: "In a general way it may be said that the Golden Age of Arabian mathematics was confined largely to the 9th and 10th centuries; that the world owes a great debt to Arab scholars for preserving and transmitting to posterity the classics of Greek mathematics; and that their work was chiefly that of transmission, although they developed considerable originality in algebra and showed some genius in their work in trigonometry." - Katz, Victor J. (1993). A History of Mathematics: An Introduction. HarperCollins college publishers. ISBN 0-673-38039-4. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Lumpkin, Beatrice; Zitler, Siham (1992). "Cairo: Science Academy of the Middle Ages". In Van Sertima, Ivan (ed.). Golden age of the Moor, Volume 11. Transaction Publishers. p. 394. ISBN 1-56000-581-5. "The Islamic mathematicians exercised a prolific influence on the development of science in Europe, enriched as much by their own discoveries as those they had inherited by the Greeks, the Indians, the Syrians, the Babylonians, etc." <a href="1-56000-581-5" target="_blank">1-56000-581-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">ben Musa, Mohammed (2013-03-28). The Algebra of Mohammed ben Musa. Cambridge University Press. ISBN 978-1-108-05507-9. <a href="978-1-108-05507-9" target="_blank">978-1-108-05507-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">ben Musa, Mohammed (2013-03-28). The Algebra of Mohammed ben Musa. Cambridge University Press. ISBN 978-1-108-05507-9. <a href="978-1-108-05507-9" target="_blank">978-1-108-05507-9</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Swetz, Frank J. (2012-08-15). Mathematical Treasures: Mesopotamian Accounting Tokens (Report). Washington, DC: The MAA Mathematical Sciences Digital Library. <a href="https://dx.doi.org/10.4169/loci003901" target="_blank">https://dx.doi.org/10.4169/loci003901</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Swetz, Frank J. (2012-08-15). Mathematical Treasures: Mesopotamian Accounting Tokens (Report). Washington, DC: The MAA Mathematical Sciences Digital Library. <a href="https://dx.doi.org/10.4169/loci003901" target="_blank">https://dx.doi.org/10.4169/loci003901</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">"Extending al-Karaji's Work on Sums of Odd Powers of Integers - Introduction | Mathematical Association of America". maa.org. Retrieved 2023-12-15. <a href="https://maa.org/press/periodicals/convergence/extending-al-karajis-work-on-sums-of-odd-powers-of-integers-introduction" target="_blank">https://maa.org/press/periodicals/convergence/extending-al-karajis-work-on-sums-of-odd-powers-of-integers-introduction</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">"algebra". Online Etymology Dictionary. <a href="http://www.etymonline.com/index.php?term=algebra&allowed_in_frame=0" target="_blank">http://www.etymonline.com/index.php?term=algebra&allowed_in_frame=0</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Boyer 1991, p. 228. - Boyer, Carl B. (1991), "Greek Trigonometry and Mensuration, and The Arabic Hegemony", A History of Mathematics (2nd ed.), New York City: John Wiley & Sons, ISBN 0-471-54397-7 <a href="https://archive.org/details/historyofmathema00boye" target="_blank">https://archive.org/details/historyofmathema00boye</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Swetz, Frank J. (1993). Learning Activities from the History of Mathematics. Walch Publishing. p. 26. ISBN 978-0-8251-2264-4. <a href="978-0-8251-2264-4" target="_blank">978-0-8251-2264-4</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Gullberg, Jan (1997). Mathematics: From the Birth of Numbers. W. W. Norton. p. 298. ISBN 0-393-04002-X. <a href="0-393-04002-X" target="_blank">0-393-04002-X</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">O'Connor, John J.; Robertson, Edmund F., "al-Marrakushi ibn Al-Banna", MacTutor History of Mathematics Archive, University of St Andrews <a href="/wiki/Edmund_F._Robertson" target="_blank">/wiki/Edmund_F._Robertson</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Gullberg, Jan (1997). Mathematics: From the Birth of Numbers. W. W. Norton. p. 298. ISBN 0-393-04002-X. <a href="0-393-04002-X" target="_blank">0-393-04002-X</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics Archive, University of St Andrews <a href="/wiki/Edmund_F._Robertson" target="_blank">/wiki/Edmund_F._Robertson</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Struik 1987, p. 96. - Struik, Dirk J. (1987), A Concise History of Mathematics (4th rev. ed.), Dover Publications, ISBN 0-486-60255-9 <a href="https://archive.org/details/concisehistoryof0000stru_m6j1" target="_blank">https://archive.org/details/concisehistoryof0000stru_m6j1</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Boyer 1991, pp. 241–242. - Boyer, Carl B. (1991), "Greek Trigonometry and Mensuration, and The Arabic Hegemony", A History of Mathematics (2nd ed.), New York City: John Wiley & Sons, ISBN 0-471-54397-7 <a href="https://archive.org/details/historyofmathema00boye" target="_blank">https://archive.org/details/historyofmathema00boye</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Struik 1987, p. 97. - Struik, Dirk J. (1987), A Concise History of Mathematics (4th rev. ed.), Dover Publications, ISBN 0-486-60255-9 <a href="https://archive.org/details/concisehistoryof0000stru_m6j1" target="_blank">https://archive.org/details/concisehistoryof0000stru_m6j1</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Boyer 1991, pp. 241–242. - Boyer, Carl B. (1991), "Greek Trigonometry and Mensuration, and The Arabic Hegemony", A History of Mathematics (2nd ed.), New York City: John Wiley & Sons, ISBN 0-471-54397-7 <a href="https://archive.org/details/historyofmathema00boye" target="_blank">https://archive.org/details/historyofmathema00boye</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Berggren, J. Lennart; Al-Tūsī, Sharaf Al-Dīn; Rashed, Roshdi (1990). "Innovation and Tradition in Sharaf al-Dīn al-Ṭūsī's al-Muʿādalāt". Journal of the American Oriental Society. 110 (2): 304–309. doi:10.2307/604533. JSTOR 604533. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Sesiano, Jacques (2000). "Islamic Mathematics". In Selin, Helaine (ed.). Mathematics Across Cultures: The History of Non-Western Mathematics. Springer Netherlands. pp. 137–165. doi:10.1007/978-94-011-4301-1_9. ISBN 978-94-011-4301-1. <a href="978-94-011-4301-1" target="_blank">978-94-011-4301-1</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">O'Connor, John J.; Robertson, Edmund F., "Abu Mansur ibn Tahir Al-Baghdadi", MacTutor History of Mathematics Archive, University of St Andrews <a href="/wiki/Edmund_F._Robertson" target="_blank">/wiki/Edmund_F._Robertson</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Allen, G. Donald (n.d.). "The History of Infinity" (PDF). Texas A&M University. Retrieved 7 September 2016. <a href="http://www.math.tamu.edu/~dallen/history/infinity.pdf" target="_blank">http://www.math.tamu.edu/~dallen/history/infinity.pdf</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Struik 1987, p. 93 - Struik, Dirk J. (1987), A Concise History of Mathematics (4th rev. ed.), Dover Publications, ISBN 0-486-60255-9 <a href="https://archive.org/details/concisehistoryof0000stru_m6j1" target="_blank">https://archive.org/details/concisehistoryof0000stru_m6j1</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Rosen 1831, p. v–vi. - Rosen, Fredrick (1831). The Algebra of Mohammed Ben Musa. Kessinger Publishing. ISBN 1-4179-4914-7. <a href="https://archive.org/details/algebraofmohamme00khuwrich" target="_blank">https://archive.org/details/algebraofmohamme00khuwrich</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Toomer, Gerald (1990). "Al-Khwārizmī, Abu Ja'far Muḥammad ibn Mūsā". In Gillispie, Charles Coulston (ed.). Dictionary of Scientific Biography. Vol. 7. New York: Charles Scribner's Sons. ISBN 0-684-16962-2 – via Encyclopedia.com. <a href="0-684-16962-2" target="_blank">0-684-16962-2</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
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<li id="fn:27">Sesiano, Jacques (2000). "Islamic Mathematics". In Selin, Helaine (ed.). Mathematics Across Cultures: The History of Non-Western Mathematics. Springer Netherlands. pp. 137–165. doi:10.1007/978-94-011-4301-1_9. ISBN 978-94-011-4301-1. <a href="978-94-011-4301-1" target="_blank">978-94-011-4301-1</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">O'Connor, John J.; Robertson, Edmund F., "Abu Abd Allah Muhammad ibn Muadh Al-Jayyani", MacTutor History of Mathematics Archive, University of St Andrews <a href="/wiki/Edmund_F._Robertson" target="_blank">/wiki/Edmund_F._Robertson</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Berggren 2007, p. 518. - Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Victor J. Katz (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-11485-9. <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Rashed, R. (1994-06-30). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Springer. pp. 36–37. ISBN 9780792325659. <a href="9780792325659" target="_blank">9780792325659</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Rashed, R. (1994-06-30). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Springer. pp. 36–37. ISBN 9780792325659. <a href="9780792325659" target="_blank">9780792325659</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Mat Rofa Bin Ismail (2008), "Algebra in Islamic Mathematics", in Helaine Selin (ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, vol. 1 (2nd ed.), Springer, p. 115, ISBN 9781402045592 <a href="9781402045592" target="_blank">9781402045592</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">Rashed, R. (1994-06-30). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Springer. pp. 36–37. ISBN 9780792325659. <a href="9780792325659" target="_blank">9780792325659</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Mat Rofa Bin Ismail (2008), "Algebra in Islamic Mathematics", in Helaine Selin (ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, vol. 1 (2nd ed.), Springer, p. 115, ISBN 9781402045592 <a href="9781402045592" target="_blank">9781402045592</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">Rashed, R. (1994-06-30). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Springer. pp. 36–37. ISBN 9780792325659. <a href="9780792325659" target="_blank">9780792325659</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">Rashed, R. (1994-06-30). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Springer. pp. 36–37. ISBN 9780792325659. <a href="9780792325659" target="_blank">9780792325659</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">Schwartz, R. K. (2004). Issues in the Origin and Development of Hisab al-Khata'ayn (Calculation by Double False Position) (PDF). Eighth North African Meeting on the History of Arab Mathematics. Radès, Tunisia. Archived from the original (PDF) on 2014-05-16. Retrieved 2012-06-08. "Issues in the Origin and Development of Hisab al-Khata'ayn (Calculation by Double False Position)". Archived from the original (.doc) on 2011-09-15. <a href="https://web.archive.org/web/20140516012137/http://www.ub.edu/islamsci/Schwartz.pdf" target="_blank">https://web.archive.org/web/20140516012137/http://www.ub.edu/islamsci/Schwartz.pdf</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">Masters, Barry R. (2011-06-08). "Biomedical ethics, 7th edition David DeGrazia, Thomas A. Mappes, Jeffrey Brand-Ballard: 2010, Softcover, 732pp, ISBN-9780073407456 £171.15 McGraw-Hill Incorporated". Graefe's Archive for Clinical and Experimental Ophthalmology. 250 (1): 159–160. doi:10.1007/s00417-011-1640-x. ISSN 0721-832X. <a href="https://dx.doi.org/10.1007/s00417-011-1640-x" target="_blank">https://dx.doi.org/10.1007/s00417-011-1640-x</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">Masters, Barry R. (2011-06-08). "Biomedical ethics, 7th edition David DeGrazia, Thomas A. Mappes, Jeffrey Brand-Ballard: 2010, Softcover, 732pp, ISBN-9780073407456 £171.15 McGraw-Hill Incorporated". Graefe's Archive for Clinical and Experimental Ophthalmology. 250 (1): 159–160. doi:10.1007/s00417-011-1640-x. ISSN 0721-832X. <a href="https://dx.doi.org/10.1007/s00417-011-1640-x" target="_blank">https://dx.doi.org/10.1007/s00417-011-1640-x</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">"Edited by", Contributions to Non-Standard Analysis, Elsevier, pp. iii, 1972, retrieved 2023-12-15 <a href="https://dx.doi.org/10.1016/s0049-237x(08)71546-5" target="_blank">https://dx.doi.org/10.1016/s0049-237x(08)71546-5</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">Rashed, Roshdi (2014-08-21). Classical Mathematics from Al-Khwarizmi to Descartes. Routledge. ISBN 978-1-317-62239-0. <a href="978-1-317-62239-0" target="_blank">978-1-317-62239-0</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
<li id="fn:42">Rashed, Roshdi (2014-08-21). Classical Mathematics from Al-Khwarizmi to Descartes. Routledge. ISBN 978-1-317-62239-0. <a href="978-1-317-62239-0" target="_blank">978-1-317-62239-0</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></li>
<li id="fn:43">Masters, Barry R. (2011-06-08). "Biomedical ethics, 7th edition David DeGrazia, Thomas A. Mappes, Jeffrey Brand-Ballard: 2010, Softcover, 732pp, ISBN-9780073407456 £171.15 McGraw-Hill Incorporated". Graefe's Archive for Clinical and Experimental Ophthalmology. 250 (1): 159–160. doi:10.1007/s00417-011-1640-x. ISSN 0721-832X. <a href="https://dx.doi.org/10.1007/s00417-011-1640-x" target="_blank">https://dx.doi.org/10.1007/s00417-011-1640-x</a> <a href="#fnref:43" class="footnote-back-ref">↩</a></li>
<li id="fn:44">Masters, Barry R. (2011-06-08). "Biomedical ethics, 7th edition David DeGrazia, Thomas A. Mappes, Jeffrey Brand-Ballard: 2010, Softcover, 732pp, ISBN-9780073407456 £171.15 McGraw-Hill Incorporated". Graefe's Archive for Clinical and Experimental Ophthalmology. 250 (1): 159–160. doi:10.1007/s00417-011-1640-x. ISSN 0721-832X. <a href="https://dx.doi.org/10.1007/s00417-011-1640-x" target="_blank">https://dx.doi.org/10.1007/s00417-011-1640-x</a> <a href="#fnref:44" class="footnote-back-ref">↩</a></li>
<li id="fn:45">Masters, Barry R. (2011-06-08). "Biomedical ethics, 7th edition David DeGrazia, Thomas A. Mappes, Jeffrey Brand-Ballard: 2010, Softcover, 732pp, ISBN-9780073407456 £171.15 McGraw-Hill Incorporated". Graefe's Archive for Clinical and Experimental Ophthalmology. 250 (1): 159–160. doi:10.1007/s00417-011-1640-x. ISSN 0721-832X. <a href="https://dx.doi.org/10.1007/s00417-011-1640-x" target="_blank">https://dx.doi.org/10.1007/s00417-011-1640-x</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></li>
<li id="fn:46">Rashed, Roshdi (1994). "The Development of Arabic Mathematics: Between Arithmetic and Algebra". Boston Studies in the Philosophy of Science. doi:10.1007/978-94-017-3274-1. ISSN 0068-0346. <a href="https://dx.doi.org/10.1007/978-94-017-3274-1" target="_blank">https://dx.doi.org/10.1007/978-94-017-3274-1</a> <a href="#fnref:46" class="footnote-back-ref">↩</a></li>
<li id="fn:47">Rashed, Roshdi (1994). "The Development of Arabic Mathematics: Between Arithmetic and Algebra". Boston Studies in the Philosophy of Science. doi:10.1007/978-94-017-3274-1. ISSN 0068-0346. <a href="https://dx.doi.org/10.1007/978-94-017-3274-1" target="_blank">https://dx.doi.org/10.1007/978-94-017-3274-1</a> <a href="#fnref:47" class="footnote-back-ref">↩</a></li>
<li id="fn:48">Rashed, Roshdi (1994). "The Development of Arabic Mathematics: Between Arithmetic and Algebra". Boston Studies in the Philosophy of Science. doi:10.1007/978-94-017-3274-1. ISSN 0068-0346. <a href="https://dx.doi.org/10.1007/978-94-017-3274-1" target="_blank">https://dx.doi.org/10.1007/978-94-017-3274-1</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></li>
<li id="fn:49">Rashed, Roshdi (1994). "The Development of Arabic Mathematics: Between Arithmetic and Algebra". Boston Studies in the Philosophy of Science. doi:10.1007/978-94-017-3274-1. ISSN 0068-0346. <a href="https://dx.doi.org/10.1007/978-94-017-3274-1" target="_blank">https://dx.doi.org/10.1007/978-94-017-3274-1</a> <a href="#fnref:49" class="footnote-back-ref">↩</a></li>
<li id="fn:50">Rashed, Roshdi (1994). "The Development of Arabic Mathematics: Between Arithmetic and Algebra". Boston Studies in the Philosophy of Science. doi:10.1007/978-94-017-3274-1. ISSN 0068-0346. <a href="https://dx.doi.org/10.1007/978-94-017-3274-1" target="_blank">https://dx.doi.org/10.1007/978-94-017-3274-1</a> <a href="#fnref:50" class="footnote-back-ref">↩</a></li>
<li id="fn:51">Rashed, Roshdi (1994). "The Development of Arabic Mathematics: Between Arithmetic and Algebra". Boston Studies in the Philosophy of Science. doi:10.1007/978-94-017-3274-1. ISSN 0068-0346. <a href="https://dx.doi.org/10.1007/978-94-017-3274-1" target="_blank">https://dx.doi.org/10.1007/978-94-017-3274-1</a> <a href="#fnref:51" class="footnote-back-ref">↩</a></li>
<li id="fn:52">Rashed, Roshdi (1994). "The Development of Arabic Mathematics: Between Arithmetic and Algebra". Boston Studies in the Philosophy of Science. doi:10.1007/978-94-017-3274-1. ISSN 0068-0346. <a href="https://dx.doi.org/10.1007/978-94-017-3274-1" target="_blank">https://dx.doi.org/10.1007/978-94-017-3274-1</a> <a href="#fnref:52" class="footnote-back-ref">↩</a></li>
</ol>

Mathematics in the medieval Islamic world open-in-new

Mathematics in the medieval Islamic world