Al-Mahani

<h2 id="biography">Biography</h2>
Historians know little of Al-Mahani's life due to lack of sources.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> He was born in <a href="/facts/Mahan,_Iran/9TwBzr5x">Mahan</a>, <a href="/facts/Persia/24yukge8">Persia</a> (hence the <a href="/facts/Nisba_(onomastics)/056hj9rC">Nisba</a> Al-Mahani).<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> He was active in the 9th century CE or 3rd century <a href="/facts/Islamic_calendar/7n4UBAYy">AH</a>, lived in Baghdad c. 860 and died c. 880.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> From a reference in <a href="/facts/Ibn_Yunus/et5L3YHW">Ibn Yunus</a>' Hakimite Tables, he was known to make astronomical observations between 853 and 866, allowing historians to estimate the time of his life and activities.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a><a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<h2 id="works">Works</h2>
<h3>Mathematics</h3>
His works on mathematics covered the topics of geometry, arithmetic, and algebra. Some of his mathematical work might have been motivated by problems he encountered in astronomy. The 10th century catalogue <a href="/facts/Al-Fihrist/BNILWphP">Al-Fihrist</a> mentions al-Mahani's contributions in mathematics but not those in astronomy.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>
He also worked on current mathematical problems at his time.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> He wrote commentaries on Greek mathematical works: Euclid's Elements, Archimedes' On the Sphere and Cylinder and Menelaus of Alexandria's Sphaerica.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> In his commentaries he added explanations, updated the language to use "modern" terms of his time, and reworked some of the proofs.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> He also wrote a standalone treatise Fi al-Nisba ("On Relationship") and another on the squaring of parabola.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>
His commentaries on the Elements covered Books I, V, X and XII; only those on Book V and parts of those on book X and XII survive today. In the Book V commentary, he worked on ratio, proposing a theory on the definition of ratio based on <a href="/facts/Continued_fraction/Wg1cMPxN">continued fractions</a> that was later discovered independently by <a href="/facts/Al-Nayrizi/w0IjgSNX">al-Nayrizi</a>.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a><a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>
In the Book X commentary, he worked on irrational numbers, including <a href="/facts/Quadratic_irrational_number/QXKP2PnT">quadratic irrational numbers</a> and cubic ones. He expanded Euclid's definition of magnitudes—which included only geometrical <a href="/facts/Line_(geometry)/gJqsr4Gj">lines</a>—by adding integers and fractions as rational magnitudes as well as square and cubic roots as irrational magnitudes. He called square roots "plane irrationalities" and cubic roots "solid irrationalities", and classified the sums or differences of these roots, as well as the results of the roots' additions or subtractions from rational magnitudes, also as irrational magnitudes. He then explained Book X using those rational and irrational magnitudes instead of geometric magnitudes like in the original.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a><a class="footnote-ref" id="fnref:19" href="#fn:19">19</a><a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>
His commentaries of the Sphaerica covered book I and parts of book II, none of which survive today. His edition was later updated by Ahmad ibn Abi Said al-Harawi (10th century). Later, <a href="/facts/Nasir_al-Din_al-Tusi/BFScoqbf">Nasir al-Din al-Tusi</a> (1201–1274) dismissed Al-Mahani and Al-Harawi's edition and wrote his own treatment of the Sphaerica, based on the works on <a href="/facts/Abu_Nasr_Mansur/bLDY9TEg">Abu Nasr Mansur</a>. Al-Tusi's edition became the most widely known edition of the Sphaerica in the Arabic-speaking world.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a><a class="footnote-ref" id="fnref:22" href="#fn:22">22</a>
Al-Mahani also attempted to solve a problem posed by Archimedes in On the Sphere and Cylinder, book II, chapter 4: how to divide a sphere by a plane into two volumes of a given ratio. His work led him to an equation, known as "Al-Mahani's equation" in the Muslim world: 
 
 
 
 
 x
 
 3
 
 
 +
 
 c
 
 2
 
 
 b
 =
 c
 
 x
 
 2
 
 
 
 
 {\displaystyle x^{3}+c^{2}b=cx^{2}}
 
. However, as documented later by <a href="/facts/Omar_Khayyam/M9ARtmAe">Omar Khayyam</a>, "after giving it lengthy meditation", he eventually failed to solve the problem. The problem was then considered unsolvable until 10th century Persian mathematician Abu Ja'far al-Khazin solved it using <a href="/facts/Conic_section/teqgLptX">conic sections</a>.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a><a class="footnote-ref" id="fnref:24" href="#fn:24">24</a><a class="footnote-ref" id="fnref:25" href="#fn:25">25</a>

<h3>Astronomy</h3>
His astronomical observations of <a href="/facts/Conjunction_(astronomy)/TgujumHI">conjunctions</a> as well as solar and lunar eclipses was cited in the <a href="/facts/Zij/U5fUtgmu">zij</a> (astronomical tables) of <a href="/facts/Ibn_Yunus/et5L3YHW">Ibn Yunus</a> (c. 950 – 1009). Ibn Yunus quoted Al-Mahani as saying that he calculated their timings with an <a href="/facts/Astrolabe/BctVIEm4">astrolabe</a>. He claimed his estimates of the start times of three consecutive lunar eclipses were accurate to within half an hour.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a><a class="footnote-ref" id="fnref:27" href="#fn:27">27</a>
He also wrote a treatise, Maqala fi ma'rifat as-samt li-aiy sa'a aradta wa fi aiy maudi aradta ("On the Determination of the Azimuth for an Arbitrary Time and an Arbitrary Place"), his only known surviving work on astronomy. In it, he provided two graphical methods and an arithmetic one of calculating the <a href="/facts/Azimuth/kEyt9nJ3">azimuth</a>—the angular measurement of a heavenly object's location. The arithmetic method corresponds to the <a href="/facts/Spherical_law_of_cosines/DYT3Er58">cosine rule</a> in <a href="/facts/Spherical_trigonometry/3G3Fhk2h">spherical trigonometry</a>, and was later used by <a href="/facts/Al-Battani/mzsBlt1T">Al-Battani</a> (c. 858 – 929).<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a><a class="footnote-ref" id="fnref:29" href="#fn:29">29</a>
He wrote another treatise, whose title, On the Latitude of the Stars, is known but its content is entirely lost. According to later astronomer <a href="/facts/Ibrahim_ibn_Sinan/7zdoE3Ku">Ibrahim ibn Sinan</a> (908–946), Al-Mahani also wrote a treatise on calculating the <a href="/facts/Ascendant/VaET7Mjb">ascendant</a> using a <a href="/facts/Solar_clock/RWM823m2">solar clock</a>.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/List_of_pre-modern_Iranian_scientists_and_scholars/2QCymHox">List of pre-modern Iranian scientists and scholars</a></li></ul>

<h3>Citations</h3>

<h3>Work cited</h3>
<ul><li><a href="/facts/Yvonne_Dold-Samplonius/KHaF5sUV">Dold-Samplonius, Yvonne</a> (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. <a href="/facts/Encyclopaedia_of_the_History_of_Science,_Technology,_and_Medicine_in_Non-Western_Cultures/SY6vMB4d">Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures</a>. <a href="/facts/New_York_City/YlyvFmVC">New York</a>: <a href="/facts/Springer_Science%2BBusiness_Media/nAesf6nT">Springer</a>. pp. 141–142. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320">10.1007/978-1-4020-4425-0_9320</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4020-4559-2.</li>
<li>Dold-Samplonius, Yvonne (2008b) [1970-80]. <a href="http://www.encyclopedia.com/doc/1G2-2830902765.html">"Al-Māhānī, Abū 'Abd Allāh Muḥammad Ibn 'Īsā"</a>. <a href="/facts/Dictionary_of_Scientific_Biography/qMA96GhQ">Complete Dictionary of Scientific Biography</a>. <a href="/facts/Charles_Scribner%27s_Sons/VRo5mFWQ">Charles Scribner's Sons</a> and <a href="/facts/Encyclopedia.com/kPn1Sqbp">Encyclopedia.com</a>.</li>
<li>Matvievskaya, Galina (1987). "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics". <a href="/facts/Annals_of_the_New_York_Academy_of_Sciences/ThS2KRMa">Annals of the New York Academy of Sciences</a>. 500 (1): 253–277. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1987NYASA.500..253M">1987NYASA.500..253M</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1111%2Fj.1749-6632.1987.tb37206.x">10.1111/j.1749-6632.1987.tb37206.x</a>.</li>
<li>O'Connor, J.J.; Robertson, E.F (1999). <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Mahani.html">"Abu Abd Allah Muhammad ibn Isa Al-Mahani"</a>. <a href="/facts/MacTutor_History_of_Mathematics_archive/xtwLmN5q">MacTutor History of Mathematics archive</a>. School of Mathematics and Statistics, <a href="/facts/University_of_St_Andrews/IwXjmljw">University of St Andrews</a>.</li>
<li><a href="/facts/George_Sarton/rhAkYlKf">Sarton, George</a> (1927). "Al-Mahani". Introduction to the History of Science. Vol. I: From Homer to Omar Khayyam. <a href="/facts/Baltimore/ncJvTzBU">Baltimore</a>: William & Wilkins Company for <a href="/facts/Carnegie_Institution_of_Washington/zS5MqNUP">Carnegie Institution of Washington</a>. pp. 597–598.</li>
<li>Sesiano, J. (1993). "Muhammad b. Isa b. Ahmad al-Mahani". In C.E. Bosworth; E. von Donzel; W.P. Heinrichs; Ch. Pellat (eds.). <a href="/facts/The_Encyclopaedia_of_Islam/oSUIAA8e">The Encyclopaedia of Islam</a>, New Edition. Vol. VII: Mif—Naz. <a href="/facts/Leiden/rs70mHlG">Leiden</a> and <a href="/facts/London/KozCCsy9">London</a>: <a href="/facts/Brill_Publishers/UwYwgUu0">Brill</a>. p. 405. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-90-04-09419-2.</li>
<li>Roshdi Rashed and Athanase Papadopoulos, Menelaus' Spherics: Early Translation and al-Mahani'/al-Harawi's version (Critical edition of Menelaus' Spherics from the Arabic manuscripts, with historical and mathematical commentaries), De Gruyter, Series: Scientia Graeco-Arabica, 21, 2017, 890 pages. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-11-057142-4</li></ul>

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

<ol>
<li id="fn:1">Meri, Josef W. (2005-10-31). Medieval Islamic Civilization: An Encyclopedia. Routledge. p. 32. ISBN 978-1-135-45603-0. <a href="978-1-135-45603-0" target="_blank">978-1-135-45603-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">On science and the construction of identities : remembering Ibn al-Haytham (965–1039) page 99 : "He neatly resolved the problem of al-Mahanī, a Persian mathematician of the 9th century" <a href="https://www.raco.cat/index.php/Contributions/article/viewFile/313116/403229/" target="_blank">https://www.raco.cat/index.php/Contributions/article/viewFile/313116/403229/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">* Roshdi Rashed and Athanase Papadopoulos, 2017 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Dold-Samplonius 2008, p. 141. - Dold-Samplonius, Yvonne (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York: Springer. pp. 141–142. doi:10.1007/978-1-4020-4425-0_9320. ISBN 978-1-4020-4559-2. <a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320" target="_blank">https://doi.org/10.1007%2F978-1-4020-4425-0_9320</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Dold-Samplonius 2008, p. 141. - Dold-Samplonius, Yvonne (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York: Springer. pp. 141–142. doi:10.1007/978-1-4020-4425-0_9320. ISBN 978-1-4020-4559-2. <a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320" target="_blank">https://doi.org/10.1007%2F978-1-4020-4425-0_9320</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Dold-Samplonius 2008, p. 141. - Dold-Samplonius, Yvonne (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York: Springer. pp. 141–142. doi:10.1007/978-1-4020-4425-0_9320. ISBN 978-1-4020-4559-2. <a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320" target="_blank">https://doi.org/10.1007%2F978-1-4020-4425-0_9320</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Sesiano 1993, p. 141. - Sesiano, J. (1993). "Muhammad b. Isa b. Ahmad al-Mahani". In C.E. Bosworth; E. von Donzel; W.P. Heinrichs; Ch. Pellat (eds.). The Encyclopaedia of Islam, New Edition. Vol. VII: Mif—Naz. Leiden and London: Brill. p. 405. ISBN 978-90-04-09419-2. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Dold-Samplonius 2008, p. 141. - Dold-Samplonius, Yvonne (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York: Springer. pp. 141–142. doi:10.1007/978-1-4020-4425-0_9320. ISBN 978-1-4020-4559-2. <a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320" target="_blank">https://doi.org/10.1007%2F978-1-4020-4425-0_9320</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">O'Connor & Robertson 1999. - O'Connor, J.J.; Robertson, E.F (1999). "Abu Abd Allah Muhammad ibn Isa Al-Mahani". MacTutor History of Mathematics archive. School of Mathematics and Statistics, University of St Andrews. <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Mahani.html" target="_blank">http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Mahani.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">O'Connor & Robertson 1999. - O'Connor, J.J.; Robertson, E.F (1999). "Abu Abd Allah Muhammad ibn Isa Al-Mahani". MacTutor History of Mathematics archive. School of Mathematics and Statistics, University of St Andrews. <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Mahani.html" target="_blank">http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Mahani.html</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Dold-Samplonius 2008, p. 141. - Dold-Samplonius, Yvonne (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York: Springer. pp. 141–142. doi:10.1007/978-1-4020-4425-0_9320. ISBN 978-1-4020-4559-2. <a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320" target="_blank">https://doi.org/10.1007%2F978-1-4020-4425-0_9320</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Dold-Samplonius 2008, p. 141. - Dold-Samplonius, Yvonne (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York: Springer. pp. 141–142. doi:10.1007/978-1-4020-4425-0_9320. ISBN 978-1-4020-4559-2. <a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320" target="_blank">https://doi.org/10.1007%2F978-1-4020-4425-0_9320</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Dold-Samplonius 2008, p. 141. - Dold-Samplonius, Yvonne (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York: Springer. pp. 141–142. doi:10.1007/978-1-4020-4425-0_9320. ISBN 978-1-4020-4559-2. <a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320" target="_blank">https://doi.org/10.1007%2F978-1-4020-4425-0_9320</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Sesiano 1993, p. 405. - Sesiano, J. (1993). "Muhammad b. Isa b. Ahmad al-Mahani". In C.E. Bosworth; E. von Donzel; W.P. Heinrichs; Ch. Pellat (eds.). The Encyclopaedia of Islam, New Edition. Vol. VII: Mif—Naz. Leiden and London: Brill. p. 405. ISBN 978-90-04-09419-2. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Sesiano 1993, p. 405. - Sesiano, J. (1993). "Muhammad b. Isa b. Ahmad al-Mahani". In C.E. Bosworth; E. von Donzel; W.P. Heinrichs; Ch. Pellat (eds.). The Encyclopaedia of Islam, New Edition. Vol. VII: Mif—Naz. Leiden and London: Brill. p. 405. ISBN 978-90-04-09419-2. <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Dold-Samplonius 2008, p. 142. - Dold-Samplonius, Yvonne (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York: Springer. pp. 141–142. doi:10.1007/978-1-4020-4425-0_9320. ISBN 978-1-4020-4559-2. <a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320" target="_blank">https://doi.org/10.1007%2F978-1-4020-4425-0_9320</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Dold-Samplonius 2008b. - Dold-Samplonius, Yvonne (2008b) [1970-80]. "Al-Māhānī, Abū 'Abd Allāh Muḥammad Ibn 'Īsā". Complete Dictionary of Scientific Biography. Charles Scribner's Sons and Encyclopedia.com. <a href="http://www.encyclopedia.com/doc/1G2-2830902765.html" target="_blank">http://www.encyclopedia.com/doc/1G2-2830902765.html</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Dold-Samplonius 2008, p. 142. - Dold-Samplonius, Yvonne (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York: Springer. pp. 141–142. doi:10.1007/978-1-4020-4425-0_9320. ISBN 978-1-4020-4559-2. <a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320" target="_blank">https://doi.org/10.1007%2F978-1-4020-4425-0_9320</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Dold-Samplonius 2008b. - Dold-Samplonius, Yvonne (2008b) [1970-80]. "Al-Māhānī, Abū 'Abd Allāh Muḥammad Ibn 'Īsā". Complete Dictionary of Scientific Biography. Charles Scribner's Sons and Encyclopedia.com. <a href="http://www.encyclopedia.com/doc/1G2-2830902765.html" target="_blank">http://www.encyclopedia.com/doc/1G2-2830902765.html</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Matvievskaya 1987, p. 259. - Matvievskaya, Galina (1987). "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics". Annals of the New York Academy of Sciences. 500 (1): 253–277. Bibcode:1987NYASA.500..253M. doi:10.1111/j.1749-6632.1987.tb37206.x. <a href="https://ui.adsabs.harvard.edu/abs/1987NYASA.500..253M" target="_blank">https://ui.adsabs.harvard.edu/abs/1987NYASA.500..253M</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Dold-Samplonius 2008, p. 141. - Dold-Samplonius, Yvonne (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York: Springer. pp. 141–142. doi:10.1007/978-1-4020-4425-0_9320. ISBN 978-1-4020-4559-2. <a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320" target="_blank">https://doi.org/10.1007%2F978-1-4020-4425-0_9320</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Dold-Samplonius 2008b. - Dold-Samplonius, Yvonne (2008b) [1970-80]. "Al-Māhānī, Abū 'Abd Allāh Muḥammad Ibn 'Īsā". Complete Dictionary of Scientific Biography. Charles Scribner's Sons and Encyclopedia.com. <a href="http://www.encyclopedia.com/doc/1G2-2830902765.html" target="_blank">http://www.encyclopedia.com/doc/1G2-2830902765.html</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">O'Connor & Robertson 1999. - O'Connor, J.J.; Robertson, E.F (1999). "Abu Abd Allah Muhammad ibn Isa Al-Mahani". MacTutor History of Mathematics archive. School of Mathematics and Statistics, University of St Andrews. <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Mahani.html" target="_blank">http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Mahani.html</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Dold-Samplonius 2008, p. 142. - Dold-Samplonius, Yvonne (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York: Springer. pp. 141–142. doi:10.1007/978-1-4020-4425-0_9320. ISBN 978-1-4020-4559-2. <a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320" target="_blank">https://doi.org/10.1007%2F978-1-4020-4425-0_9320</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Sarton 1927, p. 598. - Sarton, George (1927). "Al-Mahani". Introduction to the History of Science. Vol. I: From Homer to Omar Khayyam. Baltimore: William & Wilkins Company for Carnegie Institution of Washington. pp. 597–598. <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Dold-Samplonius 2008, p. 141. - Dold-Samplonius, Yvonne (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York: Springer. pp. 141–142. doi:10.1007/978-1-4020-4425-0_9320. ISBN 978-1-4020-4559-2. <a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320" target="_blank">https://doi.org/10.1007%2F978-1-4020-4425-0_9320</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Dold-Samplonius 2008b. - Dold-Samplonius, Yvonne (2008b) [1970-80]. "Al-Māhānī, Abū 'Abd Allāh Muḥammad Ibn 'Īsā". Complete Dictionary of Scientific Biography. Charles Scribner's Sons and Encyclopedia.com. <a href="http://www.encyclopedia.com/doc/1G2-2830902765.html" target="_blank">http://www.encyclopedia.com/doc/1G2-2830902765.html</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Dold-Samplonius 2008, p. 141. - Dold-Samplonius, Yvonne (2008). "Al‐Māhānī". In Helaine Selin (ed.). Al-Mahani. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York: Springer. pp. 141–142. doi:10.1007/978-1-4020-4425-0_9320. ISBN 978-1-4020-4559-2. <a href="https://doi.org/10.1007%2F978-1-4020-4425-0_9320" target="_blank">https://doi.org/10.1007%2F978-1-4020-4425-0_9320</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Sesiano 1993, p. 405. - Sesiano, J. (1993). "Muhammad b. Isa b. Ahmad al-Mahani". In C.E. Bosworth; E. von Donzel; W.P. Heinrichs; Ch. Pellat (eds.). The Encyclopaedia of Islam, New Edition. Vol. VII: Mif—Naz. Leiden and London: Brill. p. 405. ISBN 978-90-04-09419-2. <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Sesiano 1993, p. 405. - Sesiano, J. (1993). "Muhammad b. Isa b. Ahmad al-Mahani". In C.E. Bosworth; E. von Donzel; W.P. Heinrichs; Ch. Pellat (eds.). The Encyclopaedia of Islam, New Edition. Vol. VII: Mif—Naz. Leiden and London: Brill. p. 405. ISBN 978-90-04-09419-2. <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
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Al-Mahani