2000:
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219:. He also developed a novel method for determining the conditions under which certain types of cubic equations would have two, one, or no solutions. To al-Tusi, "solution" meant "positive solution", since the possibility of zero or negative numbers being considered genuine solutions had yet to be recognised at the time. The equations in question can be written, using modern notation, in the form
802:, "apparently the idea of a function was proposed by the Persian mathematician Sharaf al-Din al-Tusi (died 1213/4), though his approach was not very explicit, perhaps because of this point that dealing with functions without symbols is very difficult. Anyhow algebra did not decisively move to the dynamic function substage until the German mathematician Gottfried Leibniz(1646–1716)."
986:
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Al-Tusi has been credited with proposing the idea of a function, however his approach being not very explicit, algebra's decisive move to the dynamic function was made 5 centuries after him, by German polymath
Gottfried Leibniz. Sharaf al-Din used what would later be known as the
402:, and setting it equal to zero. This conclusion has been challenged, however, by others, who point out that al-Tusi nowhere wrote down an expression for the derivative, and suggest other plausible methods by which he could have discovered his expressions for the maxima.
264: is positive. The Muslim mathematicians of the time divided the potentially solvable cases of these equations into five different types, determined by the signs of the other coefficients of
641:
where a and b are positive numbers. For any other values of the coefficients of x and x2, the equation f(x) = c has no positive solution.
496:
nevertheless considers that his discovery of these conditions demonstrated an understanding of the importance of the discriminant for investigating the solutions of cubic equations.
427:
of the cubic polynomials obtained by subtracting one side of the corresponding cubic equations from the other. Although al-Tusi always writes these conditions in the forms
423: which can be obtained from al-Tusi's conditions for the numbers of roots of cubic equations by subtracting one side of these conditions from the other is today called the
2019:
160:. Little is known about his life, except what is found in the biographies of other scientists and that most mathematicians today can trace their lineage back to him.
2008:
175:, where he met his most famous disciple Kamal al-Din ibn Yunus (1156-1242). Kamal al-Din would later become the teacher of another famous mathematician from Tus,
2054:
1423:
558:. This was criticized by Jeffrey Oaks who claims that Al-Tusi did not study curves by means of equations, but rather equations by means of curves (just as
2921:
391:. Some scholars have concluded that al-Tusi obtained his expressions for these maxima by "systematically" taking the derivative of the function
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had done before him) and that the study of curves by means of equations originated with
Descartes in the seventeenth century.
2911:
2202:
1893:
1256:
680:
1999:
1203:
Nasehpour, Peyman (August 2018). "A Brief
History of Algebra with a Focus on the Distributive Law and Semiring Theory".
1980:
1944:
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735:, "This was invented by Iranian mathematician Sharaf al-Din al-Tusi (d. ca. 1213), and was known as 'Al-Tusi's cane'"
554:
Sharaf al-Din al-Tusi's "Treatise on equations" has been described by Roshdi Rashed as inaugurating the beginning of
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1939:
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for the equation to have a solution. He then determined the maximum value of this expression. A value less than
820:
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1898:
1857:
1842:
958:
Katz, Victor; Barton, Bill (October 2007). "Stages in the
History of Algebra with Implications for Teaching".
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1011:
1001 Distortions: How (Not) to
Narrate History of Science, Medicine, and Technology in Non-Western Cultures
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Mentioned in the biography of the
Damascene architect and physician Abu al-Fadhl al-Harithi (d. 1202-3).
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1852:
1822:
1599:
1513:
1147:
1478:
1181:
Hogendijk, Jan P. (1989), "Sharaf al-Dīn al-Ṭūsī on the Number of
Positive Roots of Cubic Equations",
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corresponds to two solutions. Sharaf al-Din's analysis of this equation was a notable development in
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551:, but his work was not pursued any further at that time, neither in the Muslim or European world.
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1287:
1094:
Berggren, J. Lennart (1990). "Innovation and
Tradition in Sharaf al-Dīn al-Ṭūsī's Muʿādalāt".
830:
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1985:
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574:, sometimes called the "Staff of Tusi". While it was easier to construct and was known in
8:
2531:
2379:
2120:
2047:
1569:
1473:
1368:
Anbouba, Adel (2008). "Al-Ṭūsī, Sharaf Al-dīn Al-Muẓaffar Ibn Muḥammad Ibn Al-Muẓaffar".
1344:
Encyclopaedia of the
History of Science, Technology, and Medicine in Non-Western Cultures
1319:
Encyclopaedia of the
History of Science, Technology, and Medicine in Non-Western Cultures
1294:
Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures
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Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures
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1251:, translated by Armstrong, A.F.W., Dordrecht: Springer Science+Business Media,
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327:. He then concluded that the equation would have two solutions if
275:. For each of these five types, al-Tusi wrote down an expression
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1026:"Excavating the errors in the "Mathematics" chapter of 1001 Inventions"
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Al-Tusi gave no indication of how he discovered the expressions
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The Development Of Arabic Mathematics: Between Arithmetic And Algebra
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1148:"Le calcul du maximum et la 'dérivée' selon Sharaf al-Din al-Tusi"
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2145:
1125:"Al-Tūsī, Sharaf Al-Dīn Al-Muzaffar Ibn Muhammad Ibn Al-Muzaffar"
84:
750:
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2722:
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2020:
The Compendious Book on Calculation by Completion and Balancing
168:
91:
Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī
33:
Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī
2767:
1414:
1009:
Brentjes, Sonja; Edis, Taner; Richter-Bernburg, Lutz (2016).
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and the mathematical sciences, having no equal in his time".
172:
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1372:. Vol. 13. Charles Scribner's Sons. pp. 514–517.
864:
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corresponds to one solution, while a value greater than
1272:(1st ed.), Dordrecht: Kluwer Academic Publishers,
805:
2055:
Book on the Measurement of Plane and Spherical Figures
783:
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167:and taught mathematics there. He then lived in
281: for the point where the function
2279:
1400:
539:means no positive solution; a value equal to
380: for the maxima of the functions
471:, rather than the corresponding forms
1370:Complete Dictionary of Scientific Biography
1130:Complete Dictionary of Scientific Biography
1036:
2286:
2272:
1407:
1393:
957:
244: is a cubic polynomial in which the
2922:Astronomers of the medieval Islamic world
2141:Constantinople observatory of Taqi ad-Din
1416:Mathematics in the medieval Islamic world
1285:
1208:
1202:
1194:
1180:
925:
858:
826:
814:
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661:. In Hockey, Thomas; et al. (eds.).
656:
83:For other people with similar names, see
1122:
1096:Journal of the American Oriental Society
1093:
910:
870:
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664:Biographical Encyclopedia of Astronomers
296:, and gave a geometric proof that
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1083:MacTutor History of Mathematics Archive
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2062:Encyclopedia of the Brethren of Purity
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774:Mathematics Genealogy Project Extrema
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667:. New York: Springer. p. 1051.
499:Sharaf al-Din analyzed the equation
186:, Sharaf al-Din was "outstanding in
16:Iranian mathematician and astronomer
2907:13th-century Iranian mathematicians
2902:12th-century Iranian mathematicians
1313:"Arithmetic in Islamic Mathematics"
1078:"Sharaf al-Din al-Muzaffar al-Tusi"
578:, it did not gain much popularity.
99:شرفالدین مظفر بن محمد بن مظفر توسی
98:
13:
1361:
960:Educational Studies in Mathematics
171:for three years, before moving to
14:
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598:in 1990, was named in his honor.
2927:13th-century Iranian astronomers
2917:12th-century Iranian astronomers
1998:
570:Sharaf al-Din invented a linear
1017:
1002:
617:
1155:Arabic Sciences and Philosophy
793:
767:
695:
673:10.1007/978-0-387-30400-7_1268
650:
608:
315: for any positive
193:
1:
2034:Principles of Hindu Reckoning
1141:– via Encyclopedia.com.
1133:. Charles Scribner & Sons
1123:Berggren, J. Lennart (2008).
1065:
762:O'Connor & Robertson 1999
156:Al-Tusi was probably born in
112:
102:
54:
36:
2912:Medieval Iranian astrologers
1196:10.1016/0315-0860(89)90099-2
1013:. Ergon Verlag. p. 158.
657:Brummelen, Glen van (2007).
565:
321: different from
151:
7:
2224:Hindu–Arabic numeral system
2156:University of al-Qarawiyyin
10:
2968:
2126:Al-Mustansiriya University
2105:Islamic geometric patterns
1853:Shams al-Din al-Samarqandi
1336:Smith, Julian A. (1997a),
1311:Smith, Julian A. (1997b),
1286:Hogendijk, Jan P. (1997),
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1167:10.1017/s0957423900002034
1044:"7058 Al-Tusi (1990 SN1)"
972:10.1007/s10649-006-9023-7
163:Around 1165, he moved to
71:
61:
50:
28:
21:
2937:13th-century astrologers
2932:12th-century astrologers
1879:Nizam al-Din al-Nisapuri
1838:Muhyi al-Din al-Maghribi
1088:University of St Andrews
601:
342:, one solution if
248:of the cubic term
2866:Ahmad ibn Nizam al-Mulk
2846:Minhaj al-Siraj Juzjani
2248:History of trigonometry
2243:Trigonometric functions
2237:Western Arabic numerals
2233:Eastern Arabic numerals
1798:Ibn al‐Ha'im al‐Ishbili
1288:"Sharaf al-Dīn al-Ṭūsī"
1146:Farès, Nicolas (1995),
703:"Sharaf ad-Dīn aṭ-Ṭūsī"
659:"Sharaf al-Dīn al-Ṭūsī"
586:The main-belt asteroid
2952:12th-century inventors
2947:13th-century inventors
2796:Abu'l-Ma'ali Nasrallah
2791:Abu'l-Hasan Isfarayini
2477:Abu Dawud al-Sijistani
2172:Babylonian mathematics
1863:Kamāl al-Dīn al-Fārisī
1848:Qutb al-Din al-Shirazi
1843:al-Hasan al-Marrakushi
1767:Al-Samawal al-Maghribi
1696:Abu Mansur al-Baghdadi
1439:'Abd al-Hamīd ibn Turk
1024:Oaks, Jeffrey (2016).
121:) known more often as
2942:People from Tus, Iran
2693:Abu Sa'id Abu'l-Khayr
2577:Ibn Tayfour Sajawandi
2405:Sharaf al-Din al-Tusi
2315:Abu Ma'shar al-Balkhi
2198:Byzantine mathematics
1986:Ibn Hamza al-Maghribi
1813:Alam al-Din al-Hanafi
1777:Sharaf al-Din al-Tusi
623:The five types were:
405:The quantities
127:Sharaf ad-Dīn aṭ-Ṭūsī
123:Sharaf al-Dīn al-Ṭūsī
23:Sharaf al-Dīn al-Ṭūsī
2801:Abu Muslim Khorasani
2507:Abu Qasim Samarqandi
2497:Abu Layth Samarqandi
2395:Nasir al-Din al-Tusi
2370:Abu Ja'far al-Khazin
2203:European mathematics
2151:Maragheh observatory
1976:Muhammad Baqir Yazdi
1955:Ibn Ghazi al-Miknasi
1828:Nasir al-Din al-Tusi
1736:Muhammad al-Baghdadi
1183:Historia Mathematica
1074:Robertson, Edmund F.
635:−b x + a x2 − x3 = c
292: attained its
177:Nasir al-Din al-Tusi
2121:Al-Azhar University
2048:The Book of Healing
1454:Al-Ḥajjāj ibn Yūsuf
1220:2018arXiv180711704N
1072:O'Connor, John J.;
1048:Minor Planet Center
873:, pp. 307–308.
638:b x + a x2 − x3 = c
632:b x − a x2 − x3 = c
596:Palomar Observatory
549:Islamic mathematics
357:, or none if
2786:Abu'l-Fadl Bayhaqi
2667:Tha'labi Nishapuri
2482:Abu Barakat Nasafi
2310:Abu Hatam Isfizari
2253:History of algebra
2208:Indian mathematics
2182:Indian mathematics
2131:House of Knowledge
1585:Brethren of Purity
1479:Banū Mūsā brothers
1347:, pp. 74–75,
1322:, pp. 68–70,
556:algebraic geometry
142:Islamic Golden Age
76:Islamic Golden Age
45:, present-day Iran
2879:
2878:
2841:Khalid ibn Barmak
2816:Ata-Malik Juvayni
2779:political figures
2686:Poets and artists
2542:Fatima Samarqandi
2446:Haji Bektash Veli
2325:Abu Ubayd Juzjani
2261:
2260:
2177:Greek mathematics
2100:Alhazen's problem
1994:
1993:
1615:Ibrahim ibn Sinan
1565:Sinān ibn al-Fatḥ
1258:978-90-481-4338-2
861:, pp. 71–72.
682:978-0-387-31022-0
81:
80:
2959:
2871:Shihab al-Nasawi
2856:Tahir ibn Husayn
2728:Farrukhi Sistani
2703:Aruzi Samarqandi
2502:Abu Mu'in Nasafi
2470:Islamic scholars
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2076:Tabula Rogeriana
2002:
1950:Sibt al-Maridini
1610:Sinan ibn Thabit
1519:Abu Said Gorgani
1509:Thābit ibn Qurra
1499:Ishaq ibn Hunayn
1484:Hunayn ibn Ishaq
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711:(Author Profile)
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211:approximate the
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2811:Ali-Shir Nava'i
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2562:Hakim Nishapuri
2492:Abu Hafs Nasafi
2465:
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2330:Abu Zayd Balkhi
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2229:Arabic numerals
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2136:House of Wisdom
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2003:
1990:
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1889:Ibn al-Durayhim
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1362:Further reading
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2604:
2599:
2594:
2589:
2584:
2579:
2574:
2569:
2564:
2559:
2557:Hakim Tirmidhi
2554:
2549:
2544:
2539:
2534:
2529:
2524:
2519:
2514:
2509:
2504:
2499:
2494:
2489:
2484:
2479:
2473:
2471:
2467:
2466:
2464:
2463:
2458:
2453:
2448:
2443:
2438:
2433:
2428:
2422:
2420:
2416:
2415:
2413:
2412:
2407:
2402:
2397:
2392:
2387:
2382:
2377:
2372:
2367:
2362:
2357:
2352:
2347:
2342:
2337:
2332:
2327:
2322:
2317:
2312:
2306:
2304:
2300:
2299:
2291:
2290:
2283:
2276:
2268:
2259:
2258:
2256:
2255:
2250:
2245:
2240:
2226:
2220:
2218:
2214:
2213:
2211:
2210:
2205:
2200:
2194:
2192:
2188:
2187:
2185:
2184:
2179:
2174:
2168:
2166:
2162:
2161:
2159:
2158:
2153:
2148:
2143:
2138:
2133:
2128:
2123:
2117:
2115:
2111:
2110:
2108:
2107:
2102:
2096:
2094:
2090:
2089:
2087:
2086:
2079:
2072:
2069:Toledan Tables
2065:
2058:
2051:
2044:
2041:Book of Optics
2037:
2030:
2023:
2015:
2013:
2005:
2004:
1997:
1995:
1992:
1991:
1989:
1988:
1983:
1978:
1973:
1967:
1965:
1961:
1960:
1958:
1957:
1952:
1947:
1942:
1937:
1932:
1927:
1922:
1917:
1911:
1909:
1905:
1904:
1902:
1901:
1896:
1891:
1886:
1881:
1875:
1873:
1869:
1868:
1866:
1865:
1860:
1855:
1850:
1845:
1840:
1835:
1830:
1825:
1820:
1815:
1810:
1805:
1800:
1794:
1792:
1788:
1787:
1785:
1784:
1782:Ibn al-Yasamin
1779:
1774:
1769:
1764:
1759:
1754:
1748:
1746:
1742:
1741:
1739:
1738:
1733:
1728:
1723:
1718:
1713:
1708:
1703:
1698:
1693:
1688:
1683:
1681:Kushyar Gilani
1678:
1673:
1667:
1665:
1661:
1660:
1658:
1657:
1652:
1647:
1642:
1637:
1632:
1627:
1625:Nazif ibn Yumn
1622:
1617:
1612:
1607:
1602:
1597:
1592:
1587:
1582:
1577:
1572:
1567:
1562:
1557:
1552:
1547:
1542:
1537:
1531:
1529:
1525:
1524:
1522:
1521:
1516:
1511:
1506:
1504:Na'im ibn Musa
1501:
1496:
1494:Yusuf al-Khuri
1491:
1486:
1481:
1476:
1471:
1466:
1464:Qusta ibn Luqa
1461:
1456:
1451:
1446:
1441:
1435:
1433:
1426:
1424:Mathematicians
1420:
1419:
1412:
1411:
1404:
1397:
1389:
1383:
1382:
1363:
1360:
1359:
1358:
1353:
1333:
1328:
1308:
1303:
1283:
1278:
1268:, ed. (1997),
1266:Selin, Helaine
1262:
1257:
1243:Rashed, Roshdi
1239:
1200:
1178:
1161:(2): 219–317,
1143:
1120:
1108:10.2307/604533
1102:(2): 304–309.
1091:
1067:
1064:
1061:
1060:
1035:
1016:
1001:
985:
950:
930:
926:Hogendijk 1989
915:
903:
891:
875:
863:
859:Hogendijk 1989
851:
835:
827:Hogendijk 1997
819:
815:Hogendijk 1989
804:
800:Nasehpour 2018
792:
777:
766:
749:
745:Nasehpour 2018
737:
721:
694:
681:
648:
647:
644:
643:
640:
639:
636:
633:
630:
627:
616:
606:
605:
603:
600:
583:
580:
567:
564:
233:, where
217:cubic equation
195:
192:
153:
150:
79:
78:
73:
69:
68:
63:
59:
58:
52:
48:
47:
41:
32:
30:
26:
25:
22:
15:
9:
6:
4:
3:
2:
2964:
2953:
2950:
2948:
2945:
2943:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2913:
2910:
2908:
2905:
2903:
2900:
2898:
2895:
2893:
2890:
2889:
2887:
2872:
2869:
2867:
2864:
2862:
2861:Yahya Barmaki
2859:
2857:
2854:
2852:
2851:Nizam al-Mulk
2849:
2847:
2844:
2842:
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2783:
2781:
2775:
2769:
2766:
2764:
2761:
2759:
2756:
2754:
2751:
2749:
2748:Nasir Khusraw
2746:
2744:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2724:
2721:
2719:
2716:
2714:
2711:
2709:
2706:
2704:
2701:
2699:
2696:
2694:
2691:
2690:
2688:
2684:
2678:
2675:
2673:
2670:
2668:
2665:
2663:
2660:
2658:
2655:
2653:
2650:
2648:
2645:
2643:
2640:
2638:
2635:
2633:
2630:
2628:
2625:
2623:
2620:
2618:
2615:
2613:
2612:Mulla al-Qari
2610:
2608:
2605:
2603:
2600:
2598:
2595:
2593:
2590:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2563:
2560:
2558:
2555:
2553:
2550:
2548:
2545:
2543:
2540:
2538:
2535:
2533:
2530:
2528:
2525:
2523:
2520:
2518:
2515:
2513:
2510:
2508:
2505:
2503:
2500:
2498:
2495:
2493:
2490:
2488:
2485:
2483:
2480:
2478:
2475:
2474:
2472:
2468:
2462:
2459:
2457:
2454:
2452:
2451:Nasir Khusraw
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2423:
2421:
2417:
2411:
2408:
2406:
2403:
2401:
2398:
2396:
2393:
2391:
2388:
2386:
2383:
2381:
2378:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2360:Hasib Marwazi
2358:
2356:
2353:
2351:
2348:
2346:
2343:
2341:
2338:
2336:
2333:
2331:
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2311:
2308:
2307:
2305:
2301:
2297:
2289:
2284:
2282:
2277:
2275:
2270:
2269:
2266:
2254:
2251:
2249:
2246:
2244:
2241:
2238:
2234:
2230:
2227:
2225:
2222:
2221:
2219:
2215:
2209:
2206:
2204:
2201:
2199:
2196:
2195:
2193:
2189:
2183:
2180:
2178:
2175:
2173:
2170:
2169:
2167:
2163:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2118:
2116:
2112:
2106:
2103:
2101:
2098:
2097:
2095:
2091:
2085:
2084:
2080:
2078:
2077:
2073:
2071:
2070:
2066:
2064:
2063:
2059:
2057:
2056:
2052:
2050:
2049:
2045:
2043:
2042:
2038:
2036:
2035:
2031:
2029:
2028:
2024:
2022:
2021:
2017:
2016:
2014:
2012:
2006:
2001:
1987:
1984:
1982:
1979:
1977:
1974:
1972:
1969:
1968:
1966:
1962:
1956:
1953:
1951:
1948:
1946:
1943:
1941:
1938:
1936:
1933:
1931:
1928:
1926:
1923:
1921:
1918:
1916:
1913:
1912:
1910:
1906:
1900:
1897:
1895:
1892:
1890:
1887:
1885:
1884:Ibn al-Shatir
1882:
1880:
1877:
1876:
1874:
1870:
1864:
1861:
1859:
1858:Ibn al-Banna'
1856:
1854:
1851:
1849:
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1804:
1803:Ahmad al-Buni
1801:
1799:
1796:
1795:
1793:
1789:
1783:
1780:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1749:
1747:
1743:
1737:
1734:
1732:
1729:
1727:
1724:
1722:
1719:
1717:
1714:
1712:
1709:
1707:
1704:
1702:
1699:
1697:
1694:
1692:
1689:
1687:
1684:
1682:
1679:
1677:
1674:
1672:
1669:
1668:
1666:
1662:
1656:
1653:
1651:
1648:
1646:
1643:
1641:
1638:
1636:
1633:
1631:
1628:
1626:
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1571:
1568:
1566:
1563:
1561:
1558:
1556:
1553:
1551:
1548:
1546:
1543:
1541:
1538:
1536:
1533:
1532:
1530:
1526:
1520:
1517:
1515:
1512:
1510:
1507:
1505:
1502:
1500:
1497:
1495:
1492:
1490:
1487:
1485:
1482:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1460:
1457:
1455:
1452:
1450:
1447:
1445:
1444:Sanad ibn Ali
1442:
1440:
1437:
1436:
1434:
1430:
1427:
1425:
1421:
1417:
1410:
1405:
1403:
1398:
1396:
1391:
1390:
1387:
1379:
1375:
1371:
1366:
1365:
1356:
1354:9780792340669
1350:
1346:
1345:
1339:
1334:
1331:
1329:9780792340669
1325:
1321:
1320:
1314:
1309:
1306:
1304:9780792340669
1300:
1296:
1295:
1289:
1284:
1281:
1279:0-7923-4066-3
1275:
1271:
1267:
1263:
1260:
1254:
1250:
1249:
1244:
1240:
1237:
1233:
1229:
1225:
1221:
1217:
1211:
1206:
1201:
1197:
1192:
1188:
1184:
1179:
1176:
1172:
1168:
1164:
1160:
1156:
1149:
1144:
1132:
1131:
1126:
1121:
1117:
1113:
1109:
1105:
1101:
1097:
1092:
1089:
1085:
1084:
1079:
1075:
1070:
1069:
1049:
1045:
1039:
1031:
1027:
1020:
1012:
1005:
998:
994:
989:
981:
977:
973:
969:
965:
961:
954:
947:
943:
939:
934:
927:
922:
920:
912:
911:Berggren 1990
907:
900:
895:
888:
884:
879:
872:
871:Berggren 1990
867:
860:
855:
848:
844:
839:
832:
828:
823:
817:, p. 71.
816:
811:
809:
801:
796:
789:
788:Berggren 2008
784:
782:
775:
770:
763:
758:
756:
754:
746:
741:
734:
730:
725:
710:
709:
704:
698:
684:
678:
674:
670:
666:
665:
660:
653:
649:
637:
634:
631:
628:
626:a x2 − x3 = c
625:
624:
620:
611:
607:
599:
597:
593:
592:Henry E. Holt
589:
579:
577:
573:
563:
561:
557:
552:
550:
546:
542:
538:
534:
530:
526:
522:
518:
514:
510:
506:
502:
497:
495:
494:Roshdi Rashed
489:
482:
475:
469:
465:
461:
454:
450:
446:
439:
435:
431:
426:
421:
417:
413:
409:
403:
399:
395:
388:
384:
378:
372:
369:
365:
361:
354:
350:
346:
339:
335:
331:
325:
319:
312:
308:
304:
300:
295:
289:
285:
279:
272:
268:
262:
252:
247:
241:
237:
231:
227:
223:
218:
214:
210:
206:
202:
191:
189:
185:
182:According to
180:
178:
174:
170:
166:
161:
159:
149:
147:
143:
139:
135:
134:mathematician
132:
128:
124:
120:
110:
96:
92:
86:
77:
74:
70:
67:
66:Mathematician
64:
60:
53:
49:
44:
31:
27:
20:
2892:1130s births
2753:Rabia Balkhi
2419:Philosophers
2404:
2400:Omar Khayyam
2081:
2074:
2067:
2060:
2053:
2046:
2039:
2032:
2025:
2018:
2009:Mathematical
1964:16th century
1915:Ibn al-Majdi
1908:15th century
1872:14th century
1791:13th century
1776:
1745:12th century
1731:Omar Khayyam
1664:11th century
1560:Aṣ-Ṣaidanānī
1528:10th century
1489:Al-Khwarizmi
1378:CX2830904401
1369:
1341:
1316:
1291:
1269:
1247:
1232:ResearchGate
1186:
1182:
1158:
1154:
1135:. Retrieved
1128:
1099:
1095:
1081:
1051:. Retrieved
1047:
1038:
1030:Academia.edu
1029:
1019:
1010:
1004:
988:
963:
959:
953:
933:
906:
894:
878:
866:
854:
838:
822:
795:
769:
740:
724:
713:. Retrieved
706:
697:
686:. Retrieved
662:
652:
629:b x − x3 = c
619:
610:
588:7058 Al-Ṭūsī
585:
569:
553:
544:
540:
536:
532:
528:
524:
520:
516:
515:in the form
512:
508:
504:
500:
498:
487:
480:
473:
467:
463:
459:
452:
448:
444:
437:
433:
429:
425:discriminant
419:
415:
411:
407:
404:
397:
393:
386:
382:
376:
373:
367:
363:
359:
352:
348:
344:
337:
333:
329:
323:
317:
310:
306:
302:
298:
287:
283:
277:
270:
266:
260:
258:, and
250:
239:
235:
229:
225:
221:
197:
181:
162:
155:
144:(during the
126:
122:
90:
89:
2897:1213 deaths
2831:Gawhar Shad
2677:Zamakhshari
2657:Shaykh Tusi
2572:Ibn Mubarak
2461:Shahrastani
2027:De Gradibus
1981:Taqi ad-Din
1971:Al-Birjandi
1945:al-Qalaṣādī
1726:Al-Isfizari
1691:Ibn al-Samh
1620:Al-Isfahani
1600:al-Uqlidisi
1570:al-Khojandi
1535:Abu al-Wafa
1474:al-Dinawari
1432:9th century
1338:"Astrolabe"
1053:21 November
995:, pp.
993:Rashed 1994
940:, pp.
938:Rashed 1994
883:Rashed 1994
843:Smith 1997b
729:Smith 1997a
708:zbMATH Open
485:, or
457:, or
246:coefficient
209:numerically
207:method" to
194:Mathematics
146:Middle Ages
116: 1213
106: 1135
2886:Categories
2708:Asadi Tusi
2602:Marghinani
2567:Ibn Hibban
2487:Abu Hanifa
2365:Ibn Hayyan
2340:Ali Qushji
2335:Alfraganus
2303:Scientists
2294:People of
2191:Influenced
2165:Influences
1935:Ali Qushji
1894:Al-Khalili
1762:Al-Khazini
1757:Al-Kharaqī
1716:al-Zarqālī
1706:al-Jayyānī
1650:al-Majriti
1635:Abu al-Jud
1605:Al-Battani
1580:Al-Saghani
1575:Al-Nayrizi
1514:al-Marwazi
1449:al-Jawharī
1210:1807.11704
1066:References
966:(2): 192.
899:Farès 1995
885:, p.
845:, p.
829:, p.
731:, p.
715:2023-06-18
688:2023-06-18
576:al-Andalus
560:al-Khayyam
138:astronomer
62:Occupation
2662:Taftazani
2647:Sajawandi
2456:Sijistani
2385:Khwarizmi
1940:al-Wafa'i
1930:Ulugh Beg
1833:al-Abhari
1818:Ibn Adlan
1808:Ibn Munim
1772:al-Hassar
1711:al-Nasawī
1686:Al-Biruni
1655:al-Jabali
1645:Al-Karaji
1595:Ibn Yunus
1550:Abu Kamil
1545:Al-Qabisi
1540:al-Khazin
1469:Al-Mahani
1236:326732377
1228:119176936
1189:: 69–85,
1175:170242949
1137:March 21,
980:120363574
572:astrolabe
566:Astronomy
478:,
158:Tus, Iran
152:Biography
129:, was an
109:Tus, Iran
2733:Ferdowsi
2672:Tirmidhi
2652:Sarakhsi
2632:Qushayri
2607:Maturidi
2552:Ghaznawi
2436:Avicenna
2380:Khojandi
2350:Birjandi
2345:Avicenna
2320:Abu Wafa
2296:Khorasan
2093:Concepts
1925:al-Kāshī
1899:al-Umawi
1701:Avicenna
1640:Al-Sijzi
1590:Ibn Sahl
1459:Al-Kindi
1245:(1994),
1076:(1999),
442:,
188:geometry
165:Damascus
2806:Gardizi
2743:Kashifi
2617:Muqatil
2597:Lamishi
2592:Kashifi
2582:Juwayni
2547:Ghazali
2532:Bukhari
2527:Bazdawi
2522:Bayhaqi
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2426:Algazel
2375:Khazini
2217:Related
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2114:Centers
1920:al-Rūmī
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