Abstract
Muḥammad ibn Mūsā al-Khwārazmı̄ is today arguably the most well-known medieval Arabic mathematician. His significant work on arithmetic, algebra, and astronomical tables were written in Baghdad in the first half of the third/ninth century. This paper traces the impact of al-Khwārazmı̄’s Kitāb al-jabr wa-l-muqābala (Book of algebra) on the field of algebra.
This article began as a lecture I delivered at the one-day seminar “al-Khwārizmı̄’s contribution to civilization” held November 21, 2016 at Hamid bin Khalifa University in Doha, Qatar. All translations are mine unless noted otherwise.
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Notes
- 1.
Until recently the standard transliteration of his name was “al-Khwārizmı̄”. Those who deem what is proper have recently adjusted the rule, so we now write “al-Khwārazmı̄”.
- 2.
For dates relating to Islamic civilization we give both Muslim and Christian dates. Here, the third century AH corresponds roughly to the ninth century CE.
- 3.
- 4.
- 5.
- 6.
The Arabic text with English translation is published in (al-Khwārazmı̄ 2009), which is translated from the French edition (al-Khwārazmı̄ 2007). (al-Khwārazmı̄ 2009b) contains a facsimile of the Oxford manuscript with a Spanish translation.
- 7.
The Arabic original is lost, but much of this zı̄j can be reconstructed from later books that preserve parts of the original through borrowings and commentaries. The main sources are: (a) Adelard of Bath’s twelfth-century Latin translation of an Arabic recension made in the late tenth century in Cordova by Maslama al-Majrı̄ṭı̄, (b) twelfth-century Latin and Hebrew translations of the tenth-century Arabic commentary by al-Muthannā, (c) a tenth-century commentary by Ibn Masrūr, and (d) the eleventh-century Toledan Tables of al-Zarqālı̄.
- 8.
See (Abdeljaouad & Oaks 2021, 25ff; Oaks 2023) for explanations of the different techniques.
- 9.
The translation takes into account the critical apparatus. In nearly all instances of confrontation in Arabic texts, the word qabala is not invoked: the authors simply make the subtraction.
- 10.
(al-Khwārazmı̄ 2009, 97.17, 101.3). I label the two groups as “simple” and “composite” following al-Karajı̄. Al-Khwārazmı̄ did not give names to the two kinds.
- 11.
Here “two māls and ten roots equal forty-eight dirhams” is the algebraic equation, in which the words māl and jidhr (root) are the algebraic names of the powers. The meaning of the equation is then explained by re-expressing it as an arithmetic problem, “what two māls, if added together and the same as ten of its roots are added to it, amounts to forty-eight dirhams?”, where māl and jidhr assume their arithmetical meanings of “quantity” and “square root”. Several other Arabic authors explain the meanings of equations this way.
- 12.
I.e., the equation. At this point in time there was no word for algebraic “equation” in Arabic. That would come with the translation of Diophantus’s Arithmetica later in the century.
- 13.
In this problem al-Khwārazmı̄ performs this step out of order from the norm. Setting the number of māls to 1 should be the first step in stage 3, as he himself instructed when he explained the six equations.
- 14.
(al-Khwārazmı̄ 2009, 153), taking into account the critical apparatus.
- 15.
There are 23 such problems in al-Khwārazmı̄’s book. Of the three remaining problems, two deal with an amount of money distributed among some men. The interpolated problem (7) in the Oxford manuscript deals with wheat and barley.
- 16.
- 17.
(Rashed & Vahabzadeh 1999, 117.5). In applications of algebra to mensuration (misāḥa) it is the numerical measures of the lines and areas that are found. Al-Khayyām presumes the reader is already familiar with the basics of algebraic problem-solving. His book is devoted to geometric solutions to simplified cubic equations (Oaks 2011b).
- 18.
(al-Fārisı̄ 1994, 461.19). The “proportional numbers” are the powers of the unknown: the unit, the thing, the māl, the cube, etc.
- 19.
See (al-Fārisı̄ 1994, 463.1) for the Arabic text, and (Christianidis & Oaks 2023, 54, 60) for English translations of part of it.
- 20.
(Ibn Khaldūn 1967 III, 124), Rosenthal’s translation.
- 21.
The Greek books are edited in (Diophantus 1893–5) and translated into French in (Diophantus 1959). The Arabic books are edited and translated into English in (Sesiano 1982), and edited and translated into French in (Diophantus 1986). A complete English translation of all ten extant books has recently been published in (Christianidis & Oaks 2023).
- 22.
Two of Diophantus’s derivations are preserved in al-Karajı̄’s al-Fakhrı̄, and all three are transformed into proofs in another short work of al-Karajı̄ as well as in a work of Fakhr al-Dı̄n al-Ḥilāṭı̄ (Oaks 2018; Christianidis & Oaks 2023, 63–6).
- 23.
(Ibn al-Nadı̄m 2009, II/1, 219, 259.11; Saidan 1971, 126.7).
- 24.
Only one Arabic manuscript of Abū Kāmil’s Algebra is extant: Istanbul, Beyazid Library, Kara Muṣṭafa Paşa collection 379 (new number 19046), published in facsimile in (Abū 1986) and edited with French translation in (Abū Kāmil 2012). See also (Levey 1966) for an edition of a medieval Hebrew translation with English translation. More recent is the edition and English translation of the Hebrew translation on the Mispar project website: https://mispar.ethz.ch/wiki/Main_Page.
- 25.
For the nine examples of simple equations, only \(x^2=5x\) and \({1\over 2}x=10\) are the same.
- 26.
Abū Kāmil’s problems (T2) and (T3) are the same as al-Khwārazmı̄’s, and his (T5), (11), and (14) are al-Khwārazmı̄’s (1), (5), and (31) respectively. Other problems are modifications of al-Khwārazmı̄’s.
- 27.
The earliest known authors of books on anwā' are Sahl ibn Nawbakht (fl. ca. 200/800), Abū Ma`shar (171–272/787–886), and Thābit ibn Qurra. Al-Khwārazmı̄’s chapter on mensuration in his Algebra is the earliest extant Arabic treatment of the subject. The earliest extant book on finger reckoning is by Abū-l-Wafā' (328–388/940–998), though it is likely that some lost ninth century books reported by other authors dealt with the same topic, perhaps even al-Khwārazmı̄’s Book of adding and subtracting. The earliest known treatises on double false position are by Abū Kāmil (late 9th c.) and Qusṭā ibn Lūqā (d. ca. 910). For surveys of Arabic arithmetic, see (Oaks 2023) and the introductions to (al-Uqlı̄disı̄ 1978) and (Abdeljaouad & Oaks 2021).
- 28.
- 29.
(King 1988, 26–7; King 2012, 228), his translation. A manuscript of al-Khuzā`ı̄’s book, British Library, MS Delhi Arabic 1897/1, ff. 1a-124b, is available online: http://www.qdl.qa/en/archive/81055/vdc~ 100042392200.0x000001. The passage quoted begins at fol. 1b.20.
- 30.
- 31.
The same may be true of the brief book of geometrical problems of the late third/ninth century Nu`aim ibn Muḥammad ibn Mūsā. Problem 4 gives geometric proofs based in Euclid for the solutions to the types 4 and 5 equations, and Problems 37 to 42 are geometrical variations of al-Khwārazmı̄’s Problem (9). There is no clear dependence on al-Khwārazmı̄ among these problems (Hogendijk 2003).
- 32.
These are Problems (14)–(17) in Abū Kāmil’s Book on algebra (Abū Kāmil 2012, 385.8–387.18) (Rashed numbers these as \(\langle \)20\(\rangle \) to \(\langle \)23\(\rangle \)); Problem III.22 in al-Karajı̄’s al-Fakhrı̄ (Saidan 1986, 221.15); Problem III.2 in Ibn al-Bannā'’s Book on the fundamentals and preliminaries in algebra (Saidan 1986, 575.1); and Problem (21) in Ibn al-Khawwām’s Magnificent benefits of the rules of calculation, which was copied in al-Fārisı̄’s commentary Foundation of the rules on elements of benefits (al-Fārisı̄ 1994, 549.16). Al-Khwārazmı̄’s Problem (31) is the same as Abū Kāmil’s (14), al-Karajı̄’s III.22, and Ibn al-Bannā'’s III.2. The other problems are all different. In all of these authors other nearby problems are of the same basic type, and sometimes their algebraic solutions have no explicit assignment statement.
- 33.
Regardless what method is used to work out a problem, the solution to a problem in Arabic arithmetic usually begins “By the rule (qiyās)” or “Its rule is”.
- 34.
The other three problems with a second solution by algebra are Problems (14) and (16) in Abū Kāmil, where the algebraic solutions are attributed to one Abū Yūsuf and are written in the margin of the Istanbul manuscript, and Ibn al-Bannā'’s problem.
- 35.
Translation adjusted from (al-Khwārazmı̄ 2009, 140).
- 36.
(al-Khwārazmı̄ 2009, 101.5), taking into account the critical apparatus and the commentary of al-Khuzā`ı̄.
- 37.
(al-Khwārazmı̄ 2009, 113) misplaces point B.
- 38.
- 39.
(Saidan 1986, 509.8). We can write this in modern algebraic notation as \((x+y)(x-y)=x^2-y^2=(x-y)^2+2(x-y)y\), but to Ibn al-Bannā' this rule and the others he states are not part of al-jabr wa-l-muqābala since they are not stated in algebraic language and they do not ask for the value of an unknown number.
- 40.
Al-Muqaddima al-kāfiyya fı̄ ḥisāb al-jabr wa’l-muqābala wa mā yu`rafu bihi qiyāsuhū min al-amthila, Vatican MS Sbath 5.
- 41.
- 42.
(Rashed & Vahabzadeh 2000, 120-1).
- 43.
MS Cairo, Riyadāt 260/4, ff. 95r-104v.
- 44.
(Ibn Khaldūn 1967 III, 125–26). Rosenthal’s translation, with adjustments to the transliterated names.
- 45.
In what follows, the first number is the number of lines devoted to the six equations, and the second number is the number of lines devoted to other examples and rules. Diagrams are not taken into account: Al-Khwārazmı̄ (185, 185), Abū Kāmil (345, 574) (some rules are stated and proven among the worked out problems (2), (7), (61), (62), and (63)), al-Karajı̄ (471, 1118), Ibn al-Bannā' (333, 827), `Alı̄ al-Sulamı̄ (135, 871) (Not counting three pages of tables on the names of the powers. Numbers are from the Vatican manuscript), and Ibn Badr (85, 169).
- 46.
- 47.
- 48.
- 49.
- 50.
Hissette writes: “la traduction latine de l’al-Jabr d’Abū Kāmil, dont P est assurément une copie, n’a pu être réalisée après la fin du XII\({ }^{\mathrm {e}}\) siècle.” (1999, 313). For this date he cites an article by André Allard, which in turn cites (Karpinski 1911). But the only indication of the date of the translation I found there is merely implied: “The notation of fractions employed by this translator of Abu Kamil resembles the peculiar system employed later by Leonard [i.e., Fibonacci]” (p. 53).
- 51.
Marc Moyon published the critical edition (2019) and a French translation (2017). The word restauracionis is a translation of the Arabic al-jabr. Several scholars have attributed this reworking to Gugielmo de Lunis based in the gloss in the Florence manuscript of Gerard’s translation and remarks in two medieval Italian texts (to be investigated below), but Jens Høyrup has convincingly argued against Guglielmo’s authorship (Høyrup 2010, 13).
- 52.
- 53.
- 54.
Proposition I-3 is from al-Khwārazmı̄’s problem (1), I-4 from (T4), I-14 from (2), I-15 from (3), I-17 from (10), I-19 from (T3), I-20 from (4), and I-29 from (6). Of these eight propositions, all but I-4 and I-15 are also in Abū Kāmil.
- 55.
(Jordanus de Nemore 1981, 52), translated in (Høyrup 1988, 333).
- 56.
- 57.
Three more problems are common to both books so we cannot identify their source. These three are also in Abū Kāmil, which is Fibonacci’s source.
- 58.
“Rendiamo gratie all’Altissimo, chosì chomincia el testo de l’Aghabar arabico nella reghola del geber la quale noi diciamo algebra. La quale reghola d’algebra, secondo Guglielmo de Lunis translatatore, inporta di questi 7 nomi cioè: geber, el melchel, el chal, el chelif, el fatiar, diffar el buram, el termem. É qualj nomi, secondo el detto Guglielmo, sono chosì interpretati. Geber è quanto a dire recuperatione inperoché, chome per lo seguente si chonprenderà, nella recuperatione di 2 parti igualj s’asolve il chaso [\(\ldots \)]” (Benedetto da Firenze 1982, 1).
- 59.
(Anonimo Fiorentino 1992). Algebra was often called in Italian “regola della cosa” (“rule of the thing”) because of the name cosa (“thing”), translated from the Arabic shay' for the first-degree unknown.
- 60.
“l Aregola dellargibra la quale reghola ghuolelmo a lunis latraslato darabicho anostra linghua e sechondo el detto ghuol.mo e altri dichono questa esser chomposta dauno maestro arabo invero di grande intelligenza benche alchuno altrj dichono esser stati uno del quale il nome era Geber a che lionardo pisano dice che algebra muchalbile ella interpretatione della reghola inq.lla linghua el testo (the edition reads “resto”) della detta reghola inchominca ’andano gratie allaltissimo ess.do el ditto ghuol.mo la ditta reghola inqu.lla linghua chontiene sette nomi coe sette parti chosi nella ditta linghua nominati: Geber el melchel Elchal Elchelis Elfatiar diffarel buran eltiemin e quali nomi sechondo il ditto ghuol.mo chosi sono interpretati [\(\ldots \)]” (Procissi 1954, 302).
- 61.
The common phrase “and which names, according to the said Guglielmo, are interpreted like this” speaks for direct copying, as do other common elements not taken from Guglielmo.
- 62.
- 63.
(Piero della Francesca 1970, 133). Translating from Problem (5) of al-Khwāriazmı̄, it is: “If [someone] says, ten: you divide it into two parts and you multiply one of the parts by five and you divide it by the other. Then you cast away half of what you gathered and you add it to the multiplication by five, so it gives fifty dirhams” (al-Khwārazmı̄ 2009, 165.7). Converted to modern algebra, the problem asks for a and b such that \(a+b=10\) and \({1\over 2}({5a\over b})+5a=50\).
- 64.
(M\({ }^{\mathrm {o}}\) Biagio 1983, 23.11). The “plus” translates the Italian più, which takes a meaning like “more” or “further”. This is Problem (T4) in al-Khwārazmı̄: “A māl: you multiply its third and a dirham by its fourth and a dirham to get twenty” (al-Khwārazmı̄ 2009, 151.2). Converted to modern algebra, it asks for a such that \(({1\over 3}a+1)({1\over 4}a+1)=20\).
- 65.
(Gori 1984, 27). Al-Khwārazmı̄’s Problem (21) asks: “If [someone] says, a māl: you remove its third and three dirhams, then you multiply what remains by itself, it brings back the māl.” (al-Khwārazmı̄ 2009, 181.7). The other problem, at (Gori 1984, 7), is about the buying of cloth. It ultimately comes from al-Khwārazmı̄’s Problem (T6).
- 66.
The Arabic title of this book is Kitāb al-bayān wa-l-tadhkār fı̄ ṣan`at `amal al-ghubār (Book of demonstration and recollection in the art of dust-board reckoning). An Arabic manuscript is available online: https://openn.library.upenn.edu/Data/0001/html/ljs293.html (accessed August 14 2022). For an English translation of one of the problems solved three different ways, see (Abdeljaouad & Oaks 2021, 250-1).
- 67.
The text with English translation is published in (Wartenberg 2015). Some selections from this translation appear in (Wagner 2016, 362–374).
- 68.
The algebra proper is edited and translated into English in (Levey 1966), the second part, On the pentagon and decagon, was translated into Italian by Sacerdote in 1896, and all three parts are edited and translated into English on the Mispar project website, https://mispar.ethz.ch/wiki/Main_Page. The third part extends only to problem (50) by Rashed’s numbering (Abū Kāmil 2012, 702), omitting the last 20 problems and the section on arithmetical and geometric progressions. Even some of the first 50 are skipped.
- 69.
- 70.
- 71.
- 72.
The words rei and census are Latin translations of shay' and māl respectively, the names of the the first two powers of the unknown in Arabic.
- 73.
(Regiomontanus 1537, 4th page of the oration, which is the first item in the book after the letter to the reader). Translated in (Morse 1981, 58–9).
- 74.
- 75.
- 76.
“Dice Benedecto la regola dellarcibra quale Guglielmo Delunis la traslato Darabo a nostra lingua & sicondo decto Guglielmo decta regola e composta da uno nome Arabo di grande intelligentia: & che alcuni dicono essere stato uno elquale nome era Geber: & Lionardo Pisano dice ch Algebra amuchabole e la interpretatione della reghola in quella lingua.” (Ghaligai 1521, fol. 71b), with corrections from the 1548 printing.
- 77.
“Segue el Testo di Guglielmo. Rendiamo gratie allo altissimo cosi comincia el testo dellaghabar Arabico nella regola del Geber quale noi diciamo Arcibra:& sicondo decto Guglielmo importa 7 nomi cioe [\(\ldots \)]”.
- 78.
- 79.
- 80.
- 81.
“Le premier inuanteur de cet art, selon aucuns, fùt Geber Arabe [\(\ldots \)] Selon les autres, fùt vn Mahommet fiz de Moïse Arabe : Lequel, comme dìt Gerome Cardan Millãnoes, apres vn Leonard de Pesare [\(\ldots \)] I’è ancores vù le liure de Ian Scheubel, Mathematicien de Tubingue : lequel attribue l’inuancion de cet art a vn Diophante Grec [\(\ldots \)] je ne panse point que cet Art, ni la plus part des autres, doeuet leur inuancion a vn seul auteur.” (Peletier 1554, 1–2).
- 82.
- 83.
“El inuentor desta arte, segun Leonardo Pisano, fue vn Maumetho hijo de Mosis Arauigo. Alfragano (come refiere Iuan de MonteRegio) dize que Diophanto, y que escriuio treze libros della. Otros dizen que el inuentor, fue vn Arauigo, dicho Geber, y que deste nombre se deriuo Algebra” (Pérez de Moya 1573, 429).
- 84.
(Jayawardene 1973, 511; Christianidis & Oaks 2023, 197ff).
- 85.
- 86.
- 87.
“Hie hebet sich an das Buch Algebrae, des grossen Arismetristens, geschrieben zu den zeithen Alexandri vnd Nectanebi, des grossen Grecken vnnd Nigromantis, geschrieben zu Ylem, dem grossen Geometer jn Egypten, jn Arabischer Sprach genant Gebra vnnd Almuchabola, das dann bey vns wirdt genant das Buch von dem Dinge der vnwissenden zall. Vnd ist aus Arabischer Sprach jn kriechisch transferirt von Archimede, vnnd aus kriechisch jn das Latein von Apuleio, vnd wird genandt bey den Welschen das Buch de la cosa, das dann aber wird gesprochen das Buch von dem ding; wann aus einem vnbekanten dinge findet man das wesen der zal vnd gantzen essentz, das dann gewesen ist die frage ze wissen. Vnnd aus disem Buch finden wir, das der Machomet in seinem Alkoran vermeldet vin disen Regeln, vnnd nennet sie auch Gebram vnd Almuchabolam. Sie werden auch gebraucht von den Indiern, vnnd nennen sie Aliabra vnd Aluoreth, das ist das Buch, das Aliabras zu den zeiten Alexandri aus Arabischer sprache jn jndische gesatzt hat, vnd wird bey jnen gesagtt das Buch Aluoreth, das ist von dem Dinge abermals, oder das Buch der Coniecturation [\(\ldots \)]”
- 88.
The origin of algebra in Egypt in the time of Alexander speaks for a connection with alchemy, especially the reference to Nectanebo, the last native Egyptian pharaoh (reigned 358–340 BCE). A fictitious story in the ancient Alexander romance relates that after he was deposed by the Persians, Nectanebo traveled to Macedonia where he posed as a magician and seduced Olympias to became Alexander’s father. The German commentary mentions after this passage that (the fictional) Yles taught geometry to Euclid. Of course, there is no evidence that either Archimedes or Apuleius translated any algebra book. But Italian books were called books “of the thing (cosa)”. There is no record of any person named Aliabra who translated Arabic algebra into Sanskrit, and algebra was not called Aliabra or Aluoreth in India. The term coniecturation comes from the commented Latin text and refers to the stage of simplifying an equation. I have not seen this term applied in any other book.
- 89.
- 90.
(Tartaglia 1578 Part 1, 2nd page of letter “Au Lecteur”). The passage begins: “ont esté pour l’Arabie, Moyse, Mammeth son fils, qu’on dit estre inventeur de l’Algebre, Alguc, Rabbi Abraham, & Rabbi-Issac. Pour la Grece, Diophante, qu’aucuns aultres disent estre inventeur de l’Algebre, Planude, Pythagore, [\(\ldots \)]”. In medieval European legend, Alguc, also Algus or Algos, is the name of the ancient Indian philosopher who devised the decimal “Arabic” numeration system. I do not know who François Peuenel is.
- 91.
- 92.
- 93.
- 94.
- 95.
- 96.
- 97.
(van Roomen 1597, 22–32; Bockstaele 2009). The example of “kind-crossing” at stake for this book is the use of arithmetic to prove a result or solve a problem in geometry. The capital letters in Chapter 7 representing numbers or magnitudes (or any other kind of quantifiable objects) are not an indication of any kind of algebra. They agree with the use of letters in Pappus of Alexandria’s Book II, in al-Fārisı̄, and in Jordanus de Nemore (Oaks 2018c, section 5.4). In particular, the letters are not operated on to form new expressions, and no equations are set up.
- 98.
The visit is reported in (De Thou 1734, 163–4). Bockstaele (2009, 455) notes that van Roomen was not yet familiar with Viète’s algebra when he wrote his Apologia, but by 1598 he had in his possession some of Viète’s works.
- 99.
- 100.
Alexander reigned 356–323 BCE. The first year of the 112th Olympiad was 328 BCE, which was 1909 years before 1582. The playwright Aristophanes died in 386 BCE.
- 101.
“Almuchabola: Ist Arabisch und bedeut so viel als ein Buch von dem unwissenden ding Nemlich der Zahl welchs vor 1909 jahren Algebra der hocherfarn Mathematicus dem König Alexandro dedicirt, und das durch den Aristophanem welcher der zeit Nemlich in dem 1 Jahr der 112 Olympiadum Stadtvogt oder Oberster zu Athen war dem König der domalen das fïnffte jahr seiner Regierung besasz besasz zubringen lassen und derhalb ben ihm grossen gunst und gnad erlanget hat. Es wird auff Indische sprach do es noch hoch gehalten wird Aliabra ben etlichen die besser hinein in Indiam wohnen Alboreth (das ist das Buch der Coniectuation genant die Welschen heissen das libel de lacosa.
“Es gedenckt Mahometh der Berführer Asiae dieser mysterien in seinem Alcoran und nennet sie Almuchabolam weil er seine Sect wie ben den Alten im brauch was durch die Zahlen” (Thurneisser 1583, 11), his emphases and unmatched parenthesis.
- 102.
Poking around Google Books I found other books relating this basic story by different authors published in 1588, 1613, 1626, 1646, 1655, 1670, and 1795, not counting reprints.
- 103.
- 104.
(van Roomen n.d., 7). This manuscript is now lost. Hageccius (1525–1600) was a Czech physician.
- 105.
“Qui hactenus de Algebra egerunt, ita Algebrica tractârunt, tamquam si pars esset Arithmeticae, cum tamen non minus Geometriae quam Arithmeticae possit accommodari. Imo demonstrationes propositionum Algebricarum quas hactenus videre contigit, omnes ex Geometria desumptae sunt; ut potus Geometriae, quam Arithmeticae debeat dici pars. Hinc etiam primus inventor Algebrae Mahumed suam Algebram fecit communem numeris & magnitudinibus [\(\ldots \)] Nos itaque maluimus Algebricam sive Analyticam scientiam revocare ad Mathesin primam, quae quantitatem universalem considerat.” (van Roomen n.d., 2).
- 106.
“quae author noster, tanquam ei qui ad Algebram accedit notissima, omisit, ea nostri muneris erit hic supplere” (van Roomen n.d., 17).
- 107.
- 108.
- 109.
- 110.
“Que de faits curieux, & peut-être intéressans à d’autres égards, n’y auroit-il pas à recueillir dans plusieurs de ces manuscrits! Qu’il est à regretter de ce que parmi ceux qui sont à portée de les consulter, & qui connoissent la langue dans laquelle ils sont écrits, il n’y ait personne qui ait le zele d’aller au delà di titre.” (Montucla 1758 I, 369).
- 111.
Error for L.IV.21.
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Error for L.IV.21.
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Oaks, J.A. (2024). The Legacy of Al-Khwārazmı̄ in History of Algebra. In: Zack, M., Waszek, D. (eds) Research in History and Philosophy of Mathematics. Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-46193-4_8
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