Jeffrey Oaks

University of Indianapolis, Mathematics and Computer Science, Faculty Member

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I study the history of medieval mathematics in Arabic, Latin, and Italian.

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Andrej Dujella

University of Zagreb

Armando Marques-Guedes

UNL - New University of Lisbon

David Klotz

Independent Scholar

Prof. Dr. Raffaele Pisano, MSc, HDR (Habil.)

Université des Sciences et Technologies de Lille (Lille-1)

Thomás A S Haddad

Universidade de São Paulo

Mathieu Ossendrijver

Freie Universität Berlin

Aurélien Robert

Centre National de la Recherche Scientifique / French National Centre for Scientific Research

Omran Kouba

Higher Institute of Applied Sciences and Technology (HIAST)

Umberto Cerruti

Università degli Studi di Torino

Nasreen Kausar

Yildiz Technical University

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Papers by Jeffrey Oaks

The Arithmetica of Diophantus

Actes COMHISMA XI Alger

by Mahdi M O H A M E D Abdeljaouad, Laabid Ezzaiem, and Jeffrey Oaks

َActes du Onzième Colloque Maghrébin sur l'Histoire des mathématiques arabes, 2016

Communications sur l'histoire des mathématiques arabes présentées les 26, 27 et 28 octobre 2013 à... more

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The Arithmetica of Diophantus

Actes COMHISMA14 Final anglais français

by Mahdi M O H A M E D Abdeljaouad, Nejib Boulqhiq, and Jeffrey Oaks

COMHISMA 14, 2022

The 14th Maghreb Colloquium on the History of Arab Mathematics (COMHISMA 14) took place in Sousse... more The 14th Maghreb Colloquium on the History of Arab Mathematics (COMHISMA 14) took place in Sousse (Tunisia) between May 6 and May 8, 2022.
The topics covered are
- Ibn Khaldûn and mathematics,
- The syllogism of equality among Muslim logicians
- ‘Ali ibn Mâmî Karbâsha’s use of sine quadrant (17th c.)
- The contributions of colophons to a contextual History of mathematics
- Calendar material in 'Ali al-Sharafi's atlases (1551, 1571)
- Al-Antâki's Ziyâdât to Euclid's Elements
- Mathematics in Mahmûd Maqdîsh (d.1813)
- New elements in al-Uqbânî (d. 1408)
- A Latin intermediate between Abu Kamil and Fibonacci's algebra
- Al-Qurashi (14th c.) in heritage sciences
- Mathematical Geography in Western Musulman (9th - 16th centuries)
- On the magic squares of order 3 of Ahmad al-Bûnî (d. 1225)
- Terms of powers in Arabic algebra
- An unknown text of al-Qalasadi: Madkhal alrashad ilâ 'ilm al-'ada

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Arithmetic and algebra

Routledge Handbook on the Sciences in Islamicate Societies: Practices from the 2nd/8th to the 13th/19th Centuries, 2023

This is a handbook article giving a brief introduction to medieval Arabic arithmetic and algebra

Terms for powers in Arabic algebra and arithmetic

The word ḥisāb in medieval Arabic mathematics can be translated in different circumstances with w... more The word ḥisāb in medieval Arabic mathematics can be translated in different circumstances with words like "arithmetic", "calculation", and "reckoning". It encompassed numerical calculation broadly conceived, beginning with operations performed either mentally in finger-reckoning or worked out in Indian, sexagesimal, or some other notation. It also included techniques of problem-solving, business calculations, mensuration, basic trigonometry, and combinatorics. 1 Among the different techniques of problem-solving, the most common were the rule of three, single false position, double false position, and algebra. The name given to algebra was al-jabr wa-l-muqābala, often abbreviated as al-jabr. Its place within ḥisāb as a particular technique of problem-solving is corroborated by the many brief descriptions of algebra made by medieval authors, such asʿAlī al-Sulamī (4th/10th c.), who wrote that "the goal of the science of algebra (al-jabr wa-l-muqābala) is to find unknown numbers". 2 A few authors, like al-Karajī (fl. late 4th/10th-early 5th/11thc.) and al-Fārisī (d. 718/1319), go into more detail by explaining characteristics of algebra that differentiate it from other problem-solving techniques (Christianidis & Oaks 2023, 53ff). Very briefly, what makes a solution algebraic is that an unknown is named in terms of the names of the powers, an equation is set up in terms of those names, and this equation is then simplified and solved. There are some topics covered in Arabic books on ḥisāb that were not part of al-jabr wa-l-muqābala, but which today are routinely regarded as being part of algebra, including arithmetical identities, rules for root extraction, and rules for series summation. Historians are cognizant of this difference even when translating these non-algebraic rules into modern algebraic notation. For example, in his Fiqh al-ḥisāb (Understanding calculation, written sometime 595/1199-610/1213), Ibn Munʿim (d. 626/1228) states this arithmetical identity: For any number divided into two parts in whatever manner, its cube is equal to a cube of each of its parts and the product of three times a square of each of its parts by the other. (Ibn Munʿim 2005, 68.12) His editor, Driss Lamrabet, correctly summarized the rule in a footnote as (+) ! = ! + 3 " + 3 " + ! , but without associating it with the medieval al-jabr. After all, the statement does not concern the finding of unknown numbers and it exhibits no algebraic language. The word for "equal", musāwin, is not related to the algebraic ʿadala, and the 1 For an introduction to Arabic arithmetic, see the introduction to (Abdeljaouad and Oaks 2021). 2 (ʿAlī al-Sulamī MS, 2b.14). Other authors who gave similar descriptions include Muḥammad ibn Aḥmad al-Khwārazmī (the lexicographer, latter 4th/10th c.), al-Karajī (ca. 401/1011/12), al-Khayyām (latter 5th/11th c.), al-Fārisī (late 7th/13th c.), al-ʿUqbānī (d. 811/1408), al-Kāshī (d. 832/1429), and Ḥājjī Khalīfa (11th/17th c.). terms for "square" and "cube", murabbaʿ and mukaʿʿab, are not the algebraic terms māl and kaʿb for the second and third powers. But through borrowing in one direction or the other, many of the terms for powers in arithmetic do coincide with their algebraic counterparts, which has caused some confusion among people who do not specialize in Arabic mathematics. For arithmetic these are the words for perfect th-powers and the th powers of numbers, while for algebra they are the names of the powers of the unknown. We also find variations within both the algebraic terms and the arithmetical terms, as well as a built-in flexibility for the terms of higher powers. In this article I cover the terms in both algebra and arithmetic, with attention to their different meanings. For algebra I outline their evolution up to al-Karajī, whose names then became the Arabic standard. Then I describe three sets of terms used by different authors for arithmetical powers, and the two ways that these terms were applied. Next I address one key difference between the algebraic and arithmetical series which lies with the meanings of the first-degree terms, and I conclude with an investigation into why many authors introduce the arithmetical and algebraic powers mixed together, even if they differentiate them in practice. 2. Names of the powers in algebra It is well known that the terms jidhr ("root", pl. judhūr)/shayʾ ("thing", pl. ashyāʾ) for the first degree, māl (pl. amwāl) for the second degree, followed by kaʿb ("cube", pl. kiʿāb, kuʿūb), māl māl, māl kaʿb, kaʿb kaʿb, etc., were used in Arabic algebra as the names of the powers of the unknown. For example, in his Kitāb fī l-jabr wa-l-muqābala (Book on algebra, late 3rd/9th c.) Abū Kāmil sets up this equation in his solution to one problem: "four amwāl māl and four kiʿāb equal seven amwāl", which in modern notation would be expressed as 4 # + 4 ! = 7 " (Abū Kāmil 2012, 447.1). Here the māl, the kaʿb, and the māl māl are the names of particular numbers that will become known when the equation is solved. Abū Kāmil finds that the māl is "two dirhams and a fourth less a root of two", or 2 ! " − √2 in our notation, which makes the kaʿb /15 ! # − 3 ! # and the māl māl 7 ! !$ − /40 ! %. Sometimes algebraic expressions are manipulated outside the context of an equation, so the values of the powers will not be found. But even there the unknowns are presumed to take particular values. This is most common in introductions to books and chapters on algebra where operations on expressions are explained in terms of particular examples. To pick just one, Abū Bakr al-Karajī explains in his al-Fakhrī fī ṣināʿat al-jabr wa-lmuqābala ([Book] for Fakhr [al-Mulk] on the art of algebra, ca. 401/1011), henceforth al-Fakhrī, how to "cast away ten things and four units less a māl from eight things and twenty dirhams and two māls" to get "sixteen dirhams and three māls less two things" (in modern notation, to subtract 10 + 4 − " from 8 + 20 + 2 " to get 16 + 3 " − 2) (Saidan 1986, 120.13). Such examples give the student the practice required to set up and simplify equations, so it is assumed that the unknown thing and its powers would take specific values. And although the more elaborate sample calculations worked out through tables for the multiplication, division, and root extraction of polynomials presented by al-Karajī and al-Samawʾal (d. 570/1174-5) would rarely be encountered in the course of solving problems, there, too, the names of the powers are understood to be specific numbers. 3 From Kitāb fīmā yaḥtāju ilayhi al-kuttāb wa'l-ʿummāl wa ghayruhum minʿilm al-ḥisāb (Book of what is necessary for scribes, dealers, and others from the science of arithmetic, written 961-976 CE) (Abū l-Wafāʾ 1971, 361.13). 4 (Qusṭā ibn Lūqā MS, 273b.6). The book is titled Kitāb al-burhānʿalāʿamal ḥisāb al-khaṭaʿayn (Book on proof of the method of double false position). 5 (Abū Kāmil 2012, 385.8). The solution is short, and never transitions into algebraic terminology: "You know that if you drop from the māl its third, then what remains is its two thirds, and that this two thirds is multiplied by a dirham and a half in order to bring the māl back to what it was originally. Thus the dirham and a half are three roots of the māl. So a root of the māl is a half and the māl is a fourth."

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Zero and nothing in medieval Arabic arithmetic

British Journal for the History of Mathematics, 2022

Whether explaining calculations with decimal or sexagesimal notation, arithmetic books composed i... more Whether explaining calculations with decimal or sexagesimal notation, arithmetic books composed in Arabic beginning in the ninth century CE consistently describe the zero (ṣifr) as a sign indicating an empty place where there is no number. And yet we find that some arithmeticians explicitly performed operations on this zero. To understand how the zero was conceived and manipulated in medieval Arabic texts we first address the way that numbers themselves were conceived and how 'nothing' entered into arithmetical problem-solving. From there we examine arithmetic books for their treatment of zero. We find that there is no inconsistency in operating on what is literally nothing, and thus there was no motive for arithmeticians to regard zero as a number.

Jacques Sesiano, An Ancient Greek Treatise on Magic Squares

Aestimatio: Sources and Studies in the History of Science

The two earliest Arabic treatises explaining the construction of magic squares date from the 10th... more The two earliest Arabic treatises explaining the construction of magic squares date from the 10th century ad. One is found in the Commentary on the Arithmetical [Introduction] (Kitāb tafsīr al-Arithmāṭīqī) by ʿAlī ibn Aḥmad al-Anṭākī (d. 376 H/ad 987). Only book 3 of the original three books is extant, and that in a single manuscript. This book is divided into three chapters: the first is a collection of arithmetical definitions, statements of propositions, and identities assembled from Greek and Arabic sources; the second is on magic squares; and the third deals with “hidden numbers”, in which a person thinks of a number and another discovers it after operations are performed on it. Curiously, none of these chapters have anything to do with Nicomachus’ Arithmetical Introduction, on which the book is supposed to be a commentary. Reviewed by: Jeffrey A. Oaks Published Online (2022-07-31)Copyright © 2022 by Jeffrey A. Oaks Article PDF Link: https://jps.library.utoronto.ca/index.php/ae...

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Al-Hawārī's Essential Commentary

Al-Hawārī completed his Essential Commentary in Marrakesh in 1305. This modest book was written w... more Al-Hawārī completed his Essential Commentary in Marrakesh in 1305. This modest book was written with the goal of supplying numerical examples to the rules of calculation explained in the famous Condensed Book on the Operations of Arithmetic of his teacher Ibn al-Bannāʾ. It is partly because of its modesty that al-Hawārī's book is a good vehicle for presenting the various facets of practical Arabic arithmetic, which in al-Hawārī's time included not just computational techniques but also algebra and other problem-solving methods. Al-Hawārī completed his Essential Commentary in Marrakesh in 1305. This modest book was written with the goal of supplying numerical examples to the rules of calculation explained in the famous Condensed Book on the Operations of Arithmetic of his teacher Ibn al-Bannāʾ. It is partly because of its modesty that al-Hawārī's book is a good vehicle for presenting the various facets of practical Arabic arithmetic, which in al-Hawārī's time included...

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Algebraic symbolism in medieval Arabic algebra

Philosophica, 2012

Medieval Arabic books read more or less like transcriptions of lectures, a fact that stems from t... more Medieval Arabic books read more or less like transcriptions of lectures, a fact that stems from the fundamentally oral nature of medieval Islamic civilization. Books covering algebra are no exception. There we find algebraic solutions to problems written all out in words, with no notation even for numbers. But problems were not solved rhetorically. Algebraists would work out a problem in some kind of notation on a dust board or other ephemeral surface, and to record it in a book a rhetorical version of the calculations would be composed. From early texts it appears that only the coefficients of polynomials were written in Hindi notation. But in the Maghreb in the twelfth century a notation specific to algebra developed in which the power was indicated also. This symbolic notation is not presented as a scientific development in the manuscripts, but is instead shown in textbooks to instruct students in its use.

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Fermat and Descartes in Light of Premodern Algebra and Viète

Handbook of the History and Philosophy of Mathematical Practice, 2021

There were two radically different algebras practiced in Europe in the time of Fermat and Descart... more There were two radically different algebras practiced in Europe in the time of Fermat and Descartes. One was the traditional cossic algebra grounded in arithmetic whose roots stretch back to al-Khwārizmī and Diophantus, and the other was the new geometrical algebra of Franois Viète. Both Fermat and Descartes chose to work in the latter. To understand their choice, and to determine how they understood their algebraic terms, we first outline the conceptual foundations of the two algebras, one that regards a monomial as a premodern number in which the coefficient is a multitude and the power is a kind, and the other that regards a monomial as the product of two magnitudes, one known and the other unknown. We find that cossic algebra could not have been modified to include undetermined coefficients, and thus would have been inadequate for the purposes of Fermat and Descartes. Further, both authors, when solving problems in geometry, worked with precisely the same non-arithmetized algebra as Viète, without a unit measure, respecting dimension, and using magnitudes of arbitrarily high dimension for which no justification is provided.

Diophantus, al-Karajī, and Quadratic Equations

Revolutions and Continuity in Greek Mathematics, 2018

In the beginning of his Arithmetica, Diophantus promises to show how to solve three-term quadrati... more In the beginning of his Arithmetica, Diophantus promises to show how to solve three-term quadratic equations. The rules are not in the extant part of his work, but al-Karajī (early eleventh century) provides clues as to how Diophantus may have derived and presented them. In his algebra book al-Fakhrī al-Karajī gives both the standard Arabic rule for solving three-term equations and the variation practiced by Diophantus. After proofs of the rules he gives a derivation that he calls &quot;the method of Diophantus&quot;. By examining proofs and problems in several of al-Karajī&#39;s works we propose that Diophantus derived his rules for solving three term equations by this &quot;method of Diophantus&quot;, and that al-Karajī borrowed both the derivations and the rules from the Arithmetica. That we are able to speak of Diophantus and al-Karajī both solving quadratic equations is due to the common conceptual foundations and procedures shared by Greek and Arabic algebra. Continuity from one to the other derives not from any chain of texts, but rather from oral tradition that apparently flourished across the old world, and which has left traces in our surviving texts. It is within this conceptual setting that a variation in rules for solving equations enables us to sort out al-Karajī&#39;s rules and postulate Diophantus&#39;s lost exposition.

Al-Khayyām’s Scientific Revision of Algebra

Suhayl. International Journal for the History of the Exact and Natural Sciences in Islamic Civilisation, 2011

Geometers in medieval Islam found algebra to be a useful tool in solving problems, but the Greek ... more Geometers in medieval Islam found algebra to be a useful tool in solving problems, but the Greek restriction of “number” to positive integers, adhered to by al-Khayyām (and presumably others), is at odds with the more liberal inclusion of fractions and irrational roots in traditional numerical Arabic algebra. By regarding the known and unknown numbers of traditional algebra as the homogeneous measures of continuous magnitudes (and not as magnitudes themselves), al-Khayyām was able to give a rigorous foundation for the art. This revision helps us understand al-Khayyām’s main work on algebra in the context of geometrical and numerical problem solving, and it leads us to a reassessment of his generalized notion of “number” explained in his commentary on Euclid.

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The Birth Dates of Valentinian II and Valentinian III: A Correction to PLRE I

L’Eredit arabo-islamica nelle scienze e nelle arti del calcolodell’Europa medievale (The Arabic-Islamic heritage in the sciences and arts of calculation in Medieval Europe). By Nadia Ambrosetti. Milano (Edizioni Universitarie di Lettere Economia Diritto). 2008. ISBN 978-88-7916-388-0. 407 pp. 37 €

Episodes in the Mathematics of Medieval Islam. J. L. Berggren

Nazariyat İslam Felsefe ve Bilim Tarihi Araştırmaları Dergisi (Journal for the History of Islamic Philosophy and Sciences), Nov 15, 2017

Episodes in the Mathematics of Medieval Islam has remained the only reliable English introduction... more Episodes in the Mathematics of Medieval Islam has remained the only reliable English introduction to our field since the first edition appeared in 1985. It is unfortunate that the flurry of research produced during the past thirty years has not prompted others to take up the task of writing books on this topic that are accessible to a general audience or undergraduate students. I had a plan to write one myself, but it is now hopelessly far down the list of things I 'need' to do. Berggren should also be applauded for turning his attention away from specialist studies by issuing a new edition of Episodes. Readers of Nazariyat need only be reminded of the book's organization. The introduction gives, along with other preliminaries, the biographies of four prominent Muslim scientists: al-Khwārizmī (d. after 232/847), al-Bīrūnī (d. after 442/1050), ʿUmar al-Khayyām (d. ca. 526/1131), and al-Kāshī (d. 832/1429). Chapters 2 through 6 cover arithmetic, geometry, algebra, trigonometry, and spherical trigonometry, respectively. Chapter 7, on number theory and combinatorics, is a new chapter. An index rounds out the book. Exercises and a bibliography are given at the end of each chapter. Rather than provide sweeping overviews of all known accomplishments in each topic, Berggren takes the reader through the nuts and bolts of the calculations and constructions of samples of the mathematics from selected texts. The exercises at the end of each chapter should ensure that readers-not just students-take an active part in all of this. After all, one learns math by doing it. Only a few changes have been made to the chapter sections taken over from the 1985 edition. What makes this book a second edition is that Berggren has added new sections to most chapters and an entirely new seventh chapter. He explains in the Preface that in addition to taking into account texts and studies that have

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François Viète’s revolution in algebra

Archive for History of Exact Sciences, 2018

Françios Viète (1540–1603) was a geometer in search of better techniques for astronomical calcula... more Françios Viète (1540–1603) was a geometer in search of better techniques for astronomical calculation. Through his theorem on angular sections he found a use for higher-dimensional geometric magnitudes which allowed him to create an algebra for geometry. We show that unlike traditional numerical algebra, the knowns and unknowns in Viète’s logistice speciosa are the relative sizes of non-arithmetized magnitudes in which the “calculations” must respect dimension. Along with this foundational shift Viète adopted a radically new notation based in Greek geometric equalities. His letters stand for values rather than types, and his given values are undetermined. Where previously algebra was founded in polynomials as aggregations, Viète became the first modern algebraist in working with polynomials built from operations, and the notations reflect these conceptions. Viète’s innovations are situated in the context of sixteenth-century practice, and we examine the interpretation of Jacob Klein, the only historian to have conducted a serious inquiry into the ontology of Viète’s “species”.

Irrational "Coefficients" in Renaissance Algebra

Science in context, Jun 1, 2017

Argument From the time of al-Khwārizmī in the ninth century to the beginning of the sixteenth cen... more Argument From the time of al-Khwārizmī in the ninth century to the beginning of the sixteenth century algebraists did not allow irrational numbers to serve as coefficients. To multiply by x, for instance, the result was expressed as the rhetorical equivalent of . The reason for this practice has to do with the premodern concept of a monomial. The coefficient, or "number," of a term was thought of as how many of that term are present, and not as the scalar multiple that we work with today. Then, in sixteenth-century Europe, a few algebraists began to allow for irrational coefficients in their notation. Christoff Rudolff (1525) was the first to admit them in special cases, and subsequently they appear more liberally in Cardano (1539), Scheubel (1550), Bombelli (1572), and others, though most algebraists continued to ban them. We survey this development by examining the texts that show irrational coefficients and those that argue against them. We show that the debate took pla...

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Polynomials and equations in medieval Italian algebra

Bollettino Di Storia Delle Scienze Matematiche, 2010

A recent study (Oaks 2009) shows that the medieval Arabic concepts of polynomial and equation dif... more A recent study (Oaks 2009) shows that the medieval Arabic concepts of polynomial and equation differ from those of modern algebra. We present here a continuation of this study for medieval Italian algebra, and show that Italian abbacus authors possessed essentially the same medieval concepts as their Arabic predecessors. In short, polynomials were regarded as aggregations of the different species (powers) of the unknown, and medieval algebraists preferred to set up polynomial equations devoid of mathematical operations. These concepts are consistent with the various medieval Italian notations, and extended down to Bombelli's time.

M¿l, enunciations, and the prehistory of Arabic algebra

Historia Mathematica, 2005

The Arithmetica of Diophantus

Actes COMHISMA XI Alger

by Mahdi M O H A M E D Abdeljaouad, Laabid Ezzaiem, and Jeffrey Oaks

َActes du Onzième Colloque Maghrébin sur l'Histoire des mathématiques arabes, 2016

Communications sur l'histoire des mathématiques arabes présentées les 26, 27 et 28 octobre 2013 à... more

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The Arithmetica of Diophantus

Actes COMHISMA14 Final anglais français

by Mahdi M O H A M E D Abdeljaouad, Nejib Boulqhiq, and Jeffrey Oaks

COMHISMA 14, 2022

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Arithmetic and algebra

Routledge Handbook on the Sciences in Islamicate Societies: Practices from the 2nd/8th to the 13th/19th Centuries, 2023

This is a handbook article giving a brief introduction to medieval Arabic arithmetic and algebra

Terms for powers in Arabic algebra and arithmetic

Download

Zero and nothing in medieval Arabic arithmetic

British Journal for the History of Mathematics, 2022

Jacques Sesiano, An Ancient Greek Treatise on Magic Squares

Aestimatio: Sources and Studies in the History of Science

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Al-Hawārī's Essential Commentary

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Algebraic symbolism in medieval Arabic algebra

Philosophica, 2012

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Fermat and Descartes in Light of Premodern Algebra and Viète

Handbook of the History and Philosophy of Mathematical Practice, 2021

Diophantus, al-Karajī, and Quadratic Equations

Revolutions and Continuity in Greek Mathematics, 2018

Al-Khayyām’s Scientific Revision of Algebra

Suhayl. International Journal for the History of the Exact and Natural Sciences in Islamic Civilisation, 2011

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The Birth Dates of Valentinian II and Valentinian III: A Correction to PLRE I

Episodes in the Mathematics of Medieval Islam. J. L. Berggren

Nazariyat İslam Felsefe ve Bilim Tarihi Araştırmaları Dergisi (Journal for the History of Islamic Philosophy and Sciences), Nov 15, 2017

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François Viète’s revolution in algebra

Archive for History of Exact Sciences, 2018

Irrational "Coefficients" in Renaissance Algebra

Science in context, Jun 1, 2017

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Polynomials and equations in medieval Italian algebra

Bollettino Di Storia Delle Scienze Matematiche, 2010

M¿l, enunciations, and the prehistory of Arabic algebra

Historia Mathematica, 2005

Rashed, Roshdi: L’algèbre arithmétique au XIIe siècle: ‘Al-Bāhir’ d’al-Samawʾal. Berlin/Boston: De Gruyter 2021. LXXI, 504 S. 8° = Scientia Graeco-Arabica 32. Hartbd. € 129,95. ISBN 978-3-11-073834-6.

Orientalistische Literaturzeitung, 2021

This is a book review

Review of Nadia Ambrosetti, L'Eredità arabo-islamica nelle scienze e nelle arti del calcolo dell'Europa medievale

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Episodes in the Mathematics of Medieval Islam, J. L. Berggre

by Nazariyat Journal and Jeffrey Oaks

Volume III

Date Nails and Railroad Tie Preservation, 1999

This book contains a history of railroad tie preservation in North America, with special attentio... more

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Volume I

Date Nails and Railroad Tie Preservation, 1999

This book contains a history of railroad tie preservation in North America, with special attentio... more

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Diophantus' Arithmetica

by Jean Christianidis and Jeffrey Oaks

The Arithmetica of Diophantus. A Complete Translation and Commentary, 2023

This volume offers an English translation of all ten extant books of Diophantus of Alexandria's A... more This volume offers an English translation of all ten extant books of Diophantus of Alexandria's Arithmetica, along with a comprehensive conceptual, historical, and mathematical commentary. Before his work became the inspiration for the emerging field of number theory in the seventeenth century, Diophantus (ca. 3rd c. ce) was known primarily as an algebraist. This volume explains how his method of solving arithmetical problems agrees both conceptually and procedurally with the premodern algebra later practiced in Arabic, Latin, and European vernaculars, and how this algebra differs radically from the modern algebra initiated by François Viète and René Descartes. It also discusses other surviving traces of ancient Greek algebra and follows the influence of the Arithmetica in medieval Islam, Byzantium, and the European Renaissance down to the 1621 publication of Claude-Gaspard Bachet's edition. After the English translation the book provides problem-by-problem commentary explaining the solutions in a manner compatible with Diophantus's mode of thought.

The Arithmetica of Diophantus provides an invaluable resource for historians of mathematics, science, and technology, as well as those studying ancient Greek, medieval Islamic and Byzantine, and Renaissance history. In addition, the volume is also suitable for mathematicians and mathematics educators.

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Al-Hawārī’s Essential Commentary: Arabic Arithmetic in the Fourteenth Century

We give an edition and English translation of al-Hawārī’s book, together with a passage-by-passag... more

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