Excavating the errors in the "Mathematics" chapter of 1001 Inventions

Excerpt from 1001 Distortions How (Not) to Narrate History of Science, Medicine, and Technology in Non-Western Cultures Edited by Sonja Brentjes – Taner Edis – Lutz Richter-Bernburg BIBLIOTHECA ACADEMICA Reihe Orientalistik Band 25 ERGON VERLAG The three editors together with the other contributors to the present volume address their grateful appreciation of generous material and immaterial support to: Dr. Hans-Jürgen Dietrich and Thomas Breier at Ergon-Verlag, Würzburg Max-Planck-Institut für Wissenschaftsgeschichte, Berlin Merrimack College Department of Physics, North Andover, Massachusetts The cover features a Safavid silver-inlaid brass astrolabe, signed Muhammad Khalil and Muhammad Baqir (Isfahan, probably late 17th c; Sotheby’s. Arts of the Islamic World. London 8 October 2008, lot 169). Within its ecliptic ring, the Basmalah, the Muslim invocation of God, is, for the sake of symmetry, partially rendered in mirror image. The photographic distortion of the instrument is an ironic reference to the variously distorted representations of past knowledge cultures, which the book undertakes to debunk. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de.  2016 Ergon-Verlag GmbH • 97074 Würzburg Das Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung außerhalb des Urheberrechtsgesetzes bedarf der Zustimmung des Verlages. Das gilt insbesondere für Vervielfältigungen jeder Art, Übersetzungen, Mikroverfilmungen und für Einspeicherungen in elektronische Systeme. Gedruckt auf alterungsbeständigem Papier. Satz: Thomas Breier, Ergon-Verlag GmbH Umschlaggestaltung: Jan von Hugo www.ergon-verlag.de ISBN 978-3-95650-169-2 ISSN 1866-5071 Table of Contents Sonja Brentjes Introduction: On Narratives of Amateurs and Professionals ........................................................ 9 Part 1: Inventing Narratives of Intellectual Identities and their Modern Relevance Science as a Weapon of Cultural Competition. Interview with Lorraine Daston, director, Max-Planck-Institut für Wissenschaftsgeschichte, Berlin ..................................... 19 Dhruv Raina After Exceptionalism and Heritage: Thinking through the Multiple Histories of Knowledge ..................................... 25 Helge Wendt Becoming Global: Difficulties for European Historiography in Adopting Categories of Global History............................................................ 39 Antoni Malet Science, History, and Identity in Francoist Spain, the Early Years........................................................................................................ 53 Manuela Marín Reinventing the History of al-Andalus: Scholarship, the Media, and a Touch of Islamophobia.........................................75 Khodadad Rezakhani The Present in the Mind’s Past: Imagining the Ancients in the Iranian Popularization of Pre-Islamic History.............................................................................................97 Part 2: What is Wrong with the Narrative in 1001 Inventions’ Companion Book? Hadi Joráti Misuse and Abuse of Language, and the Perils of Amateur Historiography (of Science) ............................................................. 109 6 TABLE OF CONTENTS Lutz Richter-Bernburg Potemkin in Baghdad: The Abbasid “House of Wisdom” as Constructed by 1001 Inventions ...................................................................... 121 Sonja Brentjes Science, Religion, and Education........................................................................ 133 Jeffrey A. Oaks Excavating the Errors in the “Mathematics” Chapter of 1001 Inventions .................................................................................. 151 Petra G. Schmidl “Mirror of the Stars:” The Astrolabe and What It Tells About Pre-Modern Astronomy in Islamic Societies ...............................................................................................173 Taner Edis and Amy Sue Bix Flights of Fancy: The 1001 Inventions Exhibition and Popular Misrepresentations of Medieval Muslim Science and Technology .................................................... 189 Rainer Brömer Only What Goes Around Comes Around: A Case Study on Revisionist Priority Disputes– Circulation of the Blood ..................................................................................... 201 Part 3: Delights and Dangers of Popularizing Historical and Scientific Knowledge On the Difference Between Doing Academic Research and Directing a Museum of Islamic Art and Culture. Interview with Stefan Weber, director of the Museum für Islamische Kunst, Berlin, March 2014 and Spring 2016 .................................... 215 Aaron Adair Mass Education on Science and its History: Using Misconceptions in Physics to Teach History and Science ....................... 221 Holger Schuckelt A Museum Curator at Work................................................................................ 233 Peter Adamson Without any Gaps: Podcasting the History of Philosophy ................................ 241 TABLE OF CONTENTS 7 Mike Diboll 1001 Inventions and Failed Education Reform in Bahrain: A Case Study........................................................................................................ 249 Vidar Enebakk 1001 Pieces of Islamist Propaganda? ................................................................... 265 Postscript ...............................................................................................................279 Excavating the Errors in the “Mathematics” Chapter of 1001 Inventions Jeffrey A. Oaks University of Indianapolis, Indianapolis, IN It is generally a good idea not to trouble too much over the errors in coffee table books. People rarely read these books, and most who do retain little of the actual content. The book 1001 Inventions, however, is not typical of the genre. It is already in its third edition and it is associated with a world-touring exhibition (1001 Inventions). It is this global exposure that has prompted some of us to issue a response in the form of the present collection. Yet even with this kind of publicity the four-page, error-ridden chapter 3.6, titled “Mathematics,” could not have caused much harm. The few readers of the chapter will surely come away with the notion that medieval Muslims made great contributions to the field, which is of course true. But just what those reported contributions were will remain nebulous even to readers well-versed in mathematics. This is because the chapter’s mistakes and misrepresentations are muffled by a poor writing style and lack of meaningful context. For instance, what can the reader take away from a statement like “Al-Kashi contributed to the development of decimal fractions, not only for approximating algebraic numbers, but also for real numbers such as pi” (1001 Inventions 2012, p. 87)? The errors in this statement, which I will explain below, are effectively nullified by the fact that no explanation is given for what decimal fractions are, what is meant by “algebraic numbers,” nor how such a development fits in the broader history of mathematics. There is, then, little to be gained from taking the chapter at face value and recording its errors and distortions. And yet it has attracted my attention. Chapter 3.6 interests me because it offers a modern version of what we historians of science find enticing: it is a text mainly copied from an article which in turn condenses the contents of an already flawed book that was translated from French. It brings to mind the genealogies of so many medieval texts that were copied from other texts, some of them epitomes of longer works, and some translated from other languages. So I approach the chapter not so much as a specialist assessing its accuracy, but as one curious about the history of the text itself. I begin my analysis by identifying the sources for the chapter. From there I find it convenient to review the contents of the chapter in the mode of medieval commentators. I give quotations from 1001 Inventions in italics, and after each quotation I give my comments and corrections. In my comments I trace the origin of the statement and often the contortions it underwent in its transmission from one work to the next. Also, my corrections more often than not require additional 152 JEFFREY A. OAKS background to place the mathematics in its proper historical and philosophical context. Uncovering the Sources The chapter on mathematics covers in order the algebra, number theory, and arithmetic of medieval Islamic civilization. After only a few minutes of reading it over I discovered that two thirds of it is plagiarized from O’Connor’s and Robertson’s (henceforth O&R) article “Arabic mathematics: forgotten brilliance?” (http:// www-history.mcs.st-and.ac.uk/HistTopics/Arabic_mathematics.html). Readers would have been better served if the chapter’s author, or rather compiler, had plagiarized a more reliable source. After a promising introduction O&R’s article unfortunately turns out to be riddled with mistakes and misrepresentations. But at least they tell us where they got their information. The parts on algebra and number theory are distilled from chapters 1 and 4 of Roshdi Rashed’s 1994 book The Development of Arabic Mathematics: Between Arithmetic and Algebra, and their treatment of arithmetic is based mainly on the introduction to A. S. Saidan’s 1978 book The Arithmetic of Al-Uqlīdisī. Rashed’s book is a translation of the French original from 1984. It is not a general history of Arabic mathematics, but rather a collection of studies that the author had undertaken as part of his own research program. This accounts for why it omits many important developments and does not cover basic arithmetic at all. O&R made up for the lack of arithmetic by adding a few paragraphs at the end of their essay, rather than at the beginning where it would have fit more naturally. The 1001 Inventions chapter follows O&R’s format, to which it adds some observations on arithmetic extracted from Karl Menninger’s 1969 book Number Words and Number Symbols as well as some strikingly misguided and irrelevant paragraphs whose sources elude me. Algebra Chapter 3.6 begins with al-Khwarizmi’s book on algebra, which he wrote in Baghdad in the period 813–833 CE. In this one case I give the passage in O&R to compare with the version in 1001 Inventions: O&R: Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. EXCAVATING THE ERRORS IN THE “MATHEMATICS” CHAPTER OF 1001 INVENTIONS 153 1001 Inventions: This remarkable period in the history of mathematics began with Al-Khwarizmi’s work, when he introduced the beginnings of algebra. It is important to understand just how significant this new idea was. In fact, it was a revolutionary move away from the Greek concept of mathematics, which was essentially based on geometry (1001 Inventions 2012, p. 84). Algebra was not new with al-Khwarizmi in the early ninth century, nor was it a move away from “the Greek concept of mathematics.” To explain these two points I will need to say a few words about the place of algebra in medieval problemsolving. Arabic algebra was a technique for finding unknown numbers that was practiced alongside other methods such as double false position and the rule of three (proportion). In these other methods the unknown number is found by a direct calculation using known values. What distinguishes an algebraic solution is that an unknown is given a name (today we typically call it “x”), calculations are performed on this name, and an equation is set up and solved. Algebra had already been practiced in Greece and India before the advent of Islam, and in the Middle East it was most likely taught orally within trade groups before al-Khwarizmi and others began to write books on the topic. In fact, throughout its history Arabic algebra remained part of the practical education of future merchants, surveyors, and others who performed calculations as part of their jobs. During this time many significant scientific developments were made as well, some of which are described below. There was no singular Greek concept of mathematics. It is true that prominent Greek mathematicians excelled at geometry, the most famous being Euclid, Archimedes, and Apollonius. But some also did significant work in arithmetic and number theory. Important works that later became major influences on Arabic mathematics are Books VII to IX of Euclid’s Elements (third century BCE) and Nicomachus’s Introduction to Arithmetic (ca. 100 CE), which are both devoted to number theory, and Diophantus’s Arithmetica (ca. third century CE), a treatise on solving arithmetic problems by algebra. Algebra was a unifying theory that allowed rational numbers, irrational numbers, and geometrical magnitudes all to be treated as algebraic objects (1001 Inventions 2012, p. 84). These words, lifted from O&R, paraphrase Rashed’s book: “Since algebra is concerned with both segments and numbers, the operations of algebra can be applied to any object, be it geometrical or arithmetical. Irrationals as well as rationals may be the solution of the unknown in algebraic operations precisely because they are concerned with both numbers and geometrical magnitudes” (Rashed 1994, p. 26). Apart from one interesting exception Arabic algebra was not a unifying theory. To see why, we must back up a few centuries. In theoretical Greek arithmetic numbers were restricted to the positive integers 1, 2, 3, 4, etc., though “1” was 154 JEFFREY A. OAKS strictly speaking not considered to be a number because it is not a multitude. Of 3 course Greeks who used numbers in the trades worked with fractions (like 4 and 7 1 5 10 ) and irrational roots (like 5 and 58 16) just like their counterparts in other civilizations. This more utilitarian notion of number surfaces in some later Greek texts. Many of Diophantus’s solutions are fractions, and Hero of Alexandria (first century CE) routinely gives rational approximations to irrational numbers. In practical Arabic mathematics numbers were likewise any positive quantity that arises in calculation, including fractions and irrational roots. Algebra was based on this liberal concept of number, and could be applied to any problem whose parameters could be represented numerically. In geometry problems it was the numerical measures of the magnitudes that were entered into algebraic calculation, not the magnitudes themselves. Arabic algebra was thus grounded squarely in practical arithmetic. The interesting exception is the geometrical tradition of al-Khayyam, of which more is said below. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way that had not been possible earlier (1001 Inventions 2012, p. 84). 1001 Inventions omits the explanation of this statement provided by O&R in the form of a direct quotation from Rashed’s book (square brackets show passages omitted by O&R): Al-Khwârizmî’s successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both [of them] to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. [These applications were always the starting point for new disciplines, or at least new topics.] This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose (Rashed 1994, p. 352). The torch of algebra was taken up by the successor of al-Khwarizmi, a man called AlKaraji, born in 953 (1001 Inventions 2012, p. 84). Before plunging into a discussion of the two centuries of algebra that O&R skip over here, I should mention that the date 953 is made up. Despite what you find on the Internet, there is nothing in the manuscripts that tells us when al-Karaji was born. It would have been better for O&R to say that he wrote his major works in the early eleventh century. Rashed said very little in his book about algebraists between al-Khwarizmi and al-Karaji because he had not yet published studies on them. With O&R exclusively following Rashed and our compiler plagiarizing O&R, we should not expect the chapter to mention intervening figures such as Thabit ibn Qurra, Abu Kamil, and Qusta ibn Luqa in the ninth century, or Ibn al-Fath in the tenth. I will briefly review their contributions below, but in order to make it all intelligible I will need to first outline some basic features of Arabic algebra. EXCAVATING THE ERRORS IN THE “MATHEMATICS” CHAPTER OF 1001 INVENTIONS 155 Medieval books were by and large written all in words, without any notation. Instead of using a letter like “x” for the first degree unknown, Arabic algebraists called it a “thing” (shayʾ) or “root” (jidhr). Its square (our x 2) is called a mal, a word whose everyday meaning is “amount of money”, “property”, or “wealth.” I will leave this word untranslated. The cube of the “thing” (x 3) is called a “cube” (kaʿb), and higher powers were written as a combination of mal and kaʿb, like mal mal for the fourth power and mal kaʿb for the fifth. Units were often counted in dirhams, a silver coin. So Abu Kamil’s equation “four ninths of a mal and four dirhams less two things and two thirds of a thing equal a thing and twenty-four 4 2 dirhams” corresponds to our 9 x 2 + 4 – 2 3 x = x + 24. Today we have one simplified form for quadratic equations, ax 2 + bx + c = 0, in which the coefficients a, b, and c can be positive, negative, or zero. But in medieval mathematics numbers could only be positive, and the rules for solving simplified equations involve calculations with these coefficients. So Arabic algebraists classified six types of equation, three of them simple (ax 2 = bx, ax 2 = b, and ax = b) and three composite (ax 2 + bx = c, ax 2 + b = cx, and ax + b = cx 2). Again, these are all written out in words in the texts. The rules for solving the composite equations are a bit complex, so alKhwarizmi and many other algebraists gave proofs by geometry that the rules work. In the proofs the “thing” is represented by a line, and the mal by the literal square on that line. The validity of the rules then follows from a comparison of equal lines and areas. Al-Khwarizmi made no explicit references to Greek geometry in his book, but Thabit ibn Qurra (d. 901) later wrote a short treatise reformulating these geometrical proofs so they rest on propositions from Book II of Euclid’s Elements. In the later ninth century the Egyptian mathematician Abu Kamil wrote a more comprehensive book on algebra incorporating the ideas of both alKhwarizmi and Euclid. Among many innovations he worked with powers of the unknown up to x 8 (called a mal mal mal mal), he gave both geometrical and arithmetical proofs to propositions in arithmetic and algebra, and he gave the only premodern proofs by algebra that we know. His On the Pentagon and Decagon, appended to his work on algebra, shows applications of algebra to geometry. Following this is a collection of problems, the first 38 of which are indeterminate and belong to a native Arabic tradition that does not derive directly from Diophantus. In the tenth century Sinan ibn al-Fath wrote a twenty-page treatise on higher powers in which he studied higher degree equations that reduce in one way or another to quadratic equations. Qusta ibn Luqa translated Diophantus’s Arithmetica in the second half of the ninth century. He naturally adapted the language of Arabic algebra to Diophantus’s solutions, which use not only the higher powers up to x 6, but also the reciprocals of the powers (1/x, 1/x2, to 1/x6). 156 JEFFREY A. OAKS [Al-Karaji] is seen by many as the first person to free algebra completely from geometrical operations, and to replace it with the arithmetical type of operations, which are the core of algebra today. He was the first to define the monomials x, x 2, x 3, … and 1/x, 1/x 2, 1/x 6, … and to give rules for products of any two of these (1001 Inventions 2012, p. 84). All of this is false. To sort it out we must return to al-Karaji’s two main influences, Abu Kamil and Diophantus. Like al-Khwarizmi before him, Abu Kamil had explained only the first two powers in his introduction, though he used higher powers later among his worked-out problems. By contrast, in his opening pages Diophantus presented the powers, their reciprocals, and the rules for their products and divisions. Al-Karaji adapted Diophantus’s format in his algebra book al-Fakhri, making him the first Arabic algebraist to present the powers this way. He also provided many more example calculations than Abu Kamil, but without proofs. Rashed writes this about the treatment of monomials and polynomials in the beginning of al-Karaji’s al-Fakhri: The more or less explicit aim of this exposition was to find means of realizing the autonomy and specificity of algebra, so as to be in a position to reject, in particular, the geometrical representation of algebraic operations (Rashed 1994, p. 23). For Rashed, al-Karaji did not “free algebra completely from geometrical operations,” but put it in a position to be freed from geometrical representations of the operations in proofs of the kind we find in Abu Kamil. The operations in algebra had of course always been arithmetical. But even this interpretation is not borne out in the text. Al-Karaji appears to have moved away from geometry in his al-Fakhri simply because he omitted proofs for his sample calculations. He still gave geometrical proofs based in Euclid’s Elements for the solutions to the three-term composite equations, and just before the first of these proofs he wrote I made a determination in this book to strip it of proofs, lengthy explanations, and numerous examples. But I cannot avoid giving a brief summary of proofs for the connected problems [I. e., the composite equations]... (Saidan 1986, p. 151). And al-Karaji’s successor al-Samawʾal resumed the application of geometrical proofs for sample calculations while continuing the tradition of geometrical proofs for the composite equations. There was a move away from geometrical representation in proofs, but it had nothing to do with making algebra autonomous. After completing his al-Fakhri alKaraji wrote the short treatise Causes of Calculation in Algebra in which he gave arithmetical proofs of the rules for solving equations. He states that the reason for the shift is that many people had trouble understanding the proofs based in geometry (Saidan 1986, p. 354). Many later algebraists likewise gave proofs based entirely on arithmetic, including Ibn al-Yasamin (d. 1204), Ibn al-Bannaʾ (late 13th c.), al-Farisi (d. c. 1320), and Ibn al-Haʾim (1387). Ibn al-Haʾim explained that proofs based in geometry require knowledge of Euclid, and he wished to reserve EXCAVATING THE ERRORS IN THE “MATHEMATICS” CHAPTER OF 1001 INVENTIONS 157 them for a later book on geometry (Ibn al-Hāʾim 2003, pp. Fr. 19, Ar. 79). None of this is mentioned by Rashed. While al-Karaji cannot be said to have initiated an “arithmetization of algebra,” his investigations into monomials and polynomials did give algebra a new direction for research. In case you forgot your algebra, a monomial is a one-term 3 expression, like 3x 2 or 4x. A polynomial is an expression that can have more than one term, like 3x 2 + 2x – 1. Al-Karaji called polynomials “composite numbers,” to distinguish them from monomials, which he called “simple numbers.” What is novel in his approach is the study of operations on these expressions for their own sake, independent of their role in problem solving. For example, among his many sample calculations he applies a rule for finding the square root of the five-term polynomial “a mal mal and four cubes and ten mals and eighteen things and nine dirhams” (x 4 + 4x 3 + 10x 2 + 18x + 9), something that would hardly come up in any worked-out problem. ... He started a school of algebra, which flourished for several hundred years. Two hundred years later, 12th-century scholar Al-Samawal was an important member of Al-Karaji’s school (1001 Inventions 2012, p. 84). “School” here should mean “school of thought” and not an actual, physical school with teachers and students. Rashed makes this clear in his book, which is the ultimate source of this statement. But Rashed reads more into the sources than is warranted, for there is no evidence that any school of thought was initiated by al-Karaji. All we know is that al-Samawʾal and maybe a couple others were influenced by his work. He [al-Samawʾal] was the first to give algebra the precise description of “operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known” (1001 Inventions 2012, p. 84). This brief description covers only the first stage in algebraic problem solving, in which one performs operations on the named unknowns to set up an equation. Other algebraists characterized algebra more broadly, like al-Karaji, who wrote Know that the objective of algebra is to find unknown from known quantities (Saidan 1986, p. 145). If al-Samawʾal’s description of algebra is narrowly focused on the application of arithmetical rules to polynomials, it is because that was his own area of interest. For example, he followed the lead of al-Karaji by taking advantage of the structural similarity between the powers of ten in arithmetic (10, 100, 1000, etc.) and the powers of the unknown in algebra (x, x 2, x 3, etc.) to devise rules for the division and root extraction of polynomials. Now the discussion shifts to another tradition in algebra best known through the work of ʿUmar al-Khayyam (Omar Khayyam) from around 1075 CE. 158 JEFFREY A. OAKS He gave a complete classification of cubic equations, with geometric solutions found by means of intersecting conic sections (1001 Inventions 2012, p. 84-85). This is entirely correct! What is lacking is context. Al-Khayyam was a theoretical geometer who wanted to use algebra to solve problems that could not be worked out by ruler and compass. This meant that he needed to find solutions to cubic equations, i.e., those with a cube term (x 3). Toward this end he classified all twenty-five types of simplified equations of degree three and less. But he adhered to Euclid’s notion that “number” should be restricted to positive integers, which are inadequate for representing geometrical magnitudes. To justify the use of algebra in geometry al-Khayyam regarded the practical numbers of the algebraists as the abstraction of the measures of geometrical magnitudes in a way that does not 1 tie them to the dimension of those magnitudes. For example, a quantity like 32 is strictly speaking not a number because it has a fractional part. But it can be regarded as the length of a line, the area of a rectangle, or the volume of a body. Knowing that his geometrical algebra could also be used to find unknown numbers (i. e. integers), al-Khayyam gave both arithmetical and geometrical solutions to his equations. An arithmetical solution is found through calculation, while a geometrical solution is a line constructed by a prescribed procedure. Because his unknowns can potentially be integers or measures of magnitudes, he unified two branches of mathematical calculation with his algebra. François Viète (1591) has been credited with being the first to make algebra into such a unifying theory, but al-Khayyam accomplished it five centuries earlier. Incidentally, the Arabic works that preceded Viète were unknown in Europe in his time, so his discoveries were independent. In the mid-12th century, while Al-Samawal was studying in Al-Karaji’s school, Sharaf alDin al-Tusi was following Al-Khayyam’s application of algebra to geometry. He wrote a treatise on cubic equations, and in it said that algebra “represents an essential contribution to another field, which aimed to study curves by means of equations,” thus inaugurating the field of algebraic geometry (1001 Inventions 2012, p. 85). Passing over the reference to al-Karaji’s non-existent school, we find 1001 Inventions misunderstanding their misquotation from O&R. The quotation really extends to the end of the passage above, and it is due to Roshdi Rashed (Rashed 1994, p. 103), not to Sharaf al-Din al-Tusi. O&R’s version is a misquotation, too. Rashed really writes “represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry” (my emphases). Even if 1001 Inventions had attributed the correct quotation to the right person, Rashed’s statement is not true. Al-Tusi did not study curves by means of equations, but rather equations by means of curves, just like al-Khayyam had done. The study of curves by means of equations originated with Descartes in the seventeenth century. EXCAVATING THE ERRORS IN THE “MATHEMATICS” CHAPTER OF 1001 INVENTIONS 159 But in another chapter in his book Rashed is right when he notes that Sharaf al-Din is the first algebraist known to apply what is now called the RuffiniHorner method to find numerical solutions to polynomial equations. For example, he applies the algorithm to the equation x 3 + 300x = 30x 2 + 30,081,231 to find that x = 321. This was long before François Viète applied essentially the same rule in his 1600 book On the Numerical Resolution of Powers by Exegetics. Number theory Following O&R the chapter now shifts to number theory. This begins with a description of a theorem by the ninth-century mathematician Thabit ibn Qurra, whose work on algebra was mentioned above. He is probably best known for his contribution to number theory, where he discovered a beautiful theorem allowing pairs of amicable numbers to be found. This term refers to two numbers such that each is the sum of the proper divisors of the other (1001 Inventions 2012, p. 85). An example of amicable numbers is the pair 220 and 284. The numbers that divide evenly into 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, and they add up to the other number, 284. The numbers that divide evenly into 284 are 1, 2, 4, 71, and 142, which add up to 220. Thâbit derived a clever rule for finding other pairs of amicable numbers, and he gave a geometrical proof that the rule works. Thabit ibn Qurra was one of the finest mathematicians of medieval Islam, though he himself was not a Muslim. He made major contributions in just about every branch of mathematics and astronomy, especially geometry. He is not best known for this result in number theory. Amicable numbers played a large role in Arabic mathematics, and in the 13th-century AlFarisi gave a new proof of Thabit’s theorem…He also discovered the pair of amicable numbers 17,296 and 18,416, which have been attributed to Euler, an 18th-century Swiss mathematician. And many years before Euler, another Muslim mathematician, Muhammad Baqir Yazdi, in the 17th century discovered the pair of amicable numbers 9,363,584 and 9,437,056 (1001 Inventions 2012, p. 85). Amicable numbers were certainly interesting to number theorists, but it is a stretch to say that they “played a large role in Arabic mathematics” as O&R claim. Further, al-Farisi did not discover the pair 17,296 and 18,416. It was most likely first found by Thabit, and the thirteenth-century mathematician Ibn Fallus gave the same pair, though he gave a wrong pair before it. Thabit’s rule, and the pair 17,296 and 18,416, were later rediscovered well before Euler’s time by both Fermat and Descartes in the 1630s, and the pair 9,363,584 and 9,437,056 was also found by Descartes. 160 JEFFREY A. OAKS Muslim mathematicians excelled in the tenth century in yet another area when Ibn alHaytham was the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors), such as those of the form 2k – 1(2k – 1) where 2k – 1 is prime (1001 Inventions 2012, p. 85). An example of a perfect number is 28, which equals the sum of its divisors 1, 2, 4, 7, and 14. You can think of perfect numbers as numbers that are amicable with themselves. We do not know if Ibn al-Haytham was the first who attempted to classify amicable numbers. His text is simply the earlest we have that records it. He was also the first person that we know of to state Wilson’s theorem, namely that if p is prime, then the polynomial 1+ (p-1)! is divisible by p, but it is unclear whether he knew how to prove this (1001 Inventions 2012, p. 85). Apart from the fact that for medieval mathematicians 1+ (p – 1)! is a number and not a polynomial, this is true. It is indeed a significant result of Ibn alHaytham. Arithmetic The 1001 Inventions chapter now turns to arithmetic. Here the plagiarizing becomes sporadic. Already the first paragraph contains a nonsensical statement that is not taken from O&R: Today, most of us are only aware of one counting system, which begins with zero and carries on into the billions and trillions (1001 Inventions 2012, p. 86). What differentiates “counting systems,” if one insists on that phrase, is the ways the numbers are represented, not their actual values. The rest of this sentence is too silly to bother analyzing. But in tenth-century Muslim countries, there were three different types of arithmetic used, and by the end of the century, authors such as Al-Baghdadi were writing texts comparing them. These three systems were finger-reckoning arithmetic, the sexagesimal system, and the Arabic numeral system (1001 Inventions 2012, p. 86). These three systems were in use throughout the medieval period in Muslim countries. O&R single out the tenth century because that is when al-Baghdadi wrote his book. The descriptions of the three systems below are given correctly in O&R’s source, the introduction to Saidan’s book The Arithmetic of Al-Uqlīdisī. It is O&R who introduced the errors. Finger-reckoning arithmetic came from counting on fingers with the numerals written entirely in words, and this was used by the business community (1001 Inventions 2012, p. 86). Here again we are treated to the distorted echoes of valid observations. To begin, “finger reckoning” was a method of mental calculation and did not involve EXCAVATING THE ERRORS IN THE “MATHEMATICS” CHAPTER OF 1001 INVENTIONS 161 A finger-reckoning scheme from Luca Pacioli’s Summa de Arithmetica [1523], fol. 36v (Pacioli, Luca. Summa de arithmetica, geometria, proportioni et proportionalita [1523], Call Number: SMITH 511 1523 P11 Rare Book & Manuscript Library, Columbia University in the City of New York). Courtesy Rare Book & Manuscript Library, Columbia University. 162 JEFFREY A. OAKS counting on fingers. If you were to multiply, say, 48 by 71 in your head, you would first figure 8 by 71 to get 568, and this must be added to 40 by 71, which is 2840. But by the time you have calculated the 2840 you have likely forgotten the 568. “Finger reckoning” was a way of remembering these intermediate calculations by storing them by positioning the fingers in certain ways. The system had been in use at least since Roman times. Above is a diagram illustrating the positions in use in Italy from a 1494 arithmetic book (reprinted 1523). The second part of the sentence, on the writing of the numbers, requires a digression on the role of books in medieval culture. I mentioned above that Arabic algebra is presented rhetorically in the manuscripts. The algebra books of alKhwarizmi, Abu Kamil, al-Karaji, and al-Khayyam, for instance, contain no notation at all. Even in books which teach calculation with Arabic numerals the numbers are nearly all written out in words! The Arabic numerals typically only appear as illustrations or figures, like we see in this passage from al-Hassar’s late twelfthcentury Book of Demonstration and Recollection in the Art of Dust-Board Reckoning: For example, add eight hundred seventy-five to seven hundred ninety-eight. Write the eight hundred seventy-five on a line, then the seven hundred ninety-eight on a line under the other number so that the units are under the units, the tens under the tens, and 875 (al-Hassar 1194, fol. 7v). the hundreds under the hundreds, like in this figure: 798 He then explains in words how to calculate the answer, with instructions like add the five to the eight to get thirteen, and put the three above the units... (al-Hassar 1194, fol. 7v). As al-Hassar tells us, the numerals were intended to be drawn on a dust-board or some other ephemeral surface. A dust-board is a flat board covered in fine sand on which one wrote with a stylus. It was a medium for working through temporary calculations, much like today’s chalkboard or whiteboard. Arithmetic books like al-Hassar’s gave instuctions for carrying out these operations. Today mathematics books are filled with symbols and notation. So why not also for medieval mathematics? The reason has to do with how people related to books. Medieval Islamic learning, like medieval European learning, was fundamentally oral in nature. Books were by and large thought to properly reside in memory, and were manifested through oral performance. To learn a text a student would memorize it and recite it back to the teacher. Physical paper copies of books thus often read like transcriptions of the spoken word, and were regarded more like tools for memorization than as final repositories of knowledge. Since notation serves no purpose when reading aloud, it is not part of the running text. And while it is true that not all medieval books were meant to be memorized and recited, those on elementary arithmetic usually were. We can now return to the statement that in finger reckoning “the numerals were written entirely in words.” Numbers (not numerals) in finger reckoning have no written form at all, so of course any book that describes problem-solving methods associated with the technique would show the numbers in words. Even EXCAVATING THE ERRORS IN THE “MATHEMATICS” CHAPTER OF 1001 INVENTIONS 163 in books that explain sexagesimal (base 60) and decimal (base 10) arithmetic the numerals generally appear only in figures. The sexagesimal system had numerals denoted by letters of the Arabic alphabet. It came originally from the Babylonians and was most frequently used by Arabic mathematicians in astronomical work (1001 Inventions 2012, p. 86). The sexagesimal system of the astronomers employs a subset of the abjad system, so I will describe that first. In abjad the letters of the Arabic alphabet are assigned numerical values 1, 2, …, 9, 10, 20, …, 90, 100, 200, …, 900, and 1000. This works well because the alphabet has 28 letters. The number 560, for example, was written with the letters signifying 500 (‫ )ث‬and 60 (‫)س‬: ‫ثس‬. Ancient astronomers from India to Europe calculated with a base 60 placevalue system. This is like our base ten system, except each place can have the numbers 0 through 59 instead of just 0 through 9. When Arabic astronomers appropriated the system they wrote the numbers using the letters corresponding to 1 to 9 and 10 to 50 from the abjad system along with a sign for zero. The arithmetic of the Arabic numerals and fractions with the decimal place-value system was developed from an Indian version (1001 Inventions 2012, p. 86). The decimal place-value system did indeed originate in India, and we know that it had reached the Middle East by the middle of the seventh century CE when Severus Sebokht, a Syrian bishop, mentioned them. Muslims adapted the Indian numerals into the modern numbers, 1 to 9, known as Arabic numerals. They are believed to have been based on the number of angles each character carries, but the number 7 creates a challenge, as the medial horizontal line crossing the vertical leg is a recent 19th-century development (1001 Inventions 2012, p. 86). Starting here our compiler diverges from O&R, which was immediately a bad idea because they are right that “there was not a standard set of symbols.” The shapes of the numerals have varied greatly over time and place, and are absolutely not based on the number of angles. The crazy idea of their origin in angles has been around a long time. Florian Cajori debunks it along with many other theories in his book A History of Mathematical Notations (Cajori 1928, vol. I, pp. 65-66). These have become the numerals we use in Europe and North Africa today, as distinct from the Indian numerals that are still used in some eastern parts of the Muslim world (1001 Inventions 2012, p. 86). Contrary to what is implied here, the forms of the numerals used in the eastern part of the Islamic world are nothing like the numerals in India, and both have exhibited significant variations. Likewise, our European forms of the numerals have undergone major changes across time and place, diverging early on from the forms used in North Africa. 164 JEFFREY A. OAKS The arrival of these numerals [in the west] resolved the problems caused by Latin numerals in use until then (1001 Inventions 2012, p. 86). The compiler meant to write “Roman” instead of “Latin.” I do not know where this claim comes from, or what specific problems they mean. Nobody calculated with Roman numerals, which were used only to record numbers. In medieval European calculation Arabic numerals often competed not against Roman numerals but against the abacus. This abacus was a flat board, often a tabletop, with lines drawn typically for units, fives, tens, fifties, hundreds, and higher if necessary. “Counters” (tokens resembling coins) were placed and manipulated on these lines to perform calculations. This is the origin of our word “counter” for, say, a kitchen counter. Now this next statement will be understandable: Arabic numerals were referred to as ghubari numerals because Muslims used dust (ghubar) boards when making calculations, rather than an abacus. A great refinement by Muslim mathematicians of the Indian system was the wider definition and application of zero. Muslims gave it a mathematical property, such that zero multiplied by a number equals zero. Previously zero defined a space or a “nothing.” It was also used for decimalization, making it possible to know whether, for example, the writing down of 23 meant 230, 23, or 2,300 (1001 Inventions 2012, p. 86). Neither of these characteristics is a Muslim refinement. Both were already part of Indian calculation before the system reached the Middle East. In their textbooks Indians explicitly stated that zero multiplied by a number is zero. Further, the zero can be literally “nothing” and enter into calculations. The Arabic word for zero is sifr, which is also the word for “nothing.” Its associated verb ṣafira means “to be empty, be devoid, vacant.” Arabic arithmetic books introduce the zero as a sign designating an empty place in the writing of numbers. But in working out a multiplication like 203 times 8, one needs to multiply the 0 by the 8 at one point. The Moroccan mathematician al-Hawari explains it this way in his Essential Commentary on the Condensed [Book] on the Operations of Arithmetic (1305): The multiplication of a number by zero or zero by a number are the same. It amounts to zeroing (voiding) the number, or duplicating zero. Neither of these gives a number, so its symbol is always zero.” (Al-Hawārī 2013, p. 114). So one can duplicate “nothing” to get “nothing” and still regard this “nothing” (zero) as an empty space. It is interesting to note that if we imagined the zero sitting inside a hexagon, the ratio of the diameter of the circle to the side of the hexagon would equal the golden ratio (1001 Inventions 2012, p. 86). Our compiler should be brought back to earth! Geometrical properties of the circle have nothing to do with the numeral for zero, which in the Islamic East is written as a dot! The claim is wrong even if we replace “zero” with “circle.” The ratio as described is really 3 , and there is no way to get the golden ratio (1 +2 5 ) EXCAVATING THE ERRORS IN THE “MATHEMATICS” CHAPTER OF 1001 INVENTIONS 165 The ratio of AC to AB is the golden ratio. from any of the other lines in the regular hexagon. The compiler meant to write “regular pentagon” in place of “hexagon”, and the golden ratio is the ratio of the diameter of the pentagon to its side, where this diameter is the line between two non-adjacent vertices. It does not involve the circle at all! Skipping over a couple equally off-topic ramblings we come back to more relevant statements: Arabic numerals came into Europe from three sources. First, through Gerbert (Pope Sylvester I) in the later tenth century, who studied in Córdoba and returned to Rome. Then through Robert of Chester in the 12th century, who translated the second book of AlKhwarizmi’s, which contained the second ghubari (Arabic numerals). This route of Arabic numerals into Europe is mentioned by contemporary historian Karl Menninger in Number Words and Number Symbols (1001 Inventions 2012, p. 87). Menninger wrote popular accounts of mathematics, and though his book is quite good, it is somewhat outdated and it does contain some errors here and there. The earliest known example of Indian numerals in Europe is a Latin manuscript of Isidore of Seville’s Etymologies copied in the Monastery of Albelda in Spain in 976. The figures also appear in another Spanish manuscript from 992. The “Gerbert abacus,” promoted by the future pope just before the year 1000, involved an adaptation of the nine figures to the abacus. Instead of having, say, four counters in the units column, one would have a single counter inscribed with the numeral “4.” Gerbert had studied mathematics in the late 960s in Catalonia, and it is there that he probably learned of the Indian numerals. We know that Robert of Chester translated al-Khwarizmi’s algebra book into Latin, but we know nothing about the identity of the person who translated his book on Indian (Arabic) numerals. The phrase “second ghubari” is not in Menninger and makes no sense. 166 JEFFREY A. OAKS The third route was through Fibonacci in the 13th century, who inherited and delivered them to the population of Europe. Fibonacci learned of these methods when he was sent by his father to the city of Bougie, Algeria, to learn mathematics from a teacher called Sidi Omar, who taught the mathematics of the schools of Baghdad and Mosul, which included algebraic and simultaneous equations (1001 Inventions 2012, p. 87). Fibonacci’s mathematics does derive from Arabic sources, but we have no information on the identities of any of his teachers. Everything 1001 Inventions says about this “Sidi Omar” (“Mr. Omar”) should be regarded as spurious. Not surprisingly, there are a few web sites that mention this fictitious teacher. The last part, about “algebraic and simultaneous equations,” is likewise rubbish. Fibonacci also visited the libraries of Alexandria, Cairo, and Damascus, after which he wrote his book Liber Abaci (1001 Inventions 2012, p. 87). We know that Fibonacci travelled to Egypt and Syria, but there is nothing in the sources that says he visited any libraries. Most of the mathematics he learned circulated among merchants, and it is through them that he probably learned it. We do know two algebra books that Fibonacci relied on for his Liber Abaci. One is a Latin translation of al-Khwarizmi’s book, and the other is Abu Kamil’s book, which was also available in Latin. Latin translations would not have been found in Egyptian or Syrian libraries, so Fibonacci must have consulted them at home in Pisa. There is a fourth path by which Arabic numerals arrived in the west which may have been even more influential than the other three. Italian schools for teaching the arithmetic of Arabic numerals to future merchants began to appear in the late thirteenth century. Many of the textbooks written by the teachers of these schools survive in manuscript. At the time Menninger was writing it was thought that these books were abridgements and epitomes of Fibonacci’s works, but we now know that these teachers learned Arabic numerals through some other connection with the Islamic world. Fibonacci’s texts did have some influence on this Italian tradition, but they were not its main source. The compiler now returns to copying from O&R to finish the discussion of Arabic numerals. It was this system of calculating with Arabic numerals that allowed most of the advances in numerical methods by Muslim mathematicians. Now the extraction of roots became possible by mathematicians such as Abu al-Wafa’ and Umar al-Khayyam (1001 Inventions 2012, p. 87). To clarify, the rules for the extraction of square and cube roots date back to the Indian origin of the numeration system, and they were applied in both the decimal and the sexagesimal systems. O&R are referring here to higher roots, which we encounter in al-Samawʾal and also in the work of al-Kashi, an astronomer and mathematician who worked in the early 1400’s in the central Asian city of Samarqand. Abu l-Wafaʾ and al-Khayyam are known to have written earlier books EXCAVATING THE ERRORS IN THE “MATHEMATICS” CHAPTER OF 1001 INVENTIONS 167 on root extraction, but these books are now lost. As we shall see below, extraction of higher roots was also practiced in both the decimal and sexagesimal systems. The discovery of the binomial theorem for integer exponents by Al-Karaji was a major factor in the development of numerical analysis based on the decimal system (1001 Inventions 2012, p. 87). Rashed provides the details for this in chapter 2 of his book. In a lost work quoted by al-Samawʾal, al-Karaji presented what we now know as Pascal’s triangle showing the coefficients for the expansion of expressions of the form (a + b) n. This was the basis for al-Samawʾal’s application of what is now known as the Ruffini-Horner method to approximate the nth roots of numbers. For example, he calculated the fifth root of a fractional number expressed in base 60 which, if written in our base ten system, is given to an accuracy of eighteen significant places. Recall that Sharaf al-Din al-Tusi used the same method to solve polynomial equations at about the same time. In the 14th century, Al-Kashi contributed to the development of decimal fractions, not only for approximating algebraic numbers, but also for real numbers such as pi (1001 Inventions 2012, p. 87). As noted above, al-Kashi worked in the beginning of the 15th century, not in the 14th century. A number is written with a “decimal fraction” when the fractional part is shown in base ten (like 29.75) instead of as a common fraction (like 29 34 ). Although al-Kashi was well acquainted with decimal fractions, he worked out the two calculations hinted at above in base 60. The misnamed term “algebraic numbers” can only refer to al-Kashi’s approximation of the sine of one degree. This is just a real number found by means of algebra. Al-Kashi arrived at his approximation by setting up a cubic equation and then solving it numerically the same way Sharaf Din al-Tusi had solved his equations. Al-Kashi also found a sexagesimal approximation to pi which, if written in decimal form, would be accurate to sixteen places. He decided ahead of time to make his approximation precise enough so that if the circumference of the known universe were calculated from its diameter, the result would not be off by more than the width of a hair. That accuracy was not surpassed until the Dutch mathematician Ludolf van Ceulen took it to twenty places in 1596. His contribution to decimal fractions is so major that for many years he was considered their inventor (1001 Inventions 2012, p. 87). Al-Kashi was once considered their inventor because historians had not yet discovered that earlier texts also describe decimal fractions. Currently the earliest known text is al-Uqlidisi’s mid-tenth century arithmetic book Chapters on Indian Reckoning. 168 JEFFREY A. OAKS Although not the first to do so, Al-Kashi gave an algorithm for calculating “nth roots” that is a particular example of methods developed many centuries later by Ruffini and Horner, 19th-century mathematicians from Italy and England respectively (1001 Inventions 2012, p. 87). As mentioned above, al-Samawʾal did this two and a half centuries earlier, though it is true al-Kashi did remarkable work. Although Arab mathematicians are most well known for their work on algebra, number theory, and number systems, they also made considerable contributions to geometry, trigonometry, and mathematical astronomy (1001 Inventions 2012, p. 87). Arabic geometry, trigonometry, and astronomy are just as well known as Arabic numerical mathematics. O&R show their bias toward the latter not just in this passage, but by devoting only about a fifth of their essay at the end to the other topics. Our compiler follows along by naming chapter 3.6 “Mathematics” rather than “Numerical mathematics”, which would have been more appropriate. Geometry, trigonometry, and astronomy, which had always been considered to be part of mathematics, are covered in separate chapters in 1001 Inventions. The illustrations and quotations Three illustrations break up the text of the chapter. The first shows a Soviet postage stamp with an imaginary portrait of al-Khwarizmi. The second shows an upclose pencil calculation with the caption “The algebra studied today in school has its basis in Al-Khwârizmî’s book Algebr wal Muqabala.” That should be “al-jabr wa’lmuqabala,” and the calculation they show looks more like computer code than algebra. The third illustration is reproduced below. On the left is a table of Babylonian cuneiform numerals which takes up way too much room given its marginal association with Arabic mathematics. On the top right are three versions of Arabic numerals which all clearly show that the number of angles has nothing to do with the forms of the numerals. Below that is a modern illustration of numerals 0 to 9 in which the angles are forced onto the forms. No one in history has ever written the loop of the “9” as a hexagon of six angles with three more angles on the lower part! The other numerals are just as ridiculous. The first of two large-font quotations that break up the text is al-Samawʾal’s description of algebra. That is fine. But the second is a quotation on the importance of mathematics by the thirteenth-century English philosopher Roger Bacon. Surely a quotation by another medieval Islamicate mathematician would have been more appropriate. EXCAVATING THE ERRORS IN THE “MATHEMATICS” CHAPTER OF 1001 INVENTIONS 169 The third illustration, from page 87. The caption says of the right side “The progression of Arabic numerals from the tenth to the fourteenth centuries shows how the Muslims devised modern numerals—the numbers 1 to 9 we use today—based on the use of angles” (1001 Inventions 2012, p. 87). Sources At the very end of the book, in the section “Acknowledgements,” the editor of 1001 Inventions writes “Much of the material for this book is based on peerreviewed papers, articles, and presentations published in our academic portal, www.MuslimHeritage.com. Chief among these are written by the following scholars, arranged in alphabetical order.” Two names are of interest here: “Dr. Mahbub Gani (Mathematics and Numbers)” and “Professor Mustafa Mawaldi (Mathematics).” The articles by these two are much better than the 1001 Inventions chapter. It is clear from reading them that neither Gani (sometimes spelled “Ghani”) nor Mawaldi is responsible for the chapter, and that none of the articles was consulted by the compiler. Also in the acknowledgements the editor writes “A full list of references used in this book and in the development of the 1001 Inventions exhibition can be found at http://www.1001Inventions.com/references.” Thirty-five works are listed there under “Mathematics, Trigonometry and Geometry.” The one book that the Mathematics chapter cites explicitly, Menninger’s Number Words and Number Symbols, is not in the list, nor is O&R’s essay. In fact, the only work in the list that was consulted, though indirectly, is Rashed’s The Development of Arabic Mathematics: Between Arithmetic and Algebra. 170 JEFFREY A. OAKS Concluding remarks Let’s recap what went wrong in the writing of chapter 3.6. For algebra and number theory the problems begin with Rashed’s book, which contains many misinterpretations. These were paraphrased by O&R, who introduced further errors and misunderstandings. O&R’s passages were then copied with minor changes in wording into the 1001 Inventions chapter. For arithmetic the compiler of the chapter copied from different sources. One source is again O&R, who had misrepresented information from Saidan’s book. I have not found the sources for other passages, such as the outlandish explanation of the shapes of the numerals or the fabricated background to Fibonacci. My guess is that they were taken from some popular account or even the stories that circulate orally among people. Fortunately the resulting muddled account does not adequately explain the mathematics, nor does it provide much historical context. Chapter 3.6 stands little chance, then, of seriously misinforming readers. They won’t even know what the more distorted claims mean! But they will learn that medieval Muslims studied mathematics, and that they seem to have done remarkable work. So rather than being outraged by the mass of misinformation, I remain attracted by the transformations in the trail from one text to the next. Bibliography 1001 Inventions: Muslim Heritage in Our World, chief editor, Salim T. S. Al-Hassani; co-editors, Elizabeth Woodcock & Rabah Saoud; Manchester, UK: Foundation for Science, Technology and Civilisation 32012. Cajori, Florian. A History of Mathematical Notations. 2 vols. Chicago: Open Court 1928–29. [reprint New York: Dover 1993]. Ibn al-Hāʾim. Sharḥ al-Urjūza al-Yāsamīnīya, de Ibn al-Hāʾim. Texte établi et commenté par Mahdi Abdeljaouad. Tunis: Publication de l’Association Tunisienne des Sciences Mathématiques 2003. Al-Ḥaṣṣār, 1194. Kitāb al-bayān wa’l-tadhkār fī ṣanʿat ʿamal al-ghubār (Book of Demonstration and Recollection in the Art of Dust-Board Reckoning). Lawrence J. Schoenberg collection, MS ljs 293, copied in Ṣafar 590 (Jan/Feb 1194 CE) (at: http:// dewey.library.upenn.edu/sceti/ljs/). Al-Hawārī al-Miṣrātī, ʿAbd al-ʿAzīz b. ʿAlī. Al-Lubāb fī sharḥ Talkhīṣ aʿmāl al-ḥisāb (The Essential Commentary on Ibn al-Bannāʾ’s Condensed Book on the Operations of Arithmetic). Abdeljaouad, Mahdi – Oaks, Jeffrey. Tunis: Association Tunisienne de Didactique des Mathématiques 2013. Menninger, Karl. Number Words and Number Symbols. Cambridge, MA: MIT Press 1969. [English translation of the 1958 book Zahlwort und Ziffer]. EXCAVATING THE ERRORS IN THE “MATHEMATICS” CHAPTER OF 1001 INVENTIONS 171 O’Connor, J[ohn] J[oseph] – Robertson, E[dmund] F[rederick]. “Arabic mathematics: forgotten brilliance?” 1999 (at: http://www-history.mcs.st-and.ac.uk/Hist Topics/Arabic_mathematics.html). Pacioli, Luca. Summa de Arithmetica Geometria Proportioni & Proportionalita, Continentia de tutta lopera. Venetijs: Paganino de Paganini 1494 (reedition Toscolano: Paganino de Paganini 1523). Rashed, Roshdi. Entre Arithmétique et Algèbre. Recherches sur l’Histoire des Mathématiques Arabes. Paris: Les Belles Lettres 1984. Rashed, Roshdi. The Development of Arabic Mathematics: Between Arithmetic and Algebra. Dordrecht: Kluwer 1994. Saidan, A[hmad] S[alim]. The Arithmetic of Al-Uqlīdisī. Dordrecht: D. Reidel 1978. Saidan, A[hmad] S[alīm]. Tārīkh ʿilm al-jabr fī l-ʿālam al-ʿArabī (History of Algebra in Medieval Islam). 2 vols. Kuwait: Al-Majlis al-Waṭanī li-l-Thaqāfah wa’l-Funūn wa’l-Ādāb, Qism al-Turāth al-ʿArabī 1986.