Medieval Mathematics:
Two Figures from the Later Middle Ages

Teresa Kuo

The history of mathematics, like that of any science, has had periods of rapid devel-opment and growth balanced by periods of less vigorous development. In Europe, the period lasting from approximately the early 500s A.D. to the mid 1400s is collectively referred to as the Middle Ages, and more popularly as the Dark Ages. Nonetheless, as we will argue, this period was not devoid of scientific and specifically mathematical devel-opments. Translations of ancient Greek mathematical texts enjoyed a certain popularity, and this, along with mathematical knowledge transmitted by the Arabs, influenced medieval scholars of mathematics in the later Middle Ages. Figures like Jordanus de Nemore (referred to as either Jordanus or Nemorarius), who can be tentativel y dated to the early to mid 1200s, made vital mathematical contributions, and were familiar with both ancient Greek and contemporary Arabic material. The one great mathematician of this period is Leonardo da Pisa (or Fibonacci), who also flourished in t he early thirteenth century. A critique of the mathematical prowess of the Middle Ages can come through an analysis of their respective work, and a comparison between this brilliant, if anomalous, merchant-algebraist, and his approximate contemporary Jor danus. Both Jordanus and Fibonacci rely on ancient Greek, and Arabic (and hence, indirectly, also Indian) sources in their mathematical contributions, and the achievements of such mathematicians might not have been possible without the prior intellectual "bloss -oming" of the Middle Ages, that has its roots in the diffusion of Arabic knowledge across much of Europe, knowledge which itself began to pass to the Arabs in the mid seventh century, with the conquest of Alexandria in 641 by the followers of Mohammed [1, p. 249]; by the late eighth century, a number of astronomical, mathematical, and astrological works were translated into Arabic, starting with the Sindhand in 766, an Indian astronomical-mathematical work thought to be either the Brahmasputa Siddhanta o f the Hindu mathematician Brahmagupta [1, p. 241], or the Surya Siddhanta, a Sanskrit astronomical work dated around 400 AD [1, p. 231]. The Hindu-Arabic numerals currently used today also originated with the Indians somewhat earlier, and they reached Ar abia with this translation, to be passed on to the Europeans, who were far less receptive to this system, only from the 10th century onwards. One of the first matheamticians credited with teaching Hindu-Arabic numerals in Europe is Gerbert (ca. 940 - 10 03), later to become Pope under the name Sylvester II, who may have come into contact with the system in his travels to Moslem Spain [1, p. 275]. The earliest translations in the Arab world were followed by original compositions by notable Arabic mathematicians, such as Mohammed ibn-Musa al-Khwarizmi (or simply al-Khwarizmi), who aside from his astronomical work published two books on arithmetic an d algebra [1, p. 251]. One of these, the only surviving copy of which is the Latin translation De numero indorum (or, "Concerning the Hindu Art of Reckoning"), was based, apparently, on an Arabic translation of Brahmagupta's work (ibid). Al-Khwarizmi's book concerns use of Hindu numerals, and was translated into Latin in the twelfth century by Adelard of Bath, Robert of Chester, and John of Seville, from the latter of which the term "algorism" is derived [4, p. I182], by a misunderstanding of the role of the author al-Khwarizmi, to whom the Hindu system of numeration was mistakenly attributed [1, p. 251]. Other works translated by these scholars included al-Khwarizmi's Kitab (or Kitab al-jabr mutabilia [sic]) known also under its Latinized name as the Liber algebre [3, p. 10]. However, this period of the Middle Ages was still unprepared for a deeper study of mathematics. By the rise of universities in the later Middle Ages, after or around the thirteenth century or so, the work of both the Greeks and the Arabs yielded a consid erable influence on the scientific academia. By then, the use of Arabic terms to describe mathematical processes had increased. According to Pearl Kibre's 1984 publication Studies in Medieval Science, arithmetic at Paris taught from early texts of both the Greeks and the Arabs, translated into Latin [4, p. I181]. Most likely, these included al-Khwarizmi's De numero indorum, which by then had been translated for over a century [4, p. I182]. While major institutions of study held interest in the ancient scientific arts, the extent to which they were utilized is not clear. Roger Bacon, who was associated with the university in Paris, believed the status of mathematical studies at the universi ties to be inadequate. Kibre says, "Mathematical studies, he pointed out, had been neglected for about thirty of forty years, and this neglect had practically destroyed Latin studies in the sciences. He noted specifically that students knew only three o r four propositions of Euclid at most" [4, p. I179]. However, other data suggests that this does not seem to be the case, and that the value and importance of arithmetic was valued, even above other "classic" forms of math, such as geometry [4, p. I181]. One of the mathematicians cited by Roger Bacon was Jordanus de Nemore (or Nemorarius), who had published work on the Hindu-Arabic numerals and their operation [4, pp. I182-3]. Jordanus de Nemore, an astronomer who published on a number of topics including mathematics and mechanics, is believed to be a somewhat well known figure in his time. This partly comes from Roger Bacon's reference, as well as the survival of several of h is works dated from the 13th century. While biographical information on the life of Jordanus is sketchy, he is thought to have worked somewhere between the late twelfth century and early thirteenth century, possibly in France near Paris. Information for the former point comes through the evidence that Barnabas Hughes, in his 1981 account states, "De numeris datis presumes that its readers were familiar with elementary algebra, and this knowledge did not break upon Europe before Robert of Chester's trans lation of al-Khwarizmi's Liber algebre, in 1145" [3, pp. 1-2]. Sketchy evidence also links Jordanus to an individual, Jordanus de Saxonia, a Dominican who also worked in the early thirteenth century as well as to the University of Toulouse, where he may have given a series of lectures [3, p. 2]. Evidence for these points remain unclear and can be rejected without further support [3, p. 2]. Another possibility also links Jordanus to the vicinity of Paris. "De Nemore" may possibly refer to the town of N emours (as it is currently known), which was referred to alternatively as Nemus, Nemorosium, or Nemosum in Latin [2]. Nevertheless, again, without further evidence any suggestion is just speculation. Just as biographical details remain uncertain, his published work, while known, is similarly difficult to define. What is known is that Nemorarius published six works of mathematics discussing topics that range from arithmetic and algebra to mechanics. His six works of mathematics includes the Demonstratio de algorismo, which details the Arabic number system and its use of integers. Other treatises include the Demonstratio de minutiis, which covered fractions, and the Liber phylotegni de triangulis, wh ich highlighted geometric proofs [4, p. I182]. His work De numeris datis is one of the first advanced works of algebra published in Western Europe during the medieval period [3, p. 1]. As Hughes explains, the work included numerous mathematical developm ents including quadratic and proportional equations. Boyer describes this work as mainly "a collection of algebraic rules for finding, from a given number, other numbers related to it according to certain conditions, or for showing that a number satisfyi ng specific restrictions is determined" [1, p. 284]. Evidence suggests that Jordanus was familiar with al-Khwarizmi and used translations of his work as sources for De numeris datis. For example, De datis has the three forms of the quadratic equation from al-Khwarizmi Liber algebre (as it was known in Lat in), in the same order [3, p. 11]. Specifically, these are: "a square and roots equal to numbers", "a square and numbers equal to roots", and "roots and numbers equal to squares", which are presented in IV-8, IV-9, and IV-10 [ibid]. Another source for J ordanus in De datis are works by Greek mathematician Euclid, notably the Elements and Data [3, p. 13]. Three propositions in De datis are originally found in the Elements; however, this also suggests that this work was not used as a major source [3, p. 1 4]. Data, too, may not have been major source, besides similarities in presentation (fifteen definitions followed with ninety-four propositions); there is not much in the content of De datis that hearkens Data [ibid]. Even if there is evidence against the major use of Euclid in De datis, Jordanus' method of numerical representation is more similar to the Greeks than the Indians and Arabs. Nemorarius first used this method in his Arithmetica, whose popularity is reflec ted in the number commentaries at the University of Paris in the late 1500s. This method differs from both the Greek line segments and the use of letter-diagrams employed by al-Khwarizmi. This makes possible the algebraic expression of geometric proposi tions, which before could not have been expressed by the Arab mathematicians [1, p. 284]. Jordanus's method of representing unknowns shows more influence from the Greeks, and of Euclid. For example, the rule for determining one part of a given number, w hen divided into two, when the other is given is explained by Nemorarius thus:

Let the given number be abc and let it be divided into two parts ab and c, and let d be the given product of the parts ab and c. Let the square of ab be e and let four times d be f, and let g be the result of taking f from e. Then g is the square of the difference between ab and c. Let h be the square root of g. Then h is the difference between ab and c. Since h is known, c and ab are determined
[1, p. 284]. Nemorarius, unlike Euclid however, did not state that the variables were to be regarded as being line segments, but this was inferred. A different medieval mathematician credited with using Hindu-Arabic numerals was Leonardo de Pisa, a contemporary of Ne morarius. Leonardo de Pisa, (ca. 1180-1250), was born in Pisa (now part of Italy) and was the son of Guglielmo Bonaccio (from "Fibonacci", or "son of Bonaccio", is derived). This, and most information concerning the life of Fibonacci comes from an autobiographical passage in the beginning of the one of his works, the Liber Abbaci [3, p. xvi]. Fibonacci first came into contact with Arabic mathematics when his father worked in a customs office in Bugia, located in Northern Africa. Here a Muslim teacher taught him arithmetic and Arabic. Later, partly due to business matters and partly in pursuit of manuscripts, he traveled extensively, from Egypt to Syria and Greece, among other places [7, p. 482]. As Boyer suggests, "It therefore was natural that Fibonacci ... [ was] in Arabic algebraic methods, including, fortunately, the Hindu-Arabic numerals and, unfortunately, the rhetorical form of expression" [1, p. 280]. The rhetorical form of expression Boyer cites is different from current mathematical conventions. Fib onacci, like many mathematicians, expressed mathematical concepts and problems linguistically. Some examples of a rhetorical usage can be found in the Liber Abbaci, first published, as believed, in 1202. For example, in one problem, he states, "To one of two unequal quantities of which the one is thrice the other, I add its root. Similarly with t he other quantity. Ad I multiply the two sums together and the result is ten times the larger quantity" [3, p. 13]. In the Liber Abbaci, Fibonacci also used letters to represent numbers previously represented with line segments. Sometimes Fibonacci exc ludes the last part, as in one example where he states, "Let a, b, g, d be four numbers in proportion, namely, a is to b as g is to d. Then, conversely, b is to a as d is to g" [ibid]. This notation is similar to the one used by Jordanus, stated earlier . However, Fibonacci is also noteworthy for a number of other contributions. The Liber Abbaci is noted for covering nine Hindu-Arabic numerals (9, 8, 7, 6, 5, 4, 3, 2, 1) as well as the sign 0. It then uses this numeration to explain arithmetic (multiplication, addition, subtraction, and division), operations with fractions, calc ulation of prices and financial matters, square and cubic roots, algebra, and geometry [7, p. 484]. This book, and its title does not, however, concern the "abacus" based method of calculation of earlier mathematicians. Sigler describes, In the title Liber abbaci, abbaci has the more general meaning of mathematics and calculation or applied mathematics rather than merely of the counting machine made from stringing beads on wires. The mathematicians of Tuscany following Leonardo were call ed Maestri d'Abbaco; for more than three centuries there were masters and students trained in mathematics and calculation based on the principles established in Liber abbaci [6, p. xvii]. The medieval version of the abacus (the version used in earlier times was called a "monastic abacus"), or counting board, that this passage is referring to is a tool in which numerals take on different values based on their position in each column [7, p. 473]. This tool was used as a method of calculation, and their use is covered by a number of thirteenth century manuscripts as well as by earlier scholars, including Gerbert [4, p. I182]. Early medieval counting boards can be illustrated from Gerbert's account. Karl Menninger describes, "His monastic abacus had parallel columns, 27 of them in the case of Gerbert (3 for fractions), which were sometimes closed off at the top by an "arch." This was called the arcus Pythagoreai, the "arch of Pythagoras" be cause in the Middle Ages this Greek was erroneously believed to be the inventor of the abacus" [5, p. 323]. Fibonacci, however, rejected both the monastic abacus and the arch of Pythagoras in favor of Indian computations [5, p. 426]. Indeed, eventually use of the abacus ebbed, in favor of Indian and Arabic based methods, which were found to be easier than the style previously used [7 p. 474]. Other work by Fibonacci was conducted in the field of quadratics. A response to a challenge by Johannes of Palermo was posed in his Liber quadratorum (the "Book of Squares"), which solved the problem of finding a square number that will remain such wheth er increased or decreased by five. This problem is strikingly similar to the type of problems associated with Greek mathematician Diophantus [1, p. 282]. He solved the problem by assuming the answer (to x2 ( N) can't be square unless congruent to the ty pe N = ab (a+b)(a-b), in which a and b are prime to each other and their sum is even (Taton 484). The solution to the problem, 41/12 is also given in the book. Squaring the solution and increasing and decreasing the number by five yields the solutions ( 49/12)2 and (41/12)2 [7, p. 484]. The following formula (current terminology):
(a2 + b2)(c2+d2) = (ac + bd)2 + (bc - ad)2 = (ad + bc)2 + (ac - bd)2
was used frequently in the Liber quadratorum, and is one commonly used by Indian mathematicians [1, p. 283]. These types of equations are known today as Lagrange identities, after the eighteenth century French mathematician [6, p. 28]. Furthermore, Greek mathematician Diophantos also used the formulas implicitly and an example of the formula's application is given in Problem 19, Book III of his Arithmetica [6, p. 28]. He writes: 65 = (13)(5) = (32 + 22)(22 + 12) = 82 + 12 = 72 + 42 [cited in 6, p. 28] The formula was also mentioned by Islamic mathematicians, and was stated by al-Khazin around 950 AD, along with a discussion on its use by Diophantos. More complicated formulas in this vein were also conducted by Indian mathematician Brahmagupta approxim ately half a millennium earlier [8, p. 17]. Concrete knowledge on what Arabic sources were available to Leonardo, however, is uncertain [6, p. 28]. Other work that appeared in the Quadratorum involves methods for finding Pythagorean triples. Many of these problems involve the use of line segments, as demonstrated in Proposition 2, where he demonstrates, as quoted in Sigler, "any square number exceed s the square immediately before it by the sum of the roots" [cited in 6, p. 9]. In current terminology, this results with the formula (n+1)2 - n2 = (2n+1) [6, p. 11]. Fibonacci also demonstrates, as quoted in Sigler, "that any square exceeds any smaller squ are by the product of the difference of the roots by the sum of the roots" [qtd. in 6, p. 11]. This argument is also found in Book II of Euclid's Elements [ibid]. Further along these lines, Fibonacci gives a proof for the difference of consecutive squares that equal the sum of its roots. In the introduction, he deduces, I thought about the origin of all square numbers and discovered that they arise out of the increasing sequence of odd numbers; for the unity is a square and from it is made the first square, namely 1; to this unity is added 3, making the second square, na mely 4, with root 2; if to the sum is added the third odd number, namely 5, the third square is created, namely 9, with root 3; and thus sums of consecutive odd numbers and a sequence of squares always arise together in order [qtd. in 6, p. 4]. The proof for this notion, dated to the ancient Greek Pythagoreans, is given in Proposition 11, where it is proven by adding a list of equations to produce the sum of squares [6, p. 46]. While both Fibonacci and Jordanus are recognized today, it is uncertain, if not doubtful that their contributions were particularly noted in their day. While seven copies of De Numeris datis remain from the thirteenth century, in addition to other copies and revisions from following centuries [4, p. 8]. As for Fibonacci, some suggest that since lived before the advent of the printing press, copies of his manuscripts were limited to the Italy [6, p. xv]. However, is it possible to connect either Fibonac ci or Nemorarius to other scholars and universities, which would indicate knowledge of their work? Pearl Kibre states, In 1202 there had appeared the Liber Abaci by Leonardo Fibonacci di Pisa, incorporating the Hindu numerals. However it is not possible to connect Leonardo Fibonacci di Pisa with the northern universities. On the other hand, there is a strong likelihood that the works of another author, Jordanus Nemorarius, on the Hindu numerals were being utilized for instruction in the universities of both Paris and Oxford in the later thirteenth century. The survival of a number of thirteenth century manuscripts of h is works at Paris strongly supports this possibility [4, p. I182]. In other words, while both mathematicians incorporated Hindu numerals into their work, it is not possible to connect Fibonacci's work and influence on the Northern universities such as Oxford and Paris. As mentioned earlier, distribution of Fibonacci's L iber Abbaci was limited to Italy during the thirteenth century. Nemorarius, however, who can be identified with Paris, is more likely to have had a greater influence on that university [4, p. I182]. According to Hughes, however, despite a greater "audience" available for Nemorarius, evidence of the use of De numeris datis does not seem to beget a greater popularity per se. Use of Greek geometric methods (despite the value placed on arithmetic at the Northern universities) and practical application of knowledge was de rigueur, as Nemorarius himself relates, "Since the science of weights is subalternate both to geometry and to natural philosophy, certain things in this science need to be proved in a p hilosophical manner, certain things in this science need to be proved in a geometric manner" [3, p. 8]. That is, while De Datis may have reached a larger audience than Liber Abbaci at the time, the climate of the day excluded the possibility of more wide spread use and study of analytic algebra [4, p. 9]. De datis may have also used works by Euclid, al-Khwarizmi, and Fibonacci as sources. Evidence from comparisons between De numeris datis and translated works of al-Khwarizmi suggests, in the least, a familiarity on Nemorarius's part. For example, three f orms of the quadratic equation appear in De datis appear in the order used by al-Khwarizmi in his Kitab, or Liber algebre [4, p. 11]. Other evidence also may suggest a similarity between De datis and Fibonacci's Liber Abbaci, such as similar problems sha red by the two. The possibility remains, however, that such similarities were mutually derived by a common source, such as another Arabic work, the Kitab of abu Kamil, a successor of al-Khwarizmi, which may have influenced both Fibonacci and Nemorarius [ 4, p. 12]. However, it may also be possible Fibonacci learned of this problem sometime during the course of his education and travels. Furthermore, as Hughes points out, it is unlikely Nemorarius was influenced much by the Liber Abbaci, as copies of the book, written by a contemporary, were limited to Italy [4, p. 12]. It is also unlikely Nemorarius would have ignored other aspects of t he Liber Abbaci, such as its use of equations, had he had access to it. On the other hand, it may also be possible that it was from the Liber Abbaci that Nemorarius was inspired the to use letter variables to represent the Greek line segments, both of wh ich represent numbers. A difference between this use of letters to represent numbers: while Fibonacci mostly used one unknown in the Liber Abbaci, Nemorarius used several [4, pp. 12-13]. While the view that the Middle Ages had little important mathematical development may hold true in comparison with other periods or cultures, among them the Hellenistic age and the medieval Hindu or Islamic achievements, as well as later periods in Wester n Europe, still some notable work was carried out. Individuals such as Leonardo da Pisa and Jordanus de Nemore were stimulated by contact with Islamic mathematicians (who were influenced by Indian mathematicians) and an interest in the work of the ancien t Greeks. Thus original work such as Fibonacci's Liber Abbaci and the De numeris datis from Jordanus was published. These contributions, by helping introduce and explain the use of Hindu-Arabic numerals and computation, and incorporating Greek geometry (as in the line segments), in a context in which they were little known, helped lead to a later mathematical flourishing - a mathematical renaissance - in Europe.


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