Medieval Mathematics:
Two Figures from the Later Middle Ages
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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