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:

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