


History of Mathematics from Medieval Islam to Renaissance Europe
Org: Rob Bradley (Adelphi) and Glen van Brummelen (Bennington College) [PDF]
 CHRISTOPHER BALTUS, SUNY Oswego
When is a Negative Really a Negative?
[PDF] 
Well into the 17th century, works in algebra commonly gave the rules
for arithmetic involving negatives but did not allow negative
solutions to equations. This paradoxical situation signals a less
than full acceptance of negative numbers. I believe that the
arithmetic rules, especially for multiplication, were intended for
polynomial multiplication involving subtracted terms. I will speak of
long term developments in this line of thought, from Brahmagupta and
Islamic writers up to John Wallis.
 LAWRENCE A. D'ANTONIO, Ramapo College, New Jersey
Number Theory from Fibonacci to 17th Century Safavid Persia:
a question of transmission of knowledge
[PDF] 
How much influence did Islamic mathematics have on Renaissance Europe
and vice versa? This possible transmission of knowledge is an
interesting and important topic for the historian. In this paper the
question of transmission is examined with regard to selected problems
in number theory, in particular the problem of congruent numbers. A
congruent number k is an integer for which there exists a square
such that the sum and difference of that square with k are themselves
squares.
Congruent numbers can first be found in various works of classical
Islamic mathematics, for example, in alKaraji's early 11th century
text, the alFakhri. Congruent numbers then resurface in the
treatise Liber Quadratorum of Fibonacci. We then find
congruent numbers in the influential 17th century work,
Khulasat alHisab of Baha alDin.
Was the work of Fibonacci known in the Islamic world? This is not
easy to determine, since there is no direct reference to Fibonacci in
Islamic sources. On the other hand, Edouard Lucas, in a major essay
on Fibonacci, shows the existence of an intellectual thread, if not a
clear historical thread, connecting Fibonacci and Baha alDin.
To examine the problem of transmission, it is necessary to look at the
cultural context for mathematics during the Safavid dynasty of 17th
century Persia. The Safavid period represents, perhaps, the last
flowering of classical Islamic science. Under the reign of the
Safavid ruler Shah Abbas I, 15881629, a cultural renaissance
occurred in the capital city of Isfahan. Especially important are
Safavid accomplishments in the areas of mathematics, astronomy,
scientific instrument making, carpet weaving, medicine, and
architecture. Safavid mathematics is represented primarily through
the work of Baha alDin and Mohammad Baqir Yazdi (whose major work,
the Uyun alHisab, also includes some interesting results in number
theory).
It is wellknown that many different Europeans spent time in the court
of Shah Abbas. Adventurers, travelers, and missionaries were
attracted by this center of learning. This paper examines possible
sources of transmission. For example, it is known that the 17th
century Italian traveler, Pietro Della Valle, did discuss current
trends in astronomy with Persian scientists.
The discussion of the work of these Safavid scholars will hopefully
contribute to a more complete picture of classical Islamic
mathematics.
 JOZSEF HADARITS, Royal Ontario Museum
Diamonds, Rings, and Squares: Eastern Magic in Western Hands
[PDF] 
In medieval Islamic mathematics there were two basic methods for
constructing oddorder magic squares: the socalled "diamond"
technique and another, more sophisticated one that can be understood
in terms of a virtual torus. The West produced the first detailed
description of the latter technique during the Renaissance period.
Using historical evidence, including that of art, this paper makes an
attempt to trace some of the possible routes of this intercultural
scientific receptionin order to get closer to the understanding of
the cosmologicalspiritual background of these centuriesold
mathematical problems.
 ODILE KOUTEYNIKOFF, IREM, Université Paris VII Denis Diderot
Guillaume Gosselin, an algebraist in Renaissance France
[PDF] 
Guillaume Gosselin de Caen's treatise, known as De Arte Magna
(Paris, 1577), is a short and quite simple work written by someone who
is a typical algebraist in Renaissance France. Gosselin learned
mathematics and heard about new methods in algebra from mathematicians
who worked just before him; after making these new methods his own, he
wanted them to be taught and wrote them down. He is especially good
at solving problems with several unknown quantities and several linear
equations.
It is important to notice that Gosselin's book is very dependant both
on Italian Tartaglia's Arithmetic (Venise, 1556), which
Gosselin translated into French by the same time he wrote De
Arte Magna, and on Diophante's Arithmetics, which came to be
known exactly two years before, thanks to Xylander's translation into
Latin (Bâle, 1575). Both Gosselin and Tartaglia refer to Pacioli's
work (Summa, Venise, 1494) and Pacioli himself says he
learned much from Fibonacci, especially through Liber
Quadratorum (Pise, 1225).
According to the fact that AlKhwarizmi founded Algebra during the 9th
century, it is not surprising that, when being translated into Arabic
in the late 9th century by Lebanese Ibn Luqa whose native language was
Greek, Diophante's Arithmetics seemed to be considered as a
treatise about Algebra since algebraic vocabulary and way of thinking
were most widely shared. Only few people understood that it was
actually an arithmetic treatise: AlKhazin (900971) did, and
therefore he is one of those who laid the foundations for the integer
Diophantine analysis. We know that Jean de Palerme submitted
AlKhazin's problem about congruent numbers to Fibonacci, who then
wrote Liber Quadratorum.
These are the main ways that lead from Diophante, as both a Greek and
an Arabic source, to Renaissance Europe readers. We will show from
his text how eager to learn and respectful of what he learnt Gosselin
was, and how enthusiastic about the new algebraic methods he was too.
He wished he could retranslate and explain the complete Diophante's
Arithmetics, but he didn't. We ignore what kind of work he
would have done, either an exact arithmetic treatise as Bachet (Paris,
1621) and Fermat (16011665) did, or an uptodate algebraic one
according to what all algebraists did, such as Bombelli
(Algebra, Bologne, 1572), Stevin (Arithmetic, Leyde,
1585), Viete (l'Art Analytique, Tours, 15911593) or Girard
(L'invention nouvelle en Algèbre, Amsterdam, 1629).
 FEDERICA LA NAVE, Harvard/Dibner Institue
Bombelli and L'Algebra
[PDF] 
In L'Algebra, Bombelli was the first to recognize what we
call "imaginary numbers" as numbers, and to give them operative
definitions. L'Algebra was published in three books in 1572.
However, Bombelli wrote the first version in five books in 1550. In
the 1550 manuscript version Bombelli does not believe that the roots
born in solving cubic equations in the irreducible case are numbers.
In the published version he believes they are numbers and gives rules
for operating with them. The aim of this paper is to try to
understand what happened in these twentytwo years to Bombelli's state
of beliefhow his beliefs changed and what caused that change.
 GLEN VAN BRUMMELEN, Bennington Collge
AlSamaw'al and the Errors of the Astronomers: Where the
Mistake Really Lies
[PDF] 
Ibn Yahya alMaghribi alSamaw'al, a 12thcentury converted Jew most
known for his contributions to an arithmetical revolution in algebra,
also wrote an intriguing but rarely studied book entitled
Exposure of the Errors of the Astronomers. In it he takes
shots at many of his predecessors, as far back as Ptolemy, for choices
that they had made in their astronomical methods. Some of his
criticisms seem odd, almost off the wall to a modern reader, but
perhaps there are lessons here for an understanding of the medieval
scientific mind. We shall explore some of his criticisms and attempt
to put into historical context the rationality behind his criticisms.

