rari learned that another scholar S
CIPIONE DEL
F
ERRO
(1465–1526) had also developed methods of solving
certain types of cubic equations. Although essentially
identical to the work of Tartaglia, Ferrari and Cardano
decided to publish the solution to the quartic, attribut-
ing the work on the cubic needed to del Ferro, with
whom no promise of secrecy had been made.
Tartaglia was outraged, and a bitter dispute that
lasted for many years ensued between Tartaglia and Fer-
rari. On August 10, 1548, as was common at the time,
Tartaglia challenged Ferrari to an open contest and pub-
lic debate as an attempt to demonstrate that he was in
fact the expert on cubic equations. But it was clear from
the contest that Ferrari had a more complete under-
standing of both cubic and quartic equations. Tartaglia
left before the contest was over, and victory was given
to Ferrari. He immediately garnered national fame and
was given many offers of employment, including a
request from the emperor himself to act as royal tutor.
Ferrari, however, accepted no position offered at the
time, left mathematics, and accepted a lucrative position
as tax assessor to the governor of Milan.
Ferrari died in Bolgna, Italy, in October 1565 (the
exact date is not known) and is remembered in mathe-
matics solely for his work on the quartic equation.
Ferro, Scipione del (Ferreo, dal Ferro) (1465–1526)
Italian Algebra Born on February 6, 1465, in Bolo-
gna, Italy, Scipione del Ferro is remembered as the first
mathematician to solve the
CUBIC EQUATION
. Unfortu-
nately none of his writings survive today, and we learn
of his work chiefly through the manuscripts of G
IRO
-
LAMO
C
ARDANO
(1501–76) and L
UDOVICO
F
ERRARI
(1522–65).
Del Ferro was appointed lecturer in arithmetic and
geometry at the University of Bologna in 1496, a posi-
tion he retained for all his life. Little is known of his
academic work. Letters to other scholars at the time
suggest that del Ferro studied methods for rationalizing
rational expressions, ruler-and-compass constructions
in geometry, and methods for solving cubic equations.
Mathematicians of del Ferro’s time were familiar
with the general solution to a
QUADRATIC
equation of
the form ax2+ bx + c= 0. (It should be mentioned,
however, that 16th-century scholars did not use zero as
a number in an expression, nor permitted the use of
negative numbers. Thus the equation x2– 3x+ 2 = 0,
for instance, was written x2+ 2 = 3x.) Mathematicians
also knew that, with the appropriate use of substitu-
tion, any cubic equation could be reduced to one of
two forms: x3+ ax = bor x3= ax + b. (Here, again, a
and bare positive.) Del Ferro was the first mathemati-
cian to solve equations of the first type. Some histori-
ans suggest that he was able to solve equations of the
second type as well.
Del Ferro recorded all his results in a personal
notebook, which he bequeathed to his son-in-law Han-
nibal Nave, also a mathematician. Nave later shared
the contents of the notebook with Cardano and Fer-
rari. After seeing the method of solving the cubic fully
explained, Cardano and Ferrari realized that del Ferro
had in fact solved the famous cubic equation some 30
years before N
ICCOLÒ
T
ARTAGLIA
(ca. 1499–1557),
their contemporary, had claimed to do the same. In
1545 Cardano published Ars magna (The great art),
outlining Ferrari’s solution to the
QUARTIC EQUATION
making use of del Ferro’s methods for the cubic.
Del Ferro died in Bologna, Italy, some time between
October 29 and November 16, 1526. He is remembered
in mathematics solely for his work on cubic equations.
Fibonacci (Leonardo Fibonacci, Leonardo of Pisa)
(ca. 1170–1250) Italian Arithmetic, Number theory
Born in Pisa, Italy (the exact birth date is not known),
mathematician Leonardo of Pisa, better known by his
nickname Fibonacci, is best remembered for his help
in introducing the H
INDU
-A
RABIC NUMERAL
system to
the merchants and scholars of Europe. He strongly
advocated the system in his famous 1202 text Liber
abaci (The book of counting), a treatise on the tech-
niques and practices of arithmetic and algebra, which
proved to be extremely influential. This work also
contained a large collection of arithmetical problems,
including one that leads to the famous sequence of
numbers that bears his name. Considered the most
important mathematician of the middle ages,
Fibonacci also wrote extensively on the topics of
E
UCLIDEAN GEOMETRY
and D
IOPHANTINE EQUATION
s.
He is recognized as the first scholar in the West to
make advances in the field of
NUMBER THEORY
since
the time of D
IOPHANTUS OF
A
LEXANDRIA
.
Although born in northern Italy, Fibonacci was
raised and educated in northern Africa, where his
father, a merchant and a government representative,
Fibonacci 191