cubics of the type x3+ cx = d. Seeking fame, Fiore chal-
lenged Tartaglia to a debate and presented him with 30
questions of a type that he was sure Tartaglia would be
unable to answer. In the early hours of February 13,
1535, the day of the debate, inspiration came to
Tartaglia, and he discovered a general method that
solved both types of equations. Tartaglia won the con-
test with ease.
News of Tartaglia’s victory reached Cardano, who,
with his assistant L
UDOVICO
F
ERRARI
(1522–65), was
working on the solution to the
QUARTIC EQUATION
. This
work required knowledge of the cubic solution, and so
Cardano sought from Tartaglia details of his methods.
After much beseeching, Tartaglia eventually revealed his
solution to Cardano, under the promise that the meth-
ods remain secret. After discovering that another
scholar, Scipione del Ferro, had devised identical meth-
ods decades earlier, Cardano broke that promise and
published the solution, along with the solution to the
quartic, in his text 1545 text Ars magna (The great art).
A bitter dispute between Tartaglia, Cardano, and Fer-
rari ensued. Today, to honor the achievement of both
men, the formula for the solution to the cubic is called
the Cardano-Tartaglia formula.
Tartaglia died in Venice, Italy, on December 13,
1557.
tautochrone See
CYCLOID
.
tautology In
FORMAL LOGIC
, a compound statement
that cannot possibly be false by virtue of its structure is
called a tautology. For example, “If all the planets are
made of cheese, then Mars is made of cheese” is true
regardless of the validity of the component statements
that “all planets are made of cheese” and “Mars is
made of cheese.” Tautologies are true purely because of
the laws of logic and not because of any known facts
about the world. They are therefore statements that
contain only definitional information.
See also
LAWS OF THOUGHT
;
TRUTH TABLE
.
Taylor, Brook (1685–1731) British Calculus Born
on August 18, 1685, in Edmonton, England, mathe-
matician Brook Taylor is remembered today for his
important contributions to the development of
CALCU
-
LUS
. In his 1715 text Methodus incrementorum directa
et inversa (Direct and indirect methods of incrementa-
tion), Taylor formulated methods for expanding func-
tions as infinite series. These are today known as
T
AYLOR SERIES
. Taylor also invented the technique of
INTEGRATION BY PARTS
.
Taylor graduated from St. John’s College, Cam-
bridge, in 1709 with an advanced degree in mathemat-
ics, having already written his first mathematical paper
1 year earlier. Despite his young age Taylor quickly
developed a reputation as an expert in the field of cal-
culus. He was elected to the prestigious R
OYAL
S
OCI
-
ETY
in 1712 and was immediately appointed to a
special committee to adjudicate on the issue of whether
it was S
IR
I
SAAC
N
EWTON
(1642–1727) or G
OTTFRIED
W
ILHELM
L
EIBNIZ
(1646–1716) who had discovered
the
FUNDAMENTAL THEOREM OF CALCULUS
that unites
differentiation with integration.
Despite the attachment of his name to the tech-
nique, Taylor was not the first to develop a theory of
infinite function expansions. Several decades earlier
J
AMES
G
REGORY
(1638–75), Johann Bernoulli
(1667–1748) of the famous B
ERNOULLI FAMILY
, A
BRA
-
HAM
D
E
M
OIVRE
(1667–1754), and others had inde-
pendently discovered variants of Taylor’s expansion
theorem. (Taylor was unaware of this body of work
when he wrote his famous 1715 text.) The significance
of infinite series expansions, however, was not properly
recognized until 40 years after Taylor’s death when, in
1772, influential French mathematician J
OSEPH
-L
OUIS
L
AGRANGE
(1736–1813) proclaimed it an important
basic principle of differential calculus.
As a broad-based scholar, Taylor also wrote on top-
ics in experimental and theoretical physics. He pub-
lished articles on magnetism, thermometers, vibrating
strings, capillary action, and the mathematical laws dic-
tating the motion of the planets. He invented alternative
methods for computing logarithms and approximating
solutions to algebraic equations. In 1715 he also pub-
lished Linear Perspective, an influential text outlining
the mathematical foundations of
PROJECTIVE GEOMETRY
and the role of the vanishing point in art. He died in
London, England, on December 29, 1731.
Taylor series Many functions, such as trigonometric
functions and exponential and logarithmic functions,
are difficult to manipulate, whereas adding, subtracting,
494 tautochrone