16TH CENTURY MATHEMATICS - TARTAGLIA,
CARDANO & FERRARI
In the Renaissance Italy of the early 16th Century, Bologna University
in particular was famed for its intense public mathematics
competitions. It was in just such a competition, in 1535, that the
unlikely figure of the young Venetian Tartaglia first revealed a
mathematical finding hitherto considered impossible, and which had
stumped the best mathematicians of China, India and the Islamic
world.
Niccolò Fontana became known as Tartaglia (meaning “the
stammerer”) for a speech defect he suffered due to an injury he
received in a battle against the invading French army. He was a poor
engineer known for designing fortifications, a surveyor of topography
(seeking the best means of defence or offence in battles) and a
bookkeeper in the Republic of Venice.
But he was also a self-taught, but wildly ambitious, mathematician. He
distinguised himself by producing, among other things, the first Italian
translations of works by Archimedes and Euclid from uncorrupted
Greek texts (for two centuries, Euclid's "Elements" had been taught
from two Latin translations taken from an Arabic source, parts of which
contained errors making them all but unusable), as well as an
acclaimed compilation of mathematics of his own.
Niccolò Fontana Tartaglia (14991557)
Tartaglia's greates legacy to
mathematical history, though,
occurred when he won the 1535
Bologna University mathematics
competition by demonstrating a
general algebraic formula for solving
cubic equations (equations with
terms including x3), something which
had come to be seen by this time as
an impossibility, requiring as it does
an understanding of the square
roots of negative numbers. In the
competition, he beat Scipione del
Ferro (or at least del Ferro's
assistant, Fior), who had
coincidentally produced his own
partial solution to the cubic equation
problem not long before. Although
del Ferro's solution perhaps
predated Tartaglia’s, it was much
more limited, and Tartaglia is
usually credited with the first general
solution. In the highly competitive
and cut-throat environment of 16th
Century Italy, Tartaglia even
encoded his solution in the form of a
poem in an attempt to make it more
difficult for other mathematicians to
steal it.
Tartaglia’s definitive method was,
however, leaked to Gerolamo
Cubic equations were first solved algebraically by del Ferro and Tartaglia
Cardano (or Cardan), a rather
eccentric and confrontational
mathematician, doctor and
Renaissance man, and author
throughout his lifetime of some 131 books. Cardano published it himself in his 1545 book "Ars Magna"
(despite having promised Tartaglia that he would not), along with the work of his own brilliant student
Lodovico Ferrari. Ferrari, on seeing Tartaglia's cubic solution, had realized that he could use a similar
method to solve quartic equations (equations with terms including x4).
In this work, Tartaglia, Cardano and Ferrari between them demonstrated the first uses of what are now
known as complex numbers, combinations of real and imaginary numbers of the type a + bi, where i is the
imaginary unit √-1. It fell to another Bologna resident, Rafael Bombelli, to explain, at the end of the
1560's, exactly what imaginary numbers really were and how they could be used.
Although both of the younger men were acknowledged in the foreword
of Cardano's book, as well as in several places within its body,
Tartgalia engaged Cardano in a decade-long fight over the
publication. Cardano argued that, when he happened to see (some
years after the 1535 competition) Scipione del Ferro's unpublished
independent cubic equation solution, which was dated before
Tartaglia's, he decided that his promise to Tartaglia could legitimately
be broken, and he included Tartaglia's solution in his next publication,
along with Ferrari's quartic solution.
Ferrari eventually came to understand cubic and quartic equations
much better than Tartaglia. When Ferrari challenged Tartaglia to
another public debate, Tartaglia initially accepted, but then (perhaps
wisely) decided not to show up, and Ferrari won by default. Tartaglia
was thoroughly discredited and became effectively unemployable.
Poor Tartaglia died penniless and unknown, despite having produced
(in addition to his cubic equation solution) the first translation of
Gerolamo Cardano (1501-1576)
Euclid’s “Elements” in a modern European language, formulated
Tartaglia's Formula for the volume of a tetrahedron, devised a method
to obtain binomial coefficients called Tartaglia's Triangle (an earlier
version of Pascal's Triangle), and become the first to apply
mathematics to the investigation of the paths of cannonballs (work which was later validated by Galileo's
studies on falling bodies). Even today, the solution to cubic equations is usually known as Cardano’s
Formula and not Tartgalia’s.
Ferrari, on the other hand, obtained a prestigious teaching post while still in his teens after Cardano
resigned from it and recommended him, and was eventually able to retired young and quite rich, despite
having started out as Cardano’s servant.
Cardano himself, an accomplished gambler and chess player, wrote a book called "Liber de ludo aleae"
("Book on Games of Chance") when he was just 25 years old, which contains perhaps the first systematic
treatment of probability (as well as a section on effective cheating methods). The ancient Greeks,
Romans and Indians had all been inveterate gamblers, but none of them had ever attempted to
understand randomness as being governed by mathematical laws.
The book described the - now
obvious, but then revolutionary insight that, if a random event has
several equally likely outcomes, the
chance of any individual outcome is
equal to the proportion of that
outcome to all possible outcomes.
The book was far ahead of its time,
though, and it remained unpublished
until 1663, nearly a century after his
death. It was the only serious work
on probability until Pascal's work in
the 17th Century.
Cardano was also the first to
describe hypocycloids, the pointed
plane curves generated by the trace
of a fixed point on a small circle that
rolls within a larger circle, and the
generating circles were later named
Cardano (or Cardanic) circles.
The colourful Cardano remained
notoriously short of money
thoughout his life, largely due to his
gambling habits, and was accused
of heresy in 1570 after publishing a
horoscope of Jesus (apparently, his
own son contributed to the
prosecution, bribed by Tartaglia).
The circles used to generate hypocycloids are known as Cardano Circles