Summary
Cubics in the math history
The
earliest found information about computing cubic roots and solution of
cubic
equations is found among the Babylonians (about 2000 - 400 BC).
Hindu
mathematicians took the Babylonian methods further so that Brahmagupta
(598-665 AD) gives an, almost modern, method which admits negative quantities.
Numerical values of cubic roots were computed by Aryabhata (476 -c. 550
BC)
In
about 300 BC Euclid developed a geometrical approach
The
solution of numerical higher equations for approximate values of roots has
been
known for a long time in China. It has
been called the most characteristic Chinese
mathematical contribution. The essentials
of the method are there around
c. 100 BC - 50 CE. By using the method for finding the cube root of a number
they
were able to solve a cubic equation of the form x3 + ax2
+ bx = c, where a, b
and c are positive.
The
Arabs did not know about the advances the Hindus had made so they had neither
negative quantities nor abbreviations for their unknowns. However al'Khwarizmi
(c 800) gave a classification of different types of equations.
al'Khwarizmi
gives the rule for solving each type of equation, essentially the
familiar
quadratic formula given for a numerical example in each case, and then a proof
for
each example utilizing a geometrical method for completing the square.
Abraham
bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for
his book Liber embadorum published in 1145 which is the first book
published in
Europe to give the complete solution of the quadratic equation.
Omar Khayyam believed
this could be solved only geometrically, by using conic
sections. His method, used in Algebra (c 1079) is based on a
geometrical
construction where the solution is found at the intersection between a
parabola
and a semicircle.
He
solves the equation: x3 + Bx = C using this
method.
Maestro
Dardi of Pisa in a 1344 work extended a list of equations to 198
types of
equations of degree up to four, some involving radicals. He gave an example of
how
to solve a particular cubic equation, but the methods would not generalize.
Scipione
dal Ferro (1465-1526) held the Chair of Arithmetic and Geometry at the
University of Bologna and certainly must have met Piccoli who lectured at
Bologna
in 1501-2. Pacioli declared that cubic equations were not possible to solve.
dal Ferro
is credited with solving cubic equations algebraically but most probably dal
Ferro
could only solve cubic equation of the form x3+ mx = n.
.
dal
Ferro solved this cubic equation around 1515 but kept his work a complete
secret until just before his death, in 1526, when he revealed his method to
his
student Antonio Fior.
Niccolo
of Brescia, known as Tartaglia meaning
'the stammerer', managed to solve
equations of he form x3 + mx2 = n, around 1526, and made no secret of
his discovery.
Fior
challenged Tartaglia to a public contest. Only 8 days before the problems
were to be collected, Tartaglia had found the general method for all types of
cubics.
Cardan invited
Tartaglia to visit him and, after much persuasion, made him divulge the
secret of his solution of the cubic equation. This Tartaglia did, having made
Cardan
promise to keep it secret until Tartaglia had published it himself. Cardan did
not keep
his promise. In 1545 he published Ars Magna the first Latin
treatise on algebra.
When solving x3 = 15x + 4 he
obtained an expression involving 
(-121). Cardan
knew
that you couldn't take the square root of a negative number yet he also knew
that x = 4 was a solution to the equation.
After Tartaglia had shown Cardan how to solve
cubics, Cardan encouraged his own student, Lodovico Ferrari, to examine quartic
equations.
The irreducible case of the cubic, namely the
case where Cardan's formula leads to the square root of negative numbers, was
studied in detail by Rafael Bombelli in 1572 in his work Algebra
. Rafael Bombelli (1526--1573) published in his book Algebra 1572 a way
of calculating with complex numbers which made it possible to explain the case
with negative roots. His explanation is based on the knowledge of one positive
root.
In the years after Cardan's Ars Magna many
mathematicians contributed to the solution of cubic and quartic equations. Vičte, Harriot, Tschirnhaus, Euler, Bezout and Descartes all devised methods.
Tschirnhaus' methods were extended by the Swedish mathematician E S Bring near the end of the
18 C.
Thomas Harriot made
several contributions. One of the most elementary to us, yet showing a marked
improvement in understanding, was the observation that if x = b,
x = c, x = d are solutions of a cubic then the cubic is:
((x - b)(x
- c)(x - d) = 0.
Leibniz wrote a letter to Huygens in March 1673. In it
he made many contributions to the understanding of cubic equations. Perhaps the
most striking is a direct verification of the Cardan-Tartaglia formula.
This Leibniz did by reconstructing the cubic from its three roots (as given by
the formula) as Harriot claimed in general. Nobody before Leibniz seems to have
thought of this direct method of verification. It was the first true algebraic
proof of the formula, all previous proofs being geometrical in nature.
A description of the development in modern
times is to be found at http://www.astro.virginia.edu/~eww6n/math/