The Italian connection
The theory of cubic equations was fully developed during the Renaissance. Unfortunately it
resulted in an episode when mathematics was not always on its best behaviour. Scipione Del Ferro
found the solution to the various specialized forms of the cubic equation and, hearing of it, Niccolò
Fontana – dubbed ‘Tartaglia’ or ‘the stammerer’ – a teacher from Venice, published his own results
on algebra but kept his methods secret. Girolamo Cardano from Milan persuaded Tartaglia to tell
him of his methods but was sworn to secrecy. The method leaked out and a feud between the two
developed when Tartaglia discovered his work had been published in Cardano’s 1545 book Ars
Magna.
Origins
Algebra was a significant element in the work of Islamic scholars in the ninth
century. Al-Khwarizmi wrote a mathematical textbook which contained the Arabic
word al-jabr. Dealing with practical problems in terms of linear and quadratic
equations, al-Khwarizmi’s ‘science of equations’ gave us the word ‘algebra’. Still
later Omar Khayyam is famed for writing the Rubaiyat and the immortal lines (in
translation)
A Jug of Wine, a Loaf of Bread – and Thou Beside me singing in the
Wilderness
but in 1070, aged 22, he wrote a book on algebra in which he investigated the
solution of cubic equations.
Girolamo Cardano’s great work on mathematics, published in 1545, was a
watershed in the theory of equations for it contained a wealth of results on the
cubic equation and the quartic equation – those involving a term of the kind x^4.
This flurry of research showed that the quadratic, cubic and quartic equations
could all be solved by formulae involving only the operations +, –, ×, ÷, (the
last operation means the qth root). For example, the quadratic equation ax^2 + bx
- c = 0 can be solved using the formula:
If you want to solve the equation x^2 – 3x + 2 = 0 all you do is feed the values
a = 1, b = −3 and c = 2 into the formula.
The formulae for solving the cubic and quartic equations are long and
unwieldy but they certainly exist. What puzzled mathematicians was that they