Girolamo Cardano

Bastard, heretic, gambler, occultist, astrologer, alchemist, father of a murderer, plagiarist, committer of suicide, great
mathematician and inventor, Cardano was a man of genius.

Girolamo Cardano was born in Pavia, Italy in 1501, the illegitimate son of a lawyer from Milano. He is also known by the
variants Gerolamo and Geronimo, and often by the anglicised version of his surname, Cardan.

Cardano trained and practised as a doctor, but had a passion for mathematics which he studied in his spare time. He
travelled widely, even going as far as Scotland to tend to the Archbishop of St, Andrews who was afflicted with asthma.
He is credited by some as the first to describe typhoid fever.

He was undoubtedly one of the most gifted mathematicians of the sixteenth century and held the chairs of mathematics
at Pavia, Bologna and Milano.

Cardano has been portrayed by history as a wicked man. It is rumoured that he was so angry with one of his sons one
day that he cut off the boy’s ears. His other son was executed for murdering his wife. He also had an interest in
astrology. When he published a horoscope of the life of Christ, he was imprisoned for heresy. But later he was
appointed as an astrologer at the papal court. He used astrology to predict the date of his own death (21st September
1576) and, it is said, committed suicide so that his prediction would come true. He also wrote a book on gambling,
de ludo aleae
(written in1526, but published posthumously), which led to the development of the elementary rules of
probability. Cardano’s breaking of his promise to Tartaglia not to publish the solution of the general cubic equation is
further evidence that Cardano was not of good character.

Besides mathematics, Cardano wrote on a number of other subjects, including medicine, astronomy, physics, alchemy,
music, and he published two encyclopaedias of natural science. These include details of a number of Cardano’s
inventions. He even wrote an autobiography,
De vita propria, in 1576. His most important scientific work was his De
subtilitate rerum, a philosophical treatise on Aristotelian physics, but including some new discoveries from his own time.

In 1539 he published a work on arithmetic entitled
Practica arithmetice et mensurandi singularis. One topic discussed in
this work was the rationalisation of the denominator of an expression containing cube roots.

But it was Cardano’s writing on algebra which distinguished him as one of the foremost mathematicians of his time. His
greatest work was the treatise
Artis magnae, sive de regulis algebraicis, usually referred to as Ars Magna, and
published in 1545. This work was of such importance that the year 1545 is often considered to be the beginning of the
modern era of mathematics.
Ars Magna not only contains the controversial method for solving the cubic equation, but
also a solution of the general quartic (degree 4) equation which was first discovered by his assistant Ludovico Ferrari
(1522-1565), and an approximate method for obtaining the roots of an equation of any degree. Cardano allows for
irrational and negative roots of an equation and even hints at the possibility of imaginary roots and the existence of
complex numbers.

The most enduring contribution of this work to mathematics was not the mechanical solution of cubic and quartic
equations. These methods were cumbersome and did little to supersede approximate methods of solution that were
already well known. No, the work was important for another reason - it led to the study of polynomials of any order and
in particular a search for the solution of the general quintic (degree 5) equation. It was to be another 279 years before
Niels Henrik Abel showed that such a solution does not exist.

A contribution of Cardano to mathematics

One contribution of Cardano to mathematics that is fairly easy to understand is known today as Descartes Rule of
Signs, incorrectly attributed to René Descartes (1596-1650).
Cardano used syncopated algebra and was unable to express the rule concisely. Throughout this discussion, I will use
modern notation

Cardano was well aware that a polynomial of degree n has n roots. This may in some cases include repeated roots, but
the number is always n.

Take, for example, a polynomial like f(x) = 5x - 7x^3 - 2x^2 - 4x^5 + 1
Since the highest power of x is x^5, this has degree 5, and therefore f(x) = 0 has 5 roots. But how many of them are
positive and how many are negative? The Rule of Signs helps answer this question.

First rewrite the polynomial in correct order of terms from highest to lowest:
f(x) =  -4x^5  - 7x^3 - 2x^2 + 5x + 1

Positive Roots
Count how many times the sign changes in the polynomial.
The signs are - - - + and +
So the sign changes from - to +. That’s 1 change of sign.
The number of times the sign changes tells us the maximum number of positive roots of f(x) = 0
The equation -4x^5 - 7x^3 - 2x^2 + 5x + 1 = 0, therefore, has a maximum of 1 positive root.

Negative Roots
Replace x by -x in the polynomial:
-4(-x)^5 - 7(-x)^3 - 2(-x)^2 + 5(-x) + 1
= 4x^5 + 7x^3 - 2x^2- 5x + 1
Count how many times the sign changes in the polynomial.
The signs are + + - - and +
So the sign changes from + to - to +. That’s 2 changes of sign.
The number of times the sign changes tells us the maximum number of negative roots of f(x) = 0
The equation -4x^5 - 7x^3 + 2x^2 + 5x + 1 = 0, therefore, has a maximum of 2 negative roots.

It seems that the problem is solved - there is 1 positive root and 2 negative roots. But there should be 5 roots
altogether. What about the other 2 roots? Cardano recognised that the other two roots are complex (although he did
not use that name - he dismissed them as ‘sophistic’), and he recognised that complex roots always occur in pairs. This
was a mark of his genius.

The Rule of Signs, therefore, tells us there are two possibilities:
The equation -4x^5 - 7x^3 - 2x^2 + 5x + 1 = 0 could have 1 positive root, 2 negative roots and 2 complex roots, or it could
have  1 positive root, 0 negative roots and 4 complex roots. In fact the first possibility is true in this case.