Cardano

Mathematic may be a boring subject, but often it is so just because some teachers don’t know the thrilling history hiding behind the numbers, or they don’t know how to tell to their students. Take for instance the claim made by the Greek mathematician Proclus (410 -485) that politicians and rich landlords were cheating common people, not familiar with geometry and math, by convincing them that plots of land with a longer perimeter translates into a larger area. Just think of a flat 100 feet long and 10 feet wide. It has a longer perimeter than a flat of, say, 50 feet long and 40 feet wide. In fact the first has a perimeter of 220 feet, the second 180 feet but the surface of the first will be 1.000 square feet, while the second will be 2000.

In prehistory the most advanced mathematicians were the Sumerians. There are surviving baked clay tablets, some dating four thousand years before Christ, which are mainly concerned about “real estate” problems. In fact they were not yet able to handle square roots, for that we have to get to the Egyptian, the Chinese and the Greeks, but they knew how to handle simple quadratic linear equations.

There is a famous book by Muhammad Ibn Musa, the Kithab al-jabr wa al-muqabalah, where the second word shows the origin of the term algebra. The Arabs saved some of the old inventions for the modern world, transmitting them down to the mathematicians of the Renaissance.

The Tuscan algebrists of the fourteen century were convinced that the solution of cubic and quadratic equations was a mission impossible. In fact Luca Pacioli (1445-1517) a friend of Leonardo Da Vinci, in his popular treatise Summa de Arithmetica concluded that there was no possible solution. In spite of such conclusion the brightest mathematical minds of the time struggled to find it. The first to crack it was a mathematician from Bologna, Scipione Dal Ferro (1465-1526). He did not publish his findings but disclosed them to some of his pupils. One of them leaked it out, after his death, to Tartaglia (the stammerer in Italian) from Brescia (1500 – 1557). He boasted aloud about it, claiming to be the inventor. Here enter another odd character: Girolamo Cardano (1501 – 1576?) a physician, astrologer, historian and many more things. His autobiography had been translated in many languages and it is still a classic in his genre. He tried in vain to get the magic formula out of Tartaglia, but without success. He then tried flattery, inviting him to Milan, paying for all his expenses. Tartaglia took the bait and in 1539, perhaps due to generous offerings of fine wine and the introduction to some powerful political figures, he unveiled the secret to Cardano using a poem. He then demanded him not to publish and keep the secret. Few years later Cardan discovered that the real inventor had been Dal Ferro not Tartaglia. Then, together with his gifted student, Ludovico Ferrari (1522-1565) who will die prematurely murdered by his own sister, in 1545 Cardan published in Nuremberg a book which constitutes a milestone in modern algebra, its title is the Ars Magna. Besides the cubic equations also the solution of the quadratic equations is elegantly given there, but it is probably due to Ferrari. He mentioned Tartaglia as the man who disclosed the formula to him but despite that the answer of Tartaglia was red hot rage. A series of public letters ensued, called cartelli full of insults and with a final public showdown in Milan. The European scientific community was holding its breath, waiting for the verdict. Ferrari made mincemeat of poor Tartaglia, who had to back off. After that even the solution of quintic equations seemed close at hand, and all the most brilliant mathematical minds set to work on it. But for three centuries they did not get results.In fact the equations in X3 and X4 could be resolved by a system of scaling down them by one degree but when the same system was applied on the X5, things went wrong.

The first to have understood that there was no standard solution possible was Lagrange (1736 – 1813) from Turin, but was Paolo Ruffini (1765-1822) from Modena, who spent all his life on the quintic and finally proved, in a rather convolute book which went unnoticed, that a solution of X5 by using only additions, subtractions, multiplications, divisions and extractions of roots, will never be found. To solve them an elliptic approach was needed, and it is still used today.

The final words on such millennial problem are those of two whiz-kinds. Norwegian Niels Abel (1802-1829) and Frenchman Evariste Galois (1811-1832). Abel died in obscurity, with his diary which went lost but then resurfaced in Florence in 1952. Abel introduced the concept of reduction ad absurdum in algebra, that is, instead of looking for proofs on the solvability it is better to concentrate on the contradictions. To atone with such neglect, in 2002, the Norwegian government created the Abel prize for mathematic, a sort of Nobel.  Also Evariste Galois was a tragic figure: short tempered, a school dropout for having sneered at his math professor, a revolutionary who was shot mysteriously in a dark alley of Paris for unknown reasons. He tackled the problem by using symmetry and creating, out of nowhere, a new branch of algebra, known today as Galois Theory he remains to this day one of the most beautiful minds ever existed.