## History of complex numbers: cubic equations formula

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- Published: Saturday, 27 February 2016 16:19

Pacioli’s authority led to the problem of solving equations of higher degrees being omitted for a long time, as if it did not exist. The first daredevil who broke this taboo was a professor of mathematics at the University of the Italian city of Bologna Scipio del Ferro (1445 - 1526). He found a way of solving cubic equations of the form:

x^{3}+ax=b (1)

As negative numbers were not used at the time, equations

x^{3}=ax+b (2)

x^{3}+b=ax (3)

were significantly different both from equation (1) and from each other. Therefore, to fully resolve the problem of solving cubic equations, scientists needed to find separate ways of solving each of these three types of equations. Ferro announced his discovery to his, as it turned out, much less talented student Mario Antonio Fiore, who, after the death of his teacher, decided to benefit from the secret entrusted to him, obtaining a remuneration (significant at the time) for winning the then popular intellectual battles that replaced the medieval jousts. At the end of 1534 Fiore challenged a mathematician from Venice Piccolo Tartaglia (c. 1500-1557).

Tartaglia came from a poor family, lost his father early and was able to attend school for only two weeks because of poverty. Moreover, a throat wound received during the attack of the French made him a lifetime stutterer. Tartaglia (literally "stuttering") was the scholar’s byname; his real name – Fontano – is now little known. However, his phenomenal aptitude and titanic work allowed Tartaglia, deprived by fate, not only to become a professor of mathematics at the university, but also to excel in science.

Upon accepting Fiore’s challenge in the form of 30 problems of the same type, which was to solve equations of the form (1) at different specific values of a and b, Tartaglia decided to just expose his opponent, accusing him that he himself could not solve the problems he had proposed . The reason for this was the Luca Pacioli’s authority. When the appointed period (of 50 days) was running out, Tartaglia heard rumours that Fiore did have a mysterious way for solving cubic equations. Therefore, unhappy with the prospect of holding a grand dinner for the winner’s friends in a number equal to the number of unsolved problems (such were the rules), Tartaglia concentrated all his efforts on the problem and in the remaining week, found a way not only for solving equations of the form (1), but of the form (2). Soon all the solutions to all the proposed problems were transferred to the notary, while his contestant failed to solve any of the problems offered by Tartaglia.

Tartaglia’s remarkable victory gained publicity. Later, rumours about the mysterious method for solving cubic equations reached the ears of arguably the most amazing scientist of the epoch Gerolamo Cardano (1501-1576).

Cardano’s main activity was medicine. He was considered to be one of the most prominent doctors in Europe (probably second only to his famous friend Vesalius) and enjoyed royal patronage. In his declining years Cardano wrote the autobiographical book "On my life," in which he described his medical pursuits. And in his spare time he practiced "anything and everything" - philosophy, physics, mechanics, astrology (in particular, he composed horoscopes for the living and the dead - Jesus Christ, Petrarch, Durer, Luther, the Pope, Vesalius, and of course, himself), as well as writing and publishing books (including encyclopaedic), and finally - mathematics.

In 1538 Cardano completed his mathematical work, "Practice of General Arithmetic", which, according to his plan, was to replace Luca Pacioli’s "Sums". However, hearing about Tartaglia’s secret sparked Cardano’s desire to decorate his book with it. In January 1539 he asked Tartaglia for permission for the first time, but was denied it. Tartaglia said that when he wants to publish his discovery, he will do it in his own work (unfortunately, Tartaglia’s "General Treatise on Number and Measure", which contained his original mathematical results was published only after his death). Still, Tartaglia eventually yielded to Cardano’s request and, upon swearing him to secrecy unless granted his (Tartaglia’s) permission, told him the final result without the proof. According to this arrangement, Cardano’s "Practice of General Arithmetic" was published without the formula for the roots of cubic equations.

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