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16TH CENTURY MATHEMATICS - TARTAGLIA, CARDANO & FERRARI
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Niccolò Fontana Tartaglia (1499-1557)
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In the Renaissance Italy of the early 16th Century, Bologna University in particular was famed for its intense public mathematics competitions. It was in just such a competition, in 1535, that the unlikely figure of the young Venetian Tartaglia first revealed a mathematical findinghitherto considered impossible, and which had stumped the best mathematicians of China, India and the Islamic world.
Niccolò Fontana became known as Tartaglia (meaning “the stammerer”) for a speech defect he suffered due to an injury he received in a battle against the invading French army. He was a poor engineer known for designing fortifications, a surveyor of topography (seeking the best means ofdefence or offence in battles) and a bookkeeper in the Republic of Venice.
But he was also a self-taught, but wildly ambitious, mathematician. He distinguised himself by producing, among other things, the first Italian translations of works by Archimedes and Euclid from uncorrupted Greek texts (for two centuries, Euclid's "Elements" had been taught from two Latin translations taken from an Arabicsource, parts of which contained errors making them all but unusable), as well as an acclaimed compilation of mathematics of his own.
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Cubic equations were first solved algebraically by del Ferro and Tartaglia
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Tartaglia's greates legacy to mathematical history, though, occurred when he won the 1535 Bologna University mathematics competition by demonstrating a general algebraicformula for solving cubic equations (equations with terms including x3), something which had come to be seen by this time as an impossibility, requiring as it does an understanding of the square roots of negative numbers. In the competition, he beat Scipione del Ferro (or at least del Ferro's assistant, Fior), who had coincidentally produced his own partial solution to the cubic equation problem notlong before. Although del Ferro's solution perhaps predated Tartaglia’s, it was much more limited, and Tartaglia is usually credited with the first general solution. In the highly competitive and cut-throat environment of 16th Century Italy, Tartaglia even encoded his solution in the form of a poem in an attempt to make it more difficult for other mathematicians to steal it.
Tartaglia’s definitivemethod was, however, leaked to Gerolamo Cardano (or Cardan), a rather eccentric and confrontational mathematician, doctor and Renaissance man, and author throughout his lifetime of some 131 books. Cardano published it himself in his 1545 book "Ars Magna" (despite having promised Tartaglia that he would not), along with the work of his own brilliant student Lodovico Ferrari. Ferrari, on seeingTartaglia's cubic solution, had realized that he could use a similar method to solve quartic equations (equations with terms including x4).
In this work, Tartaglia, Cardano and Ferrari between them demonstrated the first uses of what are now known as complex numbers, combinations of real and imaginary numbers of the type a + bi, where i is the imaginary unit √-1. It fell to another Bologna resident,Rafael Bombelli, to explain, at the end of the 1560's, exactly what imaginary numbers really were and how they could be used.
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Gerolamo Cardano (1501-1576)
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Although both of the younger men were acknowledged in the foreword of Cardano's book, as well as in several places within its body, Tartgalia engaged Cardano in a decade-long fight over the publication. Cardano argued that, when hehappened to see (some years after the 1535 competition) Scipione del Ferro's unpublished independent cubic equation solution, which was dated before Tartaglia's, he decided that his promise to Tartaglia could legitimately be broken, and he included Tartaglia's solution in his next publication, along with Ferrari's quartic solution.
Ferrari eventually came to understand cubic and quartic...