In the previous episode we left Tartaglia facing the dilemma of whether to resist despite the oath of card, or give in to reveal the formula.
Cardano in his oath he deftly brought up the Christian faith. Tartaglia did not feel to impose such an affront to Cardano and so reluctantly gave the formula.
addition to the discovery of the casus irriducibilis, Cardano was also the first to produce a rigorous proof of the quadratic formula . Therefore, despite the deceptions, the contribution to the overall development of Cardano's formula is by no means negligible.
In the five years after Cardano continues his work as a doctor at the same time bringing forward the draft of his Ars Magna, which will come out in 1545. With this reputation of Cardano assurgerà quickly to the algebra of Europe's most famous and experienced. Tartaglia
he procures a copy readily and quickly notice that the Ars Magna contains the solution of the equation di terzo grado, per cui viene esplicitamente citato il suo nome, che quella dell'equazione di quarto grado, la cui paternità viene attribuita a Lodovico Ferrari : "creato" e creatura di Cardano.
At this point one might ask, but once you get to fourth grade, no one would push to find solutions for the fifth and then the sixth, maybe until you come to a generalized formula the degree n? Actually they tried many. But we'll see that from fifth grade onwards, things get a bit 'more tangled. And the final answer will come only after more than two centuries with Ruffini and Abel .
What remains after hundreds of years of these events? None. Found in textbooks "formulas of Cardano." Hardly anyone mentions Tartaglia, and Scipione dal Ferro is a perfect stranger. Perhaps it would be good to start calling formulas from the Ferro-Tartaglia-Cardan: three authors for an equation of degree three.
In the next episode will talk about the introduction of other mathematical symbols and Raffaele Bombelli, which as we have said, contributed to the development of complex numbers.
Previously ...
Index Series
Cardano in his oath he deftly brought up the Christian faith. Tartaglia did not feel to impose such an affront to Cardano and so reluctantly gave the formula.
Dopo aver decifrato l'enigma in versi, dietro il quale Tartaglia in un ultimo disperato tentativo aveva celato la formula, Cardano cominciò ad impegnarsi in ulteriori ricerche personali.
Il frutto di tali ricerche non tardò ad arrivare. Cardano scopre il celebre casus irriducibilis . Quel caso cioè in cui l'equazione di terzo grado ha tutte e tre le radici reali , ma nonostante ciò, per calcolare tali radici reali è necessario passare attraverso radici quadrate di numeri negativi . Sappiamo tutti però che un numero reale elevato al quadrato dà sempre come risultato un numero positivo (meno per meno longer, no?). So what were these numbers that they gave as a result raised to a negative number? What sense could have those days when there was much reluctance to accept even negative numbers themselves? imagine the square roots of negative numbers!
But Cardano was very pragmatic. Work? Yes! And then we use them. I am ... a device useful for algebraic calculations.
Numbers " sophistical " named them. With time, however, these numbers are very useful. Although it initially seemed really algebraic artifice "that should not exist." So were first called "imaginary numbers" and then complex numbers. Cardano then, in addition to the solutions of the third degree, is also considered one of the fathers of complex numbers . We shall see later that many other mathematicians contributed to the development of these numbers. One of these was Raffaele Bombelli . But we'll deal with in the next episodes.
Il frutto di tali ricerche non tardò ad arrivare. Cardano scopre il celebre casus irriducibilis . Quel caso cioè in cui l'equazione di terzo grado ha tutte e tre le radici reali , ma nonostante ciò, per calcolare tali radici reali è necessario passare attraverso radici quadrate di numeri negativi . Sappiamo tutti però che un numero reale elevato al quadrato dà sempre come risultato un numero positivo (meno per meno longer, no?). So what were these numbers that they gave as a result raised to a negative number? What sense could have those days when there was much reluctance to accept even negative numbers themselves? imagine the square roots of negative numbers!
But Cardano was very pragmatic. Work? Yes! And then we use them. I am ... a device useful for algebraic calculations.
addition to the discovery of the casus irriducibilis, Cardano was also the first to produce a rigorous proof of the quadratic formula . Therefore, despite the deceptions, the contribution to the overall development of Cardano's formula is by no means negligible.
he procures a copy readily and quickly notice that the Ars Magna contains the solution of the equation di terzo grado, per cui viene esplicitamente citato il suo nome, che quella dell'equazione di quarto grado, la cui paternità viene attribuita a Lodovico Ferrari : "creato" e creatura di Cardano.
La rottura del giuramento fa ovviamente infuriare Tartaglia. Il quale, da come lo riporta Lodovico Ferrari, taccia Cardano di essere " ignorante nelle matematiche, poverello, uomo che tien poco sugo e poco discorso, e altre parole ingiuriose le quali per tedio lascio da parte ".
Tuttavia Cardano, nella citazione dei contributi per la scoperta delle formule, non si era limitato a citare Tartaglia e Ferrari. Egli Scipione del Ferro was also mentioned. It is this quotation to provide clues on the ground that could have led Cardano to break the oath. The fact that he had discovered that Tartaglia could not claim primacy over the paternity of the formulas. Primacy that Tartaglia had to share with Scipione del Ferro. But perhaps pushing Cardano had contributed to breaking the oath the fact that the solution of the fourth degree was obtained from the solution of the third degree. And so the publication of the first could not disregard the publication of the second .
Wanting to take the conclusions, citing Dario Bressanini , we can say that those who still are called formulas of Cardano perhaps should more properly be called formulas Del Ferro-Tartaglia-Cardan : "three authors for equation of degree three .
However you should understand you can not surely deny that these solutions are a pure product of that great intellectual movement, artistic, scientific and cultural heritage which was the Italian Renaissance.
The problem of solution of the cubic equation, which had so unsuccessfully engaged the minds of many of the great Greek mathematicians, Chinese, Indian and Muslim, had been finally resolved. What Del Ferro-Tartaglia-Cardan is probably the biggest contribution to algebra since the Babylonians had figured out how to solve quadratic equations four thousand years before.
However you should understand you can not surely deny that these solutions are a pure product of that great intellectual movement, artistic, scientific and cultural heritage which was the Italian Renaissance.
The problem of solution of the cubic equation, which had so unsuccessfully engaged the minds of many of the great Greek mathematicians, Chinese, Indian and Muslim, had been finally resolved. What Del Ferro-Tartaglia-Cardan is probably the biggest contribution to algebra since the Babylonians had figured out how to solve quadratic equations four thousand years before.
At this point one might ask, but once you get to fourth grade, no one would push to find solutions for the fifth and then the sixth, maybe until you come to a generalized formula the degree n? Actually they tried many. But we'll see that from fifth grade onwards, things get a bit 'more tangled. And the final answer will come only after more than two centuries with Ruffini and Abel .
After the Ars Magna Cardano managed to produce another important contribution to mathematics in a book written around 1560: the Liber de Ludo Alea . Book that is likely in the game and ways to cheat. Although not exactly edifying purposes, the Liber de Ludo Alea contains the first systematic treatment of probability. Publication will, however, only posthumously in 1663.
For those wishing to investigate the history of third-degree equation solving formulas point out the wonderful article: Requiem for a formula. Drama in six acts with six characters of Dario Bressanini . From whom I also stole quotes the original texts.
For further discussion, there is also this book: The secret formula. Tartaglia, Cardano and the battle raged mathematical Renaissance Italy.
conclude with the full citation of the conclusions of the article by Bressanini:
What remains after hundreds of years of these events? None. Found in textbooks "formulas of Cardano." Hardly anyone mentions Tartaglia, and Scipione dal Ferro is a perfect stranger. Perhaps it would be good to start calling formulas from the Ferro-Tartaglia-Cardan: three authors for an equation of degree three.
Previously ...
Index Series
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