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Mathematics of the Italian Renaissance MAT 112-16 Summer 2011 Prof. Douglas Furman – Mathematics Dept. SUNY Ulster – International Programs Gambling, Betrayal & Murder: Cardano & The Italian Algebraists
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The Italian algebraists of the Renaissance were able to accomplish what had eluded humanity for over three millenia!
In Al-Khwarizmi’s book on Algebra he gives algorithms for 5 forms of quadratic equations:
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The Arabs solve the cubic equation by geometrical means.
Umar ibn Ibrahim Al-Nisaburi al-Khayyami
Known to the west as Omar Khayyam (c. 1044 – c. 1123)
Know to the west more famously as a Persian poet.
He wrote The Rubaiyat translated by Edward Fitzgerald in 1859.
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The Arabs solve the cubic equation by geometrical means.
Umar ibn Ibrahim Al-Nisaburi al-Khayyami
Known to the west as Omar Khayyam (c. 1044 – c. 1123)
Know to the west more famously as a Persian poet.
He wrote The Rubaiyat translated by Edward Fitzgerald in 1859.
Khayyam’s mathematical works first published in the West in 1851.
Not available in English until 1931.
He solved 13 different cases of the cubic equation
Through ingenious geometric reasoning Khayyam is able to find the solutions to the various cubic equations as intersections of two conic sections (hyperbolas, parabolas, & circles)
But these are not numerical solutions they can only provide geometric solutions.
Cardano hears of Tartaglia’s victory and that Tartaglia has solved a cubic equation. So Cardano sends a messenger to ask if Tartaglia will share his method.
Campaigns to be rector of the students, though he is not well liked.
“ This I recognize as unique and outstanding amongst my faults - the habit, which I persist in, of preferring to say above all things what I know to be displeasing to the ears of my hearers. I am aware of this, yet I keep it up willfully, in no way ignorant of how many enemies it makes for me.”
“ I was inordinately addicted to the chess-board and the dicing table…I gambled at both for many years; and not only every year, but – I say with shame- every day.”
The Book of My Life
He once slashed a man across the face who he thought had cheated him in cards.
He eventually writes Liber de Ludo Aleae (Book on Games of Chance), published posthumously in 1663.
“ ...in times of great anxiety and grief, it is considered to be not only allowable, but even beneficial.”
In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice.”
Recall in 1539 Tartaglia had refused Cardano’s request for the solution to the cubic.
Cardano mentions that he has been discussing Tartaglia’s ingenuity with the Governor of Milan
Tartaglia leaves Venice to visit Cardano in Milan…
Eventually Tartaglia shares his solution in the form of a poem and Cardano swears to keep it secret.
“ I swear to you, by God's holy Gospels, and as a true man of honour, not only never to publish your discoveries, if you teach me them, but I also promise you, and I pledge my faith as a true Christian, to note them down in code, so that after my death no one will be able to understand them.”
With Tartaglia’s solution to a depressed cubic, Cardano & Ferrari work for six years and discover methods to solve all the other cases of cubic equations, thus, in essence, they have discovered a general solution!
But the other cases hinge in the depressed cubic
x 3 + m x = n.
Cardano wants to publish his historic “discoveries” but is prevented by his oath to Tartaglia!
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