Transcript Sept 30 Italian Algebraists

MAT 346
September 23, 2016
The Renaissance (14th – 16th Centuries)
• Political security gradually
returned to France, Spain, and
England
• Period of reawakening of the
human intellect
• Revival of the arts and literature
• Transition from feudal culture to
urban society
Algebra in the 15th and 16th Centuries
• The main players
• Scipione del Ferro
• Tartaglia (Niccolo Fontana)
• Girolamo Cardano
• Rafael Bombelli
• The quest
• How to solve cubic equations
• Mathematicians of this period required all coefficients be
positive…so we’re still using different “forms” of equations
rather than the general:
ax3 + bx 2 + cx + d = 0
Solving the general cubic
Although many mathematicians tried to solve the general
cubic during this time, the main players all used
“depressed cubics”, where either the quadratic term or
the linear term was left out.
ax 3 + cx + d = 0
ax + bx + d = 0
3
2
Scipione del Ferro (1465 – 1526)
• First to find the solution to the “depressed” cubic
equation for the special case:
x  px  q
3
where p and q
are positive
• However, he did not publish his result (he told his
student, Antonio Fiore).
• This led to one of the most famous mathematical
disputes in history (details are in the texts!).
Mathematical secrecy
• Mathematical discoveries were treated as personal
property
• Reputations based on public contests
• Winning contests resulted in academic appointments and
monetary prizes
Then, in Venice in 1535, Fiore challenged another
mathematician to solve various kinds of cubic equations.
Nicolo Tartaglia (1499 – 1557)
• Self-taught in mathematics.
• He announced he could solve
equations of the type:
x  px  q
3
2
• Fiore challenged him to a
contest and Tartaglia won.
Girolamo Cardano (1501 – 1576)
• True Renaissance man –
physician, philosopher,
mathematician, astrologer,
occultist, writer
• Spent youth gambling, went on to
earn doctorate in medicine
• Predicted the day of his death,
then committed suicide on that
day
Cardano
• Vicious feud erupted between Cardano and Tartaglia
• Prolific writer – wrote more than 200 works on medicine,
mathematics, physics, philosophy, religion, and music
• Ars Magna (“The Great Art”) – text on algebraic
equations
• Cubics – recognized three roots, negative roots, and
complex roots (although he dismissed complex roots as
“useless”)
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