Mathematics of the
Renaissance
Regiomantus and Nicolas Chuquet
A singular figure who can be identified as the
“Renaissance Man” is Regiomantus (1436-1476)
who recognized the importance of the ancient
works and began to translate them and with the
invention of the printing press, publish his works.
In France, an 1884 manuscript by Nicolas Chuquet
called Triparty en la science des nombres explained how
to work with the Hindu-Arabic numerals.
In Italy, Luca Pacioli’s Summa also had the same
influence in the growth of mathematics.
Cardano and Ferrari’s work on cubic
and quartic equations
We have already discussed Cardano’s
solution of the cubic equation.
The quartic can also be solved and
this is due to Lodovico Ferrari(1540),
a student of Cardano.
The fact that there is no general
formula for fifth degree polynomials
and higher is a famous theorem of
Abel and Ruffini proved in 1824. It
needs Galois theory and in particular
the concept of a group to which we
will come later.
But who really solved the cubic?
Although Al-Khwarizmi solved special cases of the cubic, he developed no
general theory. In 1494, Luca Pacioli (1445-1509) wrote Summa de Arithmetica
in which he posed the problem using for the first time a name for the
“unknown” which he called “cosa” or “thing”.
Scipione del Ferro (1465-1520) of the University of Bologna (the oldest
university in Europe), discovered a formula that solved the “depressed cubic”
which we now know is essentially the full solution.
The intellectual climate of that day was competitive and mathematicians
would do scholarly battle, so del Ferro never published or disclosed his
method but would use special cases of the cubic as challenges to defeat his
opponents.
It was only on his deathbed that he disclosed the method to Antonio Fior
(1506- ?) who it is written was a “mediocre mathematician” and he used it to
challenge his colleague Niccolo Fontana (1499-1557).
Apparently, when Fontana was a child in 1512, a French soldier slashed the
face of young Niccolo that disfigured him and he could no longer speak
clearly. Tartaglia – the stammerer – became his nickname.
Tartaglia and the “depressed” cubic
Tartaglia, or Fontana took up Fior’s challenge and furiously worked on the
cubic. On the night of February 13, 1535, he finally found the answer.
Fior had to provide 30 lavish banquets as the reward, but Tartaglia, in a
gesture of magnanimity, relieved him of this commitment.
But then Gerolamo Cardano (1501-1576) enters the scene. We know much
about his life since he wrote his Autobiography in 1575. Oysten Ore has
written a modern biography of Cardano from which we have taken the details
below.
Cardano figured out how to reduce the general cubic to the “depressed cubic”
but didn’t know how to complete it. He heard that Tartaglia had solved this
and wanted to know how to do it. But Tartaglia refused saying that he will
write a book about it. Finally, on March 25, 1539, Tartaglia relented and
visited Cardano and revealed the secret after Cardano swore on the Bible that
he would never divulge this knowledge!
Ludovico Ferrari (1522-1565) became Cardano’s student and seeing him
intelligent, Cardano shared his secret with him.
The del Ferro papers So the credit should go to del Ferro,
Tartaglia and Cardano!
In 1543, Cardano and Ferrari travelled to Bologna and they
inspected the posthumous papers of Scipione del Ferro.
There, the found, in del Ferro’s own hand writing, the solution
of the depressed cubic.
Since del Ferro is the first to have solved the depressed cubic and
not Tartaglia, Cardano reasoned that he is now free to publish
since his oath was to Tartaglia on the assumption that he had
been the first to solve the depressed cubic!
So in 1545, he wrote Ars Magna, or the “Great Art”. In the
preface he writes:
Reducing the general quartic to a
“depressed” quartic
Ferrari’s solution of the quartic
Completing the square, we see:
The final step
This is called the resolvent cubic, and it can be solved by
Cardano’s method. When m is the root of this equation, our
quartic equation becomes:
Mathematics of perspective and art
In the Renaissance period, several artists, including Leonardo, were becoming
interested in projective geometry so as to introduce three dimensionality into
the two-dimensional canvas.
Paintings before this period looked “flat”. Here are some examples.
The vanishing point
Filipo Brunelleschi (1377-1446) described the four rules of perspective in art.
They are:
1. The horizon appears as a horizontal line.
2. Straight lines in space should appear as straight lines in the painting.
3. Parallel lines in space meet at a vanishing point on the canvas.
4. Lines parallel to the picture frame appear as parallel lines and do not have
a vanishing point.
Examples of perspective art
Here are some famous examples of perspective art:
The School of Athens by Raphael.
The “vanishing point” in the School
of Athens painting
The geometry of light rays
The essential idea in perspective art is to understand
the geometry of light rays with respect to the eye of
the observer.
The visual cone
The eye of the observer determines a visual cone in
the three dimensional grid of observed space.
The Renaissance painters were aware
of the visual cone
Many texts on perspective art written in the Renaissance
period reveal that the artists were aware of these facts.
Here is a sketch of Durer.
Further examples
Da Vinci’s “The Last Supper”
Mathematical contributions of the
Renaissance period
The Renaissance period was a period of blossoming
knowledge.
The printing press facilitated the spread of
knowledge as well as the translations of Greek and
Arabic works.
It was also the time when some of our modern
symbology was introduced.
For example, the + and – sign along with the = sign
were used in the works of Robert Recorde (1557).
Margarita philosophica (1503) by
Gregor Reisch
