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Infinitely Gifted: the Life of
By Stephan Heddon
Bonaventura is one of the most influential names in the mathematical community this century. With
his publication of the book Geometria indivisibilis continuorum
nova and the method of indivisibles which has changed the way
mathematicians have been looking
at irregular shapes influential is an
Ever since Francesco Cavalieri was a young
child he had a passion for discovery. Born in Milan
to a son of an Aristocrat who was not very wealthy. At
17 Cavalieri joined a the Jesuati, under the rule of St.
Augustine. Under the monastery of San Gerolamo in
Milan Cavalieri was enticed by Euclid’s works (which
included geometry) studying and exploring them. His
teacher Federico Borromeo noticed his passion and
intelligence and wrote to Galileo in 1617 introducing
Cavalieri to him. Cavalieri later exclaimed himself to
be one of Galileo’s disciples.
Cavalieri soon went to Pisa to enroll in the
University of Pisa where under Benedetto Castelli
he studied geometry. He was so good that he was
sometimes even the substitute when Castelli was not
Cavalieri was called back to Milan in 1620
where he became a deacon to and protégé of Cardinal
Federigo Borromeo. He also lectured for three year on
theology at Milan.
1 l’Epresso aprile 1647
In 1629 Giovanni Antonio Magini an
astronomer with the chair of mathematics at the
university of Bologna died. Through the help of
friends like Galileo Cavallari was able to grab the
chair of mathematics at the University of Bologna
Statue of Cavalieri in Milan. Proof that Cavalieri
has become influential
at just 31 years old. During this
time he developed the method
of indivisibles (see page 3) which
found a way to determine the size of
geometric figures and explain a new
deference to his dear friend Galileo
who was working on a similar
project he waited.
His work was not greeted
with all similes and he was
challenged. Tons of people attacked
6 years later he published
him claiming he was wrong
including Paul Guldin a Swiss
continuorum nova. Inside this book mathematician. At first Cavallari
was the method of indivisibles.
says he was discouraged but he
The method has opened new
knew that he was right so he set
possibilities in the math world.The out to work. In 1647 he released
book should have been published
Exercitationes Geometricae Sex
earlier says Cavallari but out of
(“Six Geometrical Exercises”) a
book that responded to his criticism
and which held more evidence that
showed he was correct.
Though he is growing old
Cavalieri still says he is working
on something big and that he will
continue to educate the young
minds and hold his chair in
Mathematics in the University of
aprile 1647 l’Epresso 2
The Method of Impossible
Over the past few years Cavalieri’s method of
indivisibles has been questioned by all types
of mathematicians and scholars because it is a
hard concept to grasp and understand. We are
here with a article that proves why he is correct
with his method.
To understand Cavalieri’s method
you have to understand infinity.
Infinity is a hard concept to grasp
but it appear to be in even more
places than we thought according
to Cavallari. Cavalieri’s paper
Geometria indivisibilis continuorum
nova describes an infinity that is
found in curvatures in 3D shapes.
Infinity is found in a line. In
order to create a line you need
two points that connect but if
you think about a line in a bigger
picture there must be an infinite
amount of points in a line
because a line doesn’t have a gap
or hole in it. So line is has infinite
amount of points that make it up.
This isn’t the only thing that is
made up of infinity though.
Now imagine a cylinder
made of a bendable substance.
If you kept everything the same
but you bent the cylinder giving
it a curve will the newly created
irregular shape still have the same
volume that the first cylinder had?
Cavalieri say yes.
The easiest way to picture
this is to visualize a stack of
quattrini (common italian coins
used in the 1600’s in Milan) in the
shape of a cylinder. Now push in
some of the coins in the middle
of the stack and you should now
have something that resembles a
irregular shape. You will notice
however that some coins will stick
out. But if that coin is sticking out
Both figures have the same height and they share the same plane and according to Cavalieri’s Method
these figures both have the same volume.
3 l’Epresso aprile 1647
that must mean that not
all of it is 100 in the pile
yet either. That is why the
volume stays the same.
You can argue that an
irregular shape doesn’t
have these bumps on it
though because it is flat.
Well take a step back. Do
the bumps in the curves
fade away? That’s because
they are a lot smaller now
that you are farther away.
So what Cavallari
is saying is that these
bumps exist but they are
so small they are infinitely
small. That means the
coins are being divided
continuously they are that
small. Cavalieri says that
before the coins are pushed in
an irregular shape is split
up by an infinite amount of
lines going through. Think
of the coins as lines and
you will have understood
after the coins are pushed in both shapes in the pictures have the same
aprile 1647 l’Epresso 4