Transcript Slide 1

A radical life
EVARISTE GALOIS
The long road to Galois
CHAPTER 1
Babylon
How many miles to Babylon?
Three score miles and ten.
Can I get there by candle-light?
Yes, and back again.
If your heels are nimble and light,
You may get there by candle-light.[
Babylon was the capital of Babylonia, an
ancient kingdom occupying the area of
modern Iraq.
Babylonian algebra
B.M. Tablet 13901-front
(From: The Babylonian Quadratic Equation, by A.E.
Berryman, Math. Gazette, 40 (1956), 185-192)
B.M. Tablet 13901-back
Time Passes
 Centuries and then millennia pass.
 In those years empires rose and fell.
 The Greeks invented mathematics as we
know it, and in Alexandria produced the first
scientific revolution.
 The Roman empire forgot almost all that the
Greeks had done in math and science.
 Germanic tribes put an end to the Roman
empire, Arabic tribes invaded Europe, and the
Ottoman empire began forming in the east.
Time passes
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
Wars, and wars, and wars.
Christians against Arabs.
Christians against Turks.
Christians against Christians.
The Roman empire crumbled, the Holy Roman
Germanic empire appeared.
 Nations as we know them today started to form.
 And we approach the year 1500, and the
Renaissance; but I want to mention two events
preceding it.
Al-Khwarismi (~790-850)
Abu Ja'far Muhammad ibn Musa Al-Khwarizmi was born
during the reign of the most famous of all Caliphs of the
Arabic empire with capital in Baghdad: Harun al Rashid; the
one mentioned in the 1001 Nights.
He wrote a book that was to become very influential
Hisab al-jabr w'al-muqabala
in which he studies quadratic (and linear) equations. It
seems that ``al-jabr’’ means ``completion’’ and refers to
removing negative terms. It is the origin of the word
``algebra.’’
``al-muqabala’’ means balancing, and it refers to reducing
positive terms if they appear on both sides of the equation.
Al-Khwarismi (~790-850)
Al-Khwarismi divides equations into six groups, then shows
how to solve equations in each group.
1. Squares equal to roots.
2. Squares equal to numbers.
3. Roots equal to numbers.
4. Squares and roots equal to numbers; e.g. x2 + 10 x = 39.
5. Squares and numbers equal to roots; e.g. x2 + 21 = 10 x.
6. Roots and numbers equal to squares; e.g. 3 x + 4 = x2.
Life was hard before the invention of decent mathematical
notation!
Al-Khwarismi (~790-850)
His book was widely read also by European mathematicians,
and they began to talk of doing things in the Al-Khwarismi way
or, as it came to be known, by an algorithm.
What about the cubic equation?
 The Babylonians had solved some simple
cubic equations. But the first serious attempt
was perhaps due to a mathematician who is
as famous as a mathematician as he is as a
poet.
Omar Khayyam (1048-1131).
A great Persian mathematician working in the Seljuk (Turkish)
empire. Solved cubic equations numerically by intersection of
conic sections. Stated that some of these equations could not
be solved using only straightedge and compass, a result proved
some 750 years later.
The following web pages discuss his method:
Omar Khayyam and a Geometric Solution of the Cubic
Omar Khayyam and the Cubic Equation
Important general fact
 For all of the mathematicians trying to solve cubic
equations, negative numbers and 0 were mysterious not
well understood concepts. For us, the cubic equation is
x3 + ax2 + bx + c =0.
 For us there is little difference, if any, between the
equations x3 + ax = b and x3 = ax + b. For Scipione dal Ferro,
Tartaglia et al, the difference was essential because they
could only understand the equation if a, b were positive
numbers. And, of course, the equation was never written in
the form x3 + ax + b = 0; setting to 0 just didn't make any
sense.
As the year 1500 approaches…
 Only second degree equations were known to
be solvable by radicals. And then …
 But first, a few word from your real numbers
sponsor.
Solving an equation by
radicals
 There are, of course many ways of solving
equations. Once one knows there are roots, it
is only a matter of time to find them.
 At the heart of what it means to solve an
equation is the question of the meaning of
numbers.
What is a number?
 The question is (sort of) easy to answer if the
number is an integer, or even a rational
number.
 But what really is the square root of 2. The
number π?
If Greek mathematicians had had
digital computers…
and had developed fully the atomic theory
of matter, there might not have been a
need for irrational numbers.
Chances are nature is discrete, and we
only needed irrational numbers because
we could not deal with zillions of particles
at once; we had to come up with a
continuous model and invent calculus to
be able to handle it.
In the future, we might be able to
dispense with continuous models.
Raffaelo Sanzio: The School of Athens ~ 1510
If Greek mathematicians had had
digital computers…
This purely discrete mathematics will be
to current mathematics what a music
based on only two notes would be to
Mozart’s music. But…
Personally, I prefer to think that while our
world is probably discrete, it is based on a
continuous blueprint, and we want to
study the blueprint more than the
somewhat imperfect construction based
on it.
Raffaelo Sanzio: The School of Athens ~ 1510
If Greek mathematicians had had
digital computers…
But, lacking computers, and not quite sure
about atoms, the Greeks invented a
continuous mathematics and discovered
irrational numbers. They were baffled.
Raffaelo Sanzio: The School of Athens ~ 1510
Eudoxus of Cnidus
In one of the most brilliant ``tours de force’’ in mathematics,
Eudoxus (408 BCE-355 BCE) solved the problem of the irrationals.
A semi modern interpretation is that in many ways it is
meaningless to ask, for real numbers, is a = b? By Eudoxus, a =b
simply means that both a < b and b < a are false. This idea lies
behind the finding of formulas for areas and volumes of curved
figures by Euclid and Archimedes; it is essential to the notion of
convergence.
The point is…
 That one has to be very explicit by what one
means by having a formula to solve algebraic
equations. To solve an equation by radicals
means to have a formula in which the
solutions are expressed as a function of the
coefficients; the process of going from the
coefficients of the equation to the solution
should involve only the usual arithmetic
operations (+, -, ×, /), and extraction of roots,
and should involve only a finite number of
steps.
The Italian Connection
THE CUBIC CHRONICLES
Luca Pacioli (1445-1509)
In Summa de arithmetica, geometria, proportioni
et proportionalità, published 1494 in Venice, he
summarizes all that was known on equations. He
discusses quartic equations stating that the
equation that in modern notation is written as
x4 = a + bx2
can be solved as a quadratic equation but
x4 + ax2 = b
and
x4 +a = bx2
are impossible at the present state of science;
ditto the cubic equation.
Great friend of Leonardo da Vinci, briefly a
colleague of Scipione dal Ferro.
And Then…..
THE FIRST ADVANCE OF EUROPEAN MATHEMATICS SINCE THE TIME OF THE
GREEKS: Scipione dal Ferro figures out how to solve the depressed cubic equation
(ca. 1515)
The depressed cubic equation
x  px  q
3
Says cubi: I am depressed because I am missing my quadratic term.
Scipione dal Ferro (14651526).
There may not be any
reliable portrait of
Scipione dal Ferro on the
web. One that pops up is
really Tartaglia.
Professor at Bologna. Around 1515 figured
out how to solve the equation x3 + px = q by
radicals. Kept his work a complete secret until
just before his death, then revealed it to his
student Antonio Fior.
Antonio Fior (1506-?). Very little information
seems to be available about Fior. His main claim
to fame seems to be his challenging Tartaglia to
a public equation ``solvathon,’’ and losing the
challenge.
No portrait of
Antonio Fior seems
to be available.
The two great rivals
Nicolo of Brescia who adopted the name Tartaglia
(Stutterer) (1499-1557). Hearing a rumor that cubic
equations had been solved, figured out how to solve
equations of the form x3 +mx2 = n, and made it public. This
made Fior think that Tartaglia would not know how to deal
with the equations del Ferro knew how to solve and he
challenged Tartaglia to a public duel. But Tartaglia figured
out what to do with del Ferro’s equation and won the
contest.
Girolamo Cardano (1501-1576). Mathematician, physician,
gambler (which led him to study probability), a genius and a
celebrity in his day. Hearing of Tartaglia’s triumph over Fior
convinced Tartaglia to reveal the secret of the cubic,
swearing solemnly not to publish before Tartaglia had done
so. Then he published first. But there are some possible
excuses for this behavior.
Cardano’s oath
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.
And then, in Ars Magna published in 1545, Cardano revealed the formula to the world.
Completing the Picture
Lodovico Ferrari (1522-1565). A protégé of
Cardano, he discovered how to solve the quartic
equation, by reducing it to a cubic. But his result
could not be published before making public how
to solve cubic equations. Tartaglia still not
publishing, and the discovery that Scipione dal
Ferro had already solved the cubic, where part of
the reasons why Cardano published first.
Not a picture of L. Ferrari
As Ferrari wrote: Four years ago when Cardano was going to Florence and I
accompanied him, we saw at Bologna Hannibal Della Nave, a clever and humane
man who showed us a little book in the hand of Scipione del Ferro, his father-inlaw, written a long time ago, in which that discovery was elegantly and learnedly
presented.
The simple, but ingenious
idea
Tartaglia’s Solution of the Cubic Equation
Quando chel cubo con le cose appresso
Se agguaglia à qualche numero
discreto
Trouan dui altri differenti in esso.
Dapoi terrai questo per consueto
Che'llor produtto sempre sia eguale
Alterzo cubo delle cose neto,
El residuo poi suo generale
Delli lor lati cubi ben sottratti
Varra la tua cosa principale.
In el secondo de cotestiatti
Quando che'l cubo restasse lui solo
Tu osseruarai quest'altri contratti,
Del numer farai due tal part'à uolo
Che l'una in l'altra si produca schietto
El terzo cubo delle cose in stolo
Delle qual poi, per communprecetto
Torrai li lati cubi insieme gionti
Et cotal somma sara il tuo concetto.
El terzo poi de questi nostri conti
Se solue col secondo se ben guardi
Che per natura son quasi congionti.
Questi trouai, & non con paßi tardi
Nel mille cinquecentè, quatroe trenta
Con fondamenti ben sald'è gagliardi
Nella citta dal mar'intorno centa.
01) When the cube with the cose beside it
02) Equates itself to some other whole number,
03) Find two other, of which it is the difference.
04) Hereafter you will consider this customarily
05) That their product always will be equal
06) To the third of the cube of the cose net.
07) Its general remainder then
08) Of their cube sides, well subtracted,
09) Will be the value of your principal unknown.
10) In the second of these acts,
11) When the cube remains solo
12) You will observe these other arrangements:
13) Of the number you will quickly make two such parts,
14) That the one times the other will produce
straightforward
15) The third of the cube of the cose in a multitude,
16) Of which then, per common precept,
17) You will take the cube sides joined together.
18) And this sum will be your concept.
19) The third then of these our calculations
20) Solves itself with the second, if you look well after,
21) That by nature they are quasi conjoined.
22) I found these, & not with slow steps,
23) In thousand five hundred, four and thirty
24) With very firm and strong foundations
25) In the city girded around by the sea.
Translated by:
Friedrich Katscher . For more details,
see:
http://mathdl.maa.org/mathDL/46/?p
a=content&sa=viewDocument&nodeI
d=2433&pf=1
The year was 1534, the ``city girded
around by the sea’’ is Venice.
In Symbols
An important side-effect
Rafael Bombelli (1526-1572)
Engineer and self-taught mathematician, began writing in
1557 his masterwork, Algebra. It was to be five volumes
long, but only three were ready for publication in 1572, the
year he died. Made sense of the complex expressions that
appeared as solutions to cubic equations. Because of that,
MacTutor calls him the inventor of complex numbers. But it
would take still quite a while for these numbers to make true
sense; all the way to the work of Euler some two centuries in
the future.
And now on to the QUINTIC!!
 And another couple of
centuries pass. Very important
developments during these
centuries were:
 The development of
mathematical notation, much
of it due to François Viète
(1540-1603) with final touches
by René Descartes (1596-1650)
AND THEN: Calculus, a sea
change!
Nature and nature’s laws lay hid in night
God said, `Let Newton be,’ and there was light!
(Alexander Pope-1730)
AND THE WORLD WAS NEVER THE
SAME
The Bernoulli brothers, Jacob (1654-1705)
and Johann (1667-1748)
Leonhard Euler (1707-1783)
What was the sea change?
First of all, Newton did not invent Calculus out of nothing. Fermat, Descartes, Pascal,
Barrow, among others got very close. Moreover, Leibniz also came up with similar ideas.
I think the change that occurred in those years was that mathematics changed from
being a static science to a dynamic one. The concept of function, called fluent by
Newton, became a central concept in mathematics, perhaps the central concept.
Newton did not invent calculus alone, nor did he invent all of it. The work of the
Bernoulli brothers, among others, and, above all, the work of Euler, one of the greatest
mathematicians of all time, was almost as essential. And it would take another century
before the concept of function was clearly understood.
But back to equations.
 A very important development in this area
was the statement and proof of the
Fundamental Theorem of Algebra: Every
algebraic equation of degree ≥ 1, with
complex coefficients, has at least one
complex root.
The first proof of this theorem is probably
due to Jean Robert Argand (1768-1822); in
his doctoral thesis Carl Friedrich Gauss
(1777-1855) gave six different proofs.
Gauss, in 1828
Someone who deserved more
recognition.
Paolo Ruffini (1765-1822) trained both as a mathematician
and as a physician. A professor of the Foundations of Analysis
in Modena (balsamic vinegar country), he was forced to resign
his position and was barred from teaching because, on
religious grounds, he refused to swear an oath of allegiance to
the Cisalpine Republic, an invention of Napoleon consisting of
Lombardy, Bologna, Emilia, and Modena. He dedicated
himself to the practice of medicine (he was licensed as a
physician) and to mathematical research. He discovered that
the quintic equation could not be solved by radicals.
Someone who deserved more
recognition.
Ruffini was a great admirer of Lagrange, who had worked on
trying to solve the quintic, without success. In the process
Lagrange sowed the seeds of what is now group theory,
because he worked with permutations, but never composed
them. Ruffini had to invent some of the notions of group
theory.
In 1799 he published a book with a rather descriptive title:
Teoria Generale delle Equazioni, in cui si dimostra impossibile la
soluzione algebraica delle equazioni generali di grado superiore
al quarto
(General theory of equations in which it is shown that the
algebraic solution of the general equation of degree greater
than four is impossible)
Someone who deserved more
recognition.
The book begins with:
The algebraic solution of general equations of degree greater
than four is always impossible. Behold a very important
theorem which I believe I am able to assert (if I do not err): to
present the proof of it is the main reason for publishing this
volume. The immortal Lagrange, with his sublime reflections,
has provided the basis of my proof.
And nobody paid much attention, not even the immortal
Lagrange. Most mathematicians could not really understand
Ruffini’s arguments, and did not believe them. Only Cauchy,
who was influenced by Ruffini’s ideas, in a reversal of his usual
role of stealing credit, gave Ruffini credit for an important
discovery.
Niels Henrik Abel (18021829)
At the time of Abel’s birth, Norway was part of Denmark.
When he was 12 years old, it became part of Sweden,
eventually a semi-autonomous part. All this was the result of
wars, partially the Napoleonic wars. Norway suffered mass
starvation at one point; it was not a good time to be a
Norwegian.
Abel showed no special talent in mathematics until about
age 15; then the school he attended hired a good
mathematics teacher, Berndt Holmboë, who saw Abel had
talent.
Niels Henrik Abel (18021829)
Abel’s father was both prominent inpolitics, and a drunkard.
He died in 1820 leaving the family without an income.
Thanks to Holmboë who got him a scholarship, Abel finished
his secondary studies and could enter the University of
Christiania (Christiania being now Oslo).
In 1822 he graduated and discovered, so he thought, how to
solve the quintic equation by radicals. The editor of the
journal to which he submitted the paper asked him for a
numerical example, and Abel discovered his mistake.
Niels Henrik Abel (18021829)
Abel published in 1824, at his own expense, a paper showing
that the quintic was not solvable by radicals. He went on to
do very important work in the theory of elliptic integrals, a
theory he literally turned around because he inverted the
problem, and came up with elliptic functions.
He wrote paper after paper, always poor, not quite
recognized, taking on odd jobs. While in Berlin he initiated a
close friendship with August Leopold Crelle, who in 1826
began publishing a mathematical journal, Crelle’s Journal. It
still exists with the name Journal für die reine und
angewandte Mathematic. Its first volume contained six
articles by Abel.
Niels Henrik Abel (18021829)
Crelle worked hard to get Abel a decent position in Berlin,
and he finally succeeded. He wrote Abel the good news on
April 8, 1829 with the good news. Unfortunately, Abel had
died on April 5.
In 1830 the Paris Academy of Sciences awarded Abel and
Jacobi the Grand Prix for their work on elliptic functions.
In January of 2002, the year of the second centennial of
Abel’s birth, the Norwegian government established the
Abel prize, as a counterpart to the Nobel prize.
Niels Henrik Abel (18021829)
Quotes from Abel:
My eyes have been opened in the most surprising manner. If
you disregard the very simplest cases, there is in all of
mathematics not a single infinite series whose sum had been
rigorously determined. In other words, the most important
parts of mathematics stand without foundation. It is true that
most of it is valid, but that is very surprising. I struggle to find
a reason for it, an exceedingly interesting problem.
(In a letter to Holmboë)
Niels Henrik Abel (18021829)
Quotes from Abel:
Until now the theory of infinite series in general has been very
badly grounded. One applies all the operations to infinite
series as if they were finite; but is that permissible? I think not.
Where is it demonstrated that one obtains the differential of
an infinite series by taking the differential of each term?
Nothing is easier than to give instances where this is not so.
Niels Henrik Abel (18021829)
Quotes from Abel:
The divergent series are the invention of the devil, and it is a
shame to base on them any demonstration whatsoever. By
using them, one may draw any conclusion he pleases and that
is why these series have produced so many fallacies and so
many paradoxes.
Letter to Holmboë; quoted by G.H. Hardy at the beginning
of Divergent Series.
Niels Henrik Abel (18021829)
Quotes from Abel:
By studying the masters, not the pupils.
Supposedly he said this as reply to the question on how he
got to be so good in mathematics. The only reference I have
so far is Eric Temple Bell, which means that this is perhaps
the most famous of quotes, and might not be a real quote at
all.
CHAPTER 2: THE SHORT AND
TRAGIC LIFE OF EVARISTE
GALOIS
Some facts.
 Evariste Galois was born October 25, 1811.
 He was shot in a mysterious duel, May 30,
1832; died the next day. He was not quite 21
years old.
The romantic version
 All night long he had spent the fleeting hours feverishly dashing
off his scientific last will and testament, writing against time to
glean a few of the great things in his teeming mind before the
death he saw could overtake him. Time after time he broke off to
scribble in the margin "I have not time; I have not time," and
passed on to the next frantically scrawled outline. What he
wrote in those last desperate hours before the dawn will keep
generations of mathematicians busy for hundreds of years. He
had found, once and for all, the true solution of a riddle which
had tormented mathematicians for centuries: underwhat
conditions can an equation be solved? (Eric Temple Bell, Men of
Mathematics, Simon & Schuster, 1937)
Further Facts: Publications
 April 1829: Démonstration d’un Thèoréme sur les Fractions





Continues Pèriodiques, Annales de Mathématiques Pures et
Appliqués.
April 1830: Analyse d’un Mémoire sur la Résolution
Algébrique des Équations, Bulletin des Sciences
Mathématiques, Physiques et Chimiques.
June 1830: Note sur la Résolution des Équations
Numériques, Bulletin des Sciences Mathématiques,
Physiques et Chimiques.
June 1830: Sur la Théorie des Nombres, Bulletin des
Sciences Mathématiques, Physiques et Chimiques.
December 1830: Notes sur Quelques Points d’Analyse,
Annales de Mathématiques Pures et Appliqués.
January 1831: Letter to the editor of La Gazette des Écoles.
And more:
 His most famous work, titled Mémoire sur les
conditions de résolubilité des équations par
radicaux, now known as his First Memoir, was
submitted to the Academy of Sciences of
Paris January 17, 1831.
March 31, 1831
Mr. President:
I dare to hope that Messrs Lacroix and Poisson will not take it badly that I recall to their memory a
memoir relating to the theory of equations with which they were charged three months ago.
The research contained in this memoir formed part of a work which I submitted last year in
competition for the prize in mathematics, and in which I gave, for all cases, rules to recognize
whether a given equation was or was not solvable by radicals. Since, until now, this problem had
appeared to geometers to be, if not impossible, at least very difficult, the examining committee
judged a priori that I could not have solved this problem, in the first place because I was called Galois,
and further because I was a student. And the committee lost my memoir. And someone told me that
my memoir was lost.
This lesson should have been enough for me. All the same, on the advice of an honorable member of
the Academy, I reconstructed part of my memoir and presented it to you.
You see Mr. President that my research has suffered up to now almost the same fate as that of the
circle squarers. Will the analogy be pushed to its conclusion? Be so kind Mr. President as to relieve my
disquiet by inviting Messrs. Lacroix and Poisson to declare whether they have lost my memoir or
whether they have the intention to make a report of it to the Academy.
Be assured Mr. President of the homage of your respectful servant.
E. Galois
The Verdict
 Poisson writing for both Lacroix and himself,
rejects the article on July 4, 1831. He mentions
that there is some overlap with work of Abel, but
it could perhaps go further, but it was very
obscure, and unclear, and incomplete.
 One may therefore wait until the author will have
published his work in its entirety before forming a
final opinion; but given the present state of the
part he has submitted to the Academy, we cannot
propose to you that you give it your approval.
A brief Chronology
 (1804- Napoleon becomes emperor of France. )
 10/25/1811-Born in Bourg-la-Reine, a suburb of Paris.
Second of three children, his sister Nathaly-Théodore
was two years older, brother Alfred three years younger.
 (1812-Napoleon invades Russia with an army of a million
men; only 10,000 return.)
 (1814-Napoleon is forced to abdicate, exiled to Elba. The
brother of executed Louis XVI is installed as king of
France, as Louis XVIII. )
A brief Chronology
 (1815- Napoleon escapes Elba, returns to France. Louis
XVIII and his court run away. But Napoleon is finally and
decisively defeated in the battle of Waterloo. Louis XVIII
returns as king of France.)
 10/6/1823-Entered the Collège Louis-le-Grand.
 (1824-Louis XVIII dies, his younger brother becomes king
as Charles X.)
 August 1828-Failed to be admitted to the prestigious
École Polytechnique.
A brief Chronology
 7/2/1829- Nicolas-Gabriel Galois, E.’s father, commits
suicide.
 July or August 1829-Last attempt to be admitted to the
École Polytechnique. Fails again.
 November 1829: Entered the École Preparatoire (better
knwn now as École Normale Supérieure.)
 (July 26-29, 1830. Workers, students, common folk pour
onto the streets of Paris and set up barricades. It is a full
revolution against Charles X. Charles X was deposed.)
A brief Chronology
 Students from the Polytechnique were at the forefront of
the revolution, but the students of the École Normale
were not allowed on the streets by order of the director.
Galois was furious! His letter to the Gazette des Écoles
was a criticism of this director.
 January 1831- He is expelled from the École Normale.
 (1830-A liberal minded nobleman, Louis-Philippe, is
proclaimed king of France. He would be deposed in
1848.)
A brief Chronology
 April 1831-Galois takes part in a celebration of
the acquittal of some radical republicans in a
tavern in Paris. Alexandre Dumas was there and writes:
 Suddenly the name Louis-Philippe, followed by five or six whistles,
catches my ear. I turned around. One of the most animated scenes
was taking place fifteen or twenty seats from me. A young man,
holding in the same hand a raised glass and an open dagger, was
trying to make himself heard. He was Évariste Galois…one of the
most ardent republicans. All that I could perceive was that there was
a threat, and that the name of Louis-Philippe had been pronounced;
the intention was made clear by the open knife.
A brief Chronology
 Courageously, Dumas decides to escape by jumping out
of an open window.
 5/10/1831-Arrested for offensive political behavior (the
tavern incident), but then acquitted and released.
 7/14/1831 (Bastille Day)-He leads a crowd of 600
republicans across the pont Neuf; he was dressed in the
uniform of the outlawed Artillery of the National Guard,
and was carrying a loaded carbine, two pistols and a
knife. He is arrested, held at Ste. Pélagie prison.
 10/23/1831-Convicted of carrying fire arms and wearing a
banned uniform, sentenced to 6 more months in Ste.
Pélagie.
A brief Chronology
 Late May 1832-Engaged to duel. It is not really known
with whom or why.
 5/29/1832- Writes his Lettre Testamentaire, addressed to
his friend Auguste Chevalier and revised some of his
manuscripts.
 5/30/1832- Shot in an early morning duel; died a day later
in the Côchin hospital in Paris.
 3/16/1832-Released from Ste. Pélagie during an outbreak
of cholera in Paris.