WHO WAS FOURIER?

(The contents of the next two chapters are based on the excellent
biography by Herivel, "Joseph Fourier The Man and the Physicist",
Oxford 1975.)

Joseph Fourier was born in 1768 in Auxerre, the ninth child of a
master tailor. Although the death of his father left him an
orphan at the age of ten, his intelligence gained him a free
place at the local Benedictine school. At the end of a brilliant
school career he applied to enter the artillery only to be
informed that such a profession was only open to those of noble
blood and was closed to him 'even if he were a second Newton'.

Fourier began to prepare to enter the Benedictine teaching order
but, whatever his plans may have been, the course of his life
was violently altered by the outbreak of the French Revolution.
'As the natural ideas of equality developed, it was possible to
conceive the sublime hope of establishing among us a free
government exempt from kings and priests and to free from this
double yoke the long usurped soil of Europe. I readily became
enamoured of this cause, in my opinion the greatest and the most
beautiful which any nation has undertaken.'

However, the Revolution was soon threatened by problems of its
own making. The collapse of royal authority and the effects of
revolutionary zeal and political misjudgment created administrative
chaos whilst at the same time driving France into war against
most of Europe. A succession of military defeats aggravated by
treachery, economic problems and internal revolution put the
Republic in danger and brought Robespierre's party to power. In
the next two years France was reorganised for war and central
power reasserted. Both at a national and a local level, educated
men were needed to fill the administrative vacuum caused by the
revolution.  Fourier, who was now teaching at his old school,
became first an active member of the revolutionary committee of
Auxerre and then its president.

The situation of the new Republic called for ruthless measures
which the government, conscious of its own revolutionary virtues,
was well prepared to take. Treachery was fought by a political
terror in which opponents both to the left and right were
executed and, as the definition of treachery was extended, it
became clear that no one was safe. Fourier himself was arrested,
released and then rearrested. A deputation from Auxerre which,
with considerable courage, went to Paris to plead his case, was
told - 'Yes, he speaks well, but we no longer have any need of
musical patriots.' Only the fall of Robespierre saved Fourier's
head.

However Fourier's release did not mark the end of his troubles.
As coup d'etat followed coup d'etat and the revolution swung
erratically to the right he would remain a marked man. No one
had been executed in Auxerre but Fourier had been an agent of
the terror there. His arrest was on a charge of Hebertism and
the Hebertists were to the left of Robespierre.  The word
'terrorist' then, like 'Trotskyist' now, denoted a defeated yet
feared opponent.

Luckily an opportunity to leave Auxerre now presented itself.
A new college (the Ecole Normale) was being set up in Paris to
help train teachers and Fourier was nominated as a pupil by a
neighbouring district. Fourier could now study under men like
Lagrange, Monge and Laplace and escape his terrorist past.
Fourier's talents were soon noted, but the college was not
successful and its closure was followed by further problems for
Fourier.

'We shudder when we think that the pupils of the Ecole Normale
were chosen under the reign of Robespierre and his proteges It
is only too true that Balme and Fourier, pupils of the department
of Yonne have long professed the atrocious principles and
infernal maxims of the tyrants. Nevertheless they prepare to
become teachers of our children. Is It not to vomit their poison
in the bosom of innocence?' (From an address to the National
Convention, quoted by Herivel). Fourier was again arrested,
released, rearrested and finally, following yet another political
swing, released to become a teacher at the new Ecole
Polytechnique.

Here Fourier remained for three years. That his talent was
recognised is shown by the fact that he succeeded Lagrange in
the Chair of Analysis and Mechanics. This quiet interlude was
ended by a government order to join the invasion of Egypt.
Ostensibly intended to liberate Egypt from the Turks and to
threaten the British position in India, the expedition may have
been seen by the government as a way of keeping a troublesome
general as far away as possible and by the general (Napoleon) as
the first step toward becoming Emperor of the East. Fourier was
one of a group of scientists and intellectuals intended to form
part of the immense cultural benefits that France was to bestow
on Egypt.

Unfortunately the effects of stunning French victories on land
were negated by Nelson's destruction of the French invasion
fleet. Learning of serious military and domestic problems in
France, Napoleon deserted his army and rushed home to 'save
France'. A rapid and successful coup d'etat enabled him to read
the dispatches in which General Kleber (his successor in Egypt)
explained exactly what he thought of this behaviour.

Both before and after Napoleon's departure, Fourier occupied
several important administrative and political posts in Egypt.
When the French expedition finally surrendered in 1801 and
Fourier was repatriated, Napoleon offered him the post of
Prefect of the Department of the Isere centred round Grenoble.
(France had been divided into 83 Departments and each Prefect
governed his Department on behalf of the central government.)
Although he could have continued as Professor at the
Polytechnique, Fourier accepted the offer. Herivel suggests that
Egypt had given him a taste for administration and that he hoped
to rise higher. Herivel also suggests that Fourier's close
association with Kleber after Napoleon's departure accounts
for the fact that these hopes were not fulfilled.

Fourier seems to have been a popular and efficient Prefect. His
greatest achievement during his 14 years of office was by
reconciling the conflicting interests of some forty communities
to enable the swamps of Bourgoin to be drained. The draining of
twenty thousand acres of swamps resulted in major economic and
health benefits and was achieved during a period more noted for
grandiose paper plans than for concrete achievements. Fourier's
other administrative memorial was a new road across the Alps
(now Route N91).

Apart from his prefectorial duties Fourier helped organise the
Description of Egypt. This work written by the intellectuals
attached to the Egyptian expedition did much to inspire European
interest in Egypt and was thus one of the two permanent results
of that expedition.  (The other was the discovery of the Rosetta
Stone, a trilingual inscription which was to provide the key to
the deciphering of hieroglyphics.) Fourier's main contribution
was the general introduction--a survey of Egyptian history up
to modern times. (An Egyptologist with whom I discussed this
described the introduction as a masterpiece and a turning point
in the subject. He was surprised to hear that Fourier also had a
reputation as a mathematician.) On a personal level he encouraged
Champollion, a linguistic infant prodigy, to take up Egyptology
and used his position as prefect to preserve his protege from
conscription. It was Champollion who eventually deciphered the
Rosetta Stone.

Napoleon's domestic policy carried to its natural conclusion a
political compromise which had been developing since the fall of
Robespierre. The old powers of the church and the nobility were
to be reconciled with the new power of the bourgeoisie and an
administration loyal to the state alone. Henceforward men were
to be equal before the law but the state renounced any attempt
to achieve economic equality, or indeed any form of economic
control. Fourier, who had risen through his own talent but who
'could give lessons in theology to bishops and politeness to the
pre-1790 parliamentarians' both symbolised and, in his position
as Prefect, aided the new compromise.

(part 2)

'Yesterday was my 21st birthday, at that age Newton and Pascal
had [already] acquired many claims to immortality.' Fourteen
years after he wrote the postscript above, Fourier was prefect
of Isere but had still no claim to the immortality he had craved
as a young man. The work on the zeros of algebraic polynomials
which he had pursued since his time at Auxerre still gains him a
footnote in algebra textbooks--but a footnote is not immortality.
His lecturing while at the Ecole Polytechnique had been much
praised--but a good lecture is the most ephemeral of triumphs.

However, in 1804 he took up the subject of the propagation of
heat. The field of Newtonian mechanics had already been worked
over by several masters but the physics of heat, light,
electricity and magnetism had still to be brought 'under the rule
'of mathematics. The choice of one of these fields thus requires
no explanation. The particular choice of heat may have been
connected with an obsessional need for warmth which Fourier
acquired in Egypt.

In three remarkable years Fourier found the fundamental equations
for heat conduction, developed new methods to solve them, applied
these methods in a wide variety of cases and produced experimental
evidence to. support his solutions. At the end of 1807 he had
submitted a memoir containing this work to the Academy. A
commission consisting of Lagrange, Laplace, Monge and Lacroix
was set up to examine it.

Instead of the enthusiastic acceptance that Fourier may have
hoped for, lie ran into two major criticisms. On the mathematical
side neither Laplace nor Lagrange could accept the validity of
his use of Fourier series. For example Laplace could not believe
that cos x could possibly be expressed using a sine series. On
the physical side Laplace and his pupils had already attacked
the problem of heat conduction from a different angle.  Unable
to accept that Fourier's approach was superior, Laplace, Biot
and Poisson attacked his derivation of the equations of conduction
as lacking in rigour.

In 1811 the Academy gave Fourier a second chance to promote his
theory by offering its grand prize in mathematics for work on
the theory of heat conduction.

Fourier resubmitted his earlier essay together with some further
results. (The most important addition was the introduction of
the Fourier transform and its use, in the manner of Chapter 55,
in studying the cooling of infinite solids.) However, although
Fourier gained the prize, the accompanying report made it clear
that Lagrange and Laplace had not withdrawn their objections.

The attitude of Lagrange and Laplace has been severely criticised
by writers of modern texts on mathematical methods as (to quote
one of them) 'typical in its demonstration of the difference
between pure and applied mathematicians'. Since Laplace was the
greatest mathematical physicist of the age and the author of the
Traite de Mecanique Celeste whilst Lagrange (who admittedly
often worked in pure mathematics) was the author of the equally
influential Mecanique Analytique this seems a curious judgement.

In fact Laplace and, particularly, Lagrange, did not simply
doubt the rigour but also the truth of Fourier's assertions
concerning expansions in trigonometric series. They may also
have doubted the utility of such expansions. Why should
Fourier's expansion in trigonometric series be superior to an
expansion in continued fractions, or infinite products, or
formal power series or any of the hundred and one pretty tricks
of the eighteenth century mathematicians? With hindsight we can
see (or at least persuade ourselves that we can see) why Fourier
series are much better adapted than Taylor series to a whole
class of physical problems. Laplace and Lagrange could not see
into the future and their doubts are surely more a tribute
to the originality of Fourier's methods than a reproach to
mathematicians who Fourier greatly respected (and, in Lagrange's
case, admired).

Soon new political worries were added to Fourier's academic ones.
(The Academy had published neither his first nor his second
memoir leaving his results open to plagiarism OF rediscovery).
Napoleon had continued the aggressive foreign policy of the
revolution with results depicted in vast battle pieces by a
generation of French painters and, on a smaller scale, by Goya.
So long as French forces were victorious the cost of the policy
would be borne by the defeated powers and not by France but from
1812 onwards the tide of war turned. In 1814 enemy troops
entered Paris, Napoleon was forced to abdicate and Louis XVIII
installed as a constitutional monarch.

The new king understood, even if some of his followers did not,
that the Napoleonic domestic compromise would have to form the
basis of his rule. Nor could all of France's experienced
administrators be replaced by the few men who had stayed loyal
to him in exile.  Fourier remained in Grenoble as prefect of the
Isere.

However the last act of the Napoleonic adventure was still to-be
played. The new government was short of money and made the
mistake of trying to economise on the pensions of Napoleon's old
armies, and, worse, the pension of Napoleon himself. Tired of
his role as a ruler of the little kingdom of Elba, Napoleon
returned to France and began a march towards Paris. Troops sent
out to arrest him went over to him without firing a shot.
Grenoble was the first large town on Napoleon's line of march.
Fourier's attempts to organise resistance proved fruitless and
he and the commanding general left Grenoble by one gate as
Napoleon entered by another.

Whilst at Grenoble, Fourier had acted as a loyal and indeed
zealous servant of the King. But the continuing collapse of the
Royalist position seems to have persuaded him to throw in his
lot with Napoleon and five days later we find him as Napoleon's
prefect of the Rhone.

Twenty years before when Fourier was president of the
revolutionary committee of Auxerre, the republic had lacked
everything from experienced soldiers to gunpowder; but hope and
enthusiasm had saved France. Now there was material and
organisation but only Napoleon's promise of a new regime that
was to be at once revolutionary, liberal and imperial. Passive
and active resistance increased and Fourier himself was
dismissed for failure to comply with orders.  Napoleon was
defeated at Waterloo and the King returned.

Fourier was now without a job and in deep disfavour with the new
regime.  The pension to which his years of public service would
normally have entitled him was refused on technical grounds. An
internal minute concerning his case sums up the government's
attitude. 'The prefect of Grenoble at the time of the arrival of
Bonaparte should not be surprised not to have a pension.'

The low point of Fourier's fortunes did not last long. The
prefect of Paris was an old pupil and friend and had gained
sufficient credit with the new government to be able to appoint
him as Director of the Statistical Bureau of the Seine. (There
are demographers who only know of Fourier as an important figure
in the development of French governmental statistics.) In 1816
he was elected to the Academy of Sciences but the King refused
to confirm his election. In 1817 the Academy tried again and
this time his election was allowed. The physical and mathematical
results of his theory of heat began to gain general acceptance.
In 1822 his prize essay was at last published, and he was
elected permanent mathematical secretary of the Academy.

Fourier was now able to enter upon the role of grand old man
encouraging younger talent (Liouville, Sturm, Dirichlet, Navier)
and attaining that curious summit of French distinction,
election to the Academie Francaise. He did not completely stop
mathematical work and some of his later work concerns problems
in what is now called linear programming.  (Darboux, the
nineteenth-century editor of his collected work, clearly wondered
why such a great mind should be interested in such out of the
way puzzles.)

'There was at the Academy of Sciences', wrote Victor Hugo,
seeking a contrast with the growing fame of the socialist
Charles Fourier, 'a celebrated Fourier whom posterity has
forgotten'. But posterity has not forgotten Joseph Fourier the
Egyptologist, mathematician, physicist and public servant.