Born: 25 Jan 1736
in Turin, Sardinia-Piedmont (now Italy)

Died: 10 April 1813
in Paris, France

Joseph-Louis Lagrange is usually considered to be a French
mathematician, but the Italian Encyclopaedia [40] refers to him as an Italian
mathematician. They certainly have some justification in this claim since
Lagrange was born in Turin and baptised in the name of Giuseppe Lodovico
Lagrangia. Lagrange's father was Giuseppe Francesco Lodovico Lagrangia who was
Treasurer of the Office of Public Works and Fortifications in Turin, while his
mother Teresa Grosso was the only daughter of a medical doctor from Cambiano
near Turin. Lagrange was the eldest of their 11 children but one of only two to
live to adulthood.

Turin had been the capital of the duchy of Savoy, but become the
capital of the kingdom of Sardinia in 1720, sixteen years before Lagrange's
birth. Lagrange's family had French connections on his father's side, his
great-grandfather being a French cavalry captain who left France to work for
the Duke of Savoy. Lagrange always leant towards his French ancestry, for as a
youth he would sign himself Lodovico LaGrange or Luigi Lagrange, using the
French form of his family name.

Despite the fact that Lagrange's father held a position of some
importance in the service of the king of Sardinia, the family were not wealthy
since Lagrange's father had lost large sums of money in unsuccessful financial
speculation. A career as a lawyer was planned out for Lagrange by his father,
and certainly Lagrange seems to have accepted this willingly. He studied at the
College of Turin and his favourite subject was classical Latin. At first he had
no great enthusiasm for mathematics, finding Greek geometry rather dull.

Lagrange's interest in mathematics began when he read a copy
of Halley's 1693 work on the use of algebra in optics. He was also
attracted to physics by the excellent teaching of Beccaria at the College of
Turin and he decided to make a career for himself in mathematics. Perhaps the
world of mathematics has to thank Lagrange's father for his unsound financial
speculation, for Lagrange later claimed:-

If I had been rich, I probably would not have devoted myself to
mathematics.

He certainly did devote himself to mathematics, but largely he
was self taught and did not have the benefit of studying with leading
mathematicians. On 23 July 1754 he published his first mathematical work which
took the form of a letter written in Italian to Giulio Fagnano. Perhaps
most surprising was the name under which Lagrange wrote this paper, namely
Luigi De la Grange
Tournier. This work was no masterpiece and showed to some
extent the fact that Lagrange was working alone without the advice of a
mathematical supervisor. The paper draws an analogy between the binomial
theorem and the successive derivatives of the product of functions.

Before writing the paper in Italian for publication, Lagrange
had sent the results to Euler, who at this time was working in Berlin, in
a letter written in Latin. The month after the paper was published, however,
Lagrange found that the results appeared in correspondence between Johann
Bernoulli and Leibniz. Lagrange was greatly upset by this discovery since
he feared being branded a cheat who copied the results of others. However this
less than outstanding beginning did nothing more than make Lagrange redouble
his efforts to produce results of real merit in mathematics. He began working
on the tautochrone, the curve on which a weighted particle will always
arrive at a fixed point in the same time independent of its initial
position. By the end of 1754 he had made some important discoveries on the
tautochrone which would contribute substantially to the new subject of
the calculus of variations (which mathematicians were beginning to study
but which did not receive the name 'calculus of variations' before Euler
called it that in 1766).

Lagrange sent Euler his results on the tautochrone
containing his method of maxima and minima. His letter was written on 12 August
1755 and Euler replied on 6 September saying how impressed he was with
Lagrange's new ideas. Although he was still only 19 years old, Lagrange was
appointed professor of mathematics at the Royal Artillery School in Turin on 28
September 1755. It was well deserved for the young man had already shown the
world of mathematics the originality of his thinking and the depth of his great
talents.

In 1756 Lagrange sent Euler results that he had obtained
on applying the calculus of variations to mechanics. These results generalised
results which Euler had himself obtained and Euler consulted
Maupertuis, the president of the Academy, about this remarkable young
mathematician. Not only was Lagrange an outstanding mathematician but he was
also a strong advocate for the principle of least action so Maupertuis
had no hesitation but to try to entice Lagrange to a position in Prussia. He
arranged with Euler that he would let Lagrange know that the new position
would be considerably more prestigious than the one he held in Turin. However,
Lagrange did not seek greatness, he only wanted to be able to devote his time
to mathematics, and so he shyly but politely refused the position.

Euler also proposed Lagrange for election to the Berlin
Academy and he was duly elected on 2 September 1756. The following year
Lagrange was a founding member of a scientific society in Turin, which was to
become the Royal Academy of Science of Turin. One of the major roles of this
new Society was to publish a scientific journal the Mélanges de Turin which published
articles in French or Latin. Lagrange was a major contributor to the first
volumes of the Mélanges de Turin volume 1 of which appeared in 1759, volume 2 in 1762 and volume 3 in 1766.

The papers by Lagrange which appear in these transactions cover
a variety of topics. He published his beautiful results on the calculus of
variations, and a short work on the calculus of probabilities. In a work
on the foundations of dynamics, Lagrange based his development on the principle
of least action and on kinetic energy.

In the Mélanges de Turin Lagrange also made a major study on the
propagation of sound, making important contributions to the theory of vibrating
strings. He had read extensively on this topic and he clearly had thought
deeply on the works of Newton, Daniel Bernoulli,
Taylor, Euler and d'Alembert. Lagrange used a discrete mass model
for his vibrating string, which he took to consist of n masses joined by
weightless strings. He solved the resulting system of n+1 differential
equations, then let n tend to infinity to obtain the same functional solution
as Euler had done. His different route to the solution, however, shows
that he was looking for different methods than those of Euler, for whom
Lagrange had the greatest respect.

In papers which were published in the third volume, Lagrange
studied the integration of differential equations and made various applications
to topics such as fluid mechanics (where he introduced the Lagrangian
function). Also contained are methods to solve systems of linear differential
equations which used the characteristic value of a linear substitution for the
first time. Another problem to which he applied his methods was the study the
orbits of Jupiter and Saturn.

The Académie des Sciences in Paris announced its prize
competition for 1764 in
1762. The topic was on the libration of the Moon, that is the motion of the
Moon which causes the face that it presents to the Earth to oscillate causing
small changes in the position of the lunar features. Lagrange entered the
competition, sending his entry to Paris in 1763 which arrived there not long
before Lagrange himself. In November of that year he left Turin to make his
first long journey, accompanying the Marquis Caraccioli, an ambassador from
Naples who was moving from a post in Turin to one in London. Lagrange arrived
in Paris shortly after his entry had been received but took ill while there and
did not proceed to London with the ambassador. D'Alembert was upset that
a mathematician as fine as Lagrange did not receive more honour. He wrote on
his behalf:-

Monsieur de la
Grange, a young geometer from Turin, has been here for six
weeks. He has become quite seriously ill and he needs, not financial aid, for
the Marquis de Caraccioli directed upon leaving for England that he should not
lack for anything, but rather some signs of interest on the part of his native
country ... In him Turin possesses a treasure whose worth it perhaps does not
know.

Returning to Turin in early 1765, Lagrange entered, later that
year, for the Académie des Sciences prize of 1766 on the orbits of the moons of
Jupiter. D'Alembert, who had visited the Berlin Academy and was friendly
with Frederick II of Prussia, arranged for Lagrange to be offered a position in
the Berlin Academy. Despite no improvement in Lagrange's position in Turin, he
again turned the offer down writing:-

It seems to me that Berlin would not be at all suitable for me
while M Euler is there.

By March 1766 d'Alembert knew that Euler was
returning to St Petersburg and wrote again to Lagrange to encourage him to
accept a post in Berlin. Full details of the generous offer were sent to him by
Frederick II in April, and Lagrange finally accepted. Leaving Turin in August,
he visited d'Alembert in Paris, then Caraccioli in London before arriving
in Berlin in October. Lagrange succeeded Euler as Director of Mathematics
at the Berlin Academy of Science on 6 November 1766.

Lagrange was greeted warmly by most members of the Academy and
he soon became close friends with Lambert and Johann(III)
Bernoulli. However, not everyone was pleased to see this young man in such a
prestigious position, particularly Castillon who was 32 years older than
Lagrange and considered that he should have been appointed as Director of
Mathematics. Just under a year from the time he arrived in Berlin, Lagrange
married his cousin Vittoria Conti. He wrote to d'Alembert:-

My wife, who is one of my cousins and who even lived for a long
time with my family, is a very good housewife and has no pretensions at all.

They had no children, in fact Lagrange had told d'Alembert
in this letter that he did not wish to have children.

Turin always regretted losing Lagrange and from time to time his
return there was suggested, for example in 1774. However, for 20 years Lagrange
worked at Berlin, producing a steady stream of top quality papers and regularly
winning the prize from the Académie des Sciences of Paris. He shared the 1772
prize on the three body problem with Euler, won the prize for 1774,
another one on the motion of the moon, and he won the 1780 prize on
perturbations of the orbits of comets by the planets.

His work in Berlin covered many topics: astronomy, the stability
of the solar system, mechanics, dynamics, fluid mechanics, probability,
and the foundations of the calculus. He also worked on number theory
proving in 1770 that every positive integer is the sum of four squares. In 1771
he proved Wilson's theorem (first stated without proof by Waring)
that n is prime if and only if (n -1)! + 1 is divisible by n. In 1770 he
also presented his important work Réflexions sur la résolution algébrique des
équations which made a fundamental investigation of why equations of degrees up
to 4 could be solved by radicals. The paper is the first to consider the
roots of a equation as abstract quantities rather than having numerical values.
He studied permutations of the roots and, although he does not compose
permutations in the paper, it can be considered as a first step in the
development of group theory continued by Ruffini, Galois
and Cauchy.

Although Lagrange had made numerous major contributions to
mechanics, he had not produced a comprehensive work. He decided to write a
definitive work incorporating his contributions and wrote to Laplace on
15 September 1782:-

I have almost completed a Traité de mécanique analytique, based
uniquely on the principle of virtual velocities; but, as I do not yet know when
or where I shall be able to have it printed, I am not rushing to put the
finishing touches to it.

Caraccioli, who was by now in Sicily, would have liked to see
Lagrange return to Italy and he arranged for an offer to be made to him by the
court of Naples in 1781. Offered the post of Director of Philosophy of the
Naples Academy, Lagrange turned it down for he only wanted peace to do
mathematics and the position in Berlin offered him the ideal conditions. During
his years in Berlin his health was rather poor on many occasions, and that of
his wife was even worse. She died in 1783 after years of illness and Lagrange
was very depressed. Three years later Frederick II died and Lagrange's position
in Berlin became a less happy one. Many Italian States saw their chance and
attempts were made to entice him back to Italy.

The offer which was most attractive to Lagrange, however, came
not from Italy but from Paris and included a clause which meant that Lagrange
had no teaching. On 18 May 1787 he left Berlin to become a member of the
Académie des Sciences in Paris, where he remained for the rest of his career.
Lagrange survived the French Revolution while others did not and this may to
some extent be due to his attitude which he had expressed many years before
when he wrote:-

I believe that, in general, one of the first principles of every
wise man is to conform strictly to the laws of the country in which he is
living, even when they are unreasonable.

The Mécanique analytique which Lagrange had written in Berlin,
was published in 1788. It had been approved for publication by a committee of
the Académie des Sciences comprising of Laplace, Cousin, Legendre
and Condorcet. Legendre acted as an editor for the work doing proof
reading and other tasks. The Mécanique analytique summarised all the work done
in the field of mechanics since the time of Newton and is notable for its
use of the theory of differential equations. With this work Lagrange
transformed mechanics into a branch of mathematical analysis. He wrote in the
Preface:-

One will not find figures in this work. The methods that I
expound require neither constructions, nor geometrical or mechanical arguments,
but only algebraic operations, subject to a regular and uniform course.

Lagrange was made a member of the committee of the Académie des
Sciences to standardise weights and measures in May 1790. They worked on the
metric system and advocated a decimal base. Lagrange married for a second time
in 1792, his wife being Renée-Françoise-Adélaide Le Monnier the daughter of one
of his astronomer colleagues at the Académie des Sciences. He was certainly not
unaffected by the political events. In 1793 the Reign of Terror commenced and
the Académie des Sciences, along with the other learned societies, was
suppressed on 8 August. The weights and measures commission was the only one
allowed to continue and Lagrange became its chairman when others such as the
chemist Lavoisier, Borda, Laplace, Coulomb, Brisson
and Delambre were thrown off the commission.

In September 1793
a law was passed ordering the arrest of all foreigners
born in enemy countries and all their property to be confiscated. Lavoisier
intervened on behalf of Lagrange, who certainly fell under the terms of the
law, and he was granted an exception. On 8 May 1794, after a trial that lasted
less than a day, a revolutionary tribunal condemned Lavoisier, who had saved
Lagrange from arrest, and 27 others to death. Lagrange said on the death of
Lavoisier, who was guillotined on the afternoon of the day of his trial:-

It took only a moment to cause this head to fall and a hundred
years will not suffice to produce its like.

The École Polytechnique was founded on 11 March 1794 and opened
in December 1794 (although it was called the École Centrale des Travaux Publics
for the first year of its existence). Lagrange was its first professor of
analysis, appointed for the opening in 1794. In 1795 the École Normale was founded
with the aim of training school teachers. Lagrange taught courses on elementary
mathematics there. We mentioned above that Lagrange had a 'no teaching' clause
written into his contract but the Revolution changed things and Lagrange was
required to teach. However, he was not a good lecturer as Fourier, who
attended his lectures at the École Normale in 1795 wrote:-

His voice is very feeble, at least in that he does not become
heated; he has a very pronounced Italian accent and pronounces the s like z ...
The students, of whom the majority are incapable of appreciating him, give him
little welcome, but the professors make amends for it.

Similarly Bugge who attended his lectures at the École Polytechnique
in 1799 wrote:-

... whatever this great man says, deserves the highest degree of
consideration, but he is too abstract for youth.

Lagrange published two volumes of his calculus lectures. In 1797
he published the first theory of functions of a real variable with Théorie des
fonctions analytique although he failed to give enough attention to matters of
convergence. He states that the aim of the work is to give:-

... the principles of the differential calculus, freed from all
consideration of the infinitely small or vanishing quantities, of limits or
fluxions, and reduced to the algebraic analysis of finite quantities.

Also he states:-

The ordinary operations of algebra suffice to resolve problems
in the theory of curves.

Not everyone found Lagrange's approach to the calculus the best
however, for example de Prony wrote in 1835:-

Lagrange's foundations of the calculus is assuredly a very
interesting part of what one might call purely philosophical study: but when it
is a case of making transcendental analysis an instrument of exploration for
questions presented by astronomy, marine, geodesy, and the different branches
of science of the engineer, the consideration of the infinitely small leads to
the aim in a manner which is more felicitous, more prompt, and more immediately
adapted to the nature of the questions, and that is why the Leibnizian method
has, in general, prevailed in French schools.

The second work of Lagrange on this topic Leçons sur le calcul
des fonctions appeared in 1800.

Napoleon named Lagrange to the Legion of Honour and Count of the
Empire in 1808. On 3 April 1813 he was named grand croix of the Ordre Impérial
de la Réunion. He
died a week later.