Lagrange was born Giuseppe Lodovico Lagrangia in Turin. His family had French ancestors, and so it was that Lagrangia became Lagrange at an early age.
He studied Law at University, but became interested in maths and physics after reading work by the scientist Halley (the man who discovered the comet of the same name).
Initially he taught himself from books, and worked alone, publishing his work using the name Luigi De la Grange Tournier.
He started to correspond with Euler, who was impressed by the young man and became his mentor.
At 19, Lagrange was appointed maths professor at the Royal Artillery School in Turin. The following year, thanks to Euler, Lagrange was elected to the Berlin Academy.
In 1757, he founded what was to become the Royal Academy of Science of Turin, and published much of his work in their journal.
At Berlin he studied a wide variety of maths including mechanics, the solar system, astronomy, calculus, and number theory.
His time in Berlin had its low points - his wife (a cousin, whom he had married a few years earlier) died in 1783. He became depressed and his health deteriorated.
He left Berlin in 1787 for the French Academy of Science in Paris, where in 1790 he was chairman of the committee responsible for standardising weights and measures from an antiquated and cumbersome set of measures to the decimal metric system that we use today - based on metres and grams. Curiously, they did not metricate time.
This was a time of Revolution in France, and in September 1793 it became law that all foreigners born in enemy countries were to be arrested and their property confiscated. Fortunately for Lagrange he had friends in high places and was exempted.
He continued to work in Paris, and although he had hoped to concentrate entirely on research, the revolution meant he took a teaching position at the École Normale in 1795.
In 1808, Lagrange was appointed to the Legion of Honour and named Count of the Empire. He died in 1813, just three weeks after being named Grand Croix of the Ordre Impérial de la Réunion.
An example of his work in number theory concerns writing numbers as the sum of square numbers. Some whole numbers are the sum of 2 squares, for example:
while others, such as 15, are not. Again some whole numbers are the sum of 3 squares, for example:
while others, such as 15 are not. Intuition suggests that this sort of behaviour ought to persist - some numbers can be written as the sum of 4 squares while others can not. The mighty Euler suspected otherwise but in over 40 years of effort, was unable to prove it. Enter Lagrange. He proved that every whole number can be written as the sum of no more than 4 squares. Quite how he proved this is another story.
Another result that had been suspected for some time but still awaited proof, concerns a result that is called Wilson’s theorem but bizarrely, was first stated by a man called Waring. Lagrange was the first to provide a proof and if he had been lucky it would now be called Lagrange’s Theorem.
The result says that if the number:
is divisible by p then p is prime.
For example, with p=13, we have:
and this is divisible by 13:
This is a very peculiar property of prime numbers, one that is totally unexpected. It turns out to have both theoretical and practical applications.
The study of the solar system and the precise way in which it operated was another of his mathematical interests. For example, he first proved that the motion of a planet about the Sun under gravitational attraction was stable. Very reassuring.
He achieved this by considering a planet in an orbit about the Sun and the equations that described it:
Then he perturbed the orbit of the planet by a very small impulse, factored the disturbance into his equations and then sought out the subsequent changes. He found that the planet soon slipped back into its previous orbit about the Sun.
The solar system was stable. (At least that’s what everybody believed - but there is a sequel that was discovered by another French mathematician - Poincaré.) Lagrange went on to write a book (The Mécanique Analytique) that sought to set out the mathematical theory of motion - called dynamics - including all the work of Newton and others who had made advances.