Pierre-Simon Laplace (1749 - 1827)

Laplace was born to a Catholic farming family, who hoped that their son would train for the Church on leaving school. However, while studying at University in Caen, Laplace realised his true passion was mathematics.

After a couple of years he went to Paris, where he soon found a job as a mathematics professor at the École Militaire (he was only 19). He still managed to find time to write a number of papers which he sent to the Academy of Science for publication. He even went to the length of translating his work into Latin.

After three attempts and 13 papers he was eventually admitted as a member of the Academy in 1773. He served on numerous Academy committees, including one monitoring the mortality rates of the city’s hospitals, and one to standardise weights and measures.

He became an examiner for the Royal Artillery Corps in 1784, where he examined and passed the young Napoleon.

Laplace was something of a social climber as well as an accomplished mathematician, and made every effort to keep in with whichever side was in power. His influential 1812 work Théorie Analytique des Probabilités was dedicated to Napoleon.

The Emperor appointed Laplace Minister of the Interior, although Laplace lasted only six weeks in the job - he was clearly not suited to administrative work! With his friend and colleague Lagrange Napoleon awarded Laplace the Legion of Honour in 1805 and made him Count of the Empire in 1806.

He was also named a Marquis by Louis XVIII in 1817 after the restoration of the Bourbons. His political manoeuvres did little to endear him to his colleagues. He died in 1827.

Laplace’s mathematics

His mathematical work can be regarded as having 3 parts. He worked on probability and mathematical physics, and made fundamental contributions to the study and solution of the differential equations that he developed for his mathematical physics.

The former is interesting and laid the groundwork for the emerging science of statistics and probability; the second brought together the ideas of many mathematicians, and the third was decisive in introducing powerful new techniques.

Here is a popular example of his work in probability - based on the ‘Buffon-Laplace Needle Problem’. This concerns finding the probability that when a needle is dropped on to a squared paper that it crosses one of the lines making up the grid:

This problem was first solved by Count Buffon but his derivation contained an error. Laplace corrected the error. It turns out that the probability that the needle crosses a line is,

\mbox{Prob}={2l(a+b)-l^2\over{\pi{ab}}}

This is one way of determining a value for \pi: drop a given pin on such a grid a large number of times and compare the actual probability with the formula above. From this an estimate for \pi can be made - and it can be remarkably accurate.

Laplace’s work in mathematical physics is associated with an equation named in his honour - Laplace’s equation - though it was already known before his time. It is a partial differential equation,

\Delta^2\phi=\left({\partial^2\over{\partial x^2}}+{\partial^2\over{\partial y^2}}+{\partial^2\over{\partial z^2}}\right)\phi=0

Solving such equations would become a pre-occupation of mathematics hereafter. In most cases, exact solutions are not known, but solutions in some particular cases can be found as can approximate solutions.

Laplace’s most important work was his book, Traité de Mécanique Céleste which dealt with dynamics - the study of the motion of bodies. Laplace had already discovered the basic stable nature of planetary motions. These and many other of his earlier results formed the basis for this great work, which was published in 5 volumes, the first two in 1799.

The first volume of the Mécanique Céleste is divided into two books, the first on general laws of equilibrium and motion of solids and fluids, while the second book is on the law of universal gravitation and the motions of the centres of gravity of the bodies in the solar system.

The main mathematical approach here is the setting up of differential equations and solving them to describe the resulting motions. The second volume deals with mechanics applied to a study of the planets.

In it Laplace included a study of the shape of the Earth which included a discussion of data obtained from several different expeditions, and Laplace applied his theory of errors - a statistical idea - to his theoretical results. Another topic studied here by Laplace was the theory of the tides.

One other powerful result takes up his name - Laplace transforms. Basically, this exploits an idea that is at the heart of mathematics. A problem in one area is transformed into a corresponding problem in a different area.

In the new environment, the problem is solved; converting it back to its original environment gives the required solution. This is a technique that Laplace developed for differential equations. Any unknown function f occurring in such an equation is transformed by an integral into a new function F:

F(s)=\int\limits_0^\infty f(t)e^{-st}dt

This is then inserted into the differential equation and in most cases a much simpler equation results. The solution of this may be transformed back to give a solution of the original equation.

Laplace was also a scientist interested in the origins of the solar system. He came to the view that the solar system had evolved from a mass of gases rotating around an axis through its centre of mass, splitting off to form the planets as it cooled. This is a very modern viewpoint.