Pierre-Simon Laplace

Pierre-Simon Laplace was born on March 23, 1749 in the village of Beaumont-en-Auge in Normandy, the son of a small landowner who dealt in cider and farmed a few fields between the apple orchards. The family was comfortable but unremarkable. His parents intended him for the church and sent him at sixteen to the University of Caen to study theology. He stayed long enough to discover that what he loved was not scripture but the calculus of Euler and d’Alembert, and at nineteen he left Caen for Paris with a letter of introduction in his pocket and no clear plan beyond meeting Jean le Rond d’Alembert in person.

The letter, by some accounts, never opened d’Alembert’s door. The story Laplace himself later told was that he handed it over, was politely ignored, went home, and wrote out a short essay on the foundations of mechanics. He sent the essay back to d’Alembert the same week. D’Alembert read it, recognized the unmistakable thumb-print of a serious mathematician, and arranged within days for the twenty-year-old provincial to be appointed professor of mathematics at the École Militaire in Paris. Laplace would never again work outside the highest scientific institutions of France.

The work that occupied him for the next half century was the problem Isaac Newton had left half-finished a hundred years earlier: did the Principia, applied to the actual solar system, predict a stable arrangement of planets, or did it predict a slow collapse? Newton himself had despaired of the question. The planets perturb one another. Jupiter and Saturn in particular exchange small amounts of orbital energy on cycles of centuries, and Newton concluded that without the occasional intervention of God to nudge things back into order, the system would eventually fly apart. Laplace, in a series of memoirs running from the 1770s through the 1820s, showed by direct calculation that the perturbations are bounded. The planets oscillate around their mean orbits but never run away. The solar system, on the timescale of any human civilization, takes care of itself.

The summary of this work is the five-volume Traité de mécanique céleste, published between 1799 and 1825. It is the book that made him, in his own century, the equal of Newton. The second volume, published in 1799, is the one that matters most for the present story. In a long technical appendix on the gravitational potential of an extended body, Laplace needed a complete basis of functions on the surface of a sphere: a family of expansions that could represent the potential of any mass distribution, however lumpy, as a sum of orderly terms organized by angular complexity. The functions he wrote down in 1782 and refined across the next two decades are the ones now called the spherical harmonics, the Yℓm of every quantum-mechanics textbook.

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He had no idea what he had found. To Laplace they were a bookkeeping device for the shape of Saturn and the figure of the Earth. One hundred and forty-five years after the Mécanique céleste appeared, Erwin Schrödinger would write down a differential equation for the electron in a hydrogen atom, separate the variables in spherical coordinates, and discover that the angular part of the wavefunction was precisely Laplace’s family of functions. The integers ℓ and m that label them, invented in Beaumont-en-Auge to keep track of planetary perturbations, became the quantum numbers that classify every atomic orbital in the periodic table. A function written for the heavens turned out to describe the geometry of an atom.

Laplace’s second great export to quantum mechanics is conceptual rather than mathematical. In the introduction to his Essai philosophique sur les probabilités (1814) he sketched a thought experiment that has never quite stopped being argued about. Imagine an intellect, he wrote, that knew at one instant the position and velocity of every particle in the universe, and was vast enough to submit those data to analysis. For such an intellect nothing would be uncertain. The future, like the past, would lie spread out before it as if on a single map. This is the Laplace demon, the cleanest statement of classical determinism ever written. Quantum mechanics, when it arrived, did not so much refute the demon as dissolve it. There is no joint position and velocity for the demon to know. The image survives in textbooks today mostly as the foil against which Heisenberg’s uncertainty principle defines itself.

The personal Laplace was a quieter figure than his books suggest. He was small, careful, and politically supple: he held high office under the Revolution, under the Directory, under Napoléon (who briefly made him minister of the interior and then, judging him a poor administrator, kicked him upstairs to the Senate), and under the restored Bourbons, who made him a marquis. Of his political talents Napoléon, who liked him, remarked that he carried the spirit of the infinitely small into the affairs of government.

The famous Napoléon story belongs to this period. When Laplace presented the emperor with a copy of the Mécanique céleste, Napoléon, who fancied himself a mathematician, leafed through it and asked why the author had written so many pages about the universe without once mentioning its Creator. Sire, Laplace is said to have answered, je n’ai pas eu besoin de cette hypothèse. I had no need of that hypothesis. The remark survives in several versions and may have been polished in the retelling, but the cosmogony of the Mécanique céleste really did dispense with divine intervention. Newton’s stable solar system had required God as an occasional engineer. Laplace’s, derived from the same gravitational law applied more carefully, did not.

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He died at his Paris house on March 5, 1827, at seventy-seven, having outlived the Revolution, the Empire, and the restoration that followed. He is buried at Saint-Julien-de-Mailloc in Normandy, a few miles from the orchard where he was born. His name is now scattered across mathematics and physics like seed: the Laplace transform, the Laplace equation, the Laplacian operator, the Laplace plane in celestial mechanics, the Laplace pressure in surface tension, and the Yℓm spherical harmonics that he would have called his theorem of the figures of equilibrium of fluid masses.

For this book his importance is single and large. Every shape you will see in the orbital viewer, every lobe of every p orbital, every cloverleaf of every d orbital, every angular pattern that distinguishes one electron state from another, is a picture of one of Laplace’s 1782 functions. He drew the alphabet of the atom forty years before the atom was confirmed to exist.