Pierre-Simon Laplace
The Man Who Tried to Calculate Everything
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He wrote books on probability, including a theory that helped determine how likely events are. He also worked on the least squares method, the central limit theorem, and something called the probability-generating function. His ideas shaped modern statistics and physics.
This audiobook explores his life from childhood in Normandy to his role as Minister of the Interior under Napoleon. It tells how he navigated politics while making groundbreaking discoveries in science and mathematics. Anyone interested in how one mind changed our understanding of the universe will find this worth their time.
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Pierre-Simon Laplace was born in Beaumont-en-Auge, Normandy, on March 23, 1749, a village four miles west of Pont l’Évêque. Some details of his early life are unknown, as records were destroyed in 1925 when the family château in Saint Julien de Mailloc burned down. Earlier, in 1871, his home at Arcueil near Paris had been looted. W. W. Rouse Ball claimed Laplace was the son of a poor farmer or farm-labourer and that he began as a pupil before becoming an usher in a local school. But later research found that his father, Pierre de Laplace, actually owned and farmed a small estate called Mérisier. Karl Pearson criticized Rouse Ball’s account for its inaccuracies.
Pierre-Simon Laplace was born in Beaumont, not far from Caen, and grew up in a town that was actually quite intellectually vibrant for its time. He went to the University of Caen in 1765 at just sixteen, where he studied for five years and became part of a group called the Sphinx. His parents were Pierre Laplace and Marie-Anne Sochon, both from stable families—his father even worked as a cider merchant and served as syndic in Beaumont. Laplace wrote his first published paper in 1766, appearing in the Mélanges of the Royal Society of Turin, at least two years before he moved to Paris at around twenty-two or twenty-three. By then, he was already connected with Lagrange in Turin. He didn’t arrive in Paris as a self-taught peasant; he had already begun making his mark.
Pierre-Simon Laplace attended a village school run by a Benedictine priory where his father hoped he'd become a priest. At sixteen, he studied theology at the University of Caen, where math teachers Christophe Gadbled and Pierre Le Canu inspired him with their enthusiasm. His talent became clear early on, and while still at Caen, he wrote a memoir titled Sur le Calcul integral aux differences infiniment petites et aux differences finies. This work brought him into contact with Lagrange, who was thirteen years older and had started a journal called Miscellanea Taurinensia in Turin. Laplace's paper appeared in the fourth volume of that journal. Around this time, he decided he had no calling for the priesthood and chose to become a mathematician instead. Some say he then broke with the church and became an atheist. He never graduated in theology but left for Paris with a letter from Le Canu introducing him to Jean le Rond d'Alembert, who was at the time the leading figure in science.
D'Alembert initially treated Laplace poorly, dismissing him with a thick mathematics book and telling him to return once he'd read it. A few days later, when Laplace came back, d'Alembert was even less welcoming, openly doubting that Laplace could have understood the text. But after questioning him, he discovered Laplace had indeed mastered it, and from then on took him under his wing. Another version of events says that Laplace solved a problem d'Alembert had assigned for the next week—then tackled an even harder one the very next night. Impressed, d'Alembert recommended him for a teaching position at the École Militaire.
With a stable income and a teaching job that didn’t demand much, Laplace threw himself into original research. Over the next seventeen years, from 1771 to 1787, he produced much of his groundbreaking work in astronomy. From 1780 to 1784, he worked alongside the French chemist Antoine Lavoisier on several experiments, designing their own equipment for the investigations.
In 1783, they published a joint paper called Memoir on Heat, where they explored the kinetic theory of molecular motion. In their work, they measured the specific heat of different materials and looked at how metals expand when heated. They also studied the boiling points of ethanol and ether under pressure.
Laplace impressed Condorcet, and by 1771 he felt ready for the French Academy of Sciences. That year, though, admission went to Vandermonde, and in 1772 to Cousin. Disappointed, Laplace sought help from d’Alembert, who wrote to Lagrange in Berlin asking if a position might be found. But Condorcet became permanent secretary in February, and on March 31 Laplace was elected associate member at age twenty-four. He read his paper on planetary motion that March and was soon elected full member. In 1773, he presented his work before the Academy, where he would spend most of his scientific career. Later that year, he married Marie-Charlotte de Courty de Romanges in Paris. They had a son, Charles-Émile, and a daughter, Sophie-Suzanne.
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Laplace began his published work in 1771, diving into differential equations and finite differences, but even then he was already exploring probability and statistics. Before joining the Académie in 1773, he had finished two papers that would shape his legacy. The first, Mémoire sur la probabilité des causes par les événements, appeared in 1774. A second paper, published in 1776, deepened his statistical thinking and marked the start of his systematic study of celestial mechanics and the Solar System’s stability. These fields remained closely connected in his mind. “Laplace took probability as an instrument for repairing defects in knowledge.” His later work on the analytic theory of probabilities built on this foundation.
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In 1687, Sir Isaac Newton published the Philosophiæ Naturalis Principia Mathematica, where he derived Kepler’s laws from his laws of motion and universal gravitation. Though Newton had developed calculus privately, he used geometric methods in his work, which weren’t suited to capture subtle interactions between planets. He himself doubted a full mathematical solution and even suggested that divine intervention might be needed to keep the Solar System stable. Laplace would later take up the challenge of proving stability without invoking God, though modern understanding shows his methods alone aren’t precise enough to fully confirm that stability. Today, we know the Solar System is chaotic at small scales, yet remains relatively stable on larger ones.
In 1776, Laplace published a memoir addressing a troubling puzzle from observational astronomy: Jupiter's orbit appeared to be shrinking while Saturn's was expanding. The issue had been examined before, first by Leonhard Euler in 1748 and then by Joseph Louis Lagrange in 1763, but neither could resolve it. Laplace considered whether a luminiferous ether or a version of gravitation that did not act instantly might explain the motion. He eventually returned to Newton's theory of gravity. Where Euler and Lagrange had simplified their equations by ignoring small terms, Laplace saw that those terms, when added up over time, could grow in importance. By extending his analysis to include cubic terms, he showed that the Sun and any two planets must remain in mutual equilibrium. As Gerald James Whitrow later said, the work represented "the most important advance in physical astronomy since Newton."
Pierre-Simon Laplace commanded such broad expertise across sciences that he led every discussion at the Académie. He treated mathematical analysis as simply a tool for solving physical problems, yet his skill in creating the needed methods was extraordinary. Once he reached a correct result, he rarely took the time to walk through how he got there. Elegance or symmetry never mattered to him—so long as he could solve the issue at hand, any path worked.
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In 1775, Laplace introduced a dynamic theory of tides that went beyond what Newton and Bernoulli had done before. While Newton focused on the forces that generate tides and Bernoulli looked at how water reacts statically to tidal potential, Laplace described how oceans actually move in response to those forces. His model included friction, resonance, and the natural rhythms of ocean basins. It successfully predicted the vast amphidromic systems found across Earth’s seas and helps explain the real tides we observe today.
The equilibrium theory of tides, which relied on the gravitational pull from the Sun and Moon, failed to account for key factors like Earth’s rotation or the presence of continents. Because it ignored these elements, it couldn’t accurately explain how ocean tides actually behave in the real world. That limitation led scientists to seek a more complete understanding, setting the stage for further developments in tidal science.
Since measurements have confirmed the theory, many phenomena now have possible explanations, like how tides interact with deep sea ridges and chains of seamounts to create deep eddies that transport nutrients from the ocean floor to the surface. While the equilibrium tide theory calculates tide waves at less than half a meter, the dynamic theory explains why tides can reach up to 15 meters. Satellite observations confirm the accuracy of the dynamic theory, and global tide measurements are now precise to within a few centimeters. Data from the CHAMP satellite closely aligns with models based on TOPEX data. Accurate global tide models are essential for research because tidal variations must be accounted for when calculating gravity and changes in sea levels.
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In 1776, Laplace worked out a set of equations to describe how tides move across the ocean. He imagined the flow as a thin, two-dimensional sheet and included effects from Earth’s rotation, known as the Coriolis effect. These equations came from simplifying more complex fluid dynamics, but they can also be reached by using energy principles and Lagrange's equation.
For a fluid sheet with an average depth of D, certain mathematical relationships describe how the tide moves. These relationships involve the vertical movement of water, represented by ζ, and the horizontal flow in two directions—latitude φ and longitude λ—denoted by u and v. All of these elements together follow what we now call Laplace's tidal equations.
Laplace’s tidal equations describe the movement of ocean tides using partial derivatives and trigonometric functions. The first equation involves the rate of change of tidal elevation ζ over time, along with terms that account for the horizontal velocity components u and v, and a depth function D. The second and third equations govern the changes in the horizontal velocities u and v, incorporating Coriolis effects through the rotation rate Ω and the sine of latitude φ. These equations also include gravitational potential terms gζ and U, which relate to the tidal forces acting on the ocean. Together, they form a system that models how tides behave across the globe.
Laplace's tidal equations describe how ocean levels change under gravitational forces, involving key variables: Ω (angular frequency of planetary rotation), g (gravitational acceleration at average ocean surface), a (planet radius), and U (external gravitational potential causing tides). These mathematical expressions explain the complex interaction between planetary motion and tidal behavior, advancing understanding of tidal operations across celestial bodies. His work laid critical groundwork for geophysics and oceanography, enabling more accurate modeling of tidal patterns than ever before. The equations remain central to modern studies of planetary dynamics and fluid motion, demonstrating how mathematical analysis can unlock natural phenomena secrets. Laplace's contributions continue influencing research into planetary systems and their gravitational field interactions.
William Thomson, later known as Lord Kelvin, took Laplace's momentum terms and reworked them by using the curl operator, which led him to derive an equation for vorticity. This new formulation emerged under specific conditions that simplified the original complex dynamics. The process built upon Laplace’s earlier work, refining his approach to understanding fluid motion and rotational effects in mathematical terms.
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In 1783, Adrien-Marie Legendre presented a paper to the Académie where he introduced what are now called associated Legendre functions. The work dealt with mathematical expressions involving points in a plane described by polar coordinates (r, θ) and (r ', θ'), with r ' greater than or equal to r. Using basic algebraic steps, he showed how to express the inverse of the distance d between these two points in a particular form. This development would later prove foundational in areas like physics and geodesy, where such mathematical tools help describe fields around spherical objects.
The formula for inverse distance between two points in space, expressed as a series, plays a key role in Laplace's work on gravitational and electromagnetic potentials. It involves distance $d$ between two points, with $r'$ being the distance from the origin to one point, and $r$ the distance to the other. The angle $\theta'$ represents the polar angle of the first point, while $\theta$ is that of the second. This expression appears in Laplace's studies of spherical harmonics, essential for solving problems involving fields around spheres. These functions describe how forces like gravity or electricity vary across spherical surfaces, becoming central to understanding planetary motion and atomic orbital structure. The mathematical form shown here is part of a broader framework that Laplace developed to calculate physical phenomena with precision.
The mathematical expression for inverse distance between two points in spherical coordinates involves a series expansion using Legendre polynomials, written as 1/d = (1/r')∑(k=0 to ∞) Pk(cos θ' - θ)(r/r')^k. This formula plays a key role in solving gravitational and electrostatic potential problems with spherical symmetry, simplifying complex integrals into manageable series essential in celestial mechanics and physics. Laplace's work laid the foundation for understanding force behavior in three-dimensional space, particularly when calculating fields around spherical objects. This approach became fundamental in later developments in mathematical physics and astronomy.
The set of functions P0k(cos φ) are known as the associated Legendre functions, and they play a vital role in mathematical physics. These functions allow any function defined on a circle to be expressed as a series, which makes them extremely useful for solving problems with spherical symmetry. Pierre-Simon Laplace recognized their importance when dealing with potential fields in astronomy and gravitation. By using these tools, scientists could decompose complex behaviors into simpler components, enabling more accurate modeling of natural phenomena. Their influence extended far beyond abstract mathematics, shaping how we understand forces acting across spherical surfaces. In this way, the associated Legendre functions became essential for advancing theoretical work in both physics and celestial mechanics.
Laplace extended the work beyond two dimensions, building on earlier efforts but without giving Legendre proper credit for his contributions. This led to a more comprehensive set of mathematical functions known as spherical harmonics, or sometimes Laplace coefficients. While that latter term isn't used much today, the idea remains important in fields like physics and mathematics. The extension was non-trivial, meaning it wasn’t a simple step forward but required real innovation to move from flat surfaces into three-dimensional space.
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In his work, Laplace introduced the concept of scalar potential, which was a significant step forward. He explained that the gravitational force acting on a body can be described as a vector — meaning it has both magnitude and direction. A potential function, however, is what we call a scalar function. This type of function helps define how those vectors will behave. Using a scalar function makes things easier to understand and calculate compared to working directly with vector functions.
Alexis Clairaut first proposed the idea in 1743 while tackling a similar problem, using Newtonian-style geometric methods. Laplace later praised Clairaut’s work as being "in the class of the most beautiful mathematical productions." But according to Rouse Ball, the concept was actually drawn from Joseph Louis Lagrange, who had already used it in memoirs from 1773, 1777, and 1780. The term “potential,” meanwhile, came from Daniel Bernoulli, who introduced it in his 1738 work Hydrodynamica. However, the specific phrase “potential function” didn’t appear until George Green used it in his 1828 essay on electricity and magnetism.
Laplace took the concept of the potential function and used calculus to show that it always obeys a specific differential equation. This work built on earlier ideas and helped establish the mathematical foundation for what would later be known as potential theory. His approach demonstrated how the language of calculus could be applied to understand the behavior of potential functions in general. The results he derived became essential tools in the study of gravitational and electrostatic fields. By showing that the potential function satisfies this particular differential equation, Laplace provided a powerful method for analyzing physical systems. This achievement marked an important step forward in mathematical physics. The equation itself became central to many later developments in science and engineering. His findings showed how abstract mathematical concepts could be used to describe real-world phenomena. The work laid the groundwork for future discoveries in the field of potential theory.
The equation written here is known as Laplace's equation, a fundamental tool in potential theory describing how a scalar field behaves in space when its second partial derivatives sum to zero. This mathematical statement appears in three-dimensional coordinates, with respect to x, y, and z. The symbol ∇² represents the Laplacian operator, which is central to understanding gravitational and electromagnetic potentials. Named after Pierre-Simon Laplace, this equation became essential for modeling physical systems where forces are conservative. It underpins much of classical physics and engineering, especially in fields like fluid dynamics and electrostatics. The form shown here applies to regions where no sources or sinks exist. In such cases, the potential function V satisfies this condition. This idea was crucial in Laplace's broader work on celestial mechanics and mathematical physics. His contributions helped shape how we understand forces and fields today.
Laplace used that result to continue his studies of gravitational attraction. He called the quantity ∇2V the concentration of V, and its value at any point reveals how much V surpasses the average in the surrounding area. This equation, now known as Laplace's equation, is a particular case of Poisson's equation and appears throughout mathematical physics. The idea of a potential shows up in areas like fluid dynamics and electromagnetism. Rouse Ball suggested it could be seen as "the outward sign" of something deeper in Kant’s theory of perception.
The spherical harmonics play a key role in solving Laplace's equation, especially when working with spherical coordinates—like those used for charting the night sky. By using a method called separation of variables, the equation can be broken into a radial part, which depends only on distance from the center, and an angular or spherical part. The solution to that angular portion is expressed as a series made up of Laplace's own spherical harmonics. This approach makes practical calculations much more manageable.
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In 1784, Laplace delivered a memoir on planetary inequalities, following up with two more sections in 1785 and 1786. The work focused largely on what is now called the great Jupiter–Saturn inequality. He tackled a long-standing issue in predicting how these planets move. Laplace showed that the gravitational influence of one planet on another couldn’t cause major changes in their orbital shapes or tilts. But he went further, identifying that something special happened in the Jupiter–Saturn system because their speeds around the Sun came close to matching in a simple ratio.
In the case of Jupiter and Saturn, commensurability arises when the ratio of their orbital speeds is nearly equal to a simple fraction involving small whole numbers. Specifically, two orbits of Saturn almost match five of Jupiter’s. This creates a periodic difference—represented as (2nJ − 5nS)—with a cycle of roughly 900 years. When calculating the gravitational forces between them, this period appears as a small divisor in the integration process. As a result, even tiny perturbations from this long-term interaction grow disproportionately large: about 0.8 degrees of arc in Saturn’s orbital longitude and roughly 0.3 degrees for Jupiter.
In 1788 and 1789, Laplace published two memoirs that further developed his theories on planetary motion. Thanks to his work, astronomers were finally able to create much more accurate tables for the movements of Jupiter and Saturn. It was based on Laplace’s theory that Delambre calculated his astronomical tables.
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Laplace set himself a grand challenge: to write a work that would fully solve the mechanical puzzle of our Solar System. He wanted his theory so closely aligned with observation that astronomers would no longer need empirical equations in their tables. The result of this effort is found in two major books: the Exposition du système du monde and the Mécanique céleste.
In 1796, Laplace published a work that offered a general explanation of astronomical phenomena while leaving out many details. The book included a summary of the history of astronomy, and it was this summary that earned its author recognition from the French Academy. It's widely regarded as one of the masterpieces of French literature, though it isn't fully dependable for the later parts it covers.
Laplace developed the nebular hypothesis of the Solar System’s formation, an idea first suggested by Emanuel Swedenborg and later expanded by Immanuel Kant. According to Laplace's version, the Solar System began as a rotating, globular mass of incandescent gas spinning around its center. As it cooled and contracted, rings broke off from its outer edge. These rings also cooled and condensed into planets, while the Sun formed from the central core that remained. On this model, Laplace predicted that planets farther from the Sun would be older than those closer to it. This hypothesis became the most widely accepted explanation for how planetary systems originate.
The nebular hypothesis had been laid out by Immanuel Kant in 1755, who also proposed "meteoric aggregations" and tidal friction as factors in how the Solar System came to be. Laplace likely knew of this work, though like many writers of his time, he often didn’t cite what others had said.
Laplace's work on the Solar System appears in his Mécanique céleste, published in five volumes. The first two came out in 1799, laying out methods for calculating planetary motion, figuring out their shapes, and solving tidal problems. Volumes three and four, published in 1802 and 1805, applied those techniques and included several astronomical tables. The final volume, in 1825, was mostly historical but also included appendices with Laplace’s latest findings. Though the Mécanique céleste features many of his own investigations, it also draws heavily on the work of others, often without clear credit. Historians see its conclusions as the organized result of a century of research by multiple writers, not just Laplace alone.
Jean-Baptiste Biot helped Laplace revise the work for publication, and Biot noted that Laplace often couldn’t retrace his own steps in the logic. When satisfied the conclusions were right, Laplace would simply write, “Il est aisé à voir que...,” meaning “It is easy to see that...” The Mécanique céleste translated Newton’s Principia Mathematica into the language of differential calculus and filled in gaps Newton had left unfinished. Félix Tisserand later published a more refined version titled Traité de mécanique céleste between 1889 and 1896, but Laplace’s treatise remains a standard reference.
In the years 1784 to 1787, Laplace published some powerful memoirs, and one from 1784 was later reprinted in the third volume of the Mécanique céleste. That work completely figured out the gravitational pull of a spheroid on a particle outside it. It’s especially known for introducing the concept of potential into analysis, which became a helpful tool across many areas of physical science.
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In 1812, Laplace published his Théorie analytique des probabilités, laying down fundamental statistical results. The first part dealt with probability methods and problems, while the second focused on statistical applications. Though his proofs sometimes lack rigor by modern standards and he moved between Bayesian and non-Bayesian perspectives, his conclusions remain largely sound. A year later, in 1819, he released a more accessible account of his work on probability. This later book relates to the Théorie des probabilités much like Système du monde does to Mécanique céleste. Emphasizing the analytical importance of probabilistic problems—especially in the context of approximating functions of large numbers—Laplace's approach went beyond what was common at the time. His treatise remained the most influential work in mathematical probability theory until the end of the 19th century, shaping the development of an analytically driven field even though its broader statistical relevance only became clear later.
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In his Essai philosophique sur les probabilités, published in 1814, Pierre-Simon Laplace laid out a mathematical approach to inductive reasoning using probability that we now recognize as Bayesian. He begins the work with a set of principles, the first seven defining how probability works. The first principle starts with the idea that all events have equal chances, unless proven otherwise. When that isn’t the case, we must figure out each event's individual probability. Then, to find the overall probability of a desired outcome, we add up the probabilities of all the possible favourable events.
When two events A and B are connected, the chance of both happening together equals the probability of A occurring, multiplied by the likelihood that B happens, given that A has already happened. This relationship helps us understand how events influence each other. On the flip side, if we know that event B has taken place, then the probability of A happening under those conditions is calculated by taking the chance of both A and B occurring, and dividing it by the probability of B alone. These ideas form the foundation of inductive probability, a way of thinking about how we can use what we observe to make predictions about what might happen next.
The sixth principle yields three corollaries that align with what we now call Bayesian rule. If a set of events {A1, A2, ..., An} represents all possible causes for another event B, then the probability of B can be expressed as the sum of those causes. The probability of a future event C is calculated by taking each cause Bi observed from event A, multiplying it by the chance that C occurs if Bi is true, and summing all such products. This method allows for updating beliefs based on new evidence, forming the foundation of inductive reasoning in probability theory.
The rule of succession, labeled as principle seven in his system, is one of the most recognized formulas developed by Laplace. He worked out this approach under the condition that there was little to no prior knowledge about the chances of success versus failure in a trial. The formula helps estimate the probability that the next attempt will succeed, based on how many times success occurred in earlier trials. It remains useful today when someone knows what outcomes are possible but has only a limited set of observations to go on.
The rule of succession has drawn criticism, in part because of an example Laplace used to explain it. He calculated that the chance the sun will rise tomorrow—given that it has risen every day in the past—was a certain fraction, where *d* stands for how many times it has risen. Some found this result ridiculous, and have dismissed all uses of the rule as equally absurd. But Laplace himself recognized the oddness of the conclusion. Immediately after giving the example, he wrote: “But this number... is far greater for him who, seeing in the totality of phenomena the principle regulating the days and seasons, realizes that nothing at the present moment can arrest the course of it.”
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In 1779, Laplace introduced a method for estimating ratios by treating the values of a function as coefficients in a mathematical expansion. These coefficients are linked to what’s called a probability-generating function, which is based on a different variable. Laplace demonstrated how interpolation could be used to find those coefficients from the generating function. He then turned to the reverse problem, showing how to recover the generating function when given the coefficients—accomplishing this through solving a finite difference equation.
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In 1805, Legendre published the method of least squares without connecting it to probability theory. Then in 1809, Gauss derived the normal distribution by showing that the arithmetic mean of observations yields the most probable value for a measured quantity. He later used this idea to prove that if errors are normally distributed, the least squares estimates provide the most probable regression coefficients. These works inspired Laplace to finish his treatise on probability, a project he had considered as early as 1783. The fourth chapter of that work presents an exposition of the method of least squares, demonstrating Laplace’s deep command over analysis.
In two papers from 1810 and 1811, Laplace worked on large-sample theory using characteristic functions and established the first general version of the central limit theorem. In a later supplement to his 1810 paper, after seeing Gauss's work, he argued that the central limit theorem gave a Bayesian justification for least squares: if individual observations were themselves averages of many independent measurements, then least squares estimates would maximize the likelihood function when treated as a posterior distribution but also minimize the expected posterior error—without assuming anything about the error distribution or falling into circular reasoning. Then in 1811, Laplace approached the problem differently, focusing on linear unbiased estimators within a regression framework. He showed that such estimators were approximately normally distributed with large samples and concluded that least squares gave the "best" linear estimators by minimizing asymptotic variance, which minimized expected absolute error and maximized the chance that estimates would fall within any symmetric interval around the true coefficient—no matter what the error distribution was. His work also covered the joint limiting distribution of least squares estimates for two parameters.
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If there existed an intellect vast enough to understand all the forces that move nature, and to know the exact position of every particle that makes up the world, then this mind could predict everything. Such a being would see the universe's past and future in perfect clarity, because it would grasp all movements—of the largest celestial bodies and the smallest atoms—in one unified formula. To this intellect, the future would be as clear as the past, since nothing would escape its grasp. This idea, known as Laplace’s demon, suggests that if we could know enough, everything could be calculated.
This intellect is often called Laplace's demon, a name that came later, not from Laplace himself, who never used the word "demon." The idea was inspired by a phrase he wrote: “Une intelligence ... Rien ne serait incertain pour elle, et l'avenir comme le passé, serait présent à ses yeux.” In English, that means nothing would be uncertain to it, and the future as well as the past would be present before its eyes. Some have also called it Laplace's Superman, after Hans Reichenbach. The concept imagines a being with perfect knowledge of everything in the universe, able to predict all future events based on complete understanding of the present.
Even though Laplace is most commonly credited with developing the idea of causal determinism, the notion was already circulating in philosophical thought during his time. A version of this concept appears in Maupertuis' Sur la Divination, published as early as 1756. Earlier still, in 1758, the Jesuit scientist Boscovich presented a scientific framework remarkably similar to what Laplace would later articulate in his own work.
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For certain differential equations, Laplace extended a technique of integral transforms through his own extensive study. However, the modern Laplace transform, which has the form:
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In his early years, Laplace kept himself away from politics and stayed focused on science at the Académie des sciences. During the most turbulent times of the Revolution, he wisely stayed out of Paris. He avoided getting caught up in the chaos, choosing instead to remain in his scholarly world. His caution helped him survive when others did not. He never let himself become involved in the dangerous currents of political life. That careful distance served him well through the violent years of the Revolution.
In November 1799, after Napoleon seized power in the coup of 18 Brumaire, he appointed Laplace as Minister of the Interior. This position lasted only six weeks before Napoleon handed it over to his brother Lucien Bonaparte. Clearly, once Napoleon felt secure in power, he no longer needed a well-known but untested scientist in his government. Napoleon later recalled this dismissal in his Mémoires de Sainte Hélène.
Pierre-Simon Laplace was a geometrician of the first rank, yet he quickly showed he was no ordinary administrator. His performance in office made it clear from the start that we had erred in appointing him. He failed to approach any matter with the proper perspective, instead always looking for hidden complexities, creating problems where none existed, and bringing the mindset of "infinitesimals" into the workings of government.
Grattan-Guinness describes these remarks as "tendentious", since there seems to be no doubt that Laplace "was only appointed as a short-term figurehead, a place-holder while Napoleon consolidated power".
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Laplace had been removed from his position, but there was still value in keeping his support. As a result, he was elevated to the senate. A note was added to the third volume of his Mécanique céleste, in which he declared that among all the truths presented, the most precious was the pledge of loyalty he offered here toward the peacemaker of Europe. That statement was deleted from editions sold after the Bourbon Restoration. By 1814, it was clear the empire was collapsing, so Laplace quickly offered his services to the Bourbons. Then, in 1817, during the Restoration, he was granted the title of marquis.
Paul Louis Courier wrote about the disdain his more honest colleagues held for Laplace's actions, and Rouse Ball notes that this contempt can be found in Courier's pages. Still, Laplace’s scientific expertise made him valuable on many commissions, and Rouse Ball suggests this likely explains why his political dishonesty went unnoticed.
Roger Hahn challenges the view that Laplace switched sides for personal gain, noting that like many in France, Laplace doubted Napoleon after the disastrous Russian campaign. The Laplaces were grieving—only daughter Sophie had died in childbirth in September 1813—and they feared for their son Émile, who was fighting on the eastern front. Napoleon had promised stability, but it was clear he'd overextended the nation, causing Laplace's loyalty to waver. Though he still had access to Napoleon, their personal relationship cooled. As a grieving father, he was deeply hurt by Napoleon's callous response, according to Jean-Antoine Chaptal: "On his return from the rout in Leipzig, he [Napoleon] accosted Mr Laplace: 'Oh! I see that you have grown thin—Sire, I have lost my daughter—Oh! that's not a reason for losing weight. You are a mathematician; put this event in an equation, and you will find that it adds up to zero.'"
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In the second edition of his Essai philosophique, published in 1814, Laplace added thoughts on politics and governance. He said that "the practice of the eternal principles of reason, justice and humanity that produce and preserve societies" is essential, and that deviating from them brings great risk. Noting how misery has followed when leaders ignored these truths, he subtly criticized Napoleon’s ambitions. He wrote that whenever a powerful ruler, driven by conquest, seeks dominance over all, the oppressed nations will eventually form a coalition against them—though they often fail. Laplace believed that "natural limits" exist among states, and that empires must stay within them to remain stable and prosperous. Those that go beyond those bounds will inevitably be pulled back, just like sea water returning to its level after being disturbed by storm.
After seeing the chaos of political change, Laplace began to think about how societies could evolve rather than explode. He took ideas from the natural world—what he knew from physics—and used them to shape his views on government. He believed in slow, steady progress instead of sudden, violent shifts. His thoughts were not just abstract; they came from observing how things in nature moved forward gradually, not by crashing into each other. This way of thinking shaped his approach to politics, where he supported small changes over big upheavals. He saw stability in the way systems worked, whether in the stars or in people’s lives. His method was rooted in the belief that order could grow from within, not be forced from without.
In the same way we have approached the natural sciences with observation and calculation, we must bring that same rigor to understanding politics and morality. We should not oppose the advancements brought by enlightenment, for they are inevitable and beneficial. Still, any shifts in our laws or traditions must be made carefully, with deep thought. Past experience shows us the harm old practices can cause, yet we cannot know all the consequences of change. Facing this uncertainty, probability theory tells us to resist alteration—particularly sudden shifts, which in both the physical and moral world always come at a great cost.
After living through the Revolution and the Empire, Laplace formed certain beliefs about how society should be organized. He thought that nature’s stability, as shown by scientific discoveries, offered the best example for keeping humanity safe and secure. According to Hahn, those ideas matched well with Laplace’s own unwavering personality.
In the Essai philosophique, Laplace shows how probabilities can be used in politics by supporting the Borda method of voting, which ranks candidates using whole numbers. He uses the law of large numbers to back up this approach, applying it to how new members were chosen for the Academy of Sciences. His explanation is so clear and strong that it could easily become a formal mathematical proof.
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Napoleon was told that Laplace’s book made no mention of God, so when he received it, the emperor asked Laplace directly: “M. Laplace, they tell me you have written this large book on the system of the universe, and have never even mentioned its Creator.” Laplace, known for being politically flexible but firm in his philosophical views, replied bluntly: “Je n'avais pas besoin de cette hypothèse-là.” (“I had no need of that hypothesis.”) Napoleon, amused by the response, shared it with Lagrange, who then said, “Ah! c'est une belle hypothèse; ça explique beaucoup de choses.” (“Ah, it is a fine hypothesis; it explains many things.”)
In Antommarchi's The Last Moments of Napoleon, published in 1825, there is an account of a conversation he had with someone referred to as L ..... He was discussing a recently published work and remarked on how the name of God appeared constantly in Lagrange’s writings, yet did not appear at all in this man’s. The reply came swiftly: “I had no need of that hypothesis.” Whether the identity of L ..... is known or not, the quotation preserves Laplace’s clear stance on divine intervention in science. It reflects his firm belief in natural laws and mathematical explanation, without reliance on supernatural causes. This moment, recorded in a memoir about Napoleon's final days, captures a defining aspect of his intellectual approach.
In 1884, the astronomer Hervé Faye said that the story about Laplace and Napoleon was a "strangely transformed" version of what really happened. Laplace never said God was a hypothesis. Instead, he was responding to Newton, who thought God had to step in now and then to keep the Solar System from falling apart. Newton believed this because he didn’t fully understand the stability of the system. But Laplace, having figured it out through deep analysis, would have told Napoleon that Newton wrongly called on God to fix the world’s machine. Laplace said he had no need of such an assumption. It wasn’t God himself that Laplace treated as a hypothesis, but rather God's intervention at a specific point.
Laplace’s younger colleague François Arago gave an eulogy for him before the French Academy in 1827. Arago told Faye that Laplace had tried to stop a version of his encounter with Napoleon from being published. The anecdote was about to appear in a biographical collection, and Laplace asked Arago to have it removed. It was necessary either to explain or delete it, and the easiest choice was to delete it. But the story was neither deleted nor explained.
The Swiss-American historian Florian Cajori, in 1893, reached a conclusion similar to Faye’s, though unaware of his work. Stephen Hawking, in 1999, clarified that Laplace wasn’t denying God’s existence, but rather that God doesn’t interfere with the laws of science. The only firsthand account of Laplace speaking with Napoleon comes from the diary of British astronomer Sir William Herschel, recorded on 8 August 1802.
The First Consul asked some questions about astronomy and the creation of the universe, and Laplace gave answers that pleased him. The Consul then turned to Monsieur de La Place on the same topic and argued with him at length, disagreeing with the great mathematician. The disagreement began when the Consul exclaimed, “And who is the author of all this!” Laplace tried to show that natural causes could explain how the universe came to be and continues to exist. The Consul was not convinced. Some say much can be said on the matter, and combining both viewpoints leads to "Nature and nature's God." Though the text does not mention Laplace saying, “I had no need of that hypothesis,” Daniel Johnson claims the phrase was never spoken by him. Arago’s account suggests he did, though not about God’s existence.
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Pierre-Simon Laplace was raised in the Catholic faith, but as he grew older, his views seem to have moved toward deism. That appears to be his genuine belief, since only that position shows up in his own writings. Still, people who knew him back then thought he might be an atheist. More recent scholars, though, have suggested he was actually agnostic.
Napoleon, while on Saint Helena, told General Gaspard Gourgaud that he had often asked Laplace about his views on God, and Laplace admitted he was an atheist. Roger Hahn, in his biography of Laplace, recounts a dinner party where the geologist Jean-Étienne Guettard was stunned by Laplace’s direct rejection of God’s existence. To Guettard, it seemed Laplace's disbelief was rooted in a complete materialism. Yet Jean-Baptiste Dumas, who knew Laplace well during the 1820s, said that although Laplace offered arguments commonly used by materialists, he did not personally hold those beliefs.
Hahn notes that Laplace never denied God’s existence in any of his writings, public or private. In a private letter dated 17 June 1809, Laplace wrote to his son, “Je prie Dieu qu'il veille sur tes jours. Aie-Le toujours présent à ta pensée, ainsi que ton père et ta mère,” which translates as, “I pray that God watches over your days. Let Him be always present to your mind, as also your father and your mother.” According to Ian S. Glass, who quoted Herschel’s account of Laplace’s exchange with Napoleon, Laplace was “evidently a deist like Herschel.”
In Exposition du système du monde, Laplace refers to Newton's view that the orderly arrangement of the Sun, planets, and comets must be the work of an all-powerful, intelligent Being. Laplace argues that if Newton had known what they later discovered, he would have been even more convinced: the conditions of how the planets and their satellites came to be arranged ensure their long-term stability. By demonstrating that this remarkable setup follows natural laws of motion, Laplace removes the need for divine intervention—what Newton had essentially "made" God do. Laplace agrees with Leibniz's criticism of Newton's reliance on God to fix the solar system, calling it a sign of "very narrow ideas about the wisdom and the power of God." He seems amazed that Newton believed God created a machine so poorly made that it would stop working without extraordinary interference.
In a collection of manuscripts hidden away in a black envelope at the library of the Académie des sciences, and later published by Hahn, Laplace presented a deist view of Christianity. He wrote that the “first and most infallible of principles … to reject miraculous facts as untrue.” Regarding transubstantiation, he said it “offends at the same time reason, experience, the testimony of all our senses, the eternal laws of nature, and the sublime ideas that we ought to form of the Supreme Being.” He called it “the sheerest absurdity” to believe that “the sovereign lawgiver of the universe would suspend the laws that he has established, and which he seems to have maintained invariably.”
Pierre-Simon Laplace challenged the application of probability to religious questions, particularly critiquing Pascal's wager. He argued that even if one accepted Pascal’s logic, the expected gain from believing in God was still worthless. That’s because the value of the testimonies supporting such belief is infinitesimally small, while the happiness promised is finite and significant. When you multiply these two together, the resulting hope of reward becomes infinitely small—so small it isn’t worth considering.
In old age, Laplace stayed curious about God and often spoke with the Swiss astronomer Jean-Frédéric-Théodore Maurice about Christianity. He said it was "quite a beautiful thing" and recognized its civilizing effect. Maurice noticed that though Laplace's beliefs seemed to shift gradually, he never abandoned his view that nature's laws were unchanging, making miracles impossible. After Laplace died, Poisson told Maurice, "You know that I do not share your [religious] opinions, but my conscience forces me to recount something that will surely please you." When Poisson praised Laplace's "brilliant discoveries," the dying man fixed him with a thoughtful gaze and said, "Ah! We chase after phantoms [chimères]." These were his last words, interpreted by Maurice as a realization of life's ultimate futility. The curé of Arcueil and the curé of the Missions Étrangères administered the last rites to him.
According to biographer Roger Hahn, it's not believable that Laplace had a traditional Catholic conclusion to his life, and he stayed a skeptic until his final days. In the later part of his life, people described him as an agnostic.
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