Gaspard Monge

Based on Wikipedia: Gaspard Monge

The Merchant's Son Who Revolutionized How We See

In 1764, a seventeen-year-old teacher of physics at a provincial French college spent his summer break doing something unusual. He surveyed his entire hometown of Beaune, constructing the instruments himself as he went, and produced a map so precise that the town preserved it in their library. When a military engineer happened to see this map, he recognized something extraordinary—not just draftsmanship, but a mind that could see space differently than other people.

That teenager was Gaspard Monge, and within a few decades he would invent descriptive geometry, help father differential geometry, serve as Napoleon's minister of the navy, and transform how engineers, architects, and scientists communicate three-dimensional ideas on two-dimensional surfaces. Every technical drawing, every blueprint, every computer-aided design program owes something to the methods he developed.

But Monge's story is also about the brutal reality of eighteenth-century France, where talent meant nothing without noble birth—and about how a revolution can sweep away those barriers overnight.

Locked Out by Birth

After his map caught the attention of military authorities, Monge was offered a position at the Royal School of Engineering at Mézières. This sounds like a triumph. It wasn't.

Because Monge was the son of a merchant, he could only work as a draftsman. The actual engineering school—where mathematics was taught, where theories were developed, where officers were trained—was reserved exclusively for the aristocracy. Monge could draw their pictures. He could not join their classes.

"I was a thousand times tempted," he said years later, "to tear up my drawings in disgust at the esteem in which they were held, as if I had been good for nothing better."

So he worked on his ideas in his spare time, developing mathematical techniques that no one had asked for, that no one knew they needed.

The Problem That Changed Everything

Then came the assignment that would transform military engineering—and, eventually, all technical fields.

Monge was asked to produce something called a défilement plan for a fortification. To understand what this means, imagine you're defending a star-shaped fortress against artillery attack. Your enemies can position cannons on hills, on siege towers, on any elevated ground outside your walls. The question is: where inside the fort can your soldiers stand without being in direct line of fire?

Think of it this way. If you placed a lamp at every position where an enemy cannon might fire, the walls would cast shadows. The shadowed space is where your defenders are safe. But calculating this precisely—accounting for every angle, every elevation, every segment of your complex star-shaped walls—was enormously difficult.

The established method involved lengthy calculations that took about a week to complete. The famous military engineer Vauban had proposed a laborious observational process involving soldiers stationed at critical positions outside the fort, manually measuring sight lines.

Monge solved it in two days.

Seeing in Three Dimensions

His superiors refused to accept the solution at first. How could anyone solve in two days what took a week? They suspected trickery, or perhaps that Monge had simply guessed.

But when they examined his work, they found something revolutionary. Monge had developed a way to represent three-dimensional problems entirely through two-dimensional drawings. Instead of calculating everything algebraically, he constructed visibility cones graphically—showing exactly where lines of sight would and wouldn't penetrate.

This was the birth of descriptive geometry.

The technique was so valuable that the French military classified it as a state secret for years. The ability to rapidly calculate fortification designs, artillery positions, and defensive arrangements gave France a significant military advantage.

What makes descriptive geometry so powerful? It's a systematic way of projecting three-dimensional objects onto two-dimensional planes while preserving enough information to reconstruct the original object exactly. If you've ever seen an architectural blueprint with its plan view, elevation view, and section view all carefully coordinated, you're looking at Monge's method. If you've ever used computer-aided design software that shows you front, top, and side views of your model, you're using a digital implementation of principles he developed in the 1760s.

From Draftsman to Professor

Recognition came slowly at first. In 1769, when the mathematics professor Charles Bossut left the Royal School, Monge took his place. By 1770, he was also teaching experimental physics. The merchant's son who couldn't enter the school as a student was now teaching its aristocratic pupils.

His personal life took an interesting turn in 1777 when he married Cathérine Huart, who owned a forge. This marriage sparked Monge's interest in metallurgy—the science of working with metals. It's a reminder that in the eighteenth century, the boundaries between theoretical mathematics and practical craft were more porous than we might imagine. A mathematician might become fascinated by forge work because his wife happened to own one.

In 1780, Monge was elected to the French Academy of Sciences. He formed a lasting friendship with the chemist Claude-Louis Berthollet, who would later accompany him on one of history's more unusual scientific expeditions.

There's a touching story from 1783, when Monge was offered the chance to write a complete mathematics course after the death of the mathematician Étienne Bézout. Monge declined. His reason? Bézout's widow depended on income from sales of her late husband's textbooks. If Monge wrote a competing course, she would lose her livelihood. He chose loyalty to a colleague's family over his own advancement.

Revolution

Then came 1789.

The French Revolution swept away the aristocratic barriers that had limited Monge's early career. Suddenly, talent mattered more than birth. And Monge was a fervent supporter of the revolutionary cause.

In 1792, the new Legislative Assembly created an executive council, and Monge accepted the position of Minister of the Navy. He held this office for eight months—an extraordinary leap for a mathematics professor, though perhaps less surprising when you remember that the old naval officer corps had been largely aristocratic and was now in disarray.

When the Committee of Public Safety—the revolutionary government that would soon become infamous for the Reign of Terror—called on academics to help defend the republic, Monge threw himself into the work. He wrote technical manuals: "Description of the Art of Making Cannons" and "Advice to Ironworkers on the Manufacture of Steel." The mathematician who had married into a forge-owning family was now directing France's arms production.

Creating a New Kind of School

Monge's most lasting contribution to the revolution wasn't military—it was educational.

Together with Lamblardie and Lazare Carnot (himself a brilliant mathematician and military strategist), Monge founded the École Polytechnique. This wasn't just another engineering school. It was a radical reimagining of technical education.

The École Polytechnique combined rigorous mathematical theory with practical application. It trained students not for one specific profession but for multiple technical fields. Its graduates would go on to become engineers, scientists, military officers, and administrators. The school still exists today as France's most prestigious engineering institution.

Monge also helped establish what would become Arts et Métiers ParisTech, working alongside the chemists Berthollet and Chaptal and the polymath Laplace—names that still appear throughout scientific nomenclature.

At both the short-lived École Normale and the École Polytechnique, Monge taught descriptive geometry. His lectures, transcribed and published in 1799 as "Descriptive Geometry: Lectures Given at the Normal Schools," finally revealed to the world the techniques that had been kept as military secrets for decades.

And here's the connection to the blackboard in American classrooms. French mathematicians trained in Monge's methods emigrated to the United States, bringing with them the practice of demonstrating geometric constructions on large writing surfaces. Before this influence, American education had relied primarily on slates and individual copywork. The classroom blackboard—that fixture of education for two centuries—traces part of its heritage to Monge's visual approach to mathematics.

Adventure in Egypt

In 1796, Monge was sent to Italy with Berthollet and a team of artists. Their mission: to select the paintings and sculptures that France was seizing from Italian collections as spoils of war. This was cultural looting on a grand scale, though it was common practice among conquering powers at the time. It was also where Monge first became close to a young general named Napoleon Bonaparte.

That friendship would shape the rest of Monge's life.

In 1798, Napoleon launched his Egyptian campaign, and Monge went with him. Alongside Berthollet and a remarkable assembly of scholars, Monge participated in the scientific work of the Institut d'Égypte—the Egyptian Institute of Sciences and Arts. While Napoleon's military campaign would ultimately fail, the scientific expedition produced extraordinary results, including the discovery of the Rosetta Stone (though its decipherment would come later).

Monge returned to France with Napoleon in 1799 and was appointed president of the Egyptian commission. As Napoleon's power grew, so did Monge's honors. He was made a senator, given the title Count of Péluse (Péluse being the French name for Pelusium, an ancient Egyptian city), and served as president of the senate from 1806 to 1807.

The Deeper Mathematics

Throughout these adventures, Monge continued his mathematical work. And his contributions went far beyond descriptive geometry.

In 1781, Monge published a paper with an unassuming title: "On the Theory of Cut and Fill." The problem sounds mundane—how do you efficiently move earth when building fortifications or roads? If you need to dig here and pile there, what's the optimal way to move the dirt?

But Monge's elegant analysis of this problem established something profound: the curves of curvature of a surface. The great mathematician Leonhard Euler had studied curvature before, but he had looked at curvature of plane sections through surfaces. Monge asked a different question: how do the normal lines to a surface—the lines perpendicular to it at each point—relate to each other? Where do successive normals intersect?

This question opened up the field that would become differential geometry, the mathematical study of curves and surfaces using calculus. Differential geometry would eventually become essential to Einstein's general theory of relativity, to the design of modern aircraft and automobiles, to computer graphics, and to countless other applications.

The cut-and-fill paper also contained something else remarkable: the earliest known formulation of what we now call linear optimization problems. Monge essentially asked: given piles of earth to be moved and holes to be filled, what assignment of dirt-to-hole minimizes the total transport effort? This is now known as the optimal transportation problem, and it appears everywhere—in logistics, in economics, in machine learning, in image processing.

The mathematical distance measure Monge developed to solve this problem has been rediscovered repeatedly. The Soviet mathematician Leonid Kantorovich found it again while studying industrial production. The French mathematician Paul Lévy encountered it in probability theory. Today it goes by various names: the Wasserstein distance, the Kantorovich-Rubinstein metric, the earth mover's distance. It's fundamental to modern machine learning techniques, including some used in generative artificial intelligence.

A paper about moving dirt around became a cornerstone of fields that wouldn't exist for two centuries.

Liquefying the Unliquefiable

Monge also contributed to chemistry and physics. In a paper from 1783, he demonstrated that water is produced by burning hydrogen—though the credit for this discovery usually goes to Henry Cavendish, who had reached the same conclusion slightly earlier.

More strikingly, around 1783-1784, Monge worked with a colleague named Clouet on an experiment that seemed almost alchemical: they liquefied sulfur dioxide. By passing the gas through a U-tube surrounded by a mixture of ice and salt (which can reach temperatures well below the freezing point of water), they condensed the gas into a liquid.

This made them the first to liquefy a pure gas—a milestone in the history of physics. It demonstrated that the gaseous and liquid states of matter were not absolutely distinct, that a gas could be compressed and cooled into a liquid. This insight would eventually lead to the industrial production of liquid nitrogen, liquid oxygen, and all the cryogenic technologies we use today.

The Fall

Napoleon's empire collapsed in 1814 and, after a brief return, finally in 1815. And when Napoleon fell, so did Monge.

All his honors were stripped away. His title of Count was revoked. Most painfully for a scholar, he was excluded from the list of members of the reconstituted Institute of France—the successor to the Academy of Sciences where he had spent so many productive years.

Napoleon himself recorded that Monge was an atheist, which may have contributed to his posthumous treatment. He died in 1818, three years after Waterloo, his reputation in official disgrace.

But reputations recover. Today, Monge's name appears on the base of the Eiffel Tower, one of seventy-two names of distinguished French scientists inscribed there by Gustave Eiffel. A statue of him stands in his hometown of Beaune. The French Navy operates a ship called the Monge. And his remains, originally interred at Père Lachaise Cemetery, were eventually transferred to the Panthéon in Paris—France's temple of national heroes.

The Blackboard Legacy

Perhaps Monge's most unexpected influence reaches into every classroom in the world.

Before Monge, mathematics was taught primarily through textbooks and individual work. Students copied problems; teachers lectured from notes. But descriptive geometry demanded demonstration. You couldn't explain how to project a three-dimensional object onto a two-dimensional plane by simply talking about it. You had to show it.

So Monge and his colleagues at the École Polytechnique used large writing surfaces to demonstrate constructions step by step. When French scholars emigrated to America in the early nineteenth century, they brought this practice with them. American educators adopted it, adapted it, and spread it.

The classroom blackboard—along with its descendant, the whiteboard—became universal. And while many factors contributed to its adoption, the mathematical pedagogy that Monge pioneered played a significant role.

Every time a teacher stands at a board and works through a problem while students watch, they're participating in a tradition that connects back to a merchant's son in revolutionary France, who found a way to make invisible spatial relationships visible, and who believed that showing was better than telling.

What Monge Teaches Us

Gaspard Monge's life offers several lessons that remain relevant today.

First: practical problems can lead to profound theory. Monge started with fortifications and dirt-moving. He ended up founding new branches of mathematics. The deepest insights often come from taking mundane questions seriously.

Second: visualization is thinking. Monge's great contribution was not just calculating things more efficiently—it was finding ways to see problems differently. In an age of computer graphics and data visualization, this insight is more relevant than ever.

Third: educational institutions shape the future. The École Polytechnique trained generations of scientists and engineers who transformed France and the world. Monge understood that how you teach matters as much as what you teach.

Fourth: talent can be imprisoned by social structures. Monge might never have developed his methods if the Royal School hadn't at least hired him as a draftsman. How many Monges are there today, locked out of education or opportunity by accidents of birth, whose contributions we'll never see?

And finally: political winds shift. Monge rode the revolution to power and fell with Napoleon. His mathematics endured because it was true, regardless of who was king or emperor. In the long run, the ideas outlast the politics.

The next time you look at a blueprint, use a CAD program, or watch a teacher work through a problem on a whiteboard, you're seeing the world through methods that Gaspard Monge invented—a merchant's son who could see space differently, and who found a way to share that vision with everyone.