Stand beneath the Eiffel Tower and look up at the first platform. Running around the frieze, in gold letters more than half a meter tall, are 72 names. Gustave Eiffel chose them himself: French scientists, engineers and mathematicians who had distinguished themselves since the Revolution of 1789. Lagrange is there. So are Laplace, Cauchy, Fourier, Navier and Poisson, names every engineering student eventually meets in an equation.

All 72 are men. One name that belonged among them, perhaps more than most, was missing. The tower is a lattice of iron plates and members whose behavior under load depends on the theory of elasticity, and one of the first people to attempt a mathematical theory of how elastic surfaces bend and vibrate was a woman named Sophie Germain.

A Girl in Her Father’s Library

Marie-Sophie Germain was born in Paris in 1776, into a comfortable family. Her father was a prosperous merchant who later became involved in revolutionary politics. In 1789, when she was thirteen, the Revolution made the streets of Paris dangerous, and she spent long days confined to the house. She found her way to her father’s library.

There, the story goes, she read an account of the death of Archimedes. When Roman soldiers stormed Syracuse, the old mathematician was so absorbed in a geometric figure drawn in the sand that he ignored a soldier’s command, and was killed for it. Sophie wondered what kind of subject could hold a person so completely. She decided to find out.

Her parents disapproved. Mathematics was not considered a suitable pursuit for a girl, and they tried to stop her. According to accounts handed down from her family, they took away her candles and let the fire go out in her room so she could not study at night. She waited until the house was asleep, wrapped herself in blankets, and worked by candles she had hidden. Eventually her parents gave up.

The École Polytechnique did not admit women. So she borrowed the identity of a man who had dropped out.

Monsieur LeBlanc

In 1794 the École Polytechnique opened in Paris, designed to train the engineers and scientists the new republic needed. Women were not admitted. But the school made lecture notes available, and students were invited to submit written observations to their professors.

A former student named Antoine-Auguste LeBlanc had left Paris, and the school apparently had not noticed. Sophie obtained the notes under his name and began submitting work as “Monsieur LeBlanc.” One of the professors reading it was Joseph-Louis Lagrange, among the greatest mathematicians alive. He noticed that LeBlanc, who had been a poor student, had suddenly become remarkable. He asked to meet him.

Lagrange was surprised to find a young woman. To his credit, he was also supportive, and he became something of a mentor.

A few years later, Germain used the same pseudonym to write to Carl Friedrich Gauss in Germany, sending him her own ideas in number theory. Gauss, who was famously sparing with praise, took her seriously and wrote back. The disguise ended in 1806, when Napoleon’s army occupied Gauss’s home city of Brunswick. Remembering Archimedes, Germain feared for his safety, and asked a French general who was a family friend to make sure no harm came to him. When the general explained who had sent him, Gauss was confused. He had never heard of a Sophie Germain. She wrote to explain. Gauss’s reply expressed astonishment and admiration that a woman had overcome so many obstacles to reach the hardest parts of the subject.

The Problem of the Singing Plates

Around 1808, a German physicist named Ernst Chladni came to Paris with a striking demonstration. He sprinkled sand on a thin metal plate and drew a violin bow along its edge. As the plate vibrated, the sand danced away from the moving regions and settled along the still lines, forming intricate, symmetrical patterns. Different notes produced different figures.

The patterns were beautiful, and nobody could explain them mathematically. Napoleon was intrigued, and the Paris Academy of Sciences announced a prize for a mathematical theory of the vibrations of elastic surfaces that agreed with experiment. Many of the leading mathematicians of France judged the problem too difficult, or thought the mathematics of their day was not up to it.

Germain was the only person to submit an entry in the first round. It was rejected; her reasoning contained errors, which was not surprising for someone who had never received formal training in the calculus of variations. The deadline was extended. She tried again and received an honorable mention. She tried a third time. In 1816, at the age of 39, she won.

She was the first woman to win a prize from the Paris Academy of Sciences. She did not attend the ceremony; women were not welcome at its sessions unless they were the wives of members. When the Academy did not publish her winning memoir, she paid to publish it herself, in 1821.

What She Got Right, and What Others Fixed

Germain’s theory was not complete. Some of her assumptions, particularly about the conditions at the edges of a plate, were later shown to be wrong. Gustav Kirchhoff corrected them in 1850, and the classical theory of plate bending that engineers learn today owes as much to him and to later mathematicians as it does to her.

But she asked the right question, framed it in mathematical terms, and pushed it far enough that others could finish it. That matters more than it sounds. Plate theory is how engineers analyze bridge decks and floor slabs, ship hulls and aircraft skins, steel gusset plates and concrete walls. Every time a structural engineer checks how a flat element bends under load, they are working in a field she helped open.

She asked the right question, and pushed it far enough that others could finish the answer.

She also left a lasting mark on pure mathematics. Her work on Fermat’s Last Theorem was among the most important progress on the problem in the early nineteenth century. Mathematicians still study “Sophie Germain primes”: prime numbers p for which 2p + 1 is also prime.

An Ending, and a Late Correction

Germain never married. She developed breast cancer and died in Paris in 1831, at 55. Gauss had persuaded the University of Göttingen to award her an honorary degree, but she died before it could be conferred. Her death certificate, as it is often told, recorded her not as a mathematician but as a rentière, a woman of independent means.

When the Eiffel Tower opened in 1889, she had been dead for nearly sixty years, and her name was not among the 72. In 1913, an American writer pointed out the omission and suggested the obvious reason: she was a woman.

The correction has taken a long time. In January 2026, the city of Paris unveiled a list of 72 women scientists whose names are to be engraved in gold on the tower’s first level, alongside the original men. Sophie Germain is among them. The work is expected to be completed by 2027.

Why an Engineer Should Care

It is easy to read Germain’s story as a story about exclusion, and it is that. But it is also a story about how engineering knowledge actually gets built. Her first attempt failed. Her second was only partly right. Her final theory had flaws that someone else fixed decades later. None of that made her contribution smaller. It made it normal. Almost every useful theory in engineering arrived the same way: incomplete, contested, and improved by many hands.

The difference is that most people who make that kind of contribution are allowed into the room where the work is discussed. Germain was not. She had to borrow a man’s name to get the lecture notes, and she could not attend the ceremony for her own prize.

The next time you calculate the deflection of a slab or the stress in a plate, it is worth remembering that one of the people behind the theory had to study it by hidden candlelight. Her name is finally going up on the tower. It belonged there all along.

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Every civil engineering student meets him in a fluids course, usually in the second or third week. The instructor sketches a pipe that narrows in the middle, draws a few arrows, and writes a line that looks almost too tidy to be true: pressure, plus the energy of motion, plus the energy of height, stays constant along a streamline. Where the fluid speeds up, its pressure falls. Where it slows, the pressure returns.

It is called Bernoulli’s principle, and it is everywhere. It sizes the culverts under our highways and the spillways on our dams. It explains why a Venturi meter can measure flow without a single moving part, and how a Pitot tube on an aircraft’s nose reads airspeed. Water engineers lean on it so often that they stop noticing it, the way a carpenter stops noticing a level.

Few of them know that the man who published it was thrown out of his family home for doing well, or that his father later tried to claim the idea as his own.

A Family Business in Mathematics

The Bernoullis were a family of refugees before they were a family of mathematicians. Protestant merchants from Antwerp, they had fled religious persecution in the Spanish Netherlands and eventually settled in Basel, Switzerland, where they traded spices and kept their heads down.

Then, in a single generation, the family began producing mathematicians the way other families produce lawyers. Jacob Bernoulli helped found probability theory and discovered the constant we now call e. His younger brother Johann became one of the first great masters of calculus, a teacher to Leonhard Euler and, quietly, the source of much of the first calculus textbook ever printed. Over three generations the family produced eight mathematicians of note.

They were brilliant. They were also, by most accounts, impossible. Jacob and Johann spent years attacking each other in print, each insisting the other had stolen or bungled a solution. It was a family in which talent was abundant and generosity was not.

Johann’s father had tried to force him into business. Johann then did exactly the same thing to his own son.

A Son Steered Away

Daniel Bernoulli was born in 1700 in Groningen, in the Netherlands, where his father held the chair of mathematics. From early on he showed the family gift. Johann’s response was to try to keep him away from it.

Mathematics, Johann told him, did not pay. Daniel should become a merchant. The irony is hard to miss: Johann’s own father had once tried to push him into commerce, and Johann had fought it bitterly. Now he did the same thing to his son. Daniel resisted, and the two settled on a compromise that suited neither. Daniel would study medicine, and his father would teach him mathematics on the side.

It turned out to be a useful detour. Daniel’s medical thesis was on respiration, and his interest in how blood moves through the body fed directly into his later thinking about how fluids move through pipes. In 1725 he accepted a post at the new Academy of Sciences in St. Petersburg. He was lonely there, and wrote home saying so. His father’s solution was to send one of his best students to keep him company: a young Swiss named Leonhard Euler. The two became close friends and, for a few years, the most productive pair of mathematicians in Europe.

The Prize That Broke a Family

In 1734 the Paris Academy of Sciences, the most prestigious scientific body in Europe, announced the winners of its annual prize. That year the question concerned the orbits of the planets. The prize was split between two entries. One came from Johann Bernoulli. The other came from his son.

Most fathers would have been proud. Johann was furious. He could not tolerate being ranked as his son’s equal, and he banned Daniel from the family house. Daniel would go on to win the Paris prize many more times over his career. His father never forgave him for the first one.

Hydrodynamica, and Hydraulica

For years Daniel had been working on a book about how fluids move: water flowing through pipes, the force of a jet, the pressure on the walls of a vessel, the efficiency of pumps and machines for raising water. He finished the manuscript before leaving St. Petersburg, and after long delays it was printed in 1738 under the title Hydrodynamica. It was the first book to treat flowing fluids as a unified subject, and it is the reason the field still carries that name. In one chapter, almost in passing, Daniel modeled a gas as countless tiny particles colliding with the walls of their container, an idea that would become the kinetic theory of gases a century later.

Then Johann did something that still startles historians. He wrote his own book on the same subject, called it Hydraulica, and put a date of 1732 on it. The book drew heavily on his son’s work. The false date was meant to suggest the reverse: that the father had written it first, and the son had borrowed.

The deception did not hold. Colleagues recognized what had happened, and Johann’s reputation suffered for it. But the damage to the relationship was permanent. Daniel tried to reconcile. His father refused, and carried the grudge until his death in 1748.

The father had the talent to write his own masterpiece. What he lacked was the ability to let his son have one.

What the Equation Actually Says

Strip away the family drama and the idea is beautifully simple. A moving fluid carries energy in three forms: the pressure it exerts, the energy of its motion, and the energy it has because of its height. Along a smooth path, and if friction is ignored, the total stays the same. Squeeze the flow through a narrow section and it must speed up to get the same volume through. Because more of its energy is now in motion, less is left as pressure.

It is worth adding an honest footnote. Bernoulli’s principle is often used to explain why airplanes fly, usually with a story about air “needing” to travel faster over the curved top of a wing. That story is incomplete at best. The principle is real, and pressure differences do produce lift, but the full explanation involves how the wing turns the air downward as well. Even the most famous equations get misused when we want a simple story.

In engineering practice, the equation is a starting point. Real pipes have friction, real channels have turbulence, and real pumps lose energy. Engineers add correction terms for all of it. But the skeleton of almost every hydraulic calculation, from a storm drain to a hydroelectric penstock, is still the balance Daniel Bernoulli wrote down in the 1730s.

A Life Beyond the Quarrel

Daniel lived to 82. He returned to Basel, where he held chairs in anatomy, botany, physiology and finally physics, which is roughly the career of three ordinary scholars folded into one. He used mortality data to argue, with early statistics, that inoculation against smallpox would save lives, at a time when the practice was still controversial. He worked on vibrating strings, on probability, and on the problem of how people value risk, an idea economists still cite today.

He never married and had no children. Like Hooke and Newton, he left his inheritance in equations rather than in descendants.

Why an Engineer Should Care

There is an old temptation in engineering education to treat great names as statues: Bernoulli’s equation, Euler’s buckling load, as if the ideas arrived fully formed from people who had nothing else going on. The Bernoullis are a useful corrective. These were people with families and grudges, ambitions and insecurities. One of the most important books in the history of fluid mechanics was written by a man whose own father tried to discredit it.

There is a quieter lesson too. Johann Bernoulli was a great mathematician, and nothing in his own record needed his son’s work to prop it up. His mistake was treating knowledge as a contest in which one person’s success meant another’s loss. Daniel’s book did not diminish his father. It extended him. That is how knowledge is supposed to work.

The next time you write the Bernoulli equation on a page, remember that it carries a family’s name, and a family’s wound. Behind the tidy line of symbols is a young man who was told mathematics would not pay, did it anyway, and gave the world a principle that still moves its water.

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Years later, those same students are laying out bridges, tuning the suspension of a car, or modeling how a skyscraper sways in a gale — still leaning on two short expressions, and rarely sparing a thought for the men behind them. Yet behind the symbols stand two Englishmen who spent much of their lives locked in open hostility, and whose quarrel helped shape which discoveries the world got to keep.

Two Men, One Century

Robert Hooke was born in 1635 on the Isle of Wight. Isaac Newton followed seven years later, in 1642, in a Lincolnshire farmhouse. They came of age during what historians now call the Scientific Revolution — a stretch of decades when old certainties buckled, telescopes disclosed new worlds, microscopes revealed invisible ones, and mathematics was quietly becoming the language of nature.

Hooke and Newton did not merely witness that revolution; they stood near its center. And though each is remembered today for a single equation, both ranged across nearly every science of their age.

The Forgotten Genius

Ask an engineering student what Robert Hooke discovered, and the answer comes back in three words: Hooke’s Law. It is a thin epitaph for so restless a mind.

Hooke improved the microscope and the telescope. He helped survey and rebuild London after the Great Fire of 1666. He designed instruments, puzzled over fossils, and made observations in astronomy. Peering at a sliver of cork through one of his own lenses, he saw a honeycomb of tiny compartments and gave them a name that has never left biology: cells.

His book stunned a continent. Then history filed him under a single law.

His 1665 volume, Micrographia, astonished Europe with its fold-out engravings of fleas and crystals and the compound eye of a fly. For many readers it was the first glimpse of a hidden microscopic world. Hooke’s curiosity seemed to have no floor. And yet posterity has narrowed him to one equation about springs.

The Architect of Mechanics

Newton needs less rescuing. He devised the calculus independently of Leibniz, formulated three laws of motion, explained the orbits of the planets, and bound the heavens and the earth together under a single law of universal gravitation.

His 1687 masterwork, the Philosophiæ Naturalis Principia Mathematica, is routinely called one of the most important books ever written. Few thinkers have reshaped humanity’s picture of nature so completely, or left successors so ready to say, in Newton’s own borrowed phrase, that they had seen further only by standing on the shoulders of giants.

An idea, Newton insisted, is not enough. Science also demands proof.

THE HEART OF THE QUARREL OVER GRAVITY

A Feud Conducted in Letters

It is tempting to imagine Hooke and Newton in separate eras. They were not. Both belonged to the Royal Society; both attended its meetings, traded letters, and pored over each other’s work. What began as collegial scrutiny curdled, over the years, into something closer to enmity.

The first real rupture came in 1672, when Newton presented his experiments on light and color — the claim that white light is a mixture of the spectrum’s hues. Hooke, who had his own theory of light, objected sharply. Scientific disagreement is ordinary and healthy. Personal disagreement is more corrosive. Newton took the criticism to heart, and for years afterward grew reluctant to publish at all. It is sobering to wonder how many ideas stayed locked in his notebooks because of it.

Then Came Gravity

The bitterest dispute concerned gravity itself. Hooke believed he deserved credit for suggesting that the planets are drawn toward the Sun by a force that weakens with distance. Newton allowed that Hooke had, indeed, floated the notion.

But Newton drew a hard line between a suggestion and a demonstration. Hooke had a hunch; Newton supplied the mathematics. Using geometry and his new calculus, he proved that the very law pulling an apple to the ground could also hold the Moon in its orbit and swing the planets around the Sun. That proof changed science forever.

Hooke felt robbed of his due. Newton felt Hooke was claiming ownership of mathematics he had never done. Tellingly, it was in a 1675 letter to Hooke that Newton wrote his most famous sentence — about seeing further by standing on giants’ shoulders — a line some historians read less as humility than as a quiet barb at his rival. The argument never truly ended.

The Irony of It All

Here is the twist the two men never lived to enjoy. They spent years at each other’s throats. Yet today every engineering student, unknowingly, brings them back together at the very first opportunity.

Hooke’s law tells us how an elastic material pushes back when it is stretched or squeezed. Newton’s second law tells us how a force sets mass into motion. Apart, each is powerful. Set one equal to the other — the spring’s restoring pull on one side, the resulting acceleration on the other — and out falls the equation of simple harmonic motion, the seed of vibration analysis, structural dynamics, and much of modern mechanical engineering. The rivals who could not agree in life now collaborate on every blackboard.

Beyond Springs

At a glance, Hooke’s law looks like a fact about coil springs. Its reach is far wider. Wherever engineers analyze elastic deformation — a steel beam bending, reinforced concrete before it cracks, a machine part under load, a vibration isolator absorbing a jolt — Hooke’s insight is quietly at work.

Newton’s second law travels just as far from the orchard. It governs cars and aircraft, satellites and spacecraft, bridges shaken by earthquakes, towers pushed by wind, the joints of a walking robot, even the mechanics of the human body. These two equations are among the first things an engineer learns precisely because they remain among the most useful things an engineer will ever know.

The Missing Face

One small mystery clings to Robert Hooke. Newton’s face is familiar to millions; Hooke’s is unknown. No authenticated portrait of him is known to survive.

For centuries a rumor persisted that Newton, in his years of power at the Royal Society, destroyed Hooke’s likeness out of spite. Modern historians find little to support the story; the portrait was more likely lost to ordinary neglect than to malice. A painting once hopefully identified as Hooke turned out, on closer study, to depict another natural philosopher entirely. And so one of the founders of modern science looks out at us from behind a blank frame — present in every laboratory, absent from every wall.

A Legacy Without Heirs

Neither man married; neither left children. In the ordinary sense, both family lines simply stopped. And yet each achieved something exceedingly rare. Three centuries after their deaths, millions of students still learn their ideas by heart.

Their true descendants are intellectual: every scientist, engineer, and inventor who has built on their foundations. Few inheritances outlast an equation that keeps on explaining the world.

Why an Engineer Should Care

Engineering is often taught as a catalogue of formulas to be trusted and applied. History offers a useful correction: every formula began as a question. Someone looked at the world a little differently. Someone doubted the accepted answer. Someone failed, and failed again, and kept going in the face of criticism.

Hooke and Newton were not saints. They competed, they sulked, they claimed and counter-claimed. And still, between them, they laid the foundations of modern mechanics. Perhaps that is the quiet lesson underneath the equations: progress is rarely the gift of a lone genius. It is built, argument by argument, by people who inspire and provoke and occasionally infuriate one another. That is how knowledge grows.

The next time you meet F = ma or F = −kx, pause a moment. These are more than equations. They are reminders that behind every discovery is a human story — curiosity and stubbornness, rivalry and reconciliation, and the long, relentless pursuit of understanding. Perhaps that is the truest lesson of engineering itself.

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Did you know that the two scientists behind some of the most important equations in engineering were fierce rivals?

Or that one of history’s greatest mathematicians was forced to flee his home because of religious persecution?

Or that several of the people who changed science forever left no descendants—but their ideas continue to shape our world centuries later?

I’m excited to announce a new article series:

The Giants Behind the Equations

Each article will explore the life, struggles, friendships, rivalries, failures, and discoveries of the scientists and mathematicians who built modern engineering.

This series is for:

• High school students

• College students

• Engineers

• Teachers

• Anyone who loves science and history

🎓 Student Challenge

To celebrate the launch, I’m offering a $100 prize for one outstanding high school student.

To enter:

1. Read one article from the series.

2. Write a thoughtful reflection (500–1,000 words) answering:

   • What surprised you most?

   • What did you learn?

   • How did the scientist’s story change the way you think about mathematics, physics, or engineering?

3. Submit your reflection before the deadline.

The winning entry will receive $100 and may be featured (with the student’s permission) in a future edition of this newsletter.

My goal isn’t just to teach equations.

My goal is to inspire the next generation of scientists, engineers, and curious thinkers.

Because behind every great equation is an even greater story.