Émilie du Châtelet

How to use this course

Émilie du Châtelet is often remembered through Voltaire. This course puts her own work at the centre. She wrote a textbook of natural philosophy that tried to give Newton’s physics a Leibnizian foundation, she took a public side in the sharpest dispute in eighteenth-century mechanics, and she produced the only complete French translation of Newton’s Principia, with a commentary in the language of the calculus.

Each technical lesson opens with a paragraph for the general reader, then states results precisely, derives them in modern notation and ends with exercises whose solutions are folded. The physics follows her own work: the measure of force (1740–1741), the essay on fire (1737–1738), and the Principia commentary (1745–1749). Lesson 10 follows the idea of vis viva forward to the modern concept of energy.

Timeline

Paris, 1706–1725

Gabrielle Émilie Le Tonnelier de Breteuil was born in Paris on 17 December 1706. Her father, the baron de Breteuil, held a court post as introducer of ambassadors under Louis XIV and kept a weekly salon frequented by writers and savants. She was the only daughter among six children, two of whom died young.

Her father took her education seriously, at a time when daughters of the nobility were usually taught little beyond religion and manners. She had lessons in fencing and riding, and tutors came to the house. By her teens she read Latin and Italian well and later translated Latin and Greek texts into French; she also learned English, which would matter when she turned to Newton. Bernard le Bovier de Fontenelle, the permanent secretary of the Academy of Sciences, is said to have talked astronomy with her when she was about ten. A family friend, M. de Mézières, encouraged her early interest in mathematics.

Her biographers report that as a young woman she used her skill with numbers at the card table and spent winnings on books. The habit of buying books and instruments with her own money never left her.

Marriage, mathematics and Voltaire, 1725–1735

In June 1725, at eighteen, she married Florent-Claude, marquis du Chastellet-Lomont, an army officer eleven years her senior. The match was arranged in the usual way. The couple moved to Semur-en-Auxois in Burgundy, where her husband was governor. A daughter, Françoise-Gabrielle-Pauline, was born in 1726 and a son, Louis Marie Florent, in 1727. A third child, Victor-Esprit, born in 1733, died the next year. After that the marquis and marquise agreed to lead largely separate lives while keeping one household, an arrangement common in their class.

Return to mathematics

In 1733, aged 26, she resumed serious study. Pierre-Louis Moreau de Maupertuis, the Academy’s leading Newtonian, taught her algebra and the calculus. By 1735 she was working with the young Alexis Clairaut, who treated her more as a collaborator than as a pupil. A story is told that she was turned away from the Café Gradot, where the mathematicians met, and came back in men’s clothes. Whatever its accuracy, she did insist on taking part in the scientific conversation of Paris.

Voltaire

In the same year she began a relationship with François-Marie Arouet, Voltaire, that lasted until her death. In 1734 his Lettres philosophiques, which praised English liberty and Newton’s science, brought a warrant for his arrest. He took refuge at Cirey in Champagne, a neglected château belonging to her husband’s family, close to the border of independent Lorraine. She joined him there, and the two rebuilt the house as a place of work, with a large library and a gallery of physical instruments.

Cirey and the Institutions, 1735–1740

For about a decade Cirey was one of the most productive private households in Europe. Du Châtelet worked through the night on mathematics and physics. She translated and adapted Bernard Mandeville’s Fable of the Bees, adding a preface that argued for the education of women. She studied the Bible critically and wrote an essay on optics.

Voltaire’s Éléments de la philosophie de Newton (1738), the book that popularized Newtonian optics and gravitation in France, was dedicated to her, and its technical parts owe much to her. In 1737 she entered the Academy’s prize competition on fire independently and in secret (Lesson 07).

Leibniz, Wolff and König

She was not content with Newton’s refusal to give causes for gravitation. She came to think that physics needed metaphysical foundations, and she found them in Gottfried Wilhelm Leibniz and his systematizer Christian Wolff. In 1739, during a stay in Brussels on family legal business, Samuel König, a Swiss mathematician of the Leibniz–Wolff school recommended by Maupertuis, tutored her. After the book appeared König claimed that it repeated his lessons. Her early manuscript shows Leibnizian ideas already present before he arrived, and historians judge his role as that of a tutor.

The book

Institutions de physique appeared in Paris in 1740, framed as lessons for her thirteen-year-old son. A revised edition titled Institutions physiques followed in 1742, and German and Italian translations in 1743. The first chapters set out principles of knowledge: the principle of contradiction and the principle of sufficient reason. Then come God, space and time, and the elements of matter, which she took to be simple unextended beings in Leibniz’s manner. The later chapters treat motion, gravity, Newtonian attraction, pendulums, projectiles and, in chapter 21, the force of bodies.

Her chapter on hypotheses defended their use against strict Newtonians. A hypothesis, she argued, gains probability as it explains more phenomena; it may be partly true and partly false, and must be revised by experiment. Euler praised this chapter in a letter. In this way she occupied an unusual position: a convinced Newtonian on gravitation, and a Leibnizian on force and foundations.

Two measures of force

For the general reader. When a moving body strikes something, what measures the strength of its motion? René Descartes said mass times speed. Leibniz said mass times speed squared. For half a century the learned world argued about which was right. Du Châtelet chose Leibniz and defended him in print. Today we keep both quantities, as momentum and kinetic energy.

The two candidates

Descartes held that God conserves the total quantity of motion \(\sum m|v|\) in the world. In 1686 Leibniz published in the Acta Eruditorum a short paper announcing a “memorable error” of Descartes. He proposed instead the vis viva, living force, \(\sum mv^2\). Christiaan Huygens had already found that \(\sum mv^2\) is conserved in elastic collisions.

Leibniz’s argument

Leibniz started from two premises his opponents accepted. First, the same force is needed to raise a body of 1 pound through 4 feet as to raise 4 pounds through 1 foot. Second, from Galileo Galilei, a body falling from height \(h\) acquires speed \(v\) with \(v^2\propto h\); this speed is just enough to carry it back to height \(h\).

Let body A of mass 1 fall 4 units and body B of mass 4 fall 1 unit. Their speeds are in ratio \(\sqrt4:\sqrt1=2:1\). Their momenta are \(1\cdot2=2\) and \(4\cdot1=4\), unequal, although the two bodies can do the same thing: each can lift its own weight back to where it started, which by the first premise is the same task. Their vis viva are \(1\cdot 2^2=4\) and \(4\cdot1^2=4\), equal. So, Leibniz concluded, force is measured by \(mv^2\).

Where the Cartesians had a point

The Cartesians replied that a body moving twice as fast rises four times as high but takes twice as long to do it. Measured per unit of time, the effect is proportional to \(v\). In modern terms, a constant force \(F\) brings a body to rest after a time \(t=mv/F\) and over a distance \(s=mv^2/2F\). The first expression is the impulse–momentum theorem, the second the work–energy theorem. Each side had found one of them.

The clay experiment and the Mairan dispute

For the general reader. Drop a heavy ball into soft clay and measure the dent. Drop it from four times higher, so that it hits twice as fast, and the dent is four times deeper. Du Châtelet used this simple experiment of the Dutch physicist Willem ’s Gravesande to argue for Leibniz. When the secretary of the Academy of Sciences challenged her, she answered him in print and he gave up the exchange.

’s Gravesande’s experiment

Willem ’s Gravesande, professor at Leiden and a leading Newtonian, dropped balls of different masses from different heights into a bed of soft clay. Balls whose values of \(mh\), equivalently \(mv^2\), were equal made dents of equal size, whatever their momenta. His own results led him to abandon the measure \(mv\) for the force of a moving body. Du Châtelet reported these experiments in chapter 21 of the Institutions and made them the empirical centre of her case.

Derivation. With \(m\,dv/dt=-F\), the velocity falls linearly, \(v(t)=v-Ft/m\), and reaches zero at \(\tau=mv/F\). Multiplying \(m\,dv/dt=-F\) by \(v=dx/dt\) gives \(\frac{d}{dt}\big(\tfrac12mv^2\big)=-F\frac{dx}{dt}\); integrating, \(\tfrac12mv^2=Fd\). The depth measures \(mv^2\); the duration measures \(mv\).

The exchange with Mairan

In 1728 Jean-Jacques Dortous de Mairan had published a memoir in the Academy’s proceedings defending the Cartesian measure. His argument was the one in Lesson 04: if a ball with double speed digs four times as far, it does so in twice the time, so in equal times the effects are as the speeds. In the Institutions du Châtelet criticized the memoir directly. Mairan, by then secretary of the Academy, replied with a public letter dated 18 February 1741. Her Réponse appeared the same year. Its central claim was that the force of a body is measured by the whole effect it produces in being used up, whatever time that takes; time is not part of the effect. Mairan did not continue the exchange.

Dead force and living force

Following Leibniz, the Institutions distinguishes dead force, a mere tendency to motion such as the pressure of a weight on a table, from living force, the force of a body actually in motion. Dead force is measured by mass times the initial, infinitesimal velocity it would produce. Living force arises from the accumulation of infinitely many impressions of dead force, and its measure is \(mv^2\). In modern language dead force is a force and living force is the work done by a force over a distance.

Collisions and what is conserved

For the general reader. When billiard balls collide, two things survive the impact: the total momentum, counted with direction, and the total vis viva. When lumps of putty collide, only the momentum survives. Where the lost vis viva went was a puzzle that defenders of living force had to face, and it led eventually to the idea of heat as motion.

Derivation. Write the two laws as \(m_1(u_1-v_1)=m_2(v_2-u_2)\) and \(m_1(u_1^2-v_1^2)=m_2(v_2^2-u_2^2)\). Dividing the second by the first gives \(u_1+v_1=u_2+v_2\), so \(v_2-v_1=-(u_2-u_1)\): the relative velocity reverses. Solving these two linear equations gives the formulas.

Note that Descartes’s scalar quantity \(\sum m|v|\) is not conserved; the conserved momentum is the signed sum, a correction already made by Huygens. The Cartesian error was thus double: wrong as a measure of force, and wrong as a conservation law.

Inelastic collisions

If the bodies stick together, momentum conservation gives a common velocity \(V=(m_1u_1+m_2u_2)/(m_1+m_2)\) and

Leibniz held that the vis viva is not destroyed but dispersed among the small parts of the bodies, and du Châtelet followed him. The full vindication of this view, that the missing quantity becomes heat, came a century later (Lesson 10).

Fire and light

For the general reader. In 1737 the Paris Academy asked: what is fire, and how does it spread? Du Châtelet wrote her answer at night, in secret from Voltaire, who was writing his own. Her essay treated fire as weightless, linked heat closely to light, and suggested that light from other stars might contain colours missing from sunlight. Euler won the prize, but the Academy printed her essay, the first work by a woman it published.

The essay

The Dissertation sur la nature et la propagation du feu is a survey of experiment and argument rather than a report of new measurements. On the question of weight she disagreed with Voltaire, who had followed reports that metals gain weight when heated. She argued that fire itself has no weight and that the increase came from other matter. She treated fire and light as closely related. She also suggested that stars other than the Sun might emit colours absent from sunlight. A revised version, drawing more on Leibniz, appeared in 1744.

Both points have modern counterparts. Heating a body does increase its mass by \(\Delta E/c^2\), but by an amount far below any balance. And the colour of thermal light depends on temperature.

The weights of the 1730s were real but had other causes: a metal calcined in air gains weight because it combines with oxygen, a result established by Antoine Lavoisier in the 1770s.

Translating Newton: central forces

For the general reader. Isaac Newton wrote the Principia (1687) in Latin and in the language of geometry. By the 1740s Continental mathematicians did mechanics with the calculus of Leibniz. Du Châtelet’s edition did two things: it put the whole Principia into clear French, and it added a commentary that redid the key results with differential equations. This lesson does the central one: why an inverse-square force gives conic-section orbits.

The edition

She translated from the third Latin edition (1726), the last revised by Newton. Clairaut approved the plan on 20 December 1745 and checked her calculations as she went. The first volume contains Books I and II; the second contains Book III, ending with the General Scholium, followed by her commentary in two parts: an Exposition abrégée du système du monde, a non-technical account of the system of the world, and an analytic solution of the principal problems, written with the calculus and drawing on Clairaut’s work.

Derivation. In polar coordinates the radial equation is \(\ddot r-r\dot\theta^2=-GM/r^2\), and \(r^2\dot\theta=h\) because the force is central (this is Newton’s Proposition 1, the law of areas). With \(u=1/r\), \(\dot r=-h\,du/d\theta\) and \(\ddot r=-h^2u^2\,d^2u/d\theta^2\). Substituting, \(-h^2u^2(u''+u)=-GMu^2\), which is the stated equation. It is linear with constant coefficients, so \(u=GM/h^2+A\cos(\theta-\theta_0)\).

The name records exactly the quantity du Châtelet defended: the equation says that \(\tfrac12v^2-GM/r\) is constant along the orbit.

The figure of the Earth and the tides

For the general reader. Newton predicted that the spinning Earth bulges at the equator. In the 1730s French expeditions to Lapland and Peru, the first led by her teacher Maupertuis, confirmed it. In 1743 Clairaut gave the mathematical theory for an Earth whose density varies with depth. Du Châtelet’s commentary used this new work to correct Newton, and summarized Daniel Bernoulli’s prize-winning theory of the tides.

Flattening

Let \(m=\omega^2a/g_e\) be the ratio of centrifugal acceleration to gravity at the equator; for the Earth \(m\approx1/289\). Newton argued that a homogeneous fluid Earth has flattening \(f=(a-c)/a\approx\tfrac54m\approx1/230\). The Principia assumed a homogeneous Earth, an assumption du Châtelet pointed out that Newton had not justified.

The theorem lets one find the flattening, a geometric quantity, from pendulum measurements of gravity alone. For a homogeneous body \(f=\beta=\tfrac54m\); a body more concentrated toward its centre is less flattened.

Tides

The tide-raising acceleration is the difference between the attraction of a body of mass \(M\) at distance \(d\) on the near side of the Earth and at its centre. To first order in \(R/d\) it is

For the 1740 prize of the Paris Academy, on the tides, Daniel Bernoulli, Euler and Colin Maclaurin were among the prize-winners. Du Châtelet chose Bernoulli’s treatment, based on an equilibrium figure of the oceans, for her commentary.

From vis viva to energy

For the general reader. The quarrel over living force ended without a winner. Jean le Rond d’Alembert argued in 1743 that it was a dispute about words. A century later physicists made vis viva part of a much larger idea, energy, which is conserved in all processes once heat is counted. Du Châtelet’s position, that a hidden \(mv^2\) never perishes, turned out to be the right intuition.

Both follow from \(F=m\,dv/dt\), the first after multiplying by \(v\). D’Alembert’s Traité de dynamique (1743) made this point: force accumulated over distance is measured by \(mv^2\), and force accumulated over time by \(mv\). The term energy for vis viva was introduced by Thomas Young in 1807. Gaspard-Gustave Coriolis, around 1829, defined kinetic energy with the factor \(\tfrac12\) so that it equals work exactly. In the 1840s, James Joule’s measurements of the mechanical equivalent of heat and Hermann von Helmholtz’s 1847 essay on the conservation of force established the general principle.

Lunéville, 1748–1749

From the mid-1740s the Principia took most of her time. In 1746 the Academy of Sciences of the Institute of Bologna elected her a member. In these years she also wrote the Discours sur le bonheur, a reflection on happiness that values study as a source of independence, published long after her death.

In 1748 she and Voltaire spent time at Lunéville, the court of Stanisław Leszczyński, former king of Poland and duke of Lorraine. There she began a relationship with Jean-François de Saint-Lambert, a poet and officer. Late in 1748 she found herself pregnant, at 42. Fearing that she would not survive childbirth, she worked long hours to finish the translation and commentary, consulting Clairaut in Paris in the spring of 1749 and sending the manuscript to the Royal Library.

She gave birth to a daughter on 4 September 1749 at Lunéville. Six days later, in the early hours of 10 September, she died, with her husband, Voltaire and Saint-Lambert present. The child did not survive infancy. She was buried at Lunéville.

The 1759 Principia and after

Part of the translation was printed in 1756 under Clairaut’s direction. The complete edition, Principes mathématiques de la philosophie naturelle, appeared in two volumes in 1759, with a historical preface by Voltaire. It arrived as Halley’s comet returned, in the year when Clairaut’s Newtonian prediction of its perihelion was tested. It remains the only complete French translation; I. Bernard Cohen consulted it in preparing his English translation of 1999.

Legacy

• Physics: her defence of vis viva helped keep the \(mv^2\) quantity at the centre of mechanics until it became energy. Immanuel Kant’s first publication (1747), on the estimation of living forces, discussed her arguments.
• Philosophy of science: the Institutions is now read as a serious attempt to combine Newtonian physics with a Leibnizian foundation, with an original account of hypotheses. Parts of it were used in Diderot and d’Alembert’s Encyclopédie.
• Translation: the French Principia with its analytic commentary made Newton accessible to readers trained in the Continental calculus.
• Honours: a crater on Venus and the asteroid 12059 du Châtelet bear her name. The French Physical Society awards an Émilie du Châtelet Prize, and the Institut Émilie du Châtelet, founded in 2006, supports research on women and gender.

Connections

People linked to Émilie du Châtelet across this collection. A name without a link has no page here yet. See Émilie du Châtelet in the network of all 337 lives →

Teachers and advisors

• Pierre-Louis Moreau de Maupertuis 1733-1735
• Samuel König 1739

Influenced by

• Gottfried Wilhelm Leibniz
• Isaac Newton
• Christian Wolff
• Willem 's Gravesande

Influenced

• Immanuel Kant

Colleagues and collaborators

• Alexis Clairaut 1735-1749
• Leonhard Euler
• Voltaire 1733-1749

Rivals

• Jean-Jacques Dortous de Mairan 1740-1741

Friends

• Pierre-Louis Moreau de Maupertuis

Also mentioned in this life

• Gaspard-Gustave Coriolis
• Jean le Rond d’Alembert
• Daniel Bernoulli
• René Descartes
• Galileo Galilei
• Edmond Halley
• Hermann von Helmholtz
• Christiaan Huygens
• James Prescott Joule
• Johannes Kepler
• Antoine Lavoisier
• Colin Maclaurin
• Christopher Wren
• Thomas Young

Fields

• natural philosophy 12 lives
• mechanics 27 lives
• mathematics 15 lives

On CoursesHub

• Classical Mechanics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Gabrielle Émilie Le Tonnelier de Breteuil Marquise du Châtelet, MacTutor History of Mathematics.
2. Émilie du Châtelet, Wikipedia.
3. K. Detlefsen, Émilie du Châtelet, Stanford Encyclopedia of Philosophy.
4. R. Hagengruber, The Marquise du Châtelet: a controversial woman of science, arXiv:1406.7401.
5. J. P. Zinsser, Émilie du Châtelet: Daring Genius of the Enlightenment, Penguin, 2007.
6. D. Bodanis, Passionate Minds, Crown, 2006.

Physics and primary texts

1. É. du Châtelet, Institutions de physique, Paris, Prault, 1740.
2. É. du Châtelet, Réponse à la lettre que M. de Mairan lui a écrite le 18 février 1741 sur la question des forces vives, Wikisource.
3. I. Newton, Principes mathématiques de la philosophie naturelle, trans. and comm. by Mme du Châtelet, 2 vols., Paris, 1759.
4. Vis viva, Wikipedia.
5. C. Iltis, The Leibnizian–Newtonian debates: natural philosophy and social psychology, British Journal for the History of Science 6 (1973).
6. J. Zinsser and J. C. Hayes (eds.), Émilie du Châtelet: Rewriting Enlightenment Philosophy and Science, Voltaire Foundation, 2006.
7. I. B. Cohen and A. Whitman, Isaac Newton, The Principia: A New Translation, University of California Press, 1999.