René Descartes

Chapter 1 · La Flèche, and the Room With the Stove

A Jesuit education rejected on its own terms

1596 – 1619 · La Haye · La Flèche · Breda · Neuburg

René Descartes was born in 1596 at La Haye in Touraine, the son of a councillor of the parlement of Brittany. His mother died when he was a year old. He was sent at about ten to the Jesuit college of La Flèche, one of the best schools in Europe, and given a thorough education in Aristotelian philosophy, mathematics and the classics.

He came out of it unconvinced. His complaint, set out later in the *Discourse on Method*, was not that he had been badly taught but that he had been taught a great many things about which learned men disagreed entirely — and that in mathematics alone had he found reasoning certain enough to be relied on. Everything else rested on authority.

He took a law degree at Poitiers, then joined the army of Maurice of Nassau in the Dutch Republic as a gentleman volunteer — not for the fighting but, by his account, to see the world.

At Breda in 1618 he met Isaac Beeckman, a Dutch natural philosopher who set him problems in mechanics and hydrostatics and turned him toward physics.

And in November 1619, in winter quarters in Germany, he shut himself in a room with a stove and thought all day. He reported three vivid dreams that night which he took as a sign, and emerged with the conviction that all the sciences could be unified by a single method.

Why this matters

Descartes's programme begins with a specific complaint about his own excellent education: that only mathematics produced agreement, and everything else rested on authority.

You have the stove, the dreams and the programme. What would you ask him?

Ask Descartes

• “What was wrong with the education you got at La Flèche?”
• “What happened in the room with the stove?”
• “Why did mathematics convince you when philosophy did not?”
• “What did Beeckman set you working on?”
• “Why join an army you did not intend to fight in?”

Chapter 2 · A Curve Becomes an Equation

The Geometry, 1637

1628 – 1637 · The Dutch Republic · Leiden

Descartes moved to the Dutch Republic in 1628 and stayed for twenty years, moving house constantly, because he could work there without much interference.

In 1637 he published the *Discourse on the Method*, with three essays appended to demonstrate it — on optics, on meteors, and on geometry. The *Geometry* is the one that changed mathematics.

Before it, geometry and algebra were separate subjects. A curve was a shape constructed by a procedure; an equation was a relation between quantities. Descartes joined them. Fix two lines crossing at a point; locate any point in the plane by its distances along them; then a curve becomes precisely the set of points whose distances satisfy a given equation. A circle is one equation. A parabola is another.

The consequence is enormous. Geometrical problems that had required individual ingenuity became routine algebra. And, conversely, algebraic relations acquired shapes that could be looked at. The whole of later analysis depends on it, and so does every graph ever drawn with numbered axes.

He also introduced much of the notation still used: letters near the end of the alphabet for unknowns, letters near the beginning for known quantities, and superscript numerals for powers.

Pierre de Fermat had arrived at similar ideas independently and slightly earlier. The two quarrelled bitterly, over this and over optics, and Descartes was not gracious about it.

“I think, therefore I am.”

— René Descartes, Discourse on the Method (1637), Part IV

Why this matters

Analytic geometry turned two separate subjects into one and made every later graph, curve-fitting and calculus operation possible.

You have the axes and the equation of a curve. What is your question?

Ask Descartes

• “What changes when a curve can be written as an equation?”
• “Why do we still use x and y for unknowns?”
• “Was Fermat there before you?”
• “Why publish the Geometry as an appendix rather than a book?”
• “Why did you move house so often in Holland?”

Chapter 3 · The First Conservation Law

God does not change his mind

1644 · The Dutch Republic

In the *Principia Philosophiae* of 1644 Descartes set out a complete physics from a very short list of principles. Body is nothing but extension — length, breadth and depth. There is no vacuum, because extension is what body *is*, so where there is extension there is body. The world is matter in motion, and nothing else: no natures striving toward their proper places, no forms, no qualities.

And then the claim that matters: God, being immutable, conserves always the same total quantity of motion in the world.

This is the first conservation law in physics, and the idea behind it — that some quantity stays constant while everything visibly changes — is one of the most productive in the subject. Every conservation law since is a descendant.

From it he derived three laws of nature. First, each thing remains in the state it is in as far as it can. Second, all motion is of itself in a straight line, so that a body moving in a circle is always tending to recede along the tangent. Those two are essentially Newton's first law, stated forty years early, and they stand.

The third law — the rules governing what happens when two bodies collide — does not stand, and the reason is a single definition. Descartes defined the quantity of motion as the size of a body multiplied by its *speed*. Speed is a bare magnitude. It has no direction in it.

Why this matters

Descartes introduced conservation into physics, which was the right instinct; defining the conserved quantity without direction was the error that made most of his mechanics fail.

You have the conserved quantity and the definition of it. What would you ask?

Ask Descartes

• “Why should anything at all be conserved in the world?”
• “What made you think a body keeps moving without a cause?”
• “Why did you define the quantity of motion using speed rather than direction?”
• “If there is no vacuum, what is space?”
• “Why must motion of itself be in a straight line?”

Chapter 4 · Six Rules Out of Seven Are Wrong

Two coins, and whirlpools carrying the planets

1644 – 1690 · The Dutch Republic · Paris

Descartes gave seven rules for what happens when two bodies meet. Six of them are wrong, and one is spectacularly so.

His fourth rule states that a body at rest can never be set moving by a smaller body, however fast that body arrives; the smaller body simply rebounds with its speed unchanged. Anybody with two coins of different sizes can refute this on a table in about four seconds.

The root cause is the definition. If the conserved quantity is size times *speed*, with no direction, then a body that rebounds has the same quantity of motion after the collision as before — and the arithmetic of the whole encounter goes wrong. Christiaan Huygens identified this within a generation: the conserved quantity must be reckoned with its direction, so that motion one way can cancel motion the other. That directed quantity is momentum.

Descartes also refused gravity as an attraction across a distance, calling such things occult qualities that explain nothing. Instead he filled space with subtle matter circulating in enormous vortices, carrying the planets around the Sun like straws in a whirlpool. It is a genuinely mechanical account, it is entirely wrong, and it dominated French physics for decades — delaying acceptance of Newtonian gravitation in France until Voltaire and Émilie du Châtelet campaigned for it in the 1730s and 1740s.

His method was the problem. He deduced where he should have measured, and was openly contemptuous of experiment as a source of principles.

Why this matters

Descartes's collision rules fail because of one missing idea — direction — and repairing them is the entire content of momentum as a vector quantity.

You have the rule two coins can disprove. What is your question?

Ask Descartes

• “Two coins on a table refute your fourth rule — how did you miss it?”
• “What exactly did Huygens add to your conserved quantity?”
• “Why reject attraction across empty space as an explanation?”
• “How were the vortices supposed to move the planets?”
• “Would experiment have saved your physics?”

Chapter 5 · Five in the Morning in Stockholm

A queen, a winter, and the end

1635 – 1650 · Amsterdam · Egmond · Stockholm

Descartes had a daughter, Francine, born in 1635 to Helena Jans van der Strom, a servant in the house where he lodged. He acknowledged her, arranged for her upbringing, and planned to have her educated in France. She died of scarlet fever in 1640, aged five. He called it the greatest sorrow of his life, and there is no reason to doubt him.

His published work brought sustained attack. Theologians at Utrecht and Leiden accused him of atheism — an accusation that could be dangerous — and the *Discourse* and *Meditations* were eventually placed on the Index. He was cautious afterwards, and he had suppressed an earlier treatise, *Le Monde*, on hearing of Galileo's condemnation in 1633.

In 1649 Queen Christina of Sweden invited him to Stockholm to teach her philosophy. He accepted, apparently reluctantly. Christina required her lessons at five in the morning, in an unheated library, through a Swedish winter — and Descartes had spent his life rising late.

He caught a chill, developed pneumonia, and died on 11 February 1650, aged fifty-three. He had been in Sweden less than five months.

His remains were later moved to France; the skull went astray and is now in the Musée de l'Homme in Paris, separately from the rest of him. It is an oddly Cartesian ending for the man who argued that mind and body are two different substances.

Why this matters

Descartes suppressed his own treatise on the world after Galileo's condemnation, which is a concrete measure of what the Church's authority cost seventeenth-century science.

You have the daughter, the suppressed book and the Swedish winter. What would you ask?

Ask Descartes

• “Why did you suppress Le Monde after Galileo's trial?”
• “What did Francine's death do to you?”
• “Why accept an invitation you did not want?”
• “Were the accusations of atheism fair?”
• “If mind and body are separate, what killed you?”

What Descartes changed

Descartes put conservation at the centre of physics — the idea that something must stay constant through change — and created analytic geometry, which joined algebra to curves and made every later graph and the calculus itself possible. His first two laws of nature are the ancestors of Newton's first law. His collision rules were wrong, and correcting them produced momentum as a directed quantity.

A debate that continues

Whether Descartes or Fermat first arrived at coordinate geometry is genuinely disputed, and the extent to which his vortex physics delayed the acceptance of Newtonian gravitation in France is still argued.

Keep exploring — ask Descartes

• “Which of your seven rules would you rescue?”
• “Can a physics be built without any experiment at all?”
• “What did you actually mean by doubting everything?”

Related lives

• Christiaan Huygens — Who Watched From A Moving Boat
• Marin Mersenne — Who Counted A Musical Note
• Pierre de Fermat — The Margin Was Too Narrow
• François Viète — Who Gave The Known Quantities Letters

Related themes

Momentum and collisions · Coordinate geometry · Scientific method