René Descartes

How to use this course

René Descartes is read today in three departments. Philosophers study the Meditations and the search for a certain foundation. Mathematicians owe him the marriage of algebra and geometry, much of modern algebraic notation and the rule of signs. Historians of physics meet him as the author of the sine law of refraction, the first quantitative theory of the rainbow, a law of inertia and a mechanical cosmos that dominated European science until Newton. He saw these as one project, carried out by one method.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations in modern notation and exercises with folded solutions. Where Descartes was wrong, as in his rules of collision, the lesson says so and shows how his successors corrected him.

Timeline

La Haye and La Flèche, 1596–1614

René Descartes was born on 31 March 1596 at La Haye, a small town in Touraine on the border with Poitou. The town has been called Descartes since 1967. His father Joachim was a lawyer and councillor in the Parlement of Brittany at Rennes, a position that brought minor nobility. His mother, Jeanne Brochard, died in 1597 after another childbirth. René, frail and with a persistent cough, was raised by his grandmother, while his father remarried and lived largely in Brittany. He had an elder brother, Pierre, and an elder sister, Jeanne.

Around 1607 he entered the Jesuit college of La Flèche in Anjou, founded a few years earlier under royal patronage. Because of his health he was allowed to stay in bed in the mornings, a habit he kept for life and liked to call the source of his best thinking. The curriculum ran from Latin and Greek through rhetoric to logic, physics and metaphysics in the Aristotelian manner. Mathematics was taught from the textbooks of Christoph Clavius, the Jesuit astronomer behind the Gregorian calendar.

A verdict on his schooling

In the Discourse on Method, written twenty years later, Descartes described leaving La Flèche in 1614 with more doubts than knowledge. He praised his teachers and the school, yet concluded that philosophy, after centuries of debate, still contained nothing that was not disputed. Only mathematics pleased him, for the certainty and evidence of its reasoning. He wondered that nothing loftier had been built on such firm foundations. That remark contains the programme of his life: to rebuild knowledge with the certainty of geometry.

Law, war and Beeckman, 1614–1620

To please his father, Descartes studied law at the University of Poitiers and received the baccalaureate and licence in civil and canon law in 1616. He never practised. Resolving to seek no knowledge but what he could find in himself or in the great book of the world, he joined the army of Maurice of Nassau at Breda in 1618 as a gentleman volunteer, paying his own way and studying military engineering.

Isaac Beeckman

At Breda in 1618 he met Isaac Beeckman, a Dutch schoolmaster and natural philosopher eight years his senior. Beeckman was one of the first in Europe to try to explain nature by the motions and collisions of small particles, and he believed physics should be mathematical. He set Descartes problems on falling bodies, hydrostatics and the vibrations of strings. Descartes wrote for him a short treatise on music, the Compendium musicae, as a New Year gift. Descartes later said that Beeckman had woken him from idleness.

In a letter of March 1619 Descartes told Beeckman of his plan for an entirely new science that would solve, in general, every question about quantity, continuous or discrete. He had already been studying instruments, which he called new compasses, that trace curves by linked motions. These compasses reappear in Book II of the Géométrie.

The dreams of 1619

Later in 1619 Descartes joined the Catholic army of Maximilian of Bavaria. Winter caught him in quarters on the Danube. According to his early biographer Adrien Baillet, who saw a lost notebook, on the night of 10–11 November 1619 Descartes shut himself in a stove-heated room and, after a day of intense thought, had three dreams. He interpreted them as a call to found a unified science on a new method. The episode has been read as mystical, as psychological crisis, and as literary self-fashioning. Whatever its nature, Descartes dated the beginning of his philosophical vocation to that day. He is reported to have been present at the Battle of the White Mountain near Prague in November 1620, and he left military life soon after.

Paris and the Dutch retreat, 1620–1637

For several years Descartes travelled in Germany, the Low Countries and Italy, then lived mostly in Paris. He sold inherited property and invested the proceeds, which gave him a modest independence for life. In Paris he met Marin Mersenne, a Minim friar whose cell was the clearing-house of European science, and the mathematician Claude Mydorge, who shared his interest in optics. Mersenne became his most faithful correspondent and for decades the only person who knew his address.

In these years he worked on the Regulae ad directionem ingenii, rules for directing the mind, which he abandoned unfinished after the twenty-first rule. They contain the core of his method and an early version of the algebra of magnitudes (Lesson 04). By 1626 he had the sine law of refraction, and Mydorge cut a hyperbolic lens following his design (Lesson 08).

The Dutch Republic

In 1628 he moved to the Dutch Republic in search of quiet. He enrolled at the University of Franeker in 1629 and at Leiden in 1630, and changed address often. He began Le Monde, a treatise on light and a mechanical cosmos in which the Earth moves. In 1633 he learned that Galileo had been condemned in Rome for teaching the motion of the Earth. He withheld Le Monde, which appeared only after his death. In 1630 he quarrelled with Beeckman, whom he accused of claiming credit for his ideas.

In 1635 Helena Jans van der Strom, a servant in the house where he lodged, bore him a daughter at Deventer. Francine was baptised at Deventer, and Descartes acknowledged her as his daughter. She died of scarlet fever in 1640, aged five, and he was deeply grieved.

The book of 1637

Unwilling to publish his whole physics, Descartes chose to show his method at work on three sample subjects. In 1637 the Leiden printer Jan Maire issued, in French and anonymously, the Discours de la méthode with three essays: La Dioptrique on light and lenses, Les Météores on weather and the rainbow, and La Géométrie. Writing in French rather than Latin was a deliberate appeal to readers outside the schools. Part II of this course follows the essays.

The method and the algebra of segments

For the general reader. Greek geometers multiplied two lengths and got an area; three lengths gave a volume; four gave nothing they could picture. Descartes removed the obstacle by choosing a unit length. Then the product of two lengths is again a length, and every algebraic operation becomes a ruler-and-compass construction. Algebra and geometry become two languages for the same facts.

Four rules

Part II of the Discourse reduces the method to four precepts: accept nothing as true that is not clearly and distinctly known; divide each difficulty into as many parts as possible; proceed in order from the simplest to the most complex; and make enumerations so complete that nothing is omitted. In mathematics this means: name every quantity, known or unknown, by a letter; express the conditions as equations; and reduce the equations until one unknown remains.

Notation

The Géométrie introduced conventions still in use. Known quantities are \(a,b,c,\dots\), unknowns \(x,y,z\), and powers carry raised exponents, \(x^3\), \(x^4\), although Descartes usually wrote \(xx\) for the square. He gathered all terms on one side and set the equation equal to zero, which made it natural to speak of the roots of a polynomial.

Proofs

For the product, triangles \(BAC\) and \(BDE\) are similar, so \(BE/BD=BC/BA\), that is \(BE=a\cdot b/1\). For the square root, the angle \(FIH\) is right (Thales), so triangles \(FGI\) and \(IGH\) are similar and \(GI^2=FG\cdot GH=a\). These two facts show that the field generated by the given lengths, closed under square roots, is exactly what ruler and compass can reach.

Quadratic equations

Descartes solved \(z^2=az+b^2\) as follows. Draw a right triangle \(NLM\) with \(LM=b\) and \(LN=\tfrac a2\), the right angle at \(L\). Extend \(MN\) beyond \(N\) to \(O\) with \(NO=NL\). Then \(z=OM\), since

which is the positive root. He gave similar constructions for \(z^2=-az+b^2\) and \(z^2=az-b^2\).

The Pappus problem and coordinates

For the general reader. An ancient puzzle asked for the path of a point whose distances to several fixed lines satisfy a certain proportion. The Greeks solved small cases with great effort. Descartes named two unknown lengths \(x\) and \(y\), wrote the condition as an equation, and read the answer from its degree. That move is the birth of analytic geometry.

The problem

Pappus of Alexandria, writing around 320 AD, reported a problem left incomplete by Euclid and Apollonius. Jacob Golius, professor at Leiden, put it to Descartes in 1631. Given lines \(\ell_1,\dots,\ell_n\) in a plane and, for each, a fixed angle, let \(d_i(P)\) be the length of the segment from \(P\) to \(\ell_i\) drawn at that angle. Find the locus of points \(P\) such that

with analogous conditions for odd \(n\), using the unit to keep the dimensions equal.

Descartes’s solution

He chose one of the given lines as a reference, measured \(x\) along it from a fixed point, and took \(y\) as the oblique segment from \(P\) to that line. Each \(d_i\) is then an affine function \(\alpha_i x+\beta_i y+\gamma_i\), because the triangles involved have fixed angles. The condition becomes a polynomial equation in \(x,y\).

For four lines the equation is \((\alpha_1x+\beta_1y+\gamma_1)(\alpha_2x+\beta_2y+\gamma_2)=\lambda(\alpha_3x+\beta_3y+\gamma_3)(\alpha_4x+\beta_4y+\gamma_4)\), of degree 2. Descartes then showed which conic arises from the signs of the coefficients, invoking Apollonius. Conversely, every conic arises this way, a fact Newton later proved synthetically in the Principia.

What the coordinates were

The Géométrie contains no pair of perpendicular axes, no negative coordinates plotted on purpose and no graph of a given equation. The axes were chosen for each problem and were usually oblique. The direction of thought ran from a geometric problem to an equation, not from an equation to its curve. Fermat, working independently in a manuscript circulated around 1636, took the opposite direction and studied the curves defined by given equations. The rectangular coordinate system now called Cartesian was shaped by van Schooten, Wallis, Newton and their successors.

Curves, normals and the quarrel with Fermat

For the general reader. Descartes believed that knowing a curve meant knowing its equation. To prove the point he gave an algebraic way to find the perpendicular to a curve at any point, which he called the most useful problem in geometry he knew. It turns tangents into a question about double roots, a step toward calculus taken before calculus existed.

Geometric and mechanical curves

Book II admits as geometric every curve traced by a continuous motion whose parts are governed by one another, as with his linked compasses. In effect these are the curves with a polynomial equation in \(x\) and \(y\). Curves such as the spiral and the quadratrix, which require two independent motions with an incommensurable ratio, he called mechanical and excluded. Leibniz later renamed them transcendental and brought them back with calculus.

The method of normals

Let a curve be given by \(F(x,y)=0\), and let \(P=(x_0,y_0)\) be a point on it. Seek a point \(V=(v,0)\) on the axis such that the circle of centre \(V\) through \(P\),

touches the curve at \(P\) rather than crossing it. Eliminate \(y\) between the two equations. The result is a polynomial in \(x\) that has \(x_0\) as a root. Tangency means that \(x_0\) is a double root. Descartes imposed this by equating the polynomial with \((x-x_0)^2\,R(x)\), \(R\) having undetermined coefficients, and comparing coefficients. Then \(VP\) is the normal at \(P\), and the tangent is perpendicular to it.

The quarrel of 1637–38

Mersenne sent Descartes a copy of Fermat’s method of maxima and minima, which treats an expression at \(x\) and at \(x+e\) as nearly equal, divides by \(e\), then sets \(e=0\). Descartes attacked it as lacking rigour and generality. Fermat replied with tangents to curves Descartes had proposed. In 1638 Descartes challenged him with the curve now called the folium of Descartes, \(x^3+y^3=3axy\). Fermat found its tangents easily. The two methods are equivalent in power, but Fermat’s was quicker and generalized to transcendental curves, while Descartes’s became unwieldy beyond low degree. Johannes Hudde, a student of van Schooten, later gave a simple rule for double roots that eased the computation.

The theory of equations

For the general reader. Book III of the Géométrie is a short textbook on polynomial equations. It says that a polynomial of degree \(n\) can have as many roots as its degree, explains how to read off the number of positive and negative roots from the signs of the coefficients, and shows how to solve cubic and quartic problems by intersecting a parabola with a circle.

Roots and factors

Descartes observed that \(x=a\) is a root exactly when \(x-a\) divides the polynomial, and that the product \((x-2)(x-3)(x-4)(x+5)=x^4-4x^3-19x^2+106x-120\) has roots \(2,3,4\) and the false root \(-5\). He asserted that an equation can have as many roots as its degree, counting false roots (negative) and imaginary ones. The word imaginary in this sense is his. A proof that every complex polynomial has a root came only with Gauss.

Proof sketch

Descartes stated the rule without proof; the first proofs came in the eighteenth century. Let \(Z(p)\) be the number of positive roots with multiplicity. Parity. Assume \(a_0\ne0\) (divide out powers of \(x\)). The sign of \(p\) near \(0^+\) is that of \(a_0\), and near \(+\infty\) that of \(a_n\). So \(Z(p)\) is even exactly when \(a_0a_n>0\), which is exactly when \(V(p)\) is even. Bound. Induct on the degree. By Rolle’s theorem, \(Z(p')\ge Z(p)-1\). The coefficient sequence of \(p'\) is that of \(p\) with \(a_0\) removed and the others multiplied by positive numbers, so \(V(p')\le V(p)\). Hence \(Z(p)\le Z(p')+1\le V(p')+1\le V(p)+1\), and parity forces \(Z(p)\le V(p)\).

Cubics by conics

Descartes reduced every cubic or quartic to the form \(z^4=pz^2-qz+r\) (a cubic first multiplied by \(z\)), then intersected the parabola \(y=x^2\) with a suitable circle. Substituting \(y=x^2\) into \(x^2+y^2+Dx+Ey+F=0\) gives \(x^4+(1+E)x^2+Dx+F=0\), and any reduced quartic has this form. The abscissae of the intersection points are the roots. With this construction he solved the classical problems of two mean proportionals and angle trisection.

Refraction and the Cartesian ovals

For the general reader. When light passes from air into water it bends. Descartes published the exact rule: the sines of the angles before and after keep a fixed ratio. He derived it by imagining light as a ball crossing a surface, and used it to design lenses that bring all rays to a single point, the goal of every telescope maker of his day.

The ball model

In the Dioptrique light is not literally a moving ball; it is a pressure transmitted instantaneously through a subtle medium. But Descartes argued that its tendency to move obeys the same rules as a tennis ball. He separated a motion’s speed from its determination, the direction, and split the determination into components parallel and perpendicular to the surface. At a reflecting surface the parallel component is unchanged and the perpendicular one is reversed, so the angle of reflection equals the angle of incidence.

At a refracting surface the speed changes by a fixed ratio \(v_2/v_1\) that depends on the two media, and the surface cannot affect the parallel component. So

To bend light toward the normal on entering water, Descartes’s model requires \(v_2>v_1\): light must pass more easily through the denser medium. Fermat rejected this, and in 1662 derived the same law from the principle that light takes the path of least time, assuming instead \(v_2<v_1\), so that \(n=v_1/v_2\). Both derivations give the same sine law with opposite readings of the constant. Foucault’s measurement of the speed of light in water in 1850 decided for Fermat. The law had been found earlier by Ibn Sahl in tenth-century Baghdad and by Thomas Harriot and Willebrord Snell, who did not publish. Charges that Descartes took it from Snell were made after his death; historians regard them as unproven.

Perfect lenses

A spherical lens does not bring parallel rays to a single focus. Descartes asked which surfaces do, the anaclastic problem, and answered with conics and with the ovals of Book II of the Géométrie.

In modern language, a perfect image requires equal optical path along every ray, \(n_1|PS|+n_2|SQ|=\text{const}\), which is (b). For (a), a plane wave in air reaches the surface point \(S=(x,z)\) after optical path \(z\) (measured along the axis), so \(z+n|SF|=c\), that is \(|SF|=\tfrac1n(c-z)\). Distance to \(F\) is \(1/n\) times distance to a fixed plane: a conic of eccentricity \(1/n<1\), an ellipse.

The rainbow

For the general reader. Why is the rainbow a circle of about 42 degrees around the point opposite the Sun? Descartes answered with a glass globe full of water and a long calculation. Rays entering a drop at different heights leave in different directions, but near one special ray they bunch together. That concentration of light, at a fixed angle, is the bow.

Theodoric of Freiberg had explained around 1307 that the primary bow involves two refractions and one internal reflection in each drop, and the secondary bow two reflections. What Descartes added in the Météores (Discourse 8) was the quantitative account. He measured with a large water-filled glass sphere where light emerged, then used the sine law to trace many rays numerically, and found that the emerging rays crowd near a particular angle.

Deviation

A ray meets a drop at incidence angle \(i\) and refracts to angle \(r\), with \(\sin i=n\sin r\). Each refraction turns the ray by \(i-r\), and each internal reflection by \(\pi-2r\). After \(k\) internal reflections the total deviation is

Derivation

Differentiate \(\sin i=n\sin r\): \(\cos i\,di=n\cos r\,dr\). Then \(dD_k/di=2-2(k+1)\,dr/di=0\) gives \(n\cos r=(k+1)\cos i\). Square and use \(n^2\cos^2r=n^2-\sin^2i=n^2-1+\cos^2i\): \(n^2-1+\cos^2i=(k+1)^2\cos^2i\), so \(\cos^2i=(n^2-1)/(k^2+2k)\).

Colours

Because \(n\) is slightly larger for violet than for red, the rainbow angle is slightly smaller for violet, and the primary bow has red outside. For the secondary bow the order reverses. Descartes saw that colour depended on the degree of refraction, and produced a spectrum with a prism, but he explained colour by the rotation speed of his light globules. Newton’s experiments of the 1660s replaced that explanation, keeping Descartes’s geometry.

Laws of nature, collisions and vortices

For the general reader. Descartes wanted to explain the whole physical world by matter in motion and nothing else. He stated a principle of inertia close to Newton’s first law and a conservation law for motion. Then he went wrong: his collision rules contradict experiment, and his whirlpools of matter carrying the planets could not reproduce Kepler’s laws. Still, the programme of a fully mechanical physics was his.

Matter and motion

For Descartes the essence of matter is extension. There is therefore no empty space, and every motion is part of a closed circuit of displaced matter. God created matter with a definite quantity of motion and conserves it. Quantity of motion is size times speed, a scalar.

The first two laws together are the principle of inertia, published here for the first time in its rectilinear form. Galileo had stated a horizontal, ultimately circular inertia; Beeckman had a version in his private journal. Newton’s first law is close in substance to Descartes’s first two laws, and Descartes’s second law introduced the centrifugal tendency that Christiaan Huygens later quantified as \(v^2/r\).

The rules of impact

From the third law Descartes derived seven rules for collisions of two hard bodies \(B\) and \(C\). Rule 1, two equal bodies with equal speeds meeting head-on rebound with their speeds, is correct. Rule 4 says that if \(C\) is at rest and larger than \(B\), then however fast \(B\) moves it cannot move \(C\) and simply rebounds. Huygens, in work begun in the 1650s, replaced the rules with correct ones by combining conservation of momentum with the relativity of motion and what is now conservation of kinetic energy in elastic impact.

Vortices

In Part III of the Principles space is filled with fluid matter turning in vortices. The Sun sits at the centre of one, and the planets float in it, each at the level where its tendency to recede balances that of the surrounding matter. Weight on Earth is the effect of the Earth’s own small vortex pushing bodies inward. The planets are at rest relative to the matter around them, so the Earth, strictly speaking, does not move: a way of holding Copernicanism after 1633. The theory explained qualitatively why planets move in the same direction in nearly the same plane. In Book II of the Principia (1687) Newton analysed a vortex in a viscous fluid and found periods growing like the square of the radius, against Kepler’s third law.

Meditations, Principles, Stockholm, 1637–1650

After 1637 Descartes turned to the foundations. The Meditations on First Philosophy appeared in Latin in Paris in 1641. Six meditations lead from radical doubt to the certainty of one’s own thinking, to the existence of God as guarantor of clear and distinct ideas, and to the real distinction between mind and body. Mersenne circulated the manuscript, and the book was printed with objections and Descartes’s replies, from Thomas Hobbes, Pierre Gassendi, Antoine Arnauld and others. A second edition of 1642 added further objections.

The new philosophy met resistance in the Dutch universities. At Utrecht, Henricus Regius taught Cartesian physics; the rector Gisbertus Voetius had the university condemn it in 1643. In the same year Princess Elisabeth of Bohemia, living in exile in The Hague, wrote to ask how an immaterial mind could move a body. Their correspondence, among the best philosophical letters of the century, continued until his death. Descartes dedicated to her the Principia philosophiae (Amsterdam, 1644), the textbook presentation of his whole system, from metaphysics through the laws of motion to vortices, magnets and the formation of the Earth. A French translation followed in 1647. In 1649 he published Les Passions de l’âme, an account of the emotions grounded in physiology and arising from the correspondence with Elisabeth.

Descartes visited France in 1644, 1647 and 1648. In 1647 he met the young Blaise Pascal in Paris and disputed the possibility of a vacuum.

Sweden

Queen Christina of Sweden invited him to her court to teach her philosophy and plan an academy. After hesitation he sailed in the autumn of 1649. The queen chose to hold their lessons at five in the morning, three times a week, in the Swedish winter. Descartes caught a fever, most probably pneumonia, and died on 11 February 1650, aged 53. Other explanations, including poisoning, have been proposed without convincing evidence. As a Catholic in a Lutheran country he was buried in a cemetery used for orphans, at the site of today’s Adolf Fredrik Church in Stockholm.

Cartesianism and its critics

Descartes’s body was brought back to France in 1666 and reburied in Paris; it was moved again in 1819 to the church of Saint-Germain-des-Prés. A skull kept in Paris and said to be his is of doubtful authenticity. In 1663 the Roman Congregation of the Index placed his works on the list of prohibited books.

Censure did not slow the spread of his ideas. Through van Schooten’s Latin Geometria (1649, enlarged in 1659–61 with commentaries by Hudde, Jan de Witt and others), the Géométrie became the standard entry to higher mathematics. John Wallis, Isaac Newton and Gottfried Wilhelm Leibniz all learned algebraic geometry from it. Cartesian physics became the orthodoxy of the Paris Academy, and vortex theories were defended in France well into the eighteenth century, until Maupertuis, Voltaire and the measurement of the Earth’s flattening turned French science toward Newton.

Legacy

• Mathematics: the identification of curves with equations, modern algebraic notation, the rule of signs, and the reduction of geometric construction to algebra, from which analytic geometry and the theory of equations grew.
• Optics: the first published sine law of refraction, the theory of aplanatic surfaces, and the quantitative theory of the rainbow.
• Mechanics: rectilinear inertia and a conservation principle for motion, both corrected and absorbed into Newtonian mechanics; and the ideal of a physics of contact action without occult forces, which shaped objections to Newtonian gravity for a century.
• Philosophy: methodical doubt, the cogito, mind–body dualism and the project of founding science on certainty made him the starting point of modern philosophy.

The coordinate plane that bears his name is a later construction, but the idea behind it, that a geometric problem becomes an equation once you name its unknown lengths, is his.

Sources and further reading

Biography

1. J. J. O’Connor, E. F. Robertson, René Descartes, MacTutor History of Mathematics.
2. René Descartes, Wikipedia (with extensive references).
3. G. Hatfield, René Descartes, Stanford Encyclopedia of Philosophy.
4. D. M. Clarke, Descartes: A Biography, Cambridge University Press, 2006.
5. S. Gaukroger, Descartes: An Intellectual Biography, Oxford University Press, 1995.
6. A. Baillet, La Vie de Monsieur Des-Cartes, Paris, 1691.

Mathematics and physics

1. R. Descartes, Discours de la méthode … plus la Dioptrique, les Météores et la Géométrie, Leiden, 1637.
2. R. Descartes, The Geometry of René Descartes, trans. D. E. Smith and M. L. Latham, Open Court, 1925; Dover reprint.
3. R. Descartes, Principia philosophiae, Amsterdam, 1644.
4. M. S. Mahoney, Descartes: mathematics and physics, Dictionary of Scientific Biography.
5. E. Slowik, Descartes’ Physics, Stanford Encyclopedia of Philosophy.
6. H. J. M. Bos, Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction, Springer, 2001.
7. La Géométrie, Descartes’ rule of signs, Cartesian oval, Folium of Descartes, Wikipedia.
8. Snell’s law (history section) and Rainbow, Wikipedia.
9. C. B. Boyer, The Rainbow: From Myth to Mathematics, Princeton University Press, 1987.
10. I. Newton, Philosophiae Naturalis Principia Mathematica, Book II, Section IX, 1687.