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The life of René Descartes

Timeline 1596Born on March 31, in La Haye, in the Touraine region, France. 1607Enter the Jesuit school of La Flèche, one of the best in Europe. 1616He graduated in Law from the University of Poitiers. 1618He joins the army of Maurice of Nassau, in the Netherlands, and meets Isaac Beeckman. 1619On the night of 10 November, he has three dreams that he decides to follow as a mission. 1628He moves to the Netherlands, where he lives for about 20 years. 1633He gave up publishing The World upon learning of Galileo’s conviction. 1637Publishes the Method Discourse, with the appendix Geometry. 1641Publishes the Meditations on First Philosophy. 1643He begins exchanging letters with Princess Elisabeth of Bohemia. 1649He goes to Stockholm, at the invitation of Queen Christina of Sweden. 1650Dies in 11 February, in Stockholm. ◀ Previous Follow the journey Next ▶ 1596 La HayeBorn in a small town in Touraine, now called Descartes in his honour. 1607–1614 La FlècheHe studies at the Jesuit college of La Flèche, one of the best in Europe, and is allowed to stay in bed in the mornings. 1616 PoitiersHe graduates in law from the University of Poitiers. 1618 BredaHe joins the army of Maurice of Nassau and meets Isaac Beeckman, who challenges him with problems in physics and mathematics. 1619 near UlmOn a winter night, in a room heated by a stove, he has the three dreams he decides to follow as a mission. 1628–1649 HollandHe lives for about twenty years in the United Provinces, moving house many times. In Leiden he publishes the Discourse on Method (1637), with La Géométrie. 1649–1650 StockholmHe goes at the invitation of Queen Christina, who wants lessons at five in the morning. He dies of pneumonia on 11 February 1650. Enable JavaScript to see the map. The stages are listed below. Descartes’s path, from Touraine to Holland and, at the end of his life, to Sweden. Map: © OpenStreetMap contributors

October 10, 2026 Read
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Coordinates: geometry becomes algebra

The big idea of ​​Geometry is to give each point on the plane a pair of numbers. We choose two lines perpendiculars (the axes) and a point is located by the distances $x$ and $y$ to them. Today We call this Cartesian plane, in homage to him (Cartesius was Descartes’ name in Latin). With coordinates, a figure becomes an equation. See three examples. A page from The Geometry (1637). The letters a, b appear for known values ​​and the powers written at the top.René Descartes · public domain The distance between two points. The points $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ form, with the point $(x_2, y_1)$, a right triangle of legs $x_2 - x_1$ and $y_2 - y_1$. By the theorem of Pythagoras:

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Square root with ruler and compass

Geometry begins by showing that arithmetic operations can be done with segments. The most beautiful is the square root. To find $\sqrt{a}$: Draw on a straight line a segment $AB$ of length 1 and then a segment $BC$ of length $a$. Draw a semicircle with diameter $AC$. Lift a perpendicular through $B$ until you find the semicircle at point $D$. Then $BD = \sqrt{a}$. Change the value of a: the height BD is always √a. For a = 4, 9 or 2,25 the root is exact. Ative o JavaScript para ver esta figura interativa.

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The rule of signs

How many positive roots can a polynomial have? Descartes gave a simple rule: the number of roots positive is at most the number of sign changes between consecutive coefficients (and, as proved then differs from it by an even number). See with $x^3 - 6x^2 + 11x - 6$. The signs of the coefficients are $+\ -\ +\ -$: three changes. So there is 3 or 1 positive roots. In fact,

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The angles of polyhedra and Euler's formula

At one vertex of a cube there are three squares, and their angles add up. $3 \times 90° = 270°$. There are $90°$ left to complete one turn: this is the defect of the vertex. By Around 1630, Descartes discovered that, in a convex polyhedron, the sum of the defects of all vertices is always 720°. Polyhedron Angles at a vertex Defect Vertices Sum Tetrahedron 3 × 60° = 180° 180° 4 720° Cube 3 × 90° = 270° 90° 8 720° Octahedron 4 × 60° = 240° 120° 6 720° Dodecahedron 3 × 108° = 324° 36° 20 720° Icosahedron 5 × 60° = 300° 60° 12 720° The faces around a vertex, open in the plane. The red slice is the defect, the angle left to complete 360°. Ative o JavaScript para ver esta figura interativa.

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The kissing circles theorem

In 1643’s letter to Princess Elisabeth, Descartes dealt with four tangent circles two by two. Call the curvature of a circle the inverse of the radius, $k = \frac{1}{r}$. The theorem says that the curvatures satisfy $$(k_1 + k_2 + k_3 + k_4)^2 = 2\left(k_1^2 + k_2^2 + k_3^2 + k_4^2\right).$$ Application: three equal coins of radius 1 touch each other two by two. What is the radius of the circle that fits in the hole between them? With $k_1 = k_2 = k_3 = 1$ and $k_4 = k$:

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Why does the rainbow appear at 42°

In The Meteors, Descartes explained the rainbow using the law of refraction, which he published in The Dioptric. When light passes from air to water, it bends $$\sin i = n \cdot \sin r,$$ where $i$ is the entry angle, $r$ the angle into the water and $n \approx \frac{4}{3}$ for the water. (The Dutchman Willebrord Snell had found the same law before, but did not publish it. That is why it is called Snell-Descartes law.)

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Descartes and the idea of ​​function

Descartes never used the word “function”. But it was he who made the idea possible: in Geometry, a curve stops being just a drawing and becomes a relationship between two numbers, $x$ and $y$. When one changes, the other changes along with it, following a rule. In this part, we tell you how this idea gained name, symbol and graphic. The following five topics develop this reading and the subsequent history of the idea of ​​function.

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A reading of the Cogito: listening and speaking

Cogito ergo sum drawn on the Cartesian plane.Euler’s Eyes Collection Here at Euler’s Eyes we like to read “I think, therefore I am” as a little machine. The thought hears something (in Latin, audire) and speaks another (fabulari). The rule that binds what What you hear is the function. Method words physis: nature; methode: the reasoning, the path; graphiko’s: the lines, the drawing; máthema: what is learned, the act of deciphering; functus: who fulfills a task; dominus: the lord, where the word dominion comes from. An example: “say what you hear twice”. Empty parentheses act as a waiting ear a number:

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Descartes and the problem of cannons

Cannons and artillerymen in an old engraving.Euler’s Eyes Collection Descartes experienced war up close. In 1618 he served as a volunteer in the army of Maurice of Nassau and, according to his first biographer, Adrien Baillet, he was in 1628 at the siege of La Rochelle, led by Cardinal Richelieu. On the battlefields, a practical problem occupied the artillerymen: with what angle tilt the cannon to hit a target at a certain distance?

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The word "function"

The word comes from the Latin functus, participle of the verb fungor: fulfill a task, exercise an activity. The function is what “does the work” of taking one number to another. Already defunctus is someone who no longer performs any function, and hence the word defunct. A Leibniz notation for functions, from letters to Johann Bernoulli.Euler’s Eyes Collection The name and symbol took more than a century to establish itself:

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*Physis*: deciphering the traits of nature

Nothing happens in the universe without some rule of maximum or minimum appearing. — Leonhard Euler, 1744 For Leibniz and Euler, nature is made in the best possible way, and therefore follows rules. In our language: the reasoning (méthode) of the creator is in the traits (graphiko’s) of nature (physis), and it is up to us to decipher them (máthema). The same drawings appear in the most different places: Armadillo carapace: hexagonal plates.Euler’s Eyes Collection Earth’s crust: basalt columns in hexagons, formed when lava cools.Euler’s Eyes Collection Tenon: again, the hexagonal pattern.Euler’s Eyes Collection Nautilus shell: a logarithmic spiral.Euler’s Eyes Collection Egg: The oval curve distributes pressure like an arc.Euler’s Eyes Collection Starfish: pentagonal symmetry.Euler’s Eyes Collection Why so many hexagons? Only three regular polygons cover the plane without leaving holes: the triangle, the square and the hexagon. Of these, the hexagon is the one that encloses the most area with the least border. It is the most economical way to divide a space in equal parts, and that is what mathematician Thomas Hales proved in 1999.

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