Descartes: The Geometry

Geometry is about shapes: lines, circles, triangles. Algebra is about numbers and letters, like $x + 3 = 5$. For two thousand years they were mostly separate subjects. René Descartes, a French thinker, joined them together about 400 years ago.

His trick: treat lengths like numbers. Call one length $a$, another $b$. Then you can add them, multiply them, even take square roots, by drawing lines. And the other way around: a curve, like a circle, can be described by an equation that tells you where its points are.

Today, every time you draw a graph on grid paper, with points like (3, 2), you are using an idea that grew from Descartes's book. That's why the grid is called the Cartesian plane, from his name in Latin, Cartesius.

Who, when and why

René Descartes (1596–1650) published The Geometry in 1637 as the third essay after his Discourse on Method (the rung before this one), to show his method at work in mathematics. It is in French and has three books: problems that need only straight lines and circles; the nature of curved lines; and solid problems, which lead to equations of the third and higher degree.

The problem Descartes faced

Greek geometers had solved many problems by brilliant, special constructions, one trick per problem, and some problems they could not solve at all. A famous one, from Pappus of Alexandria (about AD 320), asks for the points whose distances to several given lines stand in a fixed proportion. Descartes wanted a general method: a way to turn any such problem into equations and then solve the equations step by step, as his rules in the Discourse said.

Idea 1: lengths behave like numbers

The Greeks thought of the product of two lengths as an area, and three lengths as a volume; four lengths multiplied made no sense. Descartes chose one line as the unit. Then the product of two lines can be another line, by similar triangles, and there is no limit to how many can be multiplied.

Square roots work too. Lay the unit EG next to the line GH, draw a half-circle on the whole of EH, and raise an upright line at G. Where it meets the circle, at I, the height GI is the square root of GH.

Idea 2: letters for lengths

Descartes's notation is very close to ours. He used letters early in the alphabet ($a, b, c$) for known quantities and letters near the end ($x, y, z$) for unknowns, and small raised numbers for powers, like $a^3$, though he usually wrote $aa$ for $a^2$. Algebra in school still follows these choices.

Idea 3: solving an equation by drawing

For an equation like $z^2 = az + b^2$, Descartes draws a right triangle NLM with legs $LM = b$ and $LN = \tfrac{a}{2}$, extends the hypotenuse MN past N by a length NO equal to NL, and the whole line OM is $z$. This is the geometric twin of the quadratic formula.

Idea 4: curves as equations

In Book II Descartes measures each point of a curve by two lengths: how far along a fixed line, and how far out from it at a fixed angle. Then the curve is everything that satisfies one equation in those two unknowns. He sorted curves by the degree of their equations, showed how to find the line at right angles to a curve at any point (the key step toward tangents and calculus), and solved Pappus's problem: the answer is a conic section, a circle, ellipse, parabola or hyperbola.

Idea 5: counting the roots

\"True\" roots are positive and \"false\" roots negative. Descartes's own example is $x^4 - 4x^3 - 19x^2 + 106x - 120 = 0$: three changes of sign and one repeat, with roots 2, 3, 4 and −5. Today this is Descartes's rule of signs, which gives the most positive roots there can be (the true count can be smaller by an even number).

Getting the history right

• Descartes did not draw our x–y grid. His book has no pair of perpendicular axes with negative numbers on both sides, and few graphs of equations. That familiar picture was built over the next century by his followers, especially Frans van Schooten, who published a Latin edition with commentary, and later by John Wallis, Isaac Newton and Leonhard Euler.
• He was not alone. Pierre de Fermat, a lawyer in Toulouse, worked out the same basic idea, describing curves by equations, by about 1636, and circulated it in manuscript; it was printed only in 1679.
• The fly on the ceiling. A popular story says Descartes got the idea of coordinates by watching a fly on his bedroom ceiling. It is a legend; there is no evidence from his own time.

What later science kept, and what it changed

Kept: the marriage of algebra and geometry (analytic geometry), the use of letters for knowns and unknowns, exponents, and the rule of signs. Newton and Leibniz built calculus on this foundation, and almost all of physics is written in its language. Changed: Descartes judged that only curves with algebraic equations belonged in geometry and excluded "mechanical" curves such as spirals. Mathematicians after him, starting with Newton and Leibniz, brought those curves back in.

Two pictures of one thing. After Descartes, a circle can be thought of as a shape you draw, or as the equation $x^2 + y^2 = r^2$. Each picture makes some problems easy and others hard. Learning to switch between them is one of the most powerful habits in mathematics.

A method for everything? Descartes boasted that he had left much unsaid "so as to leave to others the pleasure of discovery." He believed his method could solve any geometry problem in principle. Later mathematicians proved that some classical problems, like trisecting any angle with compass and straightedge, are impossible, using the very algebra he introduced.

How later books answered. Newton's Principia uses geometry in the Greek style on the surface, but Newton learned his mathematics from Descartes's Geometry in van Schooten's edition, and calculus grew straight out of it.

Big questions. Why should numbers and shapes fit together so well? Is a length really a number, or are we choosing to treat it like one?