For almost two thousand years, geometry and algebra lived apart. Greek geometry worked with ruler, compass and figures. Algebra, which reached Europe through the Arabs, worked with equations. At the start of the 17th century, two people had, at the same time and without consulting each other, the idea of joining them: René Descartes and Pierre de Fermat.

A sentence that created a field

Around 1636 Fermat wrote a short Latin text, Introduction to plane and solid loci. “Locus” was the Greek name for a set of points satisfying a condition, like the circle, the locus of points at a fixed distance from the centre. Right at the start, Fermat wrote:

Whenever in a final equation two unknown quantities are found, we have a locus, the extremity of one of them describing a line, straight or curved.

— Fermat, Introduction to plane and solid loci

In other words: an equation with two unknowns is a curve. Each solution $(x,y)$ is a point, and all the solutions together form a picture.

The line

Fermat wrote in Viète’s style, with vowels for unknowns and consonants for known numbers. The line appeared as “D in A aequetur B in E”, which in today’s notation is

$$D\cdot x=B\cdot y.$$

Solving for $y$: $y=\dfrac DB\,x$. It is a line through the origin with slope $\dfrac DB$. Fermat showed that every first-degree equation in $x$ and $y$ describes a line, and that every second-degree equation describes a conic: parabola, ellipse, hyperbola or circle.

The parabola

Consider the points whose coordinates satisfy

$$y=x^2.$$

For $x=-2$ we get $y=4$; for $x=0$, $y=0$; for $x=2$, $y=4$. Plotting many points, a parabola appears. The equation does much more than produce a table: it tells us, without drawing anything, whether a point lies on the curve. The point $(3,9)$ does, because $9=3^2$. The point $(2,5)$ does not, because $5\neq2^2$.

Where two curves meet

The great advantage shows when we want to know where two figures cross. With ruler and compass this can be very hard. With equations, we simply require the point to satisfy both at once.

Where does the parabola $y=x^2$ cross the line $y=2x+3$? Setting them equal:

$$x^2=2x+3\quad\Longleftrightarrow\quad x^2-2x-3=0\quad\Longleftrightarrow\quad(x-3)(x+1)=0.$$

The solutions are $x=3$ and $x=-1$, and the meeting points are $(3,9)$ and $(-1,1)$. A question of geometry has become a quadratic equation.

What changed

With analytic geometry, any curve could be studied with algebra, and any equation could be seen as a picture. It was the first step towards calculus: to speak of the tangent to a curve or the area under it, the curve first had to be written as an equation. Fermat took the next step soon after, with his method of maxima and minima.

Compare with the Descartes trail, which tells the other half of this story.

Exercise: where does the parabola y = x² meet the line y = 4?

$x^2=4$ gives $x=2$ or $x=-2$. The points are $(2,4)$ and $(-2,4)$.

Exercise: at how many points does the line y = 2x − 1 touch the parabola y = x²?

$x^2=2x-1$ gives $x^2-2x+1=(x-1)^2=0$. There is only one solution, $x=1$: the line touches the parabola at a single point, $(1,1)$. It is the tangent to the parabola at that point.

Sources: Fermat biography, MacTutor.

All Fermat topics · ← The life of Pierre de Fermat · Maxima and minima: the rectangle with the largest area →

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