Geometry begins by showing that arithmetic operations can be done with segments. The most beautiful is the square root. To find $\sqrt{a}$:

  1. Draw on a straight line a segment $AB$ of length 1 and then a segment $BC$ of length $a$.
  2. Draw a semicircle with diameter $AC$.
  3. 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.

Why? Every triangle inscribed in a semicircle, with one side in diameter, is a right angle. So $ADC$ is rectangle in $D$, and $DB$ is the height relative to the hypotenuse. The triangles $ABD$ and $DBC$ are similar, which gives

$$\frac{BD}{AB} = \frac{BC}{BD} \;\Longrightarrow\; BD^2 = AB \cdot BC = 1 \cdot a \;\Longrightarrow\; BD = \sqrt{a}.$$

In the drawing, $a = 4$: height $BD$ measures exactly 2 units.

*Geometry*, book I: Descartes solves a quadratic equation with a circle and straight lines (the NLM triangle and the point O).
Geometry, book I: Descartes solves a quadratic equation with a circle and straight lines (the NLM triangle and the point O).René Descartes, 1637 · public domain

It was also in Geometry that Descartes established conventions that we still use today: the last letters of the alphabet ($x$, $y$, $z$) for the unknowns, the first ($a$, $b$, $c$) for the values known, and the exponents written at the top, as in $x^3$.

All topics by René Descartes · ← Coordinates: geometry becomes algebra · The rule of signs →

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