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?

A man in front of a board on the table.
A man in front of a board on the table.Crop from the original engraving; improved sharpness for the Euler’s Eyes collection
Two men next to the cannon and the bullets.
Two men next to the cannon and the bullets.Crop from the original engraving; improved sharpness for the Euler’s Eyes collection
An angel next to a balanced scale.
An angel next to a balanced scale.Crop from the original engraving; improved sharpness for the Euler’s Eyes collection

It is a relationship between two numbers: the distance to the target (what the gunner “hears”) and the angle of the gun. (what he “says”). Galileo showed in 1638 that, without air resistance, the bullet describes a parabola, and the range $R$ depends on the angle $\theta$ and the exit speed $v$:

$$R = \frac{v^2 \cdot \sin(2\theta)}{g}, \qquad g \approx 9{,}8 \text{ m/s}^2.$$
Different angles, different trajectories.
Different angles, different trajectories.Redesigned diagram for the Euler’s Eyes collection

With a bullet leaving at 100 m/s:

Angle 15° 30° 45° 60° 75°
Range 510 m 884 m 1020 m 884 m 510 m
Choose the angle and shoot. The range is maximum at 45°, and angles that add up to 90° fall in the same place.

Notice two things. Range is a function of angle: each angle gives a unique range. But the the opposite way doesn’t work like that: to hit a target at 900 meters two angles are used, about 30,9° and 59,1°. In today’s language, the function is not injective.

All topics by René Descartes · ← A reading of the Cogito: listening and speaking · The word “function” →

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