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What properties of the Archimedean spiral did Archimedes study?

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Portrait of Archimedes by Giuseppe Nogari shows an elderly man with white hair and beard, holding a compass, gazing downward, set against a dark background. — Archimedes Archimedes, oil on canvas by Giuseppe Nogari, 18th century; in the Pushkin Fine Arts Museum, Moscow. Archimedes was an ancient Greek mathematician and inventor.

In his work On Spirals, the Greek mathematician Archimedes studied the properties of the spiral that now bears his name. While he did not discover the curve, he famously employed it to solve two classical problems of antiquity: squaring the circle and trisecting an angle.1

The Archimedean spiral is defined by the equation $r = a θ$ , where $r$ is the radius from the center, $θ$ is the angular position, and $a$ is a constant. Archimedes analyzed the geometric nature of this curve, noting that it is generated by a point moving at a constant rate away from a fixed origin while the radius vector rotates at a constant angular velocity. A key property he explored is that the distance between successive turns of the spiral remains constant, specifically $2 π a$ when $θ$ is measured in radians.12