On Spirals

On Spirals (Greek: Περὶ ἑλίκων) is a treatise by Archimedes, written around 225 BC.

[1] Notably, Archimedes employed the Archimedean spiral in this book to square the circle and trisect an angle.

Archimedes begins On Spirals with a message to Dositheus of Pelusium mentioning the death of Conon as a loss to mathematics.

[1] He defines the spiral as: If a straight line one extremity of which remains fixed is made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line is revolving, a point moves at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane.

[3]To square the circle, Archimedes gave the following construction: Let P be the point on the spiral when it has completed one turn.