On Spirals
On Spirals () is a treatise by Archimedes, written around 225 BC. Notably, Archimedes employed the Archimedean spiral in this book to square the circle and trisect an angle. Contents. Preface. Archimedes begins "On Spirals" with a message to Dositheus of Pelusium mentioning the death of Conon as a loss to mathematics. He then goes on to summarize the results of "On the Sphere and Cylinder" (Περὶ σφαίρας καὶ κυλίνδρου) and "On Conoids and Spheroids" (Περὶ κωνοειδέων καὶ σφαιροειδέων). He continues to state his results of "On Spirals". Archimedean spiral. The Archimedean spiral was first studied by Conon and  was later studied by Archimedes in "On Spirals". Archimedes was able to find various tangents to the spiral. He defines the spiral as: Trisecting an angle. The construction as to how Archimedes trisected the angle is as follows: Suppose the angle ABC is to be trisected. Trisect the segment BC and find BD to be one third of BC. Draw a circle with center B and radius BD. Suppose the circle with center B intersects the spiral at point E. Angle ABE is one third angle ABC. Squaring the circle. To square the circle, Archimedes gave the following construction: Let  P be the point on the spiral when it has completed one turn. Let the tangent at P cut the line perpendicular to OP at T. OT is the length of the circumference of the circle with radius OP. Archimedes had already proved as the first proposition of "Measurement of a Circle" that the area of a circle is equal to a right-angled triangle having the legs' lengths equal to the radius of the circle and the circumference of the circle. So the area of the circle with radius OP is equal to the area of the triangle OPT