''On Spirals'' ( el, Περὶ ἑλίκων) is a treatise by

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...

, written around 225 BC. Notably, Archimedes employed the Archimedean spiral in this book to

square the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficult ...

and

trisect an angle Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge an ...

Preface

Archimedes begins ''On Spirals'' with a message to Dositheus of Pelusium mentioning the death of

Conon Conon ( el, Κόνων) (before 443 BC – c. 389 BC) was an Athenian general at the end of the Peloponnesian War, who led the Athenian naval forces when they were defeated by a Peloponnesian fleet in the crucial Battle of Aegospotami; later he ...

as a loss to mathematics. He then goes on to summarize the results of ''

On the Sphere and Cylinder ''On the Sphere and Cylinder'' ( el, Περὶ σφαίρας καὶ κυλίνδρου) is a work that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a sphere and the volume of t ...

'' (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

and was later studied by Archimedes in ''On Spirals''. Archimedes was able to find various

tangents In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

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

Archimedean spiral circle squaring triangle

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.

References

{{Spirals Works by Archimedes Euclidean geometry Ancient Greek mathematical works Spirals

Contents

Preface

Archimedean spiral

Trisecting an angle

Squaring the circle

References