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On Spirals

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Ancient Greek mathematics book

On Spirals (Greek:Περὶ ἑλίκων) is a treatise byArchimedes, written around 225 BC.^[1] Notably, Archimedes employed the Archimedean spiral in this book tosquare the circle andtrisect an angle.

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Preface

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Archimedes beginsOn Spirals with a message to Dositheus of Pelusium mentioning the death ofConon as a loss to mathematics. He then goes on to summarize the results ofOn the Sphere and Cylinder (Περὶ σφαίρας καὶ κυλίνδρου) andOn Conoids and Spheroids (Περὶ κωνοειδέων καὶ σφαιροειδέων). He continues to state his results ofOn Spirals.

Archimedean spiral

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Main article:Archimedean spiral

The Archimedean spiral with three 360° turnings on one arm

The Archimedean spiral was first studied byConon and was later studied by Archimedes inOn Spirals. Archimedes was able to find varioustangents to the spiral.^[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.^[2]

Trisecting an angle

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Example of how Archimedes trisected an angle inOn Spirals.

The construction as to how Archimedestrisected 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.^[3]

Squaring the circle

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The circle and the triangle are equal in area.

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 ofMeasurement 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.^[4]

References

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^^a ^bWeisstein, Eric W."Archimedes' Spiral".MathWorld.
^Heath, Thomas Little (1921),A History of Greek Mathematics, Boston: Adamant Media Corporation, p. 64,ISBN 0-543-96877-4, retrieved2008-08-20{{citation}}:ISBN / Date incompatibility (help)
^Tokuda, Naoyuki; Chen, Liang (1999-03-18),Trisection Angles(PDF), Utsunomiya University, Utsunomiya, Japan, pp. 5–6, archived fromthe original(PDF) on 2011-07-22, retrieved2008-08-20{{citation}}: CS1 maint: location missing publisher (link)
^"History topic: Squaring the circle". Retrieved2008-08-20.

v t e Archimedes
Written works	Measurement of a Circle The Sand Reckoner On the Equilibrium of Planes Quadrature of the Parabola On the Sphere and Cylinder On Spirals On Conoids and Spheroids On Floating Bodies Ostomachion The Method of Mechanical Theorems Book of Lemmas (apocryphal)
Discoveries and inventions	Archimedean solid Archimedes's cattle problem Archimedes' principle Archimedes's screw Claw of Archimedes
Miscellaneous	Archimedes' heat ray Archimedes Palimpsest List of things named after Archimedes Pseudo-Archimedes
Related people	Euclid Eudoxus of Cnidus Apollonius of Perga Hero of Alexandria Eutocius of Ascalon
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