In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of circles, one of which passes through the center of the other and has half the area of the other. Equivalently, it is a non-convex plane region bounded by one 180-degree circular arc and one 90-degree circular arc.
Hippocrates wanted to solve the classic problem of squaring the circle, i.e. constructing a square by means of straightedge and compass, having the same area as a given circle. Not until 1882, with Ferdinand von Lindemann's proof of the transcendence of π, was that proved to be impossible.
Hippocrates proved that the lune bounded by the arcs labeled E and F in the figure, has the same area as does triangle ABO.
This afforded some hope of solving the circle-squaring problem, since the lune is bounded only by arcs of circles.
The result can be proved as follows: The center of the circle on which the arc AEB lies is the point D, which is the midpoint of the hypotenuse of the isosceles right triangle ABO. Therefore the diameter AC of the larger circle ABC is √2 times the diameter of the smaller circle on which the arc AEB lies. Consequently the smaller circle has half the area of the larger circle. The semicircle bounded by the arc AEB and the diameter AB therefore has the same area as the quarter-circle bounded by the arc AFB and the two radii OA and OB. Subtracting from both sides of that equality the area of the region bounded by the arc AFB and the line AB yields the result.
References
- Thomas L. Heath: A Manual of Greek Mathematics. Courier Dover Publications 2003, ISBN 0486432319, pp. 121–132 (restricted online version (Google Books))
- Lucas Nicolaas Hendrik Bunt, Phillip S. Jones, Jack D. Bedient: The Historical Roots of Elementary Mathematics. Courier Dover Publications 1988, ISBN 0486255638, pp. 90–91 (restricted online version (Google Books))
- W. S. Anglin: Mathematics, a Concise History and Philosophy. Springer 1994, ISBN 0387942807, pp. 51–53 (restricted online version (Google Books))
- Hippocrates of Chios. (2009). In Encyclopædia Britannica. Retrieved August 29, 2009, from Encyclopædia Britannica Online: http://www.britannica.com/EBchecked/topic/266646/Hippocrates-of-Chios
External links
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