Lune of Hippocrates
In
History
Hippocrates wanted to solve the classic problem of
Hippocrates' book on geometry in which this result appears, Elements, has been lost, but may have formed the model for Euclid's Elements.[3] Hippocrates' proof was preserved through the History of Geometry compiled by Eudemus of Rhodes, which has also not survived, but which was excerpted by Simplicius of Cilicia in his commentary on Aristotle's Physics.[2][4]
Not until 1882, with Ferdinand von Lindemann's proof of the transcendence of π, was squaring the circle proved to be impossible.[5]
Proof
Hippocrates' 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 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, and therefore the quarter circle AFBOA is equal in area to the semicircle AEBDA. Subtracting the crescent-shaped area AFBDA from the quarter circle gives triangle ABO and subtracting the same crescent from the semicircle gives the lune. Since the triangle and lune are both formed by subtracting equal areas from equal area, they are themselves equal in area.[2][6]
Generalizations
Using a similar proof to the one above, the Arab mathematician Hasan Ibn al-Haytham (Latinized name
All lunes constructable by compass and straight-edge can be specified by the two angles formed by the inner and outer arcs on their respective circles; in this notation, for instance, the lune of Hippocrates would have the inner and outer angles (90°, 180°) with ratio 1:2. Hippocrates found two other squarable concave lunes, with angles approximately (107.2°, 160.9°) with ratio 2:3 and (68.5°, 205.6°) with ratio 1:3. Two more squarable concave lunes, with angles approximately (46.9°, 234.4°) with ratio 1:5 and (100.8°, 168.0°) with ratio 3:5 were found in 1766 by Martin Johan Wallenius and again in 1840 by Thomas Clausen. In the mid-20th century, two Russian mathematicians, Nikolai Chebotaryov and his student Anatoly Dorodnov, completely classified the lunes that are constructible by compass and straightedge and that have equal area to a given square. As Chebotaryov and Dorodnov showed, these five pairs of angles give the only constructible squarable lunes; in particular, there are no other constructible squarable lunes.[1][8]
References
- ^ JSTOR 2589121. Translated from Postnikov's 1963 Russian book on Galois theory.
- ^ ISBN 0-486-43231-9.
- ^ a b "Hippocrates of Chios", Encyclopædia Britannica, 2012, retrieved 2012-01-12.
- ^ O'Connor, John J.; Robertson, Edmund F., "Hippocrates of Chios", MacTutor History of Mathematics Archive, University of St Andrews
- ISBN 978-0-691-02528-5.
- ISBN 0-486-25563-8.
- cut-the-knot, accessed 2012-01-12.
- ^ ISBN 978-0-88385-348-1.
- ISBN 0-387-94280-7.