geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the lune of Hippocrates, named after

Hippocrates of Chios Hippocrates of Chios ( grc-gre, Ἱπποκράτης ὁ Χῖος; c. 470 – c. 410 BC) was an ancient Greek mathematician, geometer, and astronomer. He was born on the isle of Chios, where he was originally a merchant. After some misadve ...

, is a lune bounded by arcs of two

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

s, the smaller of which has as its diameter a chord spanning a right angle on the larger circle. Equivalently, it is a non-

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polyto ...

plane region bounded by one 180-degree circular arc and one 90-degree circular arc. It was the first curved figure to have its exact

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open ...

calculated mathematically.. Translated from Postnikov's 1963 Russian book on

Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...

History

Hippocrates wanted to solve the classic problem of

squaring 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 difficulty ...

, i.e. constructing a square by means of

straightedge and compass In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

, having the same area as a given

. He proved that the lune bounded by the arcs labeled ''E'' and ''F'' in the figure has the same area as triangle ''ABO''. This afforded some hope of solving the circle-squaring problem, since the lune is bounded only by arcs of circles.

Heath A heath () is a shrubland habitat found mainly on free-draining infertile, acidic soils and characterised by open, low-growing woody vegetation. Moorland is generally related to high-ground heaths with—especially in Great Britain—a cooler a ...

concludes that, in proving his result, Hippocrates was also the first to prove that the

area of a circle In geometry, the area enclosed by a circle of radius is . Here the Greek letter represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159. One method of deriving this formula, which origi ...

is proportional to the square of its diameter. Hippocrates' book on geometry in which this result appears, ''Elements'', has been lost, but may have formed the model for

Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

's ''

Elements Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of o ...

''.. Hippocrates' proof was preserved through the ''History of Geometry'' compiled by

Eudemus of Rhodes Eudemus of Rhodes ( grc-gre, Εὔδημος) was an ancient Greek philosopher, considered the first historian of science, who lived from c. 370 BCE until c. 300 BCE. He was one of Aristotle's most important pupils, editing his teacher's work and m ...

, which has also not survived, but which was excerpted by

Simplicius of Cilicia Simplicius of Cilicia (; el, Σιμπλίκιος ὁ Κίλιξ; c. 490 – c. 560 AD) was a disciple of Ammonius Hermiae and Damascius, and was one of the last of the Neoplatonists. He was among the pagan philosophers persecuted by Justinian ...

in his commentary on

Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...

's ''

Physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which re ...

''.. Not until 1882, with

Ferdinand von Lindemann Carl Louis Ferdinand von Lindemann (12 April 1852 – 6 March 1939) was a German mathematician, noted for his proof, published in 1882, that (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coeffici ...

's proof of the transcendence of ''π'', was squaring the circle proved to be impossible.

Proof

Hippocrates' result can be proved as follows: The center of the circle on which the arc lies is the point , which is the midpoint of the hypotenuse of the isosceles right triangle . Therefore, the diameter of the larger circle is times the diameter of the smaller circle on which the arc lies. Consequently, the smaller circle has half the area of the larger circle, and therefore the quarter circle is equal in area to the semicircle . Subtracting the crescent-shaped area from the quarter circle gives triangle 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..

Generalizations

Using a similar proof to the one above, the Arab mathematician Hasan Ibn al-Haytham (Latinized name

Alhazen Ḥasan Ibn al-Haytham, Latinized as Alhazen (; full name ; ), was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the prin ...

, c. 965 – c. 1040) showed that where two lunes are formed, on the two sides of a

right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right ...

, whose outer boundaries are semicircles and whose inner boundaries are formed by the

circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every poly ...

of the triangle, then the areas of these two lunes added together are equal to the area of the triangle. The lunes formed in this way from a right triangle are known as the lunes of Alhazen.. The quadrature of the lune of Hippocrates is the special case of this result for an

isosceles right triangle A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45� ...

.. In the mid-20th century, two Russian mathematicians,

Nikolai Chebotaryov Nikolai Grigorievich Chebotaryov (often spelled Chebotarov or Chebotarev, uk, Мико́ла Григо́рович Чеботарьо́в, russian: Никола́й Григо́рьевич Чеботарёв) ( – 2 July 1947) was a Ukrainia ...

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. All such lunes 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°). Hippocrates found two other squarable concave lunes, with angles approximately (107.2°, 160.9°) and (68.5°, 205.6°). Two more squarable concave lunes, with angles approximately (46.9°, 234.4°) and (100.8°, 168.0°) were found in 1766 by and again in 1840 by Thomas Clausen. As Chebotaryov and Dorodnov showed, these five pairs of angles give the only constructible squarable lunes; in particular, there are no constructible squarable convex lunes.

References

{{Ancient Greek mathematics Circles Squaring the circle