In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle. Equivalently, it is a non-convex 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 calculated mathematically. [1]
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. [2] [3] 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 concludes that, in proving his result, Hippocrates was also the first to prove that the area of a circle is proportional to the square of its diameter. [2]
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]
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]
Using a similar proof to the one above, the Arab mathematician Hasan Ibn al-Haytham (Latinized name Alhazen, c. 965 – c. 1040) showed that where two lunes are formed, on the two sides of a right triangle, whose outer boundaries are semicircles and whose inner boundaries are formed by the circumcircle 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. [7] [8] The quadrature of the lune of Hippocrates is the special case of this result for an isosceles right triangle. [9]
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 constructible squarable lenses. [1] [8]
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 constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.
In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length can be constructed with compass and straightedge in a finite number of steps. Equivalently, is constructible if and only if there is a closed-form expression for using only integers and the operations for addition, subtraction, multiplication, division, and square roots.
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry; Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.
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 idealized ruler and a pair of compasses.
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible to construct by using only a compass and straightedge, but even in ancient times solutions were known that employed other tools.
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 and a compass.
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 of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.
A Reuleaux triangle[ʁœlo] is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circular disks, each having its center on the boundary of the other two. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because its width is constant, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?"
In mathematics, a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180°. It has only one line of symmetry. In non-technical usage, the term "semicircle" is sometimes used to refer to a half-disk, which is a two-dimensional geometric shape that also includes the diameter segment from one end of the arc to the other as well as all the interior points.
In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line that contains their diameters.
In mathematics, quadrature is a historical term which means the process of determining area. This term is still used nowadays in the context of differential equations, where "solving an equation by quadrature" or "reduction to quadrature" means expressing its solution in terms of integrals.
The tomahawk is a tool in geometry for angle trisection, the problem of splitting an angle into three equal parts. The boundaries of its shape include a semicircle and two line segments, arranged in a way that resembles a tomahawk, a Native American axe. The same tool has also been called the shoemaker's knife, but that name is more commonly used in geometry to refer to a different shape, the arbelos.
Nikolai Grigorievich Chebotaryov was a Ukrainian and Soviet mathematician. He is best known for the Chebotaryov density theorem.
Hippocrates of Chios was an ancient Greek mathematician, geometer, and astronomer.
In plane geometry, a lune is the concave-convex region bounded by two circular arcs. It has one boundary portion for which the connecting segment of any two nearby points moves outside the region and another boundary portion for which the connecting segment of any two nearby points lies entirely inside the region. A convex-convex region is termed a lens.
Thomas Clausen was a Danish mathematician and astronomer.
A timeline of algebra and geometry
In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
In Euclidean geometry, the right triangle altitude theorem or geometric mean theorem is a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean of the two segments equals the altitude.
