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 ...

, an arbelos is a plane region bounded by three

semicircle In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line o ...

s 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 In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...

(the ''baseline'') that contains their

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...

s. The earliest known reference to this figure is in

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...

's ''

Book of Lemmas The ''Book of Lemmas'' or ''Book of Assumptions'' (Arabic ''Maʾkhūdhāt Mansūba ilā Arshimīdis'') is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositio ...

'', where some of its mathematical properties are stated as Propositions 4 through 8. The word ''arbelos'' is Greek for 'shoemaker's knife'. The figure is closely related to the

Pappus chain In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD. Construction The arbelos is defined by two circles, ''C''U and ''C''V, which are tangent at the point A a ...

Properties

Two of the semicircles are necessarily concave, with arbitrary diameters and ; the third semicircle is

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 polytop ...

, with diameter Arbelos diagram with points marked

Area

The

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 op ...

of the arbelos is equal to the area of a circle with diameter . Proof: For the proof, reflect the arbelos over the line through the points and , and observe that twice the area of the arbelos is what remains when the areas of the two smaller circles (with diameters , ) are subtracted from the area of the large circle (with diameter ). Since the area of a circle is proportional to the square of the diameter (

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, Book XII, Proposition 2; we do not need to know that the

constant of proportionality In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality const ...

is ), the problem reduces to showing that

2, AH, ^2 = , BC, ^2 - , AC, ^2 - , B, ^2

. The length equals the sum of the lengths and , so this equation simplifies algebraically to the statement that

, AH, ^2 = , BA, , AC,

. Thus the claim is that the length of the segment is the

geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...

of the lengths of the segments and . Now (see Figure) the triangle , being inscribed in the semicircle, has a right angle at the point (Euclid, Book III, Proposition 31), and consequently is indeed a "mean proportional" between and (Euclid, Book VI, Proposition 8, Porism). This proof approximates the ancient Greek argument; Harold P. Boas cites a paper of Roger B. Nelsen who implemented the idea as the following

proof without words In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered mo ...

Rectangle

Let and be the points where the segments and intersect the semicircles and , respectively. The

quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

is actually a

rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram contain ...

. :''Proof'': , , and are right angles because they are inscribed in semicircles (by

Thales's theorem In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved ...

). The quadrilateral therefore has three right angles, so it is a rectangle. ''Q.E.D.''

Tangents

The line is tangent to semicircle at and semicircle at . :''Proof'': Since is a right angle, equals minus . However, also equals minus (since is a right angle). Therefore triangles and are similar. Therefore equals , where is the midpoint of and is the midpoint of . But is a straight line, so and are

supplementary angles In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...

. Therefore the sum of and is π. is a right angle. The sum of the angles in any quadrilateral is 2π, so in quadrilateral , must be a right angle. But is a rectangle, so the midpoint of (the rectangle's diagonal) is also the midpoint of (the rectangle's other diagonal). As (defined as the midpoint of ) is the center of semicircle , and angle is a right angle, then is tangent to semicircle at . By analogous reasoning is tangent to semicircle at . ''Q.E.D.''

Archimedes' circles

The altitude divides the arbelos into two regions, each bounded by a semicircle, a straight line segment, and an arc of the outer semicircle. The circles

inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...

in each of these regions, known as the

Archimedes' circles In geometry, the twin circles are two special circles associated with an arbelos. An arbelos is determined by three collinear points , , and , and is the curvilinear triangular region between the three semicircles that have , , and as their dia ...

of the arbelos, have the same size.

Variations and generalisations

The

parbelos The parbelos is a figure similar to the arbelos but instead of three half circles it uses three parabola segments. More precisely the parbelos consists of three parabola segments, that have a height that is one fourth of the width at their bases ...

is a figure similar to the arbelos, that uses

parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...

segments instead of half circles. A generalisation comprising both arbelos and parbelos is the ''f''-belos, which uses a certain type of similar differentiable functions.Antonio M. Oller-Marcen
"The f-belos"
In: ''Forum Geometricorum'', Volume 13 (2013), pp. 103–111. In the

Poincaré half-plane model In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincar� ...

of the

hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ' ...

, an arbelos models an

ideal triangle In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called ''triply asymptotic triangles'' or ''trebly asymptotic triangles''. The vertices are sometime ...

Etymology

The name ''arbelos'' comes from

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

ἡ ἄρβηλος ''he árbēlos'' or ἄρβυλος ''árbylos'', meaning "shoemaker's knife", a knife used by cobblers from antiquity to the current day, whose blade is said to resemble the geometric figure.

References

Thomas Little Heath (1897), ''The Works of Archimedes''. Cambridge University Press. Proposition 4 in the ''Book of Lemmas''. Quote: ''If AB be the diameter of a semicircle and N any point on AB, and if semicircles be described within the first semicircle and having AN, BN as diameters respectively, the figure included between the circumferences of the three semicircles is "what Archimedes called arbelos"; and its area is equal to the circle on PN as diameter, where PN is perpendicular to AB and meets the original semicircle in P.''
"Arbelos - the Shoemaker's Knife"

Bibliography

* * *

American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an ...

, 120 (2013), 929-935. *

External links

* * {{wiktionary-inline