A page from Archimedes' Measurement of a Circle

''Measurement of a Circle'' or ''Dimension of the Circle'' (

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 ...

: , ''Kuklou metrēsis'') is a

treatise A treatise is a formal and systematic written discourse on some subject, generally longer and treating it in greater depth than an essay, and more concerned with investigating or exposing the principles of the subject and its conclusions."Treat ...

that consists of three propositions by

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 scientists ...

, ca. 250 BCE. The treatise is only a fraction of what was a longer work.

Propositions

Proposition one

Proposition one states: The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference of the circle. Any

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 const ...

with a

circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to ...

''c'' and a

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

''r'' is equal in

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 A shape or figure is a graphics, graphical representation of an obje ...

with 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 an ...

with the two

legs A leg is a weight-bearing and locomotive anatomical structure, usually having a columnar shape. During locomotion, legs function as "extensible struts". The combination of movements at all joints can be modeled as a single, linear element ...

being ''c'' and ''r''. This proposition is proved by the

method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area bet ...

Proposition two

Proposition two states:

The area of a circle is to the square on its diameter as 11 to 14.

This proposition could not have been placed by Archimedes, for it relies on the outcome of the third proposition.

Proposition three

Proposition three states:

The ratio of the circumference of any circle to its diameter is greater than $3\tfrac$ but less than $3\tfrac$ .

This approximates what we now call the

mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...

π. He found these bounds on the value of π by inscribing and circumscribing a circle with two similar 96-sided

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

Approximation to square roots

This proposition also contains accurate approximations to the

square root of 3 The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as \sqrt or 3^. It is more precisely called the principal square root of 3 to distinguish it from the negative nu ...

(one larger and one smaller) and other larger non-perfect

square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...

s; however, Archimedes gives no explanation as to how he found these numbers. He gives the upper and lower bounds to as > > . However, these bounds are familiar from the study of

Pell's equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinate ...

and the convergents of an associated

continued fraction In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...

, leading to much speculation as to how much of this number theory might have been accessible to Archimedes. Discussion of this approach goes back at least to

Thomas Fantet de Lagny Thomas Fantet de Lagny (7 November 1660 – 11 April 1734) was a French people, French mathematician, well known for his contributions to computational mathematics, and for calculating pi, π to 112 correct decimal places. Biography Thomas Fant ...

, FRS (compare

Chronology of computation of π The table below is a brief chronology of computed numerical values of, or bounds on, the mathematical constant pi (). For more detailed explanations for some of these calculations, see Approximations of . The last 100 decimal digits of the l ...

) in 1723, but was treated more explicitly by

Hieronymus Georg Zeuthen Hieronymus Georg Zeuthen (15 February 1839 – 6 January 1920) was a Danish mathematician. He is known for work on the enumerative geometry of conic sections, algebraic surfaces, and history of mathematics. Biography Zeuthen was born in Grimst ...

. In the early 1880s,

Friedrich Otto Hultsch Friedrich may refer to: Names *Friedrich (surname), people with the surname ''Friedrich'' *Friedrich (given name), people with the given name ''Friedrich'' Other *Friedrich (board game), a board game about Frederick the Great and the Seven Years' ...

(1833–1906) and

Karl Heinrich Hunrath Karl may refer to: People * Karl (given name), including a list of people and characters with the name * Karl der Große, commonly known in English as Charlemagne * Karl Marx, German philosopher and political writer * Karl of Austria, last Austri ...

(b. 1847) noted how the bounds could be found quickly by means of simple binomial bounds on square roots close to a perfect square modelled on Elements II.4, 7; this method is favoured by

Thomas Little Heath Sir Thomas Little Heath (; 5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College. Heath translat ...

. Although only one route to the bounds is mentioned, in fact there are two others, making the bounds almost inescapable however the method is worked. But the bounds can also be produced by an iterative geometrical construction suggested by Archimedes'

Stomachion ''Ostomachion'', also known as ''loculus Archimedius'' (Archimedes' box in Latin) and also as ''syntomachion'', is a mathematical treatise attributed to Archimedes. This work has survived fragmentarily in an Arabic version and a copy, the ''A ...

in the setting of the regular dodecagon. In this case, the task is to give rational approximations to the tangent of π/12.

References

{{Authority control Ancient Greek mathematical works Works by Archimedes Euclidean geometry