Measurement of a Circle orDimension of the Circle (Greek:Κύκλου μέτρησις,Kuklou metrēsis)[1] is atreatise that consists of three propositions, probably made byArchimedes, ca. 250 BCE.[2][3] The treatise is only a fraction of what was a longer work.[4][5]
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.Anycircle with acircumferencec and aradiusr is equal inarea with aright triangle with the twolegs beingc andr. This proposition is proved by themethod of exhaustion.[6]
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.[6]
Proposition three states:
The ratio of the circumference of any circle to its diameter is greater than but less than.
This approximates what we now call themathematical constantπ. He found these bounds on the value of π byinscribing andcircumscribing a circle with twosimilar 96-sidedregular polygons.[7]
This proposition also contains accurate approximations to thesquare root of 3 (one larger and one smaller) and other larger non-perfectsquare roots; however, Archimedes gives no explanation as to how he found these numbers.[5]He gives the upper and lower bounds to√3 as1351/780 >√3 >265/153.[6] However, these bounds are familiar from the study ofPell's equation and the convergents of an associatedsimple continued fraction, 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 toThomas Fantet de Lagny, FRS (compareChronology of computation of π) in 1723, but was treated more explicitly byHieronymus Georg Zeuthen. In the early 1880s,Friedrich Otto Hultsch (1833–1906) andKarl Heinrich Hunrath (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 byThomas Little Heath. 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 in the setting of the regular dodecagon. In this case, the task is to give rational approximations to the tangent of π/12.
Themeasurement of the circle was written by Archimedes (ca. 250 B.C.E.)
Most accounts of Archimedes' works assign this writing to a time relatively late in his career. But this view is the consequence of a plain misunderstanding.
{{citation}}:ISBN / Date incompatibility (help){{citation}}:ISBN / Date incompatibility (help)| Written works |  | |
|---|---|---|
| Discoveries and inventions | ||
| Miscellaneous | ||
| Related people | ||
| International | |
|---|---|
| National | |
