* Measurement of a Circle* or

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 with a circumference *c* and a radius *r* is equal in area with a right triangle with the two legs being *c* and *r*. This proposition is proved by the method 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 the mathematical constant π. He found these bounds on the value of π by inscribing and circumscribing a circle with two similar 96-sided regular polygons.^{ [7] }

This proposition also contains accurate approximations to the square root of 3 (one larger and one smaller) and other larger non-perfect square roots; however, Archimedes gives no explanation as to how he found these numbers.^{ [5] } He gives the upper and lower bounds to √3 as 1351/780 > √3 > 265/153.^{ [6] } However, these bounds are familiar from the study of Pell's equation and the convergents of an associated 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 to Thomas Fantet de Lagny, FRS (compare Chronology of computation of π) in 1723, but was treated more explicitly by Hieronymus Georg Zeuthen. In the early 1880s, Friedrich Otto Hultsch (1833–1906) and Karl 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 by Thomas 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.

**Archimedes of Syracuse** was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered to be the greatest mathematician of ancient history, and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including: the area of a circle; the surface area and volume of a sphere; area of an ellipse; the area under a parabola; the volume of a segment of a paraboloid of revolution; the volume of a segment of a hyperboloid of revolution; and the area of a spiral.

In geometry, the **circumference** 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 a line segment. More generally, the perimeter is the curve length around any closed figure. Circumference may also refer to the circle itself, that is, the locus corresponding to the edge of a disk.

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic).

The number **π** is a mathematical constant. It is defined as the ratio of a circle's circumference to its diameter, and it also has various equivalent definitions. It appears in many formulas in all areas of mathematics and physics. It is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, and is spelled out as "**pi**". It is also referred to as **Archimedes' constant**.

A **perimeter** is a path that encompasses/surrounds a two-dimensional shape. The term may be used either for the path, or its length—in one dimension. It can be thought of as the length of the outline of a shape. The perimeter of a circle or ellipse is called its circumference.

**Squaring the circle** is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with 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.

**Apollonius of Perga** was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today.

* The Method of Mechanical Theorems*, also referred to as

The **vesica piscis** is a type of lens, a mathematical shape formed by the intersection of two disks with the same radius, intersecting in such a way that the center of each disk lies on the perimeter of the other. In Latin, "vesica piscis" literally means "bladder of a fish", reflecting the shape's resemblance to the conjoined dual air bladders found in most fish. In Italian, the shape's name is *mandorla* ("almond").

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.

**Theodorus of Cyrene** was an ancient Libyan Greek and lived during the 5th century BC. The only first-hand accounts of him that survive are in three of Plato's dialogues: the *Theaetetus*, the *Sophist*, and the *Statesman*. In the former dialogue, he posits a mathematical theorem now known as the Spiral of Theodorus.

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 between the *n*th polygon and the containing shape will become arbitrarily small as *n* becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members.

In geometry, the area enclosed by a circle of radius r is π*r*^{2}. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.

* On the Sizes and Distances * is widely accepted as the only extant work written by Aristarchus of Samos, an ancient Greek astronomer who lived circa 310–230 BCE. This work calculates the sizes of the Sun and Moon, as well as their distances from the Earth in terms of Earth's radius.

* The Quadrature of the Parabola* is a treatise on geometry, written by Archimedes in the 3rd century BC. Written as a letter to his friend Dositheus, the work presents 24 propositions regarding parabolas, culminating in a proof that the area of a parabolic segment is 4/3 that of a certain inscribed triangle.

**Liu Hui's π algorithm** was invented by Liu Hui, a mathematician of the Cao Wei Kingdom. Before his time, the ratio of the circumference of a circle to its diameter was often taken experimentally as three in China, while Zhang Heng (78–139) rendered it as 3.1724 or as . Liu Hui was not satisfied with this value. He commented that it was too large and overshot the mark. Another mathematician Wang Fan (219–257) provided π ≈ 142/45 ≈ 3.156. All these empirical π values were accurate to two digits. Liu Hui was the first Chinese mathematician to provide a rigorous algorithm for calculation of π to any accuracy. Liu Hui's own calculation with a 96-gon provided an accuracy of five digits: π ≈ 3.1416.

The * Book of Lemmas* is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositions (lemmas) on circles.

* On Spirals* is a treatise by Archimedes, written around 225 BC. Notably, Archimedes employed the Archimedean spiral in this book to square the circle and trisect an angle.

* Project Mathematics!*, is a series of educational video modules and accompanying workbooks for teachers, developed at the California Institute of Technology to help teach basic principles of mathematics to high school students. In 2017, the entire series of videos was made available on YouTube.

The **quadratrix** or **trisectrix of Hippias** is a curve, which is created by a uniform motion. It is one of the oldest examples for a kinematic curve, that is a curve created through motion. Its discovery is attributed to the Greek sophist Hippias of Elis, who used it around 420 BC in an attempt to solve the angle trisection problem. Later around 350 BC Dinostratus used it in an attempt to solve the problem of squaring the circle.

- ↑ Knorr, Wilbur R. (1986-12-01). "Archimedes' dimension of the circle: A view of the genesis of the extant text".
*Archive for History of Exact Sciences*.**35**(4): 281–324. doi:10.1007/BF00357303. ISSN 0003-9519. - ↑ Lit, L.W.C. (Eric) van. "Naṣīr al-Dīn al-Ṭūsī's Version of The Measurement of the Circle of Archimedes from his Revision of the Middle Books".
*Tarikh-e Elm*.The

*measurement of the circle*was written by Archimedes (ca. 250 B.C.E.) - ↑ Knorr, Wilbur R. (1986).
*The Ancient Tradition of Geometric Problems*. Courier Corporation. p. 153. ISBN 9780486675329.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.

- ↑ Heath, Thomas Little (1921),
*A History of Greek Mathematics*, Boston: Adamant Media Corporation, ISBN 978-0-543-96877-7 , retrieved 2008-06-30 - 1 2 "Archimedes". Encyclopædia Britannica. 2008. Retrieved 2008-06-30.
- 1 2 3 Heath, Thomas Little (1897),
*The Works of Archimedes*, Cambridge University: Cambridge University Press., pp.*lxxvii*, 50, retrieved 2008-06-30 - ↑ Heath, Thomas Little (1931),
*A Manual of Greek Mathematics*, Mineola, N.Y.: Dover Publications, p. 146, ISBN 978-0-486-43231-1

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.