\n{{overline|DR}} is the side of the square"},"image2":{"wt":"Squaring the circle-Ramanujan-1913.png"},"width2":{"wt":"180"},"alt2":{"wt":""},"caption2":{"wt":"Sketch of \"Manuscript book 1 of Srinivasa Ramanujan\" p. 54"},"footer":{"wt":""}},"i":0}}]}" id="mwARM">
In 1914, Ramanujan gave a ruler-and-compass construction which was equivalent to taking the approximate value for π to be
giving eight decimal places of π.^{ [19] } He describes his construction till line segment OS as follows.^{ [20] }
"Let AB (Fig.2) be a diameter of a circle whose centre is O. Bisect the arc ACB at C and trisect AO at T. Join BC and cut off from it CM and MN equal to AT. Join AM and AN and cut off from the latter AP equal to AM. Through P draw PQ parallel to MN and meeting AM at Q. Join OQ and through T draw TR, parallel to OQ and meeting AQ at R. Draw AS perpendicular to AO and equal to AR, and join OS. Then the mean proportional between OS and OB will be very nearly equal to a sixth of the circumference, the error being less than a twelfth of an inch when the diameter is 8000 miles long."
In this quadrature, Ramanujan did not construct the side length of the square, it was enough for him to show the line segment OS. In the following continuation of the construction, the line segment OS is used together with the line segment OB to represent the mean proportionals (red line segment OE).
Continuation of the construction up to the desired side length a of the square:
Extend AB beyond A and beat the circular arc b_{1} around O with radius OS, resulting in S′. Bisect the line segment BS′ in D and draw the semicircle b_{2} over D. Draw a straight line from O through C up to the semicircle b_{2}, it cuts b_{2} in E. The line segment OE is the mean proportional between OS′ and OB, also called geometric mean. Extend the line segment EO beyond O and transfer EO twice more, it results F and A_{1}, and thus the length of the line segment EA1 with the above described approximation value of π, the half circumference of the circle. Bisect the line segment EA_{1} in G and draw the semicircle b_{3} over G. Transfer the distance OB from A_{1} to the line segment EA_{1}, it results H. Create a vertical from H up to the semicircle b_{3} on EA_{1}, it results B_{1}. Connect A_{1} to B_{1}, thus the sought side a of the square A_{1}B_{1}C_{1}D_{1} is constructed, which has nearly the same area as the given circle.
Examples to illustrate the errors:
Finding the area under a curve, known as integration in calculus, or quadrature in numerical analysis, was known as squaring before the invention of calculus. Since the techniques of calculus were unknown, it was generally presumed that a squaring should be done via geometric constructions, that is, by compass and straightedge. For example, Newton wrote to Oldenburg in 1676 "I believe M. Leibnitz will not dislike the Theorem towards the beginning of my letter pag. 4 for squaring Curve lines Geometrically" (emphasis added).^{ [22] } After Newton and Leibniz invented calculus, they still referred to this integration problem as squaring a curve.
The mathematical proof that the quadrature of the circle is impossible using only compass and straightedge has not proved to be a hindrance to the many people who have invested years in this problem anyway. Having squared the circle is a famous crank assertion. (See also pseudomathematics.) In his old age, the English philosopher Thomas Hobbes convinced himself that he had succeeded in squaring the circle, a claim that was refuted by John Wallis as part of the Hobbes–Wallis controversy.^{ [23] }^{ [24] }
During the 18th and 19th century, the notion that the problem of squaring the circle was somehow related to the longitude problem seems to have become prevalent among would-be circle squarers. Using "cyclometer" for circle-squarer, Augustus de Morgan wrote in 1872:
Montucla says, speaking of France, that he finds three notions prevalent among cyclometers: 1. That there is a large reward offered for success; 2. That the longitude problem depends on that success; 3. That the solution is the great end and object of geometry. The same three notions are equally prevalent among the same class in England. No reward has ever been offered by the government of either country.^{ [25] }
Although from 1714 to 1828 the British government did indeed sponsor a £20,000 prize for finding a solution to the longitude problem, exactly why the connection was made to squaring the circle is not clear; especially since two non-geometric methods (the astronomical method of lunar distances and the mechanical chronometer) had been found by the late 1760s. De Morgan goes on to say that "[t]he longitude problem in no way depends upon perfect solution; existing approximations are sufficient to a point of accuracy far beyond what can be wanted." In his book, de Morgan also mentions receiving many threatening letters from would-be circle squarers, accusing him of trying to "cheat them out of their prize".
Even after it had been proved impossible, in 1894, amateur mathematician Edwin J. Goodwin claimed that he had developed a method to square the circle. The technique he developed did not accurately square the circle, and provided an incorrect area of the circle which essentially redefined pi as equal to 3.2. Goodwin then proposed the Indiana Pi Bill in the Indiana state legislature allowing the state to use his method in education without paying royalties to him. The bill passed with no objections in the state house, but the bill was tabled and never voted on in the Senate, amid increasing ridicule from the press.^{ [26] }
The mathematical crank Carl Theodore Heisel also claimed to have squared the circle in his 1934 book, "Behold! : the grand problem no longer unsolved: the circle squared beyond refutation."^{ [27] } Paul Halmos referred to the book as a "classic crank book."^{ [28] }
In 1851, John Parker published a book Quadrature of the Circle in which he claimed to have squared the circle. His method actually produced an approximation of π accurate to six digits.^{ [29] }^{ [30] }^{ [31] }
The problem of squaring the circle has been mentioned by poets such as Dante and Alexander Pope, with varied metaphorical meanings. Its literary use dates back at least to 414 BC, when the play The Birds by Aristophanes was first performed. In it, the character Meton of Athens mentions squaring the circle, possibly to indicate the paradoxical nature of his utopian city.^{ [32] }
Dante's Paradise , canto XXXIII, lines 133–135, contain the verses:
As the geometer his mind applies
To square the circle, nor for all his wit
Finds the right formula, howe'er he tries
For Dante, squaring the circle represents a task beyond human comprehension, which he compares to his own inability to comprehend Paradise.^{ [33] }
By 1742, when Alexander Pope published the fourth book of his Dunciad , attempts at circle-squaring had come to be seen as "wild and fruitless":^{ [30] }
Mad Mathesis alone was unconfined,
Too mad for mere material chains to bind,
Now to pure space lifts her ecstatic stare,
Now, running round the circle, finds it square.
Similarly, the Gilbert and Sullivan comic opera Princess Ida features a song which satirically lists the impossible goals of the women's university run by the title character, such as finding perpetual motion. One of these goals is "And the circle – they will square it/Some fine day."^{ [34] }
The sestina, a poetic form first used in the 12th century by Arnaut Daniel, has been said to square the circle in its use of a square number of lines (six stanzas of six lines each) with a circular scheme of six repeated words. Spanos (1978) writes that this form invokes a symbolic meaning in which the circle stands for heaven and the square stands for the earth.^{ [35] } A similar metaphor was used in "Squaring the Circle", a 1908 short story by O. Henry, about a long-running family feud. In the title of this story, the circle represents the natural world, while the square represents the city, the world of man.^{ [36] }
In later works circle-squarers such as Leopold Bloom in James Joyce's novel Ulysses and Lawyer Paravant in Thomas Mann's The Magic Mountain are seen as sadly deluded or as unworldly dreamers, unaware of its mathematical impossibility and making grandiose plans for a result they will never attain.^{ [37] }^{ [38] }
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. 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 the integers 0 and 1 and the operations for addition, subtraction, multiplication, division, and square roots.
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. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in 1706. 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.
Straightedge and compass construction, also known as ruler-and-compass 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 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 compass and straightedge constructions 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.
In geometry, a heptagon is a seven-sided polygon or 7-gon.
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, a hendecagon or 11-gon is an eleven-sided polygon.
In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon.
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ABCD.
In mathematics, a quadratrix is a curve having ordinates which are a measure of the area of another curve. The two most famous curves of this class are those of Dinostratus and E. W. Tschirnhaus, which are both related to the circle.
Approximations for the mathematical constant pi in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era (Archimedes). In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century.
In geometry, a 257-gon is a polygon with 257 sides. The sum of the interior angles of any non-self-intersecting 257-gon is 45,900°.
In geometry, a 65537-gon is a polygon with 65,537 (2^{16} + 1) sides. The sum of the interior angles of any non-self-intersecting 65537-gon is 11796300°.
In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon.
A Kepler triangle is a right triangle with edge lengths in a geometric progression. The ratio of the progression is √𝜙, where 𝜙 is the golden ratio, and can be written: , or approximately 1 : 1.272 : 1.618. The squares of the edges of this triangle are also in geometric progression according to the golden ratio itself.
In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
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.
In geometry, an icositrigon or 23-gon is a 23-sided polygon. The icositrigon has the distinction of being the smallest regular polygon that is not neusis constructible.
Similarly, the story "Squaring the Circle" is permeated with the integrating image: nature is a circle, the city a square.
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