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Bhāskara I's Sine Approximation Formula
In mathematics, Bhāskara I's sine approximation formula is a rational fraction, rational expression in one Variable (mathematics), variable for the computation of the approximation, approximate values of the sine, trigonometric sines discovered by Bhāskara I (c. 600 – c. 680), a seventh-century Indian mathematician. This formula is given in his treatise titled ''Mahabhaskariya''. It is not known how Bhāskara I arrived at his approximation formula. However, several historians of mathematics have put forward different hypotheses as to the method Bhāskara might have used to arrive at his formula. The formula is elegant and simple, and it enables the computation of reasonably accurate values of trigonometric sines without the use of geometry. Approximation formula The formula is given in verses 17–19, chapter VII, Mahabhaskariya of Bhāskara I. A translation of the verses is given below: (Now) I briefly state the rule (for finding the ''bhujaphala'' and the ''kotip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometry
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. History Sumerian astronomers studied angle me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brahmagupta's Interpolation Formula
Brahmagupta's interpolation formula is a second-order polynomial interpolation formula developed by the Indian mathematician and astronomer Brahmagupta (598–668 CE) in the early 7th century CE. The Sanskrit couplet describing the formula can be found in the supplementary part of ''Khandakadyaka'' a work of Brahmagupta completed in 665 CE. The same couplet appears in Brahmagupta's earlier ''Dhyana-graha-adhikara'', which was probably written "near the beginning of the second quarter of the 7th century CE, if not earlier." Brahmagupta was one of the first to describe and use an interpolation formula using second-order differences. Brahmagupta's interpolation formula is equivalent to modern-day second-order Newton–Stirling interpolation formula. Mathematicians prior to Brahmagupta used a simple linear interpolation formula. The linear interpolation formula to compute is : f(a)=f_r+ t D_r where t=\frac. For the computation of , Brahmagupta replaces with another expre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Madhava's Sine Table
Madhava's sine table is the table of trigonometric sines constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama (c. 1340 – c. 1425). The table lists the jya-s or Rsines of the twenty-four angles from 3.75 ° to 90° in steps of 3.75° (1/24 of a right angle, 90°). Rsine is just the sine multiplied by a selected radius and given as an integer. In this table, as in Aryabhata's earlier table, ''R'' is taken as 21600 ÷ 2 ≈ 3437.75. The table is encoded in the letters of the Sanskrit alphabet using the Katapayadi system, giving entries the appearance of the verses of a poem. Madhava's original work containing the table has not been found. The table is reproduced in the ''Aryabhatiyabhashya'' of Nilakantha Somayaji''The Aryabhatiam of Aryabhattacharya with the Bhashya of Nilakantha Somasutvan, Part 1 - Gaṇitapāda,'' Edited by K. Sambasiva Sastri, Trivandrum Sanskrit Series No.101. p. 55. https://archive.org/details/Triva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radius
In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is the line segment or distance from its center to any of its Vertex (geometry), vertices. The name comes from the Latin ''radius'', meaning ray but also the spoke of a chariot wheel.Definition of Radius at dictionary.reference.com. Accessed on 2009-08-08. The typical abbreviation and mathematical symbol for radius is ''R'' or ''r''. By extension, the diameter ''D'' is defined as twice the radius:Definition of radius at mathwords.com. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (angle)
A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane (mathematics), plane angle in which one Turn (geometry), full rotation is 360 degrees. It is not an SI unit—the SI unit of angular measure is the radian—but it is mentioned in the SI Brochure, SI brochure as an Non-SI units mentioned in the SI, accepted unit. Because a full rotation equals 2 radians, one degree is equivalent to radians. History The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the ecliptic path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient calendars, such as the Iranian calendar, Persian calendar and the Babylonian calendar, used 360 days for a year. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a Disk (mathematics), disc. The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Terminology * Annulus (mathematics), Annulus: a ring-shaped object, the region bounded by two concentric circles. * Circular arc, Arc: any Connected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumference
In geometry, the circumference () is the perimeter of a circle or ellipse. The circumference is 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 (geometry), locus corresponding to the Edge (geometry), edge of a Disk (geometry), disk. The is the circumference, or length, of any one of its great circles. Circle The circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this cannot be used as a definition. Under these circumstances, the circumference of a circle may be defined as the Limit (mathematics), limit of the perimeters of inscribed regular polygons as the number of sides increases without bound. The term circumference is used when measuring physical objects, as well as when considering abstrac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kerala School Of Astronomy And Mathematics
The Kerala school of astronomy and mathematics or the Kerala school was a school of Indian mathematics, mathematics and Indian astronomy, astronomy founded by Madhava of Sangamagrama in Kingdom of Tanur, Tirur, Malappuram district, Malappuram, Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and its original discoveries seem to have ended with Melpathur Narayana Bhattathiri, Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently discovered a number of important mathematical concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called ''Tantrasangraha'' (around 1500), and again in a commentary on this work, called ''Tantrasangraha-vakhya'', of unknown authors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Madhava Of Sangamagrama
Mādhava of Sangamagrāma (Mādhavan) Availabl/ref> () was an Indian mathematician and astronomer who is considered to be the founder of the Kerala school of astronomy and mathematics in the Late Middle Ages. Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry and algebra. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity". Biography Little is known about Madhava's life with certainty. However, from scattered references to Madhava found in diverse manuscripts, historians of Kerala school have pieced together information about the mathematician. In a manuscript preserved in the Oriental Institute, Baroda, Madhava has been referred to as ''Mādhavan vēṇvārōhādīnām karttā ... Mādhavan Ilaññippaḷḷi Emprān''. It has been noted that th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
