Yuktibhasa (Malayalam:യുക്തിഭാഷ ; meaning — rationale language ) also known as Ganita Yuktibhasa (compendium of astronomical rationale) is a major treatise on Mathematics and Astronomy, written by Indian astronomer Jyesthadeva of the Kerala School of Mathematics in AD 1530.^[1] The treatise is a consolidation of the discoveries of the works of Madhava of Sangamagrama, Nilakantha Somayaji, Parameswara, Jyeshtadeva, Achyuta Panikkar and other astronomer-mathematicians at the Kerala School. Yuktibhasa is mainly based on Nilakantha's Tantra Samgraha.^[2]. It is considered as the first text on calculus.^[3][4][5][6] The work predates those of European mathematicians by over a century. However, the treatise was largely unnoticed beyond Kerala, as the book was written in the local language of Malayalam. Malayalam (മലയാളം ) is the language spoken predominantly in the state of Kerala, in southern India. ... A treatise is a systematic analysis of a certain subject. ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... Radio telescopes are among many different tools used by astronomers Astronomy (Greek: αστρονομία = άστρον + νόμος, astronomia = astron + nomos, literally, law of the stars) is the science of celestial objects (such as stars, planets, comets, and galaxies) and phenomena that originate outside the Earths atmosphere (such as auroras and cosmic background radiation). ... Jyesthadeva (1500-1575), born in Kerala, was a major mathematician, and author of the 1501 Yukti-bhasa, which was a survey of Kerala mathematics and astronomy that was unique at the time for its exacting proofs of the theorems it presented. ... The Kerala School was a school of mathematics and astronomy founded by Madhava in Kerala (in South India) which included as its prominent members Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... Madhava (माधव) of Sangamagrama (1350-1425) was a major mathematician from Kerala, South India. ... Nilakantha Somayaji (नीलकण्ठ सोमयाजि) (1444-1544), from Kerala, was a major mathematician and astronomer. ... People by name Parameshwara: The fourteenth century Indian mathematician. ... Jyestadeva (1500-1610), was an astronomer of the Kerala school founded by Madhava of Sangamagrama and a student of Damodara. ... Calculus is a central branch of mathematics, developed from algebra and geometry. ... This article is about the continent. ... Kerala ( (Anglicised) or (native); Malayalam: േകരളം, — ) is a state on the tropical Malabar Coast of southwestern India. ...

The work was unique for its time for containing exacting proofs of the theorems it presented.^[7] Look up proof in Wiktionary, the free dictionary. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...

Contents

Yuktibhasa contains most of the developments of earlier Kerala School mathematicians, particularly Madhava and Nilakantha. The text is divided into two parts — the former deals with mathematical analysis of arithmetic, algebra, trigonometry and geometry, Logistics, algebraic problems, fractions, Rule of Three, Kuttakaram, circle and disquisition on R-Sine; and the latter about astronomy.^[1] Madhava (माधव) of Sangamagrama (1350-1425) was a major mathematician from Kerala, South India. ... Nilakantha Somayaji (नीलकण्ठ सोमयाजि) (1444-1544), from Kerala, was a major mathematician and astronomer. ... Arithmetic is the current mathematics collaboration of the week! Please help improve it to featured article standard. ... Algebra (from Arabic: الجبر, al-ÄŸabr) is a branch of mathematics concerning the study of structure, relation and quantity. ... Wikibooks has more about this subject: Trigonometry Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine, cosine and tangent. ... Table of Geometry, from the 1728 Cyclopaedia. ... Logistics is the management of resources and their distribution. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In common usage a fraction is any part of a unit. ... The Rule of three is the method of finding the fourth term of a mathematical proportion when three terms are given, given that the products of the first and fourth terms are equal to the product of the second and third. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ...

Mathematics

Explanation of the sine rule in Yuktibhasa

As per the old Indian tradition of starting off new chapters with elementary content, the first four chapters of the Yuktibhasa contain elementary mathematics, such as divison, proof of Pythagorean theorem, square root determination, etc.^[8] The radical ideas are not discussed until the sixth chapter on circumference of a circle. Yuktibhasa contains the derivation and proof of the power series for inverse tangent, discovered by Madhava.^[2] In the text, Jyesthadeva describes Madhava's series in the following manner: In trigonometry, the law of sines (or sine law) is a statement about arbitrary triangles in the plane. ... In mathematics, the Pythagorean theorem or Pythagorass theorem is a relation in Euclidean geometry between the three sides of a right triangle. ... The circumference is the distance around a closed curve. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.

This yields Image File history File links Cquote1. ... Image File history File links Cquote2. ...

which further yields the theorem

popularly attributed to James Gregory, who discovered it three centuries after Madhava. This series was traditionally known as the Gregory series but scholars have recently begun referring to it as the Madhava-Gregory series, in recognition of Madhava's work.^[5] James Gregory James Gregory (November 1638 – October 1675), was a Scottish mathematician and astronomer. ...

The text also elucidates Madhava's infinite series expansion of π: In mathematics, a series is often represented as the sum of a sequence of terms. ... Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...

which he obtained from the power series expansion of the arctangent function.

Using a rational approximation of this series, he gave values of the number π as 3.14159265359 - correct to 11 decimals; and as 3.1415926535898 - correct to 13 decimals. These were the most accurate approximations of π after almost a thousand years.^{[citation needed]} Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...

The text describes that he gave two methods for computing the value of π.

• One of these methods is to obtain a rapidly converging series by transforming the original infinite series of π. By doing so, he obtained the infinite series

and used the first 21 terms to compute an approximation of π correct to 11 decimal places as 3.14159265359.

• The other method was to add a remainder term to the original series of π. The remainder term was used

in the infinite series expansion of to improve the approximation of π to 13 decimal places of accuracy when n = 75.

Apart from these, the Yukthibhasa contains many elementary, and complex mathematics, including, Elementary mathematics consists of mathematics topics frequently taught at the primary and secondary school levels. ...

• Proof for the expansion of the sine and cosine functions.
• Integer solutions of systems of first degree equations (solved using a system known as kuttakaram)
• Rules for finding the sines and the cosines of the sum and difference of two angles.
• The earliest statement of Wallis product and the Taylor series.
• Geometric derivations of series.
• Tests of convergence (often attributed to Cauchy)
• Fundamentals of calculus^[5]: differentiation, term by term integration, iterative methods for solutions of non-linear equations, and the theory that the area under a curve is its integral.

Most of these resuts were achieved centuries before their European counterparts, to whom the credit is often misattributed. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... White cliffs of Dover in England White cliffs of Rugen down the Baltic coast from Schleswig The Angles is a modern English word for a Germanic-speaking people who took their name from the cultural ancestor of Angeln, a modern district located in Schleswig, Germany. ... In mathematics, Wallis product for π, written down in 1655 by John Wallis, states that Proof First of all, consider the root of sin(x)/x is ±nπ, where n = 1, 2, 3, ... Then, we can express sine as an infinite product of linear factors given by its roots: To... As the degree of the Taylor series rises, it approaches the correct function. ... In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Differentiation can mean the following: In biology: cellular differentiation; evolutionary differentiation; In mathematics: see: derivative In cosmogony: planetary differentiation Differentiation (geology); Differentiation (logic); Differentiation (marketing). ... Integration may be any of the following: In the most general sense, integration may be any bringing together of things: the integration of two or more economies, cultures, religions (usually called syncretism), etc. ...

Astronomy

Chapters seven to seventeen of the text deals essentially with subjects of astronomy, viz. Planetary orbit, Celestial sphere, ascension, declination, directions and shadows, spherical triangles, ellipses and parallax correction. The planetary theory described in the book is similar to that later adopted by Danish astronomer Tycho Brahe.^[9] Two bodies with a slight difference in mass orbiting around a common barycenter. ... The Christian doctrine of the Ascension holds that Jesus bodily ascended to heaven by His own power in presence of His disciples, following his resurrection. ... In astronomy, declination (dec) is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... The ellipse and some of its mathematical properties. ... Parallax (Greek: παραλλαγή (parallagé) = alteration) is the change of angular position of two stationary points relative to each other as seen by an observer, due to the motion of an observer. ... Tycho Brahe Monument of Tycho Brahe and Johannes Kepler in Prague , born Tyge Ottesen Brahe (December 14, 1546 – October 24, 1601), was a Danish (Scanian) nobleman astronomer as well as an astrologer and alchemist. ...

See also

• Indian Mathematics
• Kerala School
• Possible transmission of Kerala mathematics to Europe
• Eurocentrism

The chronology of Indian mathematics spans from the Indus Valley civilization (3300-1500 BC) and Vedic civilization (1500-500 BC) to modern India (21st century AD). ... The Kerala School was a school of mathematics and astronomy founded by Madhava in Kerala (in South India) which included as its prominent members Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... The Kerala School was a school of mathematics and astronomy founded by Madhava in Kerala (in South India) which included as its prominent members Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... Eurocentrism is the practice, conscious or otherwise, of placing emphasis on European (and, generally, Western) concerns, culture and values at the expense of those of other cultures. ...

External links

• Biography of Jyesthadeva — School of Mathematics and Statistics University of St Andrews, Scotland

Notes

2006 (MMVI) is a common year starting on Sunday of the Gregorian calendar. ... July 9 is the 190th day of the year (191st in leap years) in the Gregorian Calendar, with 175 days remaining. ... 2006 (MMVI) is a common year starting on Sunday of the Gregorian calendar. ... July 9 is the 190th day of the year (191st in leap years) in the Gregorian Calendar, with 175 days remaining. ... 2006 (MMVI) is a common year starting on Sunday of the Gregorian calendar. ... July 9 is the 190th day of the year (191st in leap years) in the Gregorian Calendar, with 175 days remaining. ... 2006 (MMVI) is a common year starting on Sunday of the Gregorian calendar. ... July 7 is the 188th day of the year (189th in leap years) in the Gregorian Calendar, with 177 days remaining. ... 2006 (MMVI) is a common year starting on Sunday of the Gregorian calendar. ... July 9 is the 190th day of the year (191st in leap years) in the Gregorian Calendar, with 175 days remaining. ... 2006 (MMVI) is a common year starting on Sunday of the Gregorian calendar. ... July 7 is the 188th day of the year (189th in leap years) in the Gregorian Calendar, with 177 days remaining. ... 2006 (MMVI) is a common year starting on Sunday of the Gregorian calendar. ... July 9 is the 190th day of the year (191st in leap years) in the Gregorian Calendar, with 175 days remaining. ... 2006 (MMVI) is a common year starting on Sunday of the Gregorian calendar. ... July 9 is the 190th day of the year (191st in leap years) in the Gregorian Calendar, with 175 days remaining. ...

References