Bhāskara II ('; 1114–1185), also known as Bhāskarāchārya (), was an
Indian polymath,
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
,
astronomer
An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...
and engineer. From verses in his main work,
Siddhānta Śiromaṇi, it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Satpura mountain ranges of
Western Ghats, believed to be the town of Patana in Chalisgaon, located in present-day Khandesh region of
Maharashtra
Maharashtra () is a state in the western peninsular region of India occupying a substantial portion of the Deccan Plateau. It is bordered by the Arabian Sea to the west, the Indian states of Karnataka and Goa to the south, Telangana to th ...
by scholars. In a temple in Maharashtra, an inscription supposedly created by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for several generations before him as well as two generations after him.
Henry Colebrooke who was the first European to translate (1817) Bhaskaracharya II's mathematical classics refers to the family as
Maharashtrian Brahmin
Marathi Brahmins (also known as Maharashtrian Brahmins) are communities native to the Indian state of Maharashtra. They are classified into mainly three sub-divisions based on their places of origin, " Desh", " Karad" and "Konkan". The Brahmi ...
s residing on the banks of the
Godavari
The Godavari (, �od̪aːʋəɾiː is India's second longest river after the Ganga River and drains the third largest basin in India, covering about 10% of India's total geographical area. Its source is in Trimbakeshwar, Nashik, Maharash ...
.
Born in a Hindu
Deshastha Brahmin
Deshastha Brahmin is a Hinduism, Hindu Brahmin caste, subcaste mainly from the Indian state of Maharashtra and North Karnataka. Other than these states, according to authors K. S. Singh, Gregory Naik and Pran Nath Chopra, Deshastha Brahmins a ...
family of scholars, mathematicians and astronomers, Bhaskara II was the leader of a cosmic observatory at
Ujjain
Ujjain (, , old name Avantika, ) or Ujjayinī is a city in Ujjain district of the Indian state of Madhya Pradesh. It is the fifth-largest city in Madhya Pradesh by population and is the administrative as well as religious centre of Ujjain ...
, the main mathematical centre of ancient India. Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India. His main work, ''
Siddhānta-Śiromaṇi'' (
Sanskrit
Sanskrit (; stem form ; nominal singular , ,) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in northwest South Asia after its predecessor languages had Trans-cultural ...
for "Crown of Treatises"), is divided into four parts called ''
Līlāvatī'', ''
Bījagaṇita'', ''Grahagaṇita'' and ''Golādhyāya'', which are also sometimes considered four independent works.
[ These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named ''Karaṇā Kautūhala''.][
]
Date, place and family
Bhāskara gives his date of birth, and date of composition of his major work, in a verse in the Āryā metre
''Āryā metre'' is a metre used in Sanskrit, Prakrit and Marathi verses. A verse in metre is in four metrical lines called ''pāda''s. Unlike the majority of metres employed in classical Sanskrit, the metre is based on the number of s ( morae) ...
:[
This reveals that he was born in 1036 of the ]Shaka era
The Shaka era (IAST: Śaka, Śāka) is a historical Hindu calendar era (year numbering), the epoch (its year zero) of which corresponds to Julian year (calendar), Julian year 78.
The era has been widely used in different regions of the Indian ...
(1114 CE), and that he composed the ''Siddhānta Shiromani'' when he was 36 years old.[ ''Siddhānta Shiromani'' was completed during 1150 CE. He also wrote another work called the '' Karaṇa-kutūhala'' when he was 69 (in 1183).][ His works show the influence of ]Brahmagupta
Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...
, Śrīdhara, Mahāvīra
Mahavira (Devanagari: महावीर, ), also known as Vardhamana (Devanagari: वर्धमान, ), was the 24th ''Tirthankara'' (Supreme Preacher and Ford Maker) of Jainism. Although the dates and most historical details of his lif ...
, Padmanābha and other predecessors.[ Bhaskara lived in ]Patnadevi
Patanadevi is a historic and tourist place situated 18 km to the southwest of Chalisgaon, Maharashtra. It lies inside Gautala Autramghat Sanctuary and is surrounded by the high mountains of Sahyadri.
It consists of Chandika Devi Temple and H ...
located near Patan (Chalisgaon) in the vicinity of Sahyadri.
He was born in a Deśastha Rigvedi Brahmin family near Vijjadavida (Vijjalavida).
Munishvara (17th century), a commentator on Siddhānta Shiromani of Bhaskara has given the information about the location of Vijjadavida in his work Marīci Tīkā as follows:
This description locates Vijjalavida in Maharashtra, near the Vidarbha
Vidarbha (Pronunciation: Help:IPA/Marathi, �id̪əɾbʱə is a geographical region in the west Indian States and union territories of India, state of Maharashtra. Forming the eastern part of the state, it comprises Amravati Division, Amrav ...
region and close to the banks of Godavari river
The Godavari (, Help:IPA/Sanskrit, �od̪aːʋəɾiː is India's second longest river after the Ganges River, Ganga River and drains the third largest Drainage basin, basin in India, covering about 10% of India's total geographical area. It ...
. However scholars differ about the exact location. Many scholars have placed the place near Patan in Chalisgaon Taluka of Jalgaon district whereas a section of scholars identified it with the modern day Beed city. Some sources identified Vijjalavida as Bijapur or Bidar
Bidar ( ) is a city and headquarters of the Bidar district in Karnataka state of India. Bidar is a prominent place on the archaeological map of India, it is well known for architectural, historical religious and rich heritage sites. Pictures ...
in Karnataka
Karnataka ( ) is a States and union territories of India, state in the southwestern region of India. It was Unification of Karnataka, formed as Mysore State on 1 November 1956, with the passage of the States Reorganisation Act, 1956, States Re ...
. Identification of Vijjalavida with Basar in Telangana
Telangana is a States and union territories of India, state in India situated in the Southern India, south-central part of the Indian subcontinent on the high Deccan Plateau. It is the List of states and union territories of India by area, ele ...
has also been suggested.
Bhāskara is said to have been the head of an astronomical
Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest include ...
observatory at Ujjain
Ujjain (, , old name Avantika, ) or Ujjayinī is a city in Ujjain district of the Indian state of Madhya Pradesh. It is the fifth-largest city in Madhya Pradesh by population and is the administrative as well as religious centre of Ujjain ...
, the leading mathematical centre of medieval India. History records his great-great-great-grandfather holding a hereditary post as a court scholar, as did his son and other descendants. His father Maheśvara (Maheśvaropādhyāya[) was a mathematician, astronomer][ and astrologer, who taught him mathematics, which he later passed on to his son Lokasamudra. Lokasamudra's son helped to set up a school in 1207 for the study of Bhāskara's writings. He died in 1185 CE.
]
The ''Siddhānta-Śiromaṇi''
Līlāvatī
The first section ''Līlāvatī'' (also known as ''pāṭīgaṇita'' or ''aṅkagaṇita''), named after his daughter, consists of 277 verses.[ It covers calculations, progressions, ]measurement
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared to ...
, permutations, and other topics.[
]
Bijaganita
The second section ''Bījagaṇita''(Algebra) has 213 verses.[ It discusses zero, infinity, positive and negative numbers, and indeterminate equations including (the now called) ]Pell's equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive Square number, nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian ...
, solving it using a '' kuṭṭaka'' method.[ In particular, he also solved the case that was to elude ]Fermat
Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
and his European contemporaries centuries later
Grahaganita
In the third section ''Grahagaṇita'', while treating the motion of planets, he considered their instantaneous speeds.[ He arrived at the approximation:][ It consists of 451 verses
: for.
: close to , or in modern notation:]
: .
In his words:[
This result had also been observed earlier by Muñjalācārya (or Mañjulācārya) mānasam, in the context of a table of sines.][
Bhāskara also stated that at its highest point a planet's instantaneous speed is zero.][
]
Mathematics
Some of Bhaskara's contributions to mathematics include the following:
* A proof of the Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
by calculating the same area
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
in two different ways and then cancelling out terms to get ''a''2 + ''b''2 = ''c''2.
* In ''Lilavati'', solutions of quadratic, cubic
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
** Cubic crystal system, a crystal system w ...
and quartic indeterminate equation
In mathematics, particularly in number theory, an indeterminate system has fewer equations than unknowns but an additional a set of constraints on the unknowns, such as restrictions that the values be integers. In modern times indeterminate equati ...
s are explained.[Mathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy]
* Solutions of indeterminate quadratic equations (of the type ''ax''2 + ''b'' = ''y''2).
* Integer solutions of linear and quadratic indeterminate equations ('' Kuṭṭaka''). The rules he gives are (in effect) the same as those given by the Renaissance
The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
European mathematicians of the 17th century.
* A cyclic Chakravala method
The ''chakravala'' method () is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE)Hoiberg & Ramchandani – Students' Britannica India: Bhask ...
for solving indeterminate equations of the form ''ax''2 + ''bx'' + ''c'' = ''y''. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the ''chakravala'' method.
* The first general method for finding the solutions of the problem ''x''2 − ''ny''2 = 1 (so-called "Pell's equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive Square number, nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian ...
") was given by Bhaskara II.
* Solutions of Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name:
*Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real n ...
s of the second order, such as 61''x''2 + 1 = ''y''2. This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat
Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
, but its solution was unknown in Europe until the time of Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
in the 18th century.
* Solved quadratic equations with more than one unknown, and found negative and irrational
Irrationality is cognition, thinking, talking, or acting without rationality.
Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
solutions.
* Preliminary concept of mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
.
* Preliminary concept of differential calculus, along with preliminary ideas towards integration.
* preliminary ideas of differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
and differential coefficient.
* Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value theorem
In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin ...
. Traces of the general mean value theorem are also found in his works.
* Calculated the derivative of sine function, although he did not develop the notion of a derivative. (See Calculus section below.)
* In ''Siddhanta-Śiromaṇi'', Bhaskara developed spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the edge (geometry), sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, ge ...
along with a number of other trigonometric results. (See Trigonometry section below.)
Arithmetic
Bhaskara's arithmetic
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.
...
text '' Līlāvatī'' covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, solid geometry
Solid geometry or stereometry is the geometry of Three-dimensional space, three-dimensional Euclidean space (3D space).
A solid figure is the region (mathematics), region of 3D space bounded by a two-dimensional closed surface; for example, a ...
, the shadow of the gnomon
A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields, typically to measure directions, position, or time.
History
A painted stick dating from 2300 BC that was ...
, methods to solve indeterminate equations, and combination
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are ...
s.
''Līlāvatī'' is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and measurement. More specifically the contents include:
* Definitions.
* Properties of zero
0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
(including division, and rules of operations with zero).
* Further extensive numerical work, including use of negative number
In mathematics, a negative number is the opposite (mathematics), opposite of a positive real number. Equivalently, a negative number is a real number that is inequality (mathematics), less than 0, zero. Negative numbers are often used to represe ...
s and surds.
* Estimation of π.
* Arithmetical terms, methods of multiplication
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
, and squaring.
* Inverse rule of three, and rules of 3, 5, 7, 9, and 11.
* Problems involving interest
In finance and economics, interest is payment from a debtor or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is distinct f ...
and interest computation.
* Indeterminate equations ( Kuṭṭaka), integer solutions (first and second order). His contributions to this topic are particularly important, since the rules he gives are (in effect) the same as those given by the renaissance
The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work of Aryabhata
Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the '' Āryabhaṭīya'' (which mentions that in 3600 ' ...
and subsequent mathematicians.
His work is outstanding for its systematisation, improved methods and the new topics that he introduced. Furthermore, the ''Lilavati'' contained excellent problems and it is thought that Bhaskara's intention may have been that a student of 'Lilavati' should concern himself with the mechanical application of the method.
Algebra
His ''Bījaganita'' ("''Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
''") was a work in twelve chapters. It was the first text to recognize that a positive number has two square root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
s (a positive and negative square root).[50 Timeless Scientists von K.Krishna Murty] His work ''Bījaganita'' is effectively a treatise on algebra and contains the following topics:
* Positive and negative number
In mathematics, a negative number is the opposite (mathematics), opposite of a positive real number. Equivalently, a negative number is a real number that is inequality (mathematics), less than 0, zero. Negative numbers are often used to represe ...
s.
* The 'unknown' (includes determining unknown quantities).
* Determining unknown quantities.
* Surds (includes evaluating surds and their square roots).
* '' Kuṭṭaka'' (for solving indeterminate equation
In mathematics, particularly in number theory, an indeterminate system has fewer equations than unknowns but an additional a set of constraints on the unknowns, such as restrictions that the values be integers. In modern times indeterminate equati ...
s and Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name:
*Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real n ...
s).
* Simple equations (indeterminate of second, third and fourth degree).
* Simple equations with more than one unknown.
* Indeterminate quadratic equation
In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...
s (of the type ax2 + b = y2).
* Solutions of indeterminate equations of the second, third and fourth degree.
* Quadratic equations.
* Quadratic equations with more than one unknown.
* Operations with products of several unknowns.
Bhaskara derived a cyclic, ''chakravala'' method for solving indeterminate quadratic equations of the form ax2 + bx + c = y. Bhaskara's method for finding the solutions of the problem Nx2 + 1 = y2 (the so-called "Pell's equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive Square number, nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian ...
") is of considerable importance.
Trigonometry
The '' Siddhānta Shiromani'' (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also developed spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the edge (geometry), sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, ge ...
, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, results found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for and .
Calculus
His work, the '' Siddhānta Shiromani'', is an astronomical treatise and contains many theories not found in earlier works. Preliminary concepts of Differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
and mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, along with a number of results in 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 fiel ...
that are found in the work are of particular interest.
Evidence suggests Bhaskara was acquainted with some ideas of differential calculus. Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimal
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
s'.
* There is evidence of an early form of Rolle's theorem in his work,though it was stated without a modern formal proof .
* In this astronomical work he gave one procedure that looks like a precursor to infinitesimal methods. In terms that is if then that is a derivative of sine although he did not develop the notion on derivative.
** Bhaskara uses this result to work out the position angle of the ecliptic
The ecliptic or ecliptic plane is the orbital plane of Earth's orbit, Earth around the Sun. It was a central concept in a number of ancient sciences, providing the framework for key measurements in astronomy, astrology and calendar-making.
Fr ...
, a quantity required for accurately predicting the time of an eclipse.
* In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a '' truti'', or a of a second, and his measure of velocity was expressed in this infinitesimal unit of time.
* He was aware that when a variable attains the maximum value, its differential vanishes.
* He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero. In this result, there are traces of the general mean value theorem
In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin ...
, one of the most important theorems in analysis, which today is usually derived from Rolle's theorem. The mean value formula for inverse interpolation of the sine was later founded by Parameshvara in the 15th century in the ''Lilavati Bhasya'', a commentary on Bhaskara's ''Lilavati''.
Madhava (1340–1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara's work.
Astronomy
Using an astronomical model developed by Brahmagupta
Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...
in the 7th century, Bhāskara accurately defined many astronomical quantities, including, for example, the length of the sidereal year
A sidereal year (, ; ), also called a sidereal orbital period, is the time that Earth or another planetary body takes to orbit the Sun once with respect to the fixed stars.
Hence, for Earth, it is also the time taken for the Sun to return to t ...
, the time that is required for the Sun to orbit the Earth, as approximately 365.2588 days which is the same as in Surya siddhanta. The modern accepted measurement is 365.25636 day
A day is the time rotation period, period of a full Earth's rotation, rotation of the Earth with respect to the Sun. On average, this is 24 hours (86,400 seconds). As a day passes at a given location it experiences morning, afternoon, evening, ...
s, a difference of 3.5 minutes.[IERS EOP PC Useful constants](_blank)
An SI day or mean solar day equals 86400 SI seconds
The second (symbol: s) is a unit of time derived from the division of the day first into 24 hours, then to 60 minutes, and finally to 60 seconds each (24 × 60 × 60 = 86400). The current and formal definition in the International System of ...
.
From the mean longitude referred to the mean ecliptic and the equinox J2000 given in Simon, J. L., et al., "Numerical Expressions for Precession Formulae and Mean Elements for the Moon and the Planets" ''Astronomy and Astrophysics'' 282 (1994), 663–683.
His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
.
The twelve chapters of the first part cover topics such as:
* Mean longitude
Longitude (, ) is a geographic coordinate that specifies the east- west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lett ...
s of the planet
A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
s.
* True longitudes of the planets.
* The three problem
Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...
s of diurnal rotation. Diurnal motion refers to the apparent daily motion of stars around the Earth, or more precisely around the two celestial poles. It is caused by the Earth's rotation on its axis, so every star apparently moves on a circle that is called the diurnal circle.
* Syzygies.
* Lunar eclipse
A lunar eclipse is an astronomical event that occurs when the Moon moves into the Earth's shadow, causing the Moon to be darkened. Such an alignment occurs during an eclipse season, approximately every six months, during the full moon phase, ...
s.
* Solar eclipse
A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby obscuring the view of the Sun from a small part of Earth, totally or partially. Such an alignment occurs approximately every six months, during the eclipse season i ...
s.
* Latitude
In geography, latitude is a geographic coordinate system, geographic coordinate that specifies the north-south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from −90° at t ...
s of the planets.
* Sunrise equation.
* The Moon
The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...
's crescent
A crescent shape (, ) is a symbol or emblem used to represent the lunar phase (as it appears in the northern hemisphere) in the first quarter (the "sickle moon"), or by extension a symbol representing the Moon itself.
In Hindu iconography, Hind ...
.
* Conjunctions of the planets with each other.
* Conjunctions of the planets with the fixed star
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s.
* The paths of the Sun and Moon.
The second part contains thirteen chapters on the sphere. It covers topics such as:
*Praise of study of the sphere.
*Nature of the sphere.
* Cosmography and geography
Geography (from Ancient Greek ; combining 'Earth' and 'write', literally 'Earth writing') is the study of the lands, features, inhabitants, and phenomena of Earth. Geography is an all-encompassing discipline that seeks an understanding o ...
.
*Planetary mean motion
In orbital mechanics, mean motion (represented by ''n'') is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the a ...
.
* Eccentric epicyclic model of the planets.
*The armillary sphere.
*Spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the edge (geometry), sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, ge ...
.
*Ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
calculations.
*First visibilities of the planets.
*Calculating the lunar crescent.
*Astronomical instruments.
*The season
A season is a division of the year based on changes in weather, ecology, and the number of daylight hours in a given region. On Earth, seasons are the result of the axial parallelism of Earth's axial tilt, tilted orbit around the Sun. In temperat ...
s.
*Problems of astronomical calculations.
Engineering
The earliest reference to a perpetual motion
Perpetual motion is the motion of bodies that continues forever in an unperturbed system. A perpetual motion machine is a hypothetical machine that can do work indefinitely without an external energy source. This kind of machine is impossible ...
machine date back to 1150, when Bhāskara II described a wheel
A wheel is a rotating component (typically circular in shape) that is intended to turn on an axle Bearing (mechanical), bearing. The wheel is one of the key components of the wheel and axle which is one of the Simple machine, six simple machin ...
that he claimed would run forever.
Bhāskara II invented a variety of instruments one of which is ''Yaṣṭi-yantra''. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale.
Legends
In his book '' Lilavati'', he reasons: "In this quantity also which has zero as its divisor there is no change even when many quantities have entered into it or come out f it just as at the time of destruction and creation when throngs of creatures enter into and come out of im, there is no change inthe infinite and unchanging ishnu.
"Behold!"
It has been stated, by several authors, that Bhaskara II proved the Pythagorean theorem by drawing a diagram and providing the single word "Behold!". Sometimes Bhaskara's name is omitted and this is referred to as the ''Hindu proof'', well known by schoolchildren.
However, as mathematics historian Kim Plofker points out, after presenting a worked-out example, Bhaskara II states the Pythagorean theorem:
This is followed by:
Plofker suggests that this additional statement may be the ultimate source of the widespread "Behold!" legend.
Legacy
A number of institutes and colleges in India are named after him, including Bhaskaracharya Pratishthana in Pune, Bhaskaracharya College of Applied Sciences in Delhi, Bhaskaracharya Institute For Space Applications and Geo-Informatics in Gandhinagar.
On 20 November 1981 the Indian Space Research Organisation (ISRO) launched the Bhaskara II satellite honouring the mathematician and astronomer.
Invis Multimedia released ''Bhaskaracharya'', an Indian documentary short on the mathematician in 2015.
See also
* List of Indian mathematicians
Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians in the modern era. One of such works is Hindu numeral system which is predominantly used today and is likely ...
* Bride's Chair
* Bījapallava
Notes
References
Bibliography
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Further reading
* W. W. Rouse Ball. ''A Short Account of the History of Mathematics'', 4th Edition. Dover Publications, 1960.
* George Gheverghese Joseph. ''The Crest of the Peacock: Non-European Roots of Mathematics'', 2nd Edition. Penguin Books
Penguin Books Limited is a Germany, German-owned English publishing, publishing house. It was co-founded in 1935 by Allen Lane with his brothers Richard and John, as a line of the publishers the Bodley Head, only becoming a separate company the ...
, 2000.
* University of St Andrews
The University of St Andrews (, ; abbreviated as St And in post-nominals) is a public university in St Andrews, Scotland. It is the List of oldest universities in continuous operation, oldest of the four ancient universities of Scotland and, f ...
, 2000.
* Ian Pearce
''Bhaskaracharya II''
at the MacTutor archive. St Andrews University, 2002.
*
External links
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12th-century Indian mathematicians
12th-century Indian astronomers
People from Jalgaon
1110s births
1185 deaths
Algebraists
Scientists from Maharashtra
Scholars from Maharashtra
Acharyas
Sanskrit writers