Brahmagupta Formula
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Brahmagupta ( – ) was an Indian
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 ...
and
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 ...
. He is the author of two early works on
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 ar ...
and
astronomy 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 includ ...
: the ''
Brāhmasphuṭasiddhānta The ''Brāhma-sphuṭa-siddhānta'' ("Correctly Established Doctrine of Brahma", abbreviated BSS) is a main work of Brahmagupta, written c. 628. This text of mathematical astronomy contains significant mathematical content, including the first goo ...
'' (BSS, "correctly established
doctrine Doctrine (from , meaning 'teaching, instruction') is a codification (law), codification of beliefs or a body of teacher, teachings or instructions, taught principles or positions, as the essence of teachings in a given branch of knowledge or in a ...
of
Brahma Brahma (, ) is a Hindu god, referred to as "the Creator" within the Trimurti, the triple deity, trinity of Para Brahman, supreme divinity that includes Vishnu and Shiva.Jan Gonda (1969)The Hindu Trinity, Anthropos, Bd 63/64, H 1/2, pp. 212– ...
", dated 628), a theoretical treatise, and the ''
Khandakhadyaka ''Khaṇḍakhādyaka'' (meaning "edible bite; morsel of food") is a Sanskrit-language astronomical treatise written by Indian mathematician and astronomer Brahmagupta in 665 CE. The treatise contains eight chapters covering such topics as the lo ...
'' ("edible bite", dated 665), a more practical text. In 628 CE, Brahmagupta first described
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
as an attractive force, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to describe it. He is also credited with the first clear description of the
quadratic formula In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions. Given a general quadr ...
(the solution of the quadratic equation)Bradley, Michael. ''The Birth of Mathematics: Ancient Times to 1300'', p. 86 (Infobase Publishing 2006) in his main work, the ''Brāhma-sphuṭa-siddhānta''.


Life and career

Brahmagupta, according to his own statement, was born in 598 CE. Born in ''Bhillamāla'' in
Gurjaradesa Gurjaradesa, (, or Gurjaratra)* * is a historical region in India comprising the southern Rajasthan and northern Gujarat during the period of 6th–12th century CE. The predominant power of the region, the Gurjara-Pratiharas eventually cont ...
(modern
Bhinmal Bhinmal (previously Shrimal Nagar) is an ancient town in the Jalore District of Rajasthan, India. It is south of Jalore. Bhinmal was the early capital of Gurjaradesa, comprising modern-day southern Rajasthan and northern Gujarat. The town was ...
in
Rajasthan Rajasthan (; Literal translation, lit. 'Land of Kings') is a States and union territories of India, state in northwestern India. It covers or 10.4 per cent of India's total geographical area. It is the List of states and union territories of ...
, India) during the reign of the
Chavda dynasty The Chavda (IAST:Chávaḍá), also spelled Chawda or Chavada was a dynasty which ruled the region of modern-day Gujarat in India, from c. 690 to 942. Variants of the name for the dynasty include Chapotkatas, Chahuda and Chávoṭakas. Van ...
ruler Vyagrahamukha. He was the son of Jishnugupta and was a Hindu by religion, in particular, a
Shaivite Shaivism (, , ) is one of the major Hindu traditions, which worships Shiva as the supreme being. It is the second-largest Hindu sect after Vaishnavism, constituting about 385 million Hindus, found widely across South Asia (predominantly in ...
.: "Brahmagupta, one of the most celebrated mathematicians of the East, indeed of the world, was born in the year 598 CE, in the town of Bhillamala during the reign of King Vyaghramukh of the Chapa Dynasty." He lived and worked there for a good part of his life.
Prithudaka Svamin Chaturveda Prithudaka Swami () was an Indian mathematician best known for his work on solving equations. He also wrote an important commentary on Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astro ...
, a later commentator, called him ''Bhillamalacharya'', the teacher from Bhillamala. Bhillamala was the capital of the
Gurjaradesa Gurjaradesa, (, or Gurjaratra)* * is a historical region in India comprising the southern Rajasthan and northern Gujarat during the period of 6th–12th century CE. The predominant power of the region, the Gurjara-Pratiharas eventually cont ...
, the second-largest kingdom of Western India, comprising southern
Rajasthan Rajasthan (; Literal translation, lit. 'Land of Kings') is a States and union territories of India, state in northwestern India. It covers or 10.4 per cent of India's total geographical area. It is the List of states and union territories of ...
and northern
Gujarat Gujarat () is a States of India, state along the Western India, western coast of India. Its coastline of about is the longest in the country, most of which lies on the Kathiawar peninsula. Gujarat is the List of states and union territories ...
in modern-day India. It was also a centre of learning for mathematics and astronomy. He became an astronomer of the ''Brahmapaksha'' school, one of the four major schools of Indian astronomy during this period. He studied the five traditional ''
Siddhantas (Devanagari: ) is a Sanskrit term denoting the established and accepted view of any particular school within Indian philosophy; literally "settled opinion or doctrine, dogma, axiom, received or admitted truth; any fixed or established or canon ...
'' on Indian astronomy as well as the work of other astronomers including
Aryabhata I 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 '' ...
, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra. In the year 628, at the age of 30, he composed the ''Brāhmasphuṭasiddhānta'' ("improved treatise of Brahma") which is believed to be a revised version of the received ''Siddhanta'' of the ''Brahmapaksha'' school of astronomy. Scholars state that he incorporated a great deal of originality into his revision, adding a considerable amount of new material. The book consists of 24 chapters with 1008 verses in the ārya metre. A good deal of it is astronomy, but it also contains key chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain new insights due to Brahmagupta himself. Later, Brahmagupta moved to
Ujjaini 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 ...
,
Avanti Avanti (in Italian, meaning 'ahead', 'forward', or 'before', and also an unrelated Sanskrit name) may refer to: Vehicles * Studebaker Avanti, a model of automobile built by Studebaker * Avanti II, a successor model made by Avanti Motor Corporati ...
, a major centre for astronomy in central India. At the age of 67, he composed his next well-known work '' Khanda-khādyaka'', a practical manual of Indian astronomy in the '' karana'' category meant to be used by students. Brahmagupta died in 668 CE, and he is presumed to have died in Ujjain.


Works

Brahmagupta composed the following treatises: * ''
Brāhmasphuṭasiddhānta The ''Brāhma-sphuṭa-siddhānta'' ("Correctly Established Doctrine of Brahma", abbreviated BSS) is a main work of Brahmagupta, written c. 628. This text of mathematical astronomy contains significant mathematical content, including the first goo ...
'', composed in 628 CE. * ''
Khaṇḍakhādyaka ''Khaṇḍakhādyaka'' (meaning "edible bite; morsel of food") is a Sanskrit-language astronomical treatise written by Indian mathematician and astronomer Brahmagupta in 665 CE. The treatise contains eight chapters covering such topics as the lo ...
'', composed in 665 CE. * ''Grahaṇārkajñāna'', (ascribed in one manuscript)


Reception

Brahmagupta's mathematical advances were carried on further by
Bhāskara II Bhāskara II ('; 1114–1185), also known as Bhāskarāchārya (), was an Indian people, Indian polymath, Indian mathematicians, mathematician, astronomer and engineer. From verses in his main work, Siddhānta Śiromaṇi, it can be inferre ...
, a lineal descendant in Ujjain, who described Brahmagupta as the ''ganaka-chakra-chudamani'' (the gem of the circle of mathematicians).
Prithudaka Svamin Chaturveda Prithudaka Swami () was an Indian mathematician best known for his work on solving equations. He also wrote an important commentary on Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astro ...
wrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations.
Lalla Lalla ( 720–790 CE) was an Indian Indian mathematics, mathematician, astronomer, and astrologer who belonged to a family of astronomers. Lalla was the son of Trivikrama Bhatta and the grandson of Śâmba."Lalla." Complete Dictionary of Scientif ...
and
Bhattotpala Utpala, also known as (') was an astronomer from Kashmir region of present-day India, who lived in the 9th or the 10th century. He wrote several Sanskrit-language texts on astrology and astronomy, the best-known being his commentaries on the w ...
in the 8th and 9th centuries wrote commentaries on the ''Khanda-khadyaka''. Further commentaries continued to be written into the 12th century. A few decades after the death of Brahmagupta,
Sindh Sindh ( ; ; , ; abbr. SD, historically romanized as Sind (caliphal province), Sind or Scinde) is a Administrative units of Pakistan, province of Pakistan. Located in the Geography of Pakistan, southeastern region of the country, Sindh is t ...
came under the Arab Caliphate in 712 CE.
Expeditions Expedition may refer to: * An exploration, journey, or voyage undertaken by a group of people especially for discovery and scientific research Places * Expedition Island, a park in Green River, Wyoming, US * Expedition Range, a mountain range ...
were sent into ''Gurjaradesa'' ("''Al-Baylaman'' in ''Jurz''", as per Arab historians). The kingdom of Bhillamala seems to have been annihilated but Ujjain repulsed the attacks. The court of Caliph
Al-Mansur Abū Jaʿfar ʿAbd Allāh ibn Muḥammad al-Manṣūr (; ‎; 714 – 6 October 775) usually known simply as by his laqab al-Manṣūr () was the second Abbasid caliph, reigning from 754 to 775 succeeding his brother al-Saffah (). He is known ...
(754–775) received an embassy from Sindh, including an astrologer called Kanaka, who brought (possibly memorised) astronomical texts, including those of Brahmagupta. Brahmagupta's texts were translated into Arabic by
Muḥammad ibn Ibrāhīm al-Fazārī Muhammad ibn Ibrahim ibn Habib ibn Sulayman ibn Samra ibn Jundab al-Fazari () (died 796 or 806) was an Arab philosopher, mathematician and astronomer. Biography Al-Fazārī translated many scientific books into Arabic and Persian. He is credi ...
, an astronomer in Al-Mansur's court, under the names ''
Sindhind ''Zīj as-Sindhind'' (, ''Zīj as‐Sindhind al‐kabīr'', lit. "Great astronomical tables of the Sindhind"; from Sanskrit ''siddhānta'', "system" or "treatise") is a work of zij (astronomical handbook with tables used to calculate celestial po ...
'' and ''Arakhand''. An immediate outcome was the spread of the decimal number system used in the texts. The mathematician
Al-Khwarizmi Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
(800–850 CE) wrote a text called ''al-Jam wal-tafriq bi hisal-al-Hind'' (Addition and Subtraction in Indian Arithmetic), which was translated into Latin in the 13th century as ''Algorithmi de numero indorum''. Through these texts, the decimal number system and Brahmagupta's algorithms for arithmetic have spread throughout the world. Al-Khwarizmi also wrote his own version of ''Sindhind'', drawing on Al-Fazari's version and incorporating Ptolemaic elements. Indian astronomic material circulated widely for centuries, even making its way into medieval Latin texts. The historian of science
George Sarton George Alfred Leon Sarton (; 31 August 1884 – 22 March 1956) was a Belgian-American chemist and historian. He is considered the founder of the discipline of the history of science as an independent field of study. His most influential works were ...
called Brahmagupta "one of the greatest scientists of his race and the greatest of his time."


Mathematics


Algebra

Brahmagupta gave the solution of the general
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
in chapter eighteen of ''Brahmasphuṭasiddhānta'',
The difference between ''rupas'', when inverted and divided by the difference of the oefficientsof the nknowns is the unknown in the equation. The ''rupas'' are ubtracted on the sidebelow that from which the square and the unknown are to be subtracted.
which is a solution for the equation where ''rupas'' refers to the constants and . The solution given is equivalent to . He further gave two equivalent solutions to the general
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 ...
18.44. Diminish by the middle
umber Umber is a natural earth pigment consisting of iron oxide and manganese oxide; it has a brownish color that can vary among shades of yellow, red, and green. Umber is considered one of the oldest pigments known to humans, first used in the Ajant ...
the square-root of the ''rupas'' multiplied by four times the square and increased by the square of the middle
umber Umber is a natural earth pigment consisting of iron oxide and manganese oxide; it has a brownish color that can vary among shades of yellow, red, and green. Umber is considered one of the oldest pigments known to humans, first used in the Ajant ...
divide the remainder by twice the square. he result isthe middle
umber Umber is a natural earth pigment consisting of iron oxide and manganese oxide; it has a brownish color that can vary among shades of yellow, red, and green. Umber is considered one of the oldest pigments known to humans, first used in the Ajant ...

18.45. Whatever is the square-root of the ''rupas'' multiplied by the square ndincreased by the square of half the unknown, diminished that by half the unknown nddivide he remainderby its square. he result isthe unknown.
which are, respectively, solutions for the equation equivalent to, :x = \frac and :x = \frac He went on to solve systems of simultaneous
indeterminate equations 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 ...
stating that the desired variable must first be isolated, and then the equation must be divided by the desired variable's
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
. In particular, he recommended using "the pulverizer" to solve equations with multiple unknowns.
18.51. Subtract the colors different from the first color. he remainderdivided by the first olor's coefficientis the measure of the first.
erms The Erms () is a river of the karstified Swabian Alb range in Baden-Württemberg, Germany. It flows into the Neckar in Neckartenzlingen. On its way from the Karst spring to the next large municipality Bad Urach, a former Erms sedimented, especia ...
two by two reconsidered hen reduced tosimilar divisors, nd so onrepeatedly. If there are many olors the pulverizer s to be used
Like the algebra of
Diophantus Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. Although Jose ...
, the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms. "he was the first one to give a ''general'' solution of the linear Diophantine equation ''ax'' + ''by'' = ''c'', where ''a'', ''b'', and ''c'' are integers. ..It is greatly to the credit of Brahmagupta that he gave ''all'' integral solutions of the linear Diophantine equation, whereas Diophantus himself had been satisfied to give one particular solution of an indeterminate equation. Inasmuch as Brahmagupta used some of the same examples as Diophantus, we see again the likelihood of Greek influence in India – or the possibility that they both made use of a common source, possibly from Babylonia. It is interesting to note also that the algebra of Brahmagupta, like that of Diophantus, was syncopated. Addition was indicated by juxtaposition, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, as in our fractional notation but without the bar. The operations of multiplication and evolution (the taking of roots), as well as unknown quantities, were represented by abbreviations of appropriate words." The extent of Greek influence on this
syncopation In music, syncopation is a variety of rhythms played together to make a piece of music, making part or all of a tune or piece of music off-beat (music), off-beat. More simply, syncopation is "a disturbance or interruption of the regular flow of ...
, if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source.


Arithmetic

The four fundamental operations (addition, subtraction, multiplication, and division) were known to many cultures before Brahmagupta. This current system is based on the
Hindu–Arabic numeral system The Hindu–Arabic numeral system (also known as the Indo-Arabic numeral system, Hindu numeral system, and Arabic numeral system) is a positional notation, positional Decimal, base-ten numeral system for representing integers; its extension t ...
and first appeared in the ''Brāhmasphuṭasiddhānta''. Brahmagupta describes multiplication in the following way:
The multiplicand is repeated like a string for cattle, as often as there are integrant portions in the multiplier and is repeatedly multiplied by them and the products are added together. It is multiplication. Or the multiplicand is repeated as many times as there are component parts in the multiplier.
Indian arithmetic was known in medieval Europe as ''modus Indorum'' meaning "method of the Indians". In the ''Brāhmasphuṭasiddhānta'', four methods for multiplication were described, including ''gomūtrikā'', which is said to be close to the present-day methods. In the beginning of chapter twelve of his ''Brāhmasphuṭasiddhānta'', entitled "Calculation", he also details operations on fractions. The reader is expected to know the basic arithmetic operations as far as taking the square root, although he explains how to find the cube and cube-root of an integer and later gives rules facilitating the computation of squares and square roots. He then gives rules for dealing with five types of combinations of fractions: ; ; ; ; and .


Squares and Cubes

Brahmagupta then goes on to give the sum of the squares and cubes of the first integers.
12.20. The sum of the squares is that ummultiplied by twice the
umber of Umber is a natural earth pigment consisting of iron oxide and manganese oxide; it has a brownish color that can vary among shades of yellow, red, and green. Umber is considered one of the oldest pigments known to humans, first used in the Ajant ...
step increased by one nddivided by three. The sum of the cubes is the square of that umPiles of these with identical balls an also be computed
Here Brahmagupta found the result in terms of the ''sum'' of the first integers, rather than in terms of as is the modern practice. Here the sums of the squares and cubes of the first ''n'' integers are defined in terms of the sum of the ''n'' integers itself; He gives the sum of the squares of the first natural numbers as and the sum of the cubes of the first n natural numbers as .


Zero

Brahmagupta's ''Brahmasphuṭasiddhānta'' is the first book that provides rules for arithmetic manipulations that apply to
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 ...
and to
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 ''Brāhmasphuṭasiddhānta'' is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the
Babylonians Babylonia (; , ) was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Kuwait, Syria and Iran). It emerged as an Akkadian-populated but Amorite-ru ...
or as a symbol for lack of quantity as was done by
Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
and the
Romans Roman or Romans most often refers to: *Rome, the capital city of Italy *Ancient Rome, Roman civilization from 8th century BC to 5th century AD *Roman people, the people of Roman civilization *Epistle to the Romans, shortened to Romans, a letter w ...
. In chapter eighteen of his ''Brāhmasphuṭasiddhānta'', Brahmagupta describes operations on negative numbers. He first describes addition and subtraction,
18.30. he sumof two positives is positives, of two negatives negative; of a positive and a negative he sumis their difference; if they are equal it is zero. The sum of a negative and zero is negative,
hat A hat is a Headgear, head covering which is worn for various reasons, including protection against weather conditions, ceremonial reasons such as university graduation, religious reasons, safety, or as a fashion accessory. Hats which incorpor ...
of a positive and zero positives, nd thatof two zeros zero.
..br /> 18.32. A negative minus zero is negative, a positive inus zerois positive; zero inus zerois zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added.
He goes on to describe multiplication,
18.33. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.
But his description of
division by zero In mathematics, division by zero, division (mathematics), division where the divisor (denominator) is 0, zero, is a unique and problematic special case. Using fraction notation, the general example can be written as \tfrac a0, where a is the di ...
differs from our modern understanding:
18.34. A positive divided by a positive or a negative divided by a negative is positive; a zero divided by zero is zero; a positive divided by a negative is negative; a negative divided by a positive is lsonegative.
18.35. A negative or a positive divided by zero has that eroas its divisor, or zero divided by a negative or a positive as that negative or positive as its divisor The square of a negative or positive is positive; he squareof zero is zero. That of which he squareis the square is tssquare root.
Here Brahmagupta states that = 0 and as for the question of where ≠ 0 he did not commit himself.: However, here again Brahmagupta spoiled matters somewhat by asserting that 0 ÷ 0 = 0, and on the touchy matter of ''a'' ÷ 0, he did not commit himself. His rules for
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. ...
on
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 zero are quite close to the modern understanding, except that in modern mathematics division by zero is left
undefined Undefined may refer to: Mathematics *Undefined (mathematics), with several related meanings **Indeterminate form, in calculus Computing *Undefined behavior, computer code whose behavior is not specified under certain conditions *Undefined valu ...
.


Diophantine analysis


Pythagorean triplets

In chapter twelve of his ''Brāhmasphuṭasiddhānta'', Brahmagupta provides a formula useful for generating
Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Py ...
s:
12.39. The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey.
Or, in other words, if , then a traveller who "leaps" vertically upwards a distance from the top of a mountain of height , and then travels in a straight line to a city at a horizontal distance from the base of the mountain, travels the same distance as one who descends vertically down the mountain and then travels along the horizontal to the city. Stated geometrically, this says that if a right-angled triangle has a base of length and altitude of length , then the length, , of its hypotenuse is given by . And, indeed, elementary algebraic manipulation shows that whenever has the value stated. Also, if and are rational, so are , , and . A Pythagorean triple can therefore be obtained from , and by multiplying each of them by the
least common multiple In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two integers ''a'' and ''b'', usually denoted by , is the smallest positive integer that is divisible by both ''a'' and ...
of their
denominator A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...
s.


Pell's equation

Brahmagupta went on to give a recurrence relation for generating solutions to certain instances of Diophantine equations of the second degree such as (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 ...
) by using the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is a ...
. The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces.
The nature of squares:
18.64. ut downtwice the square-root of a given square by a multiplier and increased or diminished by an arbitrary
umber Umber is a natural earth pigment consisting of iron oxide and manganese oxide; it has a brownish color that can vary among shades of yellow, red, and green. Umber is considered one of the oldest pigments known to humans, first used in the Ajant ...
The product of the first
air An atmosphere () is a layer of gases that envelop an astronomical object, held in place by the gravity of the object. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A stellar atmosph ...
multiplied by the multiplier, with the product of the last
air An atmosphere () is a layer of gases that envelop an astronomical object, held in place by the gravity of the object. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A stellar atmosph ...
is the last computed.
18.65. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive ''rupas''.
The key to his solution was the identity, :(x^2_1 - Ny^2_1)(x^2_2 - Ny^2_2) = (x_1 x_2 + Ny_1 y_2)^2 - N(x_1 y_2 + x_2 y_1)^2 which is a generalisation of an identity that was discovered by
Diophantus Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. Although Jose ...
, :(x^2_1 - y^2_1)(x^2_2 - y^2_2) = (x_1 x_2 + y_1 y_2)^2 - (x_1 y_2 + x_2 y_1)^2. Using his identity and the fact that if and are solutions to the equations and , respectively, then is a solution to , he was able to find integral solutions to Pell's equation through a series of equations of the form . Brahmagupta was not able to apply his solution uniformly for all possible values of , rather he was only able to show that if has an integer solution for = ±1, ±2, or ±4, then has a solution. The solution of the general Pell's equation would have to wait for
Bhāskara II Bhāskara II ('; 1114–1185), also known as Bhāskarāchārya (), was an Indian people, Indian polymath, Indian mathematicians, mathematician, astronomer and engineer. From verses in his main work, Siddhānta Śiromaṇi, it can be inferre ...
in .


Geometry


Brahmagupta's formula

Brahmagupta's most famous result in geometry is his
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area,
12.21. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate
rea REA or Rea may refer to: Places * Rea, Lombardy, in Italy * Rea, Missouri, United States * Rea River, in Fiordland, New Zealand * River Rea, a river in Birmingham, England * River Rea, Shropshire, a river in Shropshire, England * Rea, Hunga ...
is the square root from the product of the halves of the sums of the sides diminished by achside of the quadrilateral.
So given the lengths , , and of a cyclic quadrilateral, the approximate area is while, letting , the exact area is : . Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case.
Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), the area is A = \sqrt. It is named after first-century ...
is a special case of this formula and it can be derived by setting one of the sides equal to zero.


Triangles

Brahmagupta dedicated a substantial portion of his work to geometry. One theorem gives the lengths of the two segments a triangle's base is divided into by its altitude:
12.22. The base decreased and increased by the difference between the squares of the sides divided by the base; when divided by two they are the true segments. The perpendicular ltitudeis the square-root from the square of a side diminished by the square of its segment.
Thus the lengths of the two segments are . He further gives a theorem on
rational triangles An integer triangle or integral triangle is a triangle all of whose side lengths are integers. A rational triangle is one whose side lengths are rational numbers; any rational triangle can be rescaled by the lowest common denominator of the s ...
. A triangle with rational sides , , and rational area is of the form: :a = \frac\left(\frac+v\right), \ \ b = \frac\left(\frac+w\right), \ \ c = \frac\left(\frac - v + \frac - w\right) for some rational numbers , , and .


Brahmagupta's theorem

Brahmagupta continues,
12.23. The square-root of the sum of the two products of the sides and opposite sides of a non-unequal quadrilateral is the diagonal. The square of the diagonal is diminished by the square of half the sum of the base and the top; the square-root is the perpendicular ltitudes
So, in a "non-unequal" cyclic quadrilateral (that is, an isosceles
trapezoid In geometry, a trapezoid () in North American English, or trapezium () in British English, is a quadrilateral that has at least one pair of parallel sides. The parallel sides are called the ''bases'' of the trapezoid. The other two sides are ...
), the length of each diagonal is . He continues to give formulas for the lengths and areas of geometric figures, such as the circumradius of an isosceles trapezoid and a scalene quadrilateral, and the lengths of diagonals in a scalene cyclic quadrilateral. This leads up to Brahmagupta's famous theorem,
12.30–31. Imaging two triangles within cyclic quadrilateralwith unequal sides, the two diagonals are the two bases. Their two segments are separately the upper and lower segments
ormed William Ormed ( fl. 1395), was an English Member of Parliament (MP). He was a Member of the Parliament of England The Parliament of England was the legislature of the Kingdom of England from the 13th century until 1707 when it was replaced ...
at the intersection of the diagonals. The two
ower segments Ower is a hamlet in the New Forest district of Hampshire, England. Its nearest towns are Totton – approximately to the southeast, and Romsey – around to the north-east. Ower lies on the A36 road northwest of Totton. It lies mostly ...
of the two diagonals are two sides in a triangle; the base
f the quadrilateral is the base of the triangle F, or f, is the sixth letter of the Latin alphabet and many modern alphabets influenced by it, including the modern English alphabet and the alphabets of all other modern western European languages. Its name in English is ''ef'' (pronounc ...
Its perpendicular is the lower portion of the entralperpendicular; the upper portion of the entralperpendicular is half of the sum of the
ides Ides or IDES may refer to: Calendar dates * Ides (calendar), a day in the Roman calendar that fell roughly in the middle of the month ** Ides of March, a day in the Roman calendar that corresponded to March 15 Music * "St. Ides Heaven", a song by ...
perpendiculars diminished by the lower ortion of the central perpendicular


Pi

In verse 40, he gives values of ,
12.40. The diameter and the square of the radius achmultiplied by 3 are espectivelythe practical circumference and the area
f a circle F, or f, is the sixth letter of the Latin alphabet and many modern alphabets influenced by it, including the modern English alphabet and the alphabets of all other modern western European languages. Its name in English is ''ef'' (pronounc ...
The accurate aluesare the square-roots from the squares of those two multiplied by ten.
So Brahmagupta uses 3 as a "practical" value of , and \sqrt \approx 3.1622\ldots as an "accurate" value of , with an error less than 1%.


Measurements and constructions

Brahmagupta gives constructions of various figures with arbitrary sides. He essentially manipulated right triangles to produce isosceles triangles, scalene triangles, rectangles, isosceles trapezoids, isosceles trapezoids with three equal sides, and a scalene cyclic quadrilateral. After giving the value of pi, he deals with the geometry of plane figures and solids, such as finding volumes and surface areas (or empty spaces dug out of solids). He finds the volume of rectangular prisms, pyramids, and the frustum of a square pyramid. He further finds the average depth of a series of pits. For the volume of a
frustum In geometry, a ; (: frusta or frustums) is the portion of a polyhedron, solid (normally a pyramid (geometry), pyramid or a cone (geometry), cone) that lies between two parallel planes cutting the solid. In the case of a pyramid, the base faces a ...
of a pyramid, he gives the "pragmatic" value as the depth times the square of the mean of the edges of the top and bottom faces, and he gives the "superficial" volume as the depth times their mean area.


Trigonometry


Sine table

In Chapter 2 of his ''Brāhmasphuṭasiddhānta'', entitled ''Planetary True Longitudes'', Brahmagupta presents a sine table:
2.2–5. The sines: The Progenitors, twins; Ursa Major, twins, the Vedas; the gods, fires, six; flavors, dice, the gods; the moon, five, the sky, the moon; the moon, arrows, suns ../blockquote> Here Brahmagupta uses names of objects to represent the digits of place-value numerals, as was common with numerical data in Sanskrit treatises. Progenitors represents the 14 Progenitors ("Manu") in Indian cosmology or 14, "twins" means 2, "Ursa Major" represents the seven stars of Ursa Major or 7, "Vedas" refers to the 4 Vedas or 4, dice represents the number of sides of the traditional die or 6, and so on. This information can be translated into the list of sines, 214, 427, 638, 846, 1051, 1251, 1446, 1635, 1817, 1991, 2156, 2312, 2459, 2594, 2719, 2832, 2933, 3021, 3096, 3159, 3207, 3242, 3263, and 3270, with the radius being 3270 (this numbers represent 3270\sin \frac for n=1,\dots,24). Brahmagupta's sine table, like much other numerical data in Sanskrit treatises, is encoded mostly in concrete-number notation that uses names of objects to represent the digits of place-value numerals, starting with the least significant. ..br />There are fourteen Progenitors ("Manu") in Indian cosmology; "twins" of course stands for 2; the seven stars of Ursa Major (the "Sages") for 7, the four Vedas, and the four sides of the traditional dice used in gambling, for 6, and so on. Thus Brahmagupta enumerates his first six sine-values as 214, 427, 638, 846, 1051, 1251. (His remaining eighteen sines are 1446, 1635, 1817, 1991, 2156, 2312, 2459, 2594, 2719, 2832, 2933, 3021, 3096, 3159, 3207, 3242, 3263, 3270). The ''Paitamahasiddhanta'', however, specifies an initial sine-value of 225 (although the rest of its sine-table is lost), implying a trigonometric radius of ''R'' = 3438 approx= C(')/2π: a tradition followed, as we have seen, by Aryabhata. Nobody knows why Brahmagupta chose instead to normalize these values to R = 3270.


Interpolation formula

In 665 Brahmagupta devised and used a special case of the Newton–Stirling interpolation formula of the second-order to
interpolate In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a ...
new values of the
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
function from other values already tabulated. The formula gives an estimate for the value of a function at a value of its argument (with and ) when its value is already known at , and . The formula for the estimate is: : f( a + x h ) \approx f(a) + x \frac + x^2 \frac. where is the first-order forward-
difference operator In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
, i.e. : \Delta f(a) \ \stackrel\ f(a+h) - f(a).


Early concept of gravity

Brahmagupta in 628 first described gravity as an attractive force, using the term " gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" to describe it:
The earth on all its sides is the same; all people on the earth stand upright, and all heavy things fall down to the earth by a law of nature, for it is the nature of the earth to attract and to keep things, as it is the nature of water to flow ... If a thing wants to go deeper down than the earth, let it try. The earth is the only ''low'' thing, and seeds always return to it, in whatever direction you may throw them away, and never rise upwards from the earth.


Astronomy

Brahmagupta directed a great deal of criticism towards the work of rival astronomers, and his ''Brāhmasphuṭasiddhānta'' displays one of the earliest schisms among Indian mathematicians. The division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself. In Brahmagupta's case, the disagreements stemmed largely from the choice of astronomical parameters and theories. Critiques of rival theories appear throughout the first ten astronomical chapters and the eleventh chapter is entirely devoted to criticism of these theories, although no criticisms appear in the twelfth and eighteenth chapters. In chapter seven of his ''Brāhmasphuṭasiddhānta'', entitled ''Lunar Crescent'', Brahmagupta rebuts the idea that the Moon is farther from the Earth than the Sun. He does this by explaining the illumination of the Moon by the Sun.
1. If the moon were above the sun, how would the power of waxing and waning, etc., be produced from calculation of the longitude of the moon? The near half would always be bright.
2. In the same way that the half seen by the sun of a pot standing in sunlight is bright, and the unseen half dark, so is
he illumination He or HE may refer to: Language * He (letter), the fifth letter of the Semitic abjads * He (pronoun), a pronoun in Modern English * He (kana), one of the Japanese kana (へ in hiragana and ヘ in katakana) * Ge (Cyrillic), a Cyrillic letter cal ...
of the moon
f it is F, or f, is the sixth letter of the Latin alphabet and many modern alphabets influenced by it, including the modern English alphabet and the alphabets of all other modern western European languages. Its name in English is ''ef'' (pronounce ...
beneath the sun.
3. The brightness is increased in the direction of the sun. At the end of a bright .e. waxinghalf-month, the near half is bright and the far half dark. Hence, the elevation of the horns
f the crescent can be derived F, or f, is the sixth letter of the Latin alphabet and many modern alphabets influenced by it, including the modern English alphabet and the alphabets of all other modern western European languages. Its name in English is ''ef'' (pronounc ...
from calculation. ..ref>
He explains that since the Moon is closer to the Earth than the Sun, the degree of the illuminated part of the Moon depends on the relative positions of the Sun and the Moon, and this can be computed from the size of the angle between the two bodies. Brahmagupta discusses the illumination of the moon by the sun, rebutting an idea maintained in scriptures: namely, that the moon is farther from the earth than the sun is. In fact, as he explains, because the moon is closer the extent of the illuminated portion of the moon depends on the relative positions of the moon and the sun, and can be computed from the size of the angular separation α between them. Further work exploring the longitudes of the planets, diurnal rotation, lunar and solar eclipses, risings and settings, the moon's crescent and conjunctions of the planets, are discussed in his treatise
Khandakhadyaka ''Khaṇḍakhādyaka'' (meaning "edible bite; morsel of food") is a Sanskrit-language astronomical treatise written by Indian mathematician and astronomer Brahmagupta in 665 CE. The treatise contains eight chapters covering such topics as the lo ...
.


See also

*
Brahmagupta–Fibonacci identity In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity sa ...
*
Brahmagupta's formula In Euclidean geometry, Brahmagupta's formula, named after the 7th century Indian mathematician, is used to find the area of any convex cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides. Its generalized vers ...
* Brahmagupta theorem *
Brahmagupta triangle A Brahmagupta triangle is a triangle whose side lengths are consecutive positive integers and area is a positive integer. The triangle whose side lengths are 3, 4, 5 is a Brahmagupta triangle and so also is the triangle whose side lengths are 13, 1 ...
*
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 ...
*
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 ...
*
History of science and technology on the Indian subcontinent The history of science and technology on the Indian subcontinent begins with the prehistoric human activity of the Indus Valley Civilisation to the early Indian states and empires. Prehistory By 5500 BCE a number of sites similar to Meh ...


References


Notes


Citations


Bibliography

* * * * * * * * * * * *


Further reading

* * * *


External links


Brahmagupta's Brahma-sphuta-siddhanta
edited by Ram Swarup Sharma, Indian Institute of Astronomical and Sanskrit Research, 1966. English introduction, Sanskrit text, Sanskrit and Hindi commentaries (PDF) * , translated by
Henry Thomas Colebrooke Henry Thomas Colebrooke FRS FRSE FLS (15 June 1765 – 10 March 1837) was an English orientalist and botanist. He has been described as "the first great Sanskrit scholar in Europe". Biography Henry Thomas Colebrooke was born on 15 June ...


{{DEFAULTSORT:Brahmagupta Brahmagupta, 590s births 7th-century deaths Scientists from Rajasthan Rajasthani people People from Jalore district 7th-century Indian mathematicians Scholars from Rajasthan