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{{Short description|Indian mathematician and astronomer (598–668)}}
{{pp-pc|small=yes}}
{{Use dmy dates|date=September 2019}}
{{Infobox scientist
| name              = Brahmagupta
| image             = 
| caption           = 
| birth_date        = {{circa|598 CE}}
| birth_place       = Bhillamala, [[Gurjaradesa]], [[Chavda dynasty|Chavda kingdom]] <br/>
(modern day  [[Bhinmal]], [[Rajasthan]], [[India]])
| death_date        = {{circa|668 CE}} (aged c. 69–70)
| death_place       = [[Ujjain]], [[Chalukya Empire]] 
<br/> (modern day [[Madhya Pradesh]], [[India]])
| fields            = [[Astronomy]], [[mathematics]]
| known_for         = {{ubl
 | Rules for computing with [[Zero]]
 | Modern [[numeral system]]
 | [[Brahmagupta's theorem]]
 | [[Brahmagupta's identity]]
 | [[Brahmagupta's problem]]
 | [[Brahmagupta–Fibonacci identity]]
 | [[Brahmagupta's interpolation formula]]
 | [[Brahmagupta's formula]]}}
}}
'''Brahmagupta'''  ({{circa|598}} – {{circa|668 [[Common Era|CE]]}}) 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|doctrine]] of [[Brahma]]", dated 628), a theoretical treatise, and the ''[[Khandakhadyaka]]'' ("edible bite", dated 665), a more practical text.

In 628 CE, Brahmagupta first described [[gravity]] as an attractive force, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to describe it.<ref name="Oxford">{{cite book|last=Pickover|first=Clifford|url=https://books.google.com/books?id=SQXcpvjcJBUC&pg=PA105|title=Archimedes to Hawking: Laws of Science and the Great Minds Behind Them|date=2008|publisher=Oxford University Press|isbn=978-0-19-979268-9|page=105}}</ref><ref name="Bose">{{cite book|last=Bose|first=Mainak Kumar|url=https://books.google.com/books?id=nbItAAAAMAAJ|title=Late classical India|date=1988|publisher=A. Mukherjee & Co.}}{{page needed|date=March 2020}}</ref><ref name="Sen">{{cite book|last=Sen|first=Amartya|title=The Argumentative Indian|date=2005|publisher=Allen Lane|isbn=978-0-7139-9687-6|page=29}}</ref><ref name="Thurston">{{cite book|last=Thurston|first=Hugh|title=Early Astronomy|publisher=Springer-Verlag|year=1993|isbn=978-0-387-94107-3|location=New York}}{{page needed|date=March 2020}}{{failed verification|date=October 2023}}</ref> He is also credited with the first clear description of the [[quadratic formula]] (the solution of the quadratic equation)<ref name=Bradley>Bradley, Michael.  ''The Birth of Mathematics: Ancient Times to 1300'', p. 86 (Infobase Publishing 2006)</ref> in his main work, the ''Brāhma-sphuṭa-siddhānta''.<ref>Mackenzie, Dana.  ''[https://books.google.com/books?id=2QYSAAAAQBAJ&q=Brahmagupta The Universe in Zero Words: The Story of Mathematics as Told through Equations]'', p. 61 (Princeton University Press, 2012).</ref>

==Life and career==
Brahmagupta, according to his own statement, was born in 598 CE. Born in ''Bhillamāla'' in [[Gurjaradesa]]<ref name="Sachau2013">{{citation|last=Sachau|first=Edward C.|title=Alberuni's India, Volume I|url=https://archive.org/details/alberunisindiaac01biru/page/152/mode/2up|publisher=Kegan Paul, Trench and Trubner |location=London|year=1910|via=archive.org|page=153|quote=''Brahma-siddhānta'', so-called from Brahman, composed by Brahmagupta, the son of Jishnu, from the town of Bhillamāla between Multān and Anhilwāra, 16 ''yojana'' from the latter place (?)}}</ref> (modern [[Bhinmal]] in [[Rajasthan]], India) during the reign of the [[Chavda dynasty]] ruler Vyagrahamukha. He was the son of Jishnugupta and was a Hindu by religion, in particular, a  [[Shaivite]].<ref name=Bhattacharyya>{{harvnb|Bhattacharyya|2011|p=185}}: "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."</ref> He lived and worked there for a good part of his life. [[Prithudaka Svamin]], a later commentator, called him ''Bhillamalacharya'', the teacher from Bhillamala.{{sfn|Gupta|2008|p=162}}

Bhillamala was the capital of the [[Gurjaratra|Gurjaradesa]], the second-largest kingdom of Western India, comprising southern [[Rajasthan]] and northern [[Gujarat]] 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]]'' on Indian astronomy as well as the work of other astronomers including [[Aryabhata I]], Latadeva, Pradyumna, [[Varahamihira]], Simha, Srisena, Vijayanandin and Vishnuchandra. {{sfn|Gupta|2008|p=162}}

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 [[arya metre|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.{{sfn|Gupta|2008|p=162}}{{sfn|Bhattacharyya|2011|pp=185–186}}{{sfn|Bose|Sen|Subbarayappa|1971}}

Later, Brahmagupta moved to [[Ujjaini]], [[Avanti (Ancient India)|Avanti]],{{sfn|Gupta|2008|p=163}} a major centre for astronomy in central India. At the age of 67, he composed his next well-known work ''[[Khandakhadyaka|Khanda-khādyaka]]'', a practical manual of Indian astronomy in the ''[[karana (astronomy)|karana]]'' category meant to be used by students.{{sfn|Gupta|2008|p=163}}

Brahmagupta died in 668 CE, and he is presumed to have died in Ujjain.

== Works ==
Brahmagupta composed the following treatises:

* ''[[Brāhmasphuṭasiddhānta]]'',<ref name=":0">{{Cite book|last=Pingree|first=David E.|url=http://archive.org/details/PingreeCESS|title=Pingree's Census of the Exact Sciences in Sanskrit|publisher=APS|year=1970–1994|pages=A4, 256 ff.; A5, 239–240 et passim}}</ref> composed in 628 CE.
* ''[[Khandakhadyaka|Khaṇḍakhādyaka]]'',<ref name=":0"/> composed in 665 CE.
* ''Grahaṇārkajñāna'',<ref name=":0"/> (ascribed in one manuscript)

==Reception==
Brahmagupta's mathematical advances were carried on further by [[Bhāskara II]], a lineal descendant in Ujjain, who described Brahmagupta as the ''ganaka-chakra-chudamani'' (the gem of the circle of mathematicians). [[Prithudaka Svamin]] wrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations. [[Lalla]] and [[Bhattotpala]] in the 8th and 9th centuries wrote commentaries on the ''Khanda-khadyaka''.{{sfn|Bhattacharyya|2011|p=185}} Further commentaries continued to be written into the 12th century.{{sfn|Gupta|2008|p=163}}

A few decades after the death of Brahmagupta, [[Sindh]] came under the Arab Caliphate in 712 CE. [[Umayyad campaigns in India|Expeditions]] were sent into ''Gurjaradesa'' ("''Al-Baylaman'' in ''Jurz''", as per Arab historians). The kingdom of Bhillamala seems to have been annihilated but Ujjain [[Umayyad campaigns in India|repulsed the attacks]]. The court of Caliph [[Al-Mansur]] (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]], an astronomer in Al-Mansur's court, under the names ''[[Sindhind]]'' and ''Arakhand''. An immediate outcome was the spread of the decimal number system used in the texts. The mathematician [[Al-Khwarizmi]] (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.{{sfn|Avari|2013|p=32}}<ref name="YoungLatham2006">
{{citation|last1=Young|first1=M. J. L.|last2=Latham|first2=J. D.|last3=Serjeant|first3=R. B.|title=Religion, Learning and Science in the 'Abbasid Period|url=https://books.google.com/books?id=cJuDafHpk3oC&pg=PA302|date=2 November 2006|publisher=Cambridge University Press|isbn=978-0-521-02887-5|pages=302–303}}</ref><ref name="van Bladel">{{citation |last=van Bladel |first=Kevin |chapter=Eighth Century Indian Astronomy in the Two Cities of Peace |editor1=Asad Q. Ahmed |editor2=Benham Sadeghi |editor3=Robert G. Hoyland |title=Islamic Cultures, Islamic Contexts: Essays in Honor of Professor Patricia Crone |chapter-url=https://books.google.com/books?id=7nSjBQAAQBAJ&pg=PA260 |date=28 November 2014 |publisher=BRILL |isbn=978-90-04-28171-4 |pages=257–294}}
</ref>

The historian of science [[George Sarton]] called Brahmagupta "one of the greatest scientists of his race and the greatest of his time."{{sfn|Gupta|2008|p=163}}

==Mathematics==
===Algebra===
Brahmagupta gave the solution of the general [[linear equation]] in chapter eighteen of ''Brahmasphuṭasiddhānta'',

<blockquote> The difference between ''rupas'', when inverted and divided by the difference of the [coefficients] of the [unknowns], is the unknown in the equation. The ''rupas'' are [subtracted on the side] below that from which the square and the unknown are to be subtracted.<ref name="Plofker Chapter 18">{{harvtxt|Plofker|2007|pp=428–434}}</ref></blockquote>

which is a solution for the equation {{math|''bx'' + ''c'' {{=}} ''dx'' + ''e''}} where ''rupas'' refers to the constants {{math|''c''}} and {{math|''e''}}. The solution given is equivalent to {{math|''x'' {{=}} {{sfrac|''e'' − ''c''|''b'' − ''d''}}}}.

He further gave two equivalent solutions to the general [[quadratic equation]]

<blockquote>18.44. Diminish by the middle [number] the square-root of the ''rupas'' multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number].<br />
18.45. Whatever is the square-root of the ''rupas'' multiplied by the square [and] increased by the square of half the unknown, diminished that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.<ref name="Plofker Chapter 18"/></blockquote>

which are, respectively, solutions for the equation {{math|''ax''<sup>2</sup> + ''bx'' {{=}} ''c''}} equivalent to,

:<math>x = \frac{\pm\sqrt{4ac+b^2}-b}{2a} </math>

and

:<math>x = \frac{\pm\sqrt{ac+\tfrac{b^2}{4}}-\tfrac{b}{2}}{a} </math>

He went on to solve systems of simultaneous [[indeterminate equations]] stating that the desired variable must first be isolated, and then the equation must be divided by the desired variable's [[coefficient]]. In particular, he recommended using "the pulverizer" to solve equations with multiple unknowns.

<blockquote>18.51. Subtract the colors different from the first color. [The remainder] divided by the first [color's coefficient] is the measure of the first. [Terms] two by two [are] considered [when reduced to] similar divisors, [and so on] repeatedly. If there are many [colors], the pulverizer [is to be used].<ref name="Plofker Chapter 18"/></blockquote>

Like the algebra of [[Diophantus]], 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.<ref name="Boyer Brahmagupta Indeterminate equations">{{harvtxt|Boyer|1991|loc="China and India" p. 221}} "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."</ref> The extent of Greek influence on this [[History of algebra|syncopation]], if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source.<ref name="Boyer Brahmagupta Indeterminate equations"/>

===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]] and first appeared in the ''Brāhmasphuṭasiddhānta''. Brahmagupta describes multiplication in the following way:

<blockquote>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.<ref name="Brahmagupta1817">{{cite book|author1=Brahmagupta|author2=Bhāskara II|title=Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara|url=https://books.google.com/books?id=ebZIAAAAcAAJ&pg=PA319|year=1817|publisher=[[John Murray (publishing house)|John Murray]]|page=319|translator=[[Henry Thomas Colebrooke]]}}</ref></blockquote>

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.<ref name="Tiwari1992">{{citation|last=Tiwari|first=Sarju|title=Mathematics in History, Culture, Philosophy, and Science: From Ancient Times to Modern Age|url=https://books.google.com/books?id=-yyDJUD0DNwC&pg=PA91|year=1992|publisher=Mittal Publications|isbn=978-81-7099-404-6|pages=91–}}</ref> 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: {{math|{{sfrac|''a''|''c''}} + {{sfrac|''b''|''c''}}}}; {{math|{{sfrac|''a''|''c''}} × {{sfrac|''b''|''d''}}}}; {{math|{{sfrac|''a''|1}} + {{sfrac|''b''|''d''}}}}; {{math|{{sfrac|''a''|''c''}} + {{sfrac|''b''|''d''}} × {{sfrac|''a''|''c''}} {{=}} {{sfrac|''a''(''d'' + ''b'')|''cd''}}}}; and {{math|{{sfrac|''a''|''c''}} − {{sfrac|''b''|''d''}} × {{sfrac|''a''|''c''}} {{=}} {{sfrac|''a''(''d'' − ''b'')|''cd''}}}}.<ref>{{harvtxt|Plofker|2007|pp=422}} The reader is apparently expected to be familiar with basic arithmetic operations as far as the square-root; Brahmagupta merely notes some points about applying them to fractions. The procedures for finding the cube and cube-root of an integer, however, are described (compared the latter to Aryabhata's very similar formulation). They are followed by rules for five types of combinations: [...]</ref>

====Squares and Cubes====
Brahmagupta then goes on to give the sum of the squares and cubes of the first {{math|''n''}} integers.
<blockquote>12.20. The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed].<ref name="Plofker Brahmagupta quote Chapter 12">{{harvtxt|Plofker|2007|pp=421–427}}</ref></blockquote>
Here Brahmagupta found the result in terms of the ''sum'' of the first {{math|''n''}} integers, rather than in terms of {{math|''n''}} as is the modern practice.<ref name="Plofker 423">{{harvtxt|Plofker|2007|p=423}} 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;</ref>

He gives the sum of the squares of the first {{math|''n''}} natural numbers as {{math|{{sfrac|''n''(''n'' + 1)(2''n'' + 1)|6}}}} and the sum of the cubes of the first n natural numbers as {{math|<big>(</big>{{sfrac|''n''(''n'' + 1)|2}}<big>)</big>{{su|p=2}}}}.

====Zero====
Brahmagupta's ''Brahmasphuṭasiddhānta'' is the first book that provides rules for arithmetic manipulations that apply to [[zero]] and to [[negative number]]s.<ref>{{cite book|first=Robert |last=Kaplan |title=The Nothing That Is: A Natural History of Zero |publisher=Allen Lane/The Penguin Press|location=London |year=1999 |pages=68–75|bibcode=2000tnti.book.....K }}</ref> 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]] or as a symbol for lack of quantity as was done by [[Ptolemy]] and the [[Ancient Rome|Romans]]. In chapter eighteen of his ''Brāhmasphuṭasiddhānta'', Brahmagupta describes operations on negative numbers. He first describes addition and subtraction,
<blockquote>18.30. [The sum] of two positives is positives, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positives, [and that] of two zeros zero.<br />
[...]<br />
18.32. A negative minus zero is negative, a positive [minus zero] is positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added.<ref name="Plofker Chapter 18"/></blockquote>

He goes on to describe multiplication,
<blockquote>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.<ref name="Plofker Chapter 18"/></blockquote>

But his description of [[division by zero]] differs from our modern understanding:
<blockquote>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 [also] negative.<br />
18.35. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square root.<ref name="Plofker Chapter 18"/></blockquote>

Here Brahmagupta states that {{sfrac|0|0}} = 0 and as for the question of {{sfrac|{{math|''a''}}|0}} where {{math|''a''}} ≠ 0 he did not commit himself.<ref name="Boyer Brahmagupta p220">{{harvtxt|Boyer|1991|p=220}}: 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.</ref> His rules for [[arithmetic]] on [[negative number]]s and zero are quite close to the modern understanding, except that in modern mathematics division by zero is left [[defined and undefined|undefined]].

===Diophantine analysis===

====Pythagorean triplets====
In chapter twelve of his ''Brāhmasphuṭasiddhānta'', Brahmagupta provides a formula useful for generating [[Pythagorean triple]]s:
<blockquote>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.<ref name="Plofker2007p426">{{harvtxt|Plofker|2007|p=426}}</ref></blockquote>
Or, in other words, if  {{math|''d'' {{=}} {{sfrac|''mx''|''x'' + 2}}}}, then a traveller who "leaps" vertically upwards a distance {{math|''d''}} from the top of a mountain of height {{math|''m''}}, and then travels in a straight line to a city at a horizontal distance {{math|''mx''}} 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.<ref name="Plofker2007p426"/> Stated geometrically, this says that if a right-angled triangle has a base of length  {{math|''a'' {{=}} ''mx''}} and altitude of length {{math|''b'' {{=}} ''m'' + ''d''}}, then the length, {{math|''c''}}, of its hypotenuse is given by {{math|''c'' {{=}} ''m''(1 + ''x'') − ''d''}}. And, indeed, elementary algebraic manipulation shows that  {{math|''a''<sup>2</sup> + ''b''<sup>2</sup> {{=}} ''c''<sup>2</sup>}} whenever {{math|''d''}} has the value stated. Also, if {{math|''m''}} and {{math|''x''}} are rational, so are {{math|''d''}}, {{math|''a''}}, {{math|''b''}} and {{math|''c''}}. A Pythagorean triple can therefore be obtained from {{math|''a''}}, {{math|''b''}} and {{math|''c''}} by multiplying each of them by the [[least common multiple]] of their [[denominator]]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 {{math|''Nx''{{sup|2}} + 1 {{=}} ''y''{{sup|2}}}} (called [[Pell's equation]]) by using the [[Euclidean algorithm]]. The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces.<ref>{{harvtxt|Stillwell|2004|pp=44–46}}: In the seventh century CE the Indian mathematician Brahmagupta gave a recurrence relation for generating solutions of ''x''{{sup|2}} − ''Dy''{{sup|2}} {{=}} 1, as we shall see in Chapter 5. The Indians called the Euclidean algorithm the "pulverizer" because it breaks numbers down to smaller and smaller pieces. To obtain a recurrence one has to know that a rectangle proportional to the original eventually recurs, a fact that was rigorously proved only in 1768 by Lagrange.</ref>

<blockquote>The nature of squares:<br />18.64. [Put down] twice the square-root of a given square by a multiplier and increased or diminished by an arbitrary [number]. The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed.<br />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''.<ref name="Plofker Chapter 18"/></blockquote>

The key to his solution was the identity,<ref name="Stillwell p. 72-74"/>

:<math>(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</math>

which is a generalisation of an identity that was discovered by [[Diophantus]],

:<math>(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.</math>

Using his identity and the fact that if {{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>)}} and {{math|(''x''<sub>2</sub>, ''y''<sub>2</sub>)}} are solutions to the equations {{math|''x''<sup>2</sup> − ''Ny''<sup>2</sup> {{=}} ''k''<sub>1</sub>}} and {{math|''x''<sup>2</sup> − ''Ny''<sup>2</sup> {{=}} ''k''<sub>2</sub>}}, respectively, then {{math|(''x''<sub>1</sub>''x''<sub>2</sub> + ''Ny''<sub>1</sub>''y''<sub>2</sub>, ''x''<sub>1</sub>''y''<sub>2</sub> + ''x''<sub>2</sub>''y''<sub>1</sub>)}} is a solution to {{math|''x''<sup>2</sup> − ''Ny''<sup>2</sup> {{=}} ''k''<sub>1</sub>''k''<sub>2</sub>}}, he was able to find integral solutions to Pell's equation through a series of equations of the form {{math|''x''<sup>2</sup> − ''Ny''<sup>2</sup> {{=}} ''k<sub>i</sub>''}}. Brahmagupta was not able to apply his solution uniformly for all possible values of {{math|''N''}}, rather he was only able to show that if {{math|''x''<sup>2</sup> − ''Ny''<sup>2</sup> {{=}} ''k''}} has an integer solution for {{math|''k''}} = ±1, ±2, or ±4, then {{math|''x''<sup>2</sup> − ''Ny''<sup>2</sup> {{=}} 1}} has a solution. The solution of the general Pell's equation would have to wait for [[Bhāskara II]] in {{circa|1150 CE}}.<ref name="Stillwell p. 72-74">{{harvtxt|Stillwell|2004|pp=72–74}}</ref>

===Geometry===

====Brahmagupta's formula====
[[Image:Brahmaguptas formula.svg|200px|thumb|right|Diagram for reference]]
{{Main|Brahmagupta's formula}}

Brahmagupta's most famous result in geometry is his [[Brahmagupta's formula|formula]] 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,
<blockquote>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 [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral.<ref name="Plofker Brahmagupta quote Chapter 12"/></blockquote>
So given the lengths {{math|''p''}}, {{math|''q''}}, {{math|''r''}} and {{math|''s''}} of a cyclic quadrilateral, the approximate area is {{math|{{sfrac|''p'' + ''r''|2}} · {{sfrac|''q'' + ''s''|2}}}} while, letting {{math|''t'' {{=}} [[Semiperimeter|{{sfrac|''p'' + ''q'' + ''r'' + ''s''|2}}]]}}, the exact area is

: {{math|{{sqrt|(''t'' − ''p'')(''t'' − ''q'')(''t'' − ''r'')(''t'' − ''s'')}}}}.

Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case.<ref>{{harvtxt|Plofker|2007|p=424}} Brahmagupta does not explicitly state that he is discussing only figures inscribed in circles, but it is implied by these rules for computing their circumradius.</ref> [[Heron's formula]] 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:
<blockquote>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 [altitude] is the square-root from the square of a side diminished by the square of its segment.<ref name="Plofker Brahmagupta quote Chapter 12"/></blockquote>
Thus the lengths of the two segments are {{math|{{sfrac|1|2}}(''b'' ± {{sfrac|''c''<sup>2</sup> − ''a''<sup>2</sup>|''b''}})}}.

He further gives a theorem on [[rational triangles]]. A triangle with rational sides {{math|''a''}}, {{math|''b''}}, {{math|''c''}} and rational area is of the form:

:<math>a = \frac{1}{2}\left(\frac{u^2}{v}+v\right), \ \ b =  \frac{1}{2}\left(\frac{u^2}{w}+w\right), \ \ c =  \frac{1}{2}\left(\frac{u^2}{v} - v + \frac{u^2}{w} - w\right) </math>

for some rational numbers {{math|''u''}}, {{math|''v''}}, and {{math|''w''}}.<ref>{{harvtxt|Stillwell|2004|p=77}}</ref>

====Brahmagupta's theorem====
{{Main|Brahmagupta theorem}}
[[Image:Brahmaguptra's theorem.svg|thumb|right|Brahmagupta's theorem states that ''AF'' = ''FD''.]]

Brahmagupta continues,
<blockquote>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 [altitudes].<ref name="Plofker Brahmagupta quote Chapter 12"/></blockquote>
So, in a "non-unequal" cyclic quadrilateral (that is, an isosceles [[trapezoid]]), the length of each diagonal is {{math|{{sqrt|''pr'' + ''qs''}}}}.

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 theorem|Brahmagupta's famous theorem]],
<blockquote>12.30–31. Imaging two triangles within [a cyclic quadrilateral] with unequal sides, the two diagonals are the two bases. Their two segments are separately the upper and lower segments [formed] at the intersection of the diagonals. The two [lower segments] of the two diagonals are two sides in a triangle; the base [of the quadrilateral is the base of the triangle]. Its perpendicular is the lower portion of the [central] perpendicular; the upper portion of the [central] perpendicular is half of the sum of the [sides] perpendiculars diminished by the lower [portion of the central perpendicular].<ref name="Plofker Brahmagupta quote Chapter 12"/></blockquote>

====Pi====
In verse 40, he gives values of [[pi|{{pi}}]],
<blockquote>12.40. The diameter and the square of the radius [each] multiplied by 3 are [respectively] the practical circumference and the area [of a circle]. The accurate [values] are the square-roots from the squares of those two multiplied by ten.<ref name="Plofker Brahmagupta quote Chapter 12"/></blockquote>
So Brahmagupta uses 3 as a "practical" value of {{pi}}, and <math>\sqrt{10} \approx 3.1622\ldots</math> as an "accurate" value of {{pi}}, 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]] 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.<ref>{{harvtxt|Plofker|2007|p=427}} After the geometry of plane figures, Brahmagupta discusses the computation of volumes and surface areas of solids (or empty spaces dug out of solids). His straightforward rules for the volumes of a rectangular prism and pyramid are followed by a more ambiguous one, which may refer to finding the average depth of a sequence of puts with different depths. The next formula apparently deals with the volume of a frustum of a square pyramid, where the "pragmatic" volume is the depth times the square of the mean of the edges of the top and bottom faces, while the "superficial" volume is the depth times their mean area.</ref>

===Trigonometry===

====Sine table====

In Chapter 2 of his ''Brāhmasphuṭasiddhānta'', entitled ''Planetary True Longitudes'', Brahmagupta presents a sine table:

<blockquote>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 [...]<ref>{{harvtxt|Plofker|2007|p=419}}</ref></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 <math>3270\sin \frac{\pi n}{48}</math> for <math>n=1,\dots,24</math>).<ref name="Plofker 419–420">{{harvtxt|Plofker|2007|pp=419–420}} 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.</ref>

====Interpolation formula====
{{Main|Brahmagupta's interpolation formula}}

In 665 Brahmagupta devised and used a special case of the Newton–Stirling interpolation formula of the second-order to [[interpolation|interpolate]]
new values of the [[Trigonometric function|sine]] function from other values already tabulated.<ref>Joseph [[#joseph-2000|(2000, pp.285–86)]].</ref> The formula gives an estimate for the value of a function {{math|''f''}} at a value {{math|''a'' + ''xh''}} of its argument (with {{math|''h'' > 0}} and {{math|−1 ≤ ''x'' ≤ 1}}) when its value is already known at {{math|''a'' − ''h''}}, {{math|''a''}} and {{math|''a'' + ''h''}}.

The formula for the estimate is:

: <math>f( a + x h ) \approx f(a) + x \frac{\Delta f(a) + \Delta f(a-h)}{2} + x^2 \frac{\Delta^2 f(a-h)}{2}.</math>

where {{math|Δ}} is the first-order forward-[[difference operator]], i.e.

: <math> \Delta f(a) \ \stackrel{\mathrm{def}}{=}\ f(a+h) - f(a).</math>

== Early concept of gravity ==
Brahmagupta in 628 first described gravity as an attractive force, using the term "[[Wikt:गुरुत्वाकर्षणम्|gurutvākarṣaṇam (गुरुत्वाकर्षणम्)]]" to describe it:<ref name="Oxford"/><ref name="Bose"/><ref name="Sen"/><ref name="Thurston"/>

<blockquote>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.<ref name="Alberuni">{{cite book|url=http://www.columbia.edu/cu/lweb/digital/collections/cul/texts/ldpd_5949073_001/pages/ldpd_5949073_001_00000328.html?toggle=image&menu=maximize&top=&left=|title=Alberuni's India|publisher=London: Kegan Paul, Trench, Trübner & Co., 1910. Electronic reproduction. Vol. 1 and 2. New York: Columbia University Libraries, 2006|page=272|access-date=3 June 2014}}</ref><ref>{{cite book|title=Kitāb al-Jawharatayn al-'atīqatayn al-mā'i'atayn min al-ṣafrā' wa-al-bayḍā': al-dhahab wa-al-fiḍḍah |location=Cairo |publisher=Maṭba'at Dār al-Kutub wa-al-Wathā'iq al-Qawmīyah bi-al-Qāhirah |script-title=ar:كتاب الجوهرتين العتيقتين المائعتين من الصفراء والبيضاء : الذهب والفضة |date=2004 |pages=43–44, 87|oclc=607846741}}</ref>{{efn|The source of this quote is ''[[Al-Biruni]]'s India'' (c.&nbsp;1030).<ref name=Alberuni/>}}</blockquote>

==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.<ref name="Plofker 418-419">{{harvtxt|Plofker|2007|pp=418–419}}</ref> 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.<ref name="Plofker 418-419"/>

In chapter seven of his ''Brāhmasphuṭasiddhānta'', entitled ''Lunar Crescent'', Brahmagupta rebuts the idea that the Moon is farther from the Earth than the Sun.{{clarify|date=November 2015}} He does this by explaining the illumination of the Moon by the Sun.<ref name="Plofker 420"/>

<blockquote>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.<br />
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 [the illumination] of the moon [if it is] beneath the sun.<br />
3. The brightness is increased in the direction of the sun. At the end of a bright [i.e. waxing] half-month, the near half is bright and the far half dark. Hence, the elevation of the horns [of the crescent can be derived] from calculation. [...]<ref>{{harvtxt|Plofker|2007|p=420}}</ref></blockquote>

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.<ref name="Plofker 420">{{harvtxt|Plofker|2007|pp=419–420 }} 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.</ref>

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]].

== See also ==
* [[Brahmagupta–Fibonacci identity]]
* [[Brahmagupta's formula]]
* [[Brahmagupta theorem]]
* [[Brahmagupta triangle]]
* [[Chakravala method]]
* [[List of Indian mathematicians]]
* [[History of science and technology on the Indian subcontinent]]

== References ==
=== Notes ===
{{notelist}}

=== Citations ===
{{Reflist|30em}}

=== Bibliography ===
* {{Cite book |last=Cai |first=Tianxin |url=https://books.google.com/books?id=DFLNEAAAQBAJ&pg=PA114 |title=A Brief History of Mathematics: A Promenade through the Civilizations of Our World |date=2023-07-25 |publisher=Springer Nature |isbn=978-3-031-26841-0 |pages= |language=en}}
* {{citation |last=Avari |first=Burjor |author-link=Burjor Avari |title=Islamic Civilization in South Asia: A history of Muslim power and presence in the Indian subcontinent |publisher=Routledge |year=2013 |isbn=978-0-415-58061-8}}
* {{citation |last1=Bose |first1=D. M. |last2=Sen |first2=S. N. |last3=Subbarayappa |first3=B. V. |year=1971 |title=A Concise History of Science in India |location=New Delhi |publisher=Indian National Academy of Science |url=http://www.dli.gov.in/scripts/FullindexDefault.htm?path1=/data_copy/upload/0080/243&first=114&last=1168&barcode=1040010080238 |pages=95–97 |url-status=dead |archive-url=https://web.archive.org/web/20151208075132/http://www.dli.gov.in/scripts/FullindexDefault.htm?path1=%2Fdata_copy%2Fupload%2F0080%2F243&first=114&last=1168&barcode=1040010080238 |archive-date=8 December 2015  }}
* {{citation |last=Bhattacharyya |first=R. K. |chapter=Brahmagupta: The Ancient Indian Mathematician |editor1=B. S. Yadav |editor2=Man Mohan |title=Ancient Indian Leaps into Mathematics |url=https://books.google.com/books?id=nwrw0Lv1vXIC |year=2011 |publisher=Springer Science & Business Media |isbn=978-0-8176-4695-0 |pages=185–192}}
* {{citation |last=Boyer |first=Carl B. |title=A History of Mathematics |publisher=John Wiley & Sons, Inc |year=1991 |isbn=0-471-54397-7 |url=https://archive.org/details/AHistoryOfMathematics}}
* {{citation |last=Cooke |first=Roger |title=The History of Mathematics: A Brief Course |publisher=Wiley-Interscience |year=1997 |isbn=0-471-18082-3 |url=https://archive.org/details/historyofmathema0000cook }}
*{{citation| first= Radha Charan | last= Gupta | author-link= Radha Charan Gupta|title=Brahmagupta|editor-first=Helaine |editor-last=Selin |encyclopedia=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |isbn=978-1-4020-4559-2 |publisher=Springer |year=2008 |url=https://books.google.com/books?id=raKRY3KQspsC&pg=PA162 |pages=162–163}}
* {{citation |last=Joseph |first=George G. |title=The Crest of the Peacock |publisher=Princeton University Press |year=2000 |isbn=0-691-00659-8 |ref=joseph-2000 |url=https://archive.org/details/crestofpeacockno00jose }}
* {{citation |last=O'Leary |first=De Lacy |year=2001 |edition=2nd |orig-year=first published 1948 |title=How Greek Science Passed to the Arabs |publisher=Goodword Books |isbn=8187570245}}
* {{citation |last=Plofker | first=Kim | author-link1=Kim Plofker | editor = Victor Katz | title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook | chapter=Mathematics in India | publisher=Princeton University Press |year=2007 |isbn=978-0-691-11485-9 |chapter-url=https://books.google.com/books?id=3ullzl036UEC}}
* {{citation |last=Stillwell |first=John |title=Mathematics and its History |edition=Second |publisher=Springer Science + Business Media Inc. |year=2004 |isbn=0-387-95336-1 |url=https://books.google.com/books?id=WNjRrqTm62QC}}
* {{citation |title=Biographical Encyclopedia of Astronomers |editor-last=Hockey |editor-first=Thomas |chapter=Brahmagupta |publisher=Springer Science & Business Media |year=2007 |isbn=978-0387304007 |page=165}}

==Further reading==
* {{cite book |title=Brāhmasphuṭasiddhānta (Chapter 21) of Brahmagupta with Commentary of Pṛthūdhakasvāmin, critically edited with English translation and notes  |author=Setsuro Ikeyama |publisher=INSA |year=2003}}
* {{cite book |title=Census of the Exact Sciences in Sanskrit (CESS)  |author=David Pingree |publisher=American Philosophical Society | at =A4, p. 254}}
* {{cite book |title=Mathematics & Astronomers of Ancient India |author=Shashi S. Sharma |publisher=Pitambar Publishing |url=https://books.google.com/books?id=g9ykYZlzV1oC|isbn=9788120914216 }}
* {{MacTutor|id=Brahmagupta|title=Brahmagupta}}

==External links==
{{Commons category|Brahmagupta}}
{{Wikisource author}}
* [http://www.wilbourhall.org/index.html#BSS 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)
* {{Internet Archive|id=algebrawitharith00brahuoft|name=Algebra, with Arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bháscara}}, translated by [[Henry Thomas Colebrooke]]. [https://books.google.com/books?id=ebZIAAAAcAAJ]

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