Brahmagupta
Brahmagupta (listen (help·info)) (born c. 598 CE, died c. 668 CE) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the Brāhmasphuṭasiddhānta (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the Khaṇḍakhādyaka ("edible bite", dated 665), a more practical text.
Brahmagupta  

Born  c. 598 CE 
Died  c. 668 CE 
Residence 

Known for  
Scientific career  
Fields  Mathematics, astronomy 
Brahmagupta was the first to give rules to compute with zero. The texts composed by Brahmagupta were composed in elliptic verse in Sanskrit, as was common practice in Indian mathematics. As no proofs are given, it is not known how Brahmagupta's results were derived.^{[2]}
Contents
Life and careerEdit
Brahmagupta was born in 598 CE according to his own statement. He lived in Bhillamala (modern Bhinmal) during the reign of the Chapa dynasty ruler, Vyagrahamukha. He was the son of Jishnugupta and was a Shaivite by religion.^{[3]} Even though most scholars assume that Brahmagupta was born in Bhillamala, there is no conclusive evidence for it. However, he lived and worked there for a good part of his life. Prithudaka Svamin, a later commentator, called him Bhillamalacharya, the teacher from Bhillamala.^{[4]} Sociologist G. S. Ghurye believed that he might have been from the Multan or Abu region.^{[5]}
Bhillamala, called pilomolo by Xuanzang, was the apparent capital of the Gurjaradesa, the second largest kingdom of Western India, comprising southern Rajasthan and northern Gujarat in modernday India. It was also a centre of learning for mathematics and astronomy. Brahmagupta became an astronomer of the Brahmapaksha school, one of the four major schools of Indian astronomy during this period. He studied the five traditional siddhanthas on Indian astronomy as well as the work of other astronomers including Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra.^{[4]}
In the year 628, at an age of 30, he composed the Brāhmasphuṭasiddhānta (the improved treatise of Brahma) which is believed to be a revised version of the received siddhanta of the Brahmapaksha school. Scholars state that he incorporated a great deal of originality to 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.^{[4]}^{[6]}^{[7]}
Later, Brahmagupta moved to Ujjain, which was also a major centre for astronomy. At the age of 67, he composed his next well known work Khandakhādyaka, a practical manual of Indian astronomy in the karana category meant to be used by students.^{[8]}
Brahmagupta lived beyond 665 CE. He is believed to have died in Ujjain.^{[citation needed]}
ControversyEdit
Brahmagupta directed a great deal of criticism towards the work of rival astronomers, and his Brahmasphutasiddhanta 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.^{[9]} 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.^{[9]}
ReceptionEdit
The historian of science George Sarton called him "one of the greatest scientists of his race and the greatest of his time."^{[8]} Brahmagupta's mathematical advances were carried on further by Bhāskara II, a lineal descendant in Ujjain, who described Brahmagupta as the ganakachakrachudamani (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 Khandakhadyaka.^{[10]} Further commentaries continued to be written into the 12th century.^{[8]}
A few decades after the death of Brahmagupta, Sindh came under the Arab Caliphate in 712 CE. Expeditions were sent into Gurjaradesa. The kingdom of Bhillamala seems to have been annihilated but Ujjain repulsed the attacks. The court of Caliph AlMansur (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 Muhammad alFazari, an astronomer in AlMansur'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 AlKhwarizmi (800–850 CE) wrote a text called alJam waltafriq bi hisalalHind (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. AlKhwarizmi also wrote his own version of Sindhind, drawing on AlFazari's version and incorporating Ptolemaic elements. Indian astronomic material circulated widely for centuries, even passing into medieval Latin texts.^{[11]}^{[12]}^{[13]}
MathematicsEdit
AlgebraEdit
Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta,
The difference between rupas, when inverted and divided by the difference 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.^{[14]}
which is a solution for the equation bx + c = dx + e equivalent to x = e − c/b − d, where rupas refers to the constants c and e. He further gave two equivalent solutions to the general quadratic equation
18.44. Diminish by the middle [number] the squareroot 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].
18.45. Whatever is the squareroot of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.^{[14]}
which are, respectively, solutions for the equation ax^{2} + bx = c equivalent to,
and
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.
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].^{[14]}
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.^{[15]} The extent of Greek influence on this syncopation, if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source.^{[15]}
ArithmeticEdit
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 number system and first appeared in Brahmasphutasiddhanta. Brahmagupta describes the multiplication as thus “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”. ^{[16]}^{[page needed]} Indian arithmetic was known in Medieval Europe as "Modus Indoram" meaning method of the Indians. In Brahmasphutasiddhanta, multiplication was named Gomutrika. In the beginning of chapter twelve of his Brahmasphutasiddhanta, entitled Calculation, Brahmagupta 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 cuberoot 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: a/c + b/c; a/c × b/d; a/1 + b/d; a/c + b/d × a/c = a(d + b)/cd; and a/c − b/d × a/c = a(d − b)/cd.^{[17]}
SeriesEdit
Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers.
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].^{[18]}
Here Brahmagupta found the result in terms of the sum of the first n integers, rather than in terms of n as is the modern practice.^{[19]}
He gives the sum of the squares of the first n natural numbers as n(n + 1)(2n + 1)/6 and the sum of the cubes of the first n natural numbers as (n(n + 1)/2)^{2}
_{}.
ZeroEdit
Brahmagupta's Brahmasphuṭasiddhanta is the first book that provides rules for arithmetic manipulations that apply to zero and to negative numbers.^{[20]} The Brahmasphutasiddhanta 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 a lack of quantity as was done by Ptolemy and the Romans. In chapter eighteen of his Brahmasphutasiddhanta, Brahmagupta describes operations on negative numbers. He first describes addition and subtraction,
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 positive, [and that] of two zeros zero.
[...]
18.32. A negative minus zero is negative, a positive [minus zero] 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.^{[14]}
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.^{[14]}
But his description of division by zero 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 a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative.
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 of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] squareroot.^{[14]}
Here Brahmagupta states that 0/0 = 0 and as for the question of a/0 where a ≠ 0 he did not commit himself.^{[21]} His rules for arithmetic on negative numbers and zero are quite close to the modern understanding, except that in modern mathematics division by zero is left undefined.
Diophantine analysisEdit
Pythagorean triplesEdit
In chapter twelve of his Brahmasphutasiddhanta, Brahmagupta provides a formula useful for generating Pythagorean triples:
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.^{[22]}
Or, in other words, if d = mx/x + 2, then a traveller who "leaps" vertically upwards a distance d from the top of a mountain of height m, and then travels in a straight line to a city at a horizontal distance 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.^{[22]} Stated geometrically, this says that if a rightangled triangle has a base of length a = mx and altitude of length b = m + d, then the length, c, of its hypotenuse is given by c = m(1 + x) − d. And, indeed, elementary algebraic manipulation shows that a^{2} + b^{2} = c^{2} whenever d has the value stated. Also, if m and x are rational, so are d, a, b and c. A Pythagorean triple can therefore be obtained from a, b and c by multiplying each of them by the least common multiple of their denominators.
Pell's equationEdit
Brahmagupta went on to give a recurrence relation for generating solutions to certain instances of Diophantine equations of the second degree such as Nx^{2} + 1 = y^{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.^{[23]}
The nature of squares:
18.64. [Put down] twice the squareroot 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.
18.65. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two squareroots, divided by the additive or the subtractive, are the additive rupas.^{[14]}
The key to his solution was the identity,^{[24]}
which is a generalisation of an identity that was discovered by Diophantus,
Using his identity and the fact that if (x_{1}, y_{1}) and (x_{2}, y_{2}) are solutions to the equations x^{2} − Ny^{2} = k_{1} and x^{2} − Ny^{2} = k_{2}, respectively, then (x_{1}x_{2} + Ny_{1}y_{2}, x_{1}y_{2} + x_{2}y_{1}) is a solution to x^{2} − Ny^{2} = k_{1}k_{2}, he was able to find integral solutions to Pell's equation through a series of equations of the form x^{2} − Ny^{2} = k_{i}. Brahmagupta was not able to apply his solution uniformly for all possible values of N, rather he was only able to show that if x^{2} − Ny^{2} = k has an integer solution for k = ±1, ±2, or ±4, then x^{2} − Ny^{2} = 1 has a solution. The solution of the general Pell's equation would have to wait for Bhaskara II in c. 1150 CE.^{[24]}
GeometryEdit
Brahmagupta's formulaEdit
Brahmagupta's most famous result in geometry is his 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,
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.^{[18]}
So given the lengths p, q, r and s of a cyclic quadrilateral, the approximate area is p + r/2 · q + s/2 while, letting t = p + q + r + s/2, the exact area is
 √(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.^{[25]} Heron's formula is a special case of this formula and it can be derived by setting one of the sides equal to zero.
TrianglesEdit
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 [altitude] is the squareroot from the square of a side diminished by the square of its segment.^{[18]}
Thus the lengths of the two segments are 1/2(b ± c^{2} − a^{2}/b).
He further gives a theorem on rational triangles. A triangle with rational sides a, b, c and rational area is of the form:
for some rational numbers u, v, and w.^{[26]}
Brahmagupta's theoremEdit
Brahmagupta continues,
12.23. The squareroot of the sum of the two products of the sides and opposite sides of a nonunequal 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 squareroot is the perpendicular [altitudes].^{[18]}
So, in a "nonunequal" cyclic quadrilateral (that is, an isosceles trapezoid), the length of each diagonal is √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 famous theorem,
12.3031. 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].^{[18]}
PiEdit
In verse 40, he gives values of π,
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 squareroots from the squares of those two multiplied by ten.^{[18]}
So Brahmagupta uses 3 as a "practical" value of π, and as an "accurate" value of π. The error in this "accurate" value is less than 1%.
Measurements and constructionsEdit
In some of the verses before verse 40, 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.^{[27]}
TrigonometryEdit
Sine tableEdit
In Chapter 2 of his Brahmasphutasiddhanta, entitled Planetary True Longitudes, Brahmagupta presents a sine table:
2.25. 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 [...]^{[28]}
Here Brahmagupta uses names of objects to represent the digits of placevalue 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 tradition 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, 1459, 2594, 2719, 2832, 2933, 3021, 3096, 3159, 3207, 3242, 3263, and 3270, with the radius being 3270.^{[29]}
Interpolation formulaEdit
In 665 Brahmagupta devised and used a special case of the Newton–Stirling interpolation formula of the secondorder to interpolate new values of the sine function from other values already tabulated.^{[30]} The formula gives an estimate for the value of a function f at a value a + xh of its argument (with h > 0 and −1 ≤ x ≤ 1) when its value is already known at a − h, a and a + h.
The formula for the estimate is:
where Δ is the firstorder forwarddifference operator, i.e.
AstronomyEdit
This section needs expansion with: Astronomical details reflecting his substantial astronomical work. You can help by adding to it. (July 2016) 
Some of the important contributions made by Brahmagupta in astronomy are his methods for calculating the position of heavenly bodies over time (ephemerides), their rising and setting, conjunctions, and the calculation of solar and lunar eclipses.^{[31]}
In chapter seven of his Brahmasphutasiddhanta, entitled Lunar Crescent, Brahmagupta rebuts the idea that the Moon is farther from the Earth than the Sun, an idea which had been suggested by Vedic scripture.^{[clarification needed]} He does this by explaining the illumination of the Moon by the Sun.^{[32]}
7.1. 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 be] always bright.
7.2. 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.
7.3. The brightness is increased in the direction of the sun. At the end of a bright [i.e. waxing] halfmonth, the near half is bright and the far half dark. Hence, the elevation of the horns [of the crescent can be derived] from calculation. [...]^{[33]}
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.^{[32]}
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 alsoEdit
Citations and footnotesEdit
 ^ Sachau, Edward C. (2013), Alberuni's India, Routledge, p. 156, ISBN 9781136383571,
Brahmasiddhānta, 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 (?)
 ^ Brahmagupta biography, Article by: J J O'Connor and E F Robertson, School of Mathematics and Statistics, University of St Andrews, Scotland, November 2000
 ^ Bhattacharyya 2011, p. 185: "Brahmagupta, one of the most celebrated mathematicians of the East, indeed of the world, was born in the year 598 c.e., in the town of Bhillamala during the reign of King Vyaghramukh of the Chapa Dynasty."
 ^ ^{a} ^{b} ^{c} Gupta 2008, p. 162.
 ^ Pillai, S. Devadas (1997), Indian Sociology Through Ghurye, a Dictionary, Popular Prakashan, p. 216, ISBN 9788171548071,
Brahmagupta (b. 598 AD) was a native of either the Multan region of the Punjab (now this areas is in Pakistan) or the Abu region of Rajasthan.
 ^ Bhattacharyya 2011, pp. 185186.
 ^ Bose, Sen & Subbarayappa 1971.
 ^ ^{a} ^{b} ^{c} Gupta 2008, p. 163.
 ^ ^{a} ^{b} Plofker (2007, pp. 418–419)
 ^ Bhattacharyya 2011, p. 185.
 ^ Avari 2013, p. 32.
 ^ Young, M. J. L.; Latham, J. D.; Serjeant, R. B. (2 November 2006), Religion, Learning and Science in the 'Abbasid Period, Cambridge University Press, pp. 302–303, ISBN 9780521028875
 ^ van Bladel, Kevin (28 November 2014), "Eighth Century Indian Astronomy in the Two Cities of Peace", in Asad Q. Ahmed; Benham Sadeghi; Robert G. Hoyland, Islamic Cultures, Islamic Contexts: Essays in Honor of Professor Patricia Crone, BRILL, pp. 257–294, ISBN 9789004281714
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} Plofker (2007, pp. 428–434)
 ^ ^{a} ^{b} Boyer (1991, "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."
 ^ Brahmasputha Siddhanta, Translated to English by H.T Colebrook, 1817 AD
 ^ Plofker (2007, pp. 422) The reader is apparently expected to be familiar with basic arithmetic operations as far as the squareroot; Brahmagupta merely notes some points about applying them to fractions. The procedures for finding the cube and cuberoot 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: [...]
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} Plofker (2007, pp. 421–427)
 ^ 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;
 ^ Kaplan, Robert (1999). The Nothing That Is: A Natural History of Zero. London: Allen Lane/The Penguin Press. pp. 68–75.
 ^ 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.
 ^ ^{a} ^{b} Plofker (2007, p. 426)
 ^ Stillwell (2004, pp. 44–46): In the seventh century CE the Indian mathematician Brahmagupta gave a recurrence relation for generating solutions of x^{2} − Dy^{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.
 ^ ^{a} ^{b} Stillwell (2004, pp. 72–74)
 ^ 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.
 ^ Stillwell (2004, p. 77)
 ^ 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.
 ^ Plofker (2007, p. 419)
 ^ Plofker (2007, pp. 419–420) Brahmagupta's sine table, like much other numerical data in Sanskrit treatises, is encoded mostly in concretenumber notation that uses names of objects to represent the digits of placevalue numerals, starting with the least significant. [...]
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 sinevalues as 214, 427, 638, 846, 1051, 1251. (His remaining eighteen sines are 1446, 1635, 1817, 1991, 2156, 2312, 1459, 2594, 2719, 2832, 2933, 3021, 3096, 3159, 3207, 3242, 3263, 3270). The Paitamahasiddhanta, however, specifies an initial sinevalue of 225 (although the rest of its sinetable 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.  ^ Joseph (2000, pp.285–86).
 ^ Teresi, Dick (2002). Lost Discoveries: The Ancient Roots of Modern Science. Simon and Schuster. p. 135. ISBN 074324379X.
 ^ ^{a} ^{b} 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.
 ^ Plofker (2007, p. 420)
ReferencesEdit
 Avari, Burjor (2013), Islamic Civilization in South Asia: A history of Muslim power and presence in the Indian subcontinent, Routledge, ISBN 9780415580618
 Bose, D. M.; Sen, S. N.; Subbarayappa, B. V. (1971), A Concise History of Science in India, New Delhi: Indian National Academy of Science, pp. 95–97, archived from the original on 8 December 2015
 Bhattacharyya, R. K. (2011), "Brahmagupta: The Ancient Indian Mathematician", in B. S. Yadav; Man Mohan, Ancient Indian Leaps into Mathematics, Springer Science & Business Media, pp. 185–192, ISBN 9780817646950
 Boyer, Carl B. (1991), A History of Mathematics, John Wiley & Sons, Inc, ISBN 0471543977
 Cooke, Roger (1997), The History of Mathematics: A Brief Course, WileyInterscience, ISBN 0471180823
 Gupta, Radha Charan (2008), "Brahmagupta", in Selin, Helaine, Encyclopaedia of the History of Science, Technology, and Medicine in NonWestern Cultures, Springer, pp. 162–163, ISBN 9781402045592
 Joseph, George G. (2000), The Crest of the Peacock, Princeton University Press, ISBN 0691006598
 O'Leary, De Lacy (2001) [first published 1948], How Greek Science Passed to the Arabs (2nd ed.), Goodword Books, ISBN 8187570245
 Plofker, Kim (2007), "Mathematics in India", in Victor Katz, The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 9780691114859
 Stillwell, John (2004), Mathematics and its History (Second ed.), Springer Science + Business Media Inc., ISBN 0387953361
 Hockey, Thomas, ed. (2007), "Brahmagupta", Biographical Encyclopedia of Astronomers, Springer Science & Business Media, p. 165, ISBN 0387304002
Further readingEdit
 Seturo Ikeyama (2003). Brāhmasphuṭasiddhānta of Brahmagupta with Commentary of Pṛthūdhaka, critically edited with English translation and notes. INSA.
 David Pingree. Census of the Exact Sciences in Sanskrit (CESS). American Philosophical Society. A4, p. 254.
 Shashi S. Sharma. Mathematics & Astronomers of Ancient India. Pitambar Publishing.
 O'Connor, John J.; Robertson, Edmund F., "Brahmagupta", MacTutor History of Mathematics archive, University of St Andrews.
External linksEdit
Wikimedia Commons has media related to Brahmagupta. 
 Brahmagupta's Brahmasphutasiddhanta edited by Ram Swarup Sharma, Indian Institute of Astronomical and Sanskrit Research, 1966. English introduction, Sanskrit text, Sanskrit and Hindi commentaries (PDF)
 Algebra, with Arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bháscara on the Internet Archive, translated by Henry Thomas Colebrooke. [1]