Brahmagupta (Sanskrit: ब्रह्मगुप्त; listen (help·info)) (597–668 CE) was an Indian mathematician and astronomer who wrote two important works on Mathematics and Astronomy: the Brāhmasphuṭasiddhānta (Extensive Treatise of Brahma) (628), a theoretical treatise, and the Khaṇḍakhādyaka, a more practical text. There are reasons to believe that Brahmagupta originated from Bhinmal.
Brahmagupta was the first to give rules to compute with zero. The texts composed by Brahmagupta were composed in elliptic verse, as was common practice in Indian mathematics, and consequently has a poetic ring to it. As no proofs are given, it is not known how Brahmagupta's mathematics was derived.
- 1 Life and work
- 2 Mathematics
- 2.1 Algebra
- 2.2 Arithmetic
- 2.3 Diophantine analysis
- 2.4 Geometry
- 2.5 Trigonometry
- 3 Astronomy
- 4 See also
- 5 Citations and footnotes
- 6 References
- 7 External links
Life and work
In the Brāhmasphuṭasiddhānta verses 7 and 8 of chapter XXIV state that Brahmagupta composed this text at the age of thirty in Śaka 550 (= 628 CE) during the reign of King Vyāghramukha, we can thus gather that he was born in 598. Commentators refer to him as a great scholar from Bhinmal, a city in the state of Rajasthan of Northwest India. In ancient times Bhillamala was the seat of power of the Gurjars. His father was Jisnugupta. He likely lived most of his life in Bhillamala (modern Bhinmal in Rajasthan) during the reign (and possibly under the patronage) of King Vyaghramukha. As a result, Brahmagupta is often referred to as Bhillamalacharya, that is, the teacher from Bhillamala. He was the head of the astronomical observatory at Ujjain, and during his tenure there wrote four texts on mathematics and astronomy: the Cadamekela in 624, the Brahmasphutasiddhanta in 628, the Khandakhadyaka in 665, and the Durkeamynarda in 672. The Brahmasphutasiddhanta (Extensive Treatise of Brahma) is arguably his most famous work. The historian al-Biruni (c. 1050) in his book Tariq al-Hind states that the Abbasid caliph al-Ma'mun had an embassy in India and from India a book was brought to Baghdad which was translated into Arabic as Sindhind. It is generally presumed that Sindhind is none other than Brahmagupta's Brahmasphuta-siddhanta.
Although Brahmagupta was familiar with the works of astronomers following the tradition of Aryabhatiya, it is not known if he was familiar with the work of Bhaskara I, a contemporary. Brahmagupta had a plethora of criticism directed towards the work of rival astronomers, and in his Brahmasphutasiddhanta is found one of the earliest attested 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.
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.
which is a solution for the equation equivalent to , 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 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].
18.45. Whatever is the square-root 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.
which are, respectively, solutions for the equation equivalent to,
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].
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. 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.
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”.  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 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 .
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].
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.
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.
Brahmagupta's Brahmasphuṭasiddhanta is the first book that mentions zero as a number, hence Brahmagupta is considered the first to formulate the concept of zero. He gave rules of using zero with negative and positive numbers. Zero plus a positive number is the positive number and negative number plus zero is a negative number etc. 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.
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 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] square-root.
Here Brahmagupta states that and as for the question of where he did not commit himself. 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.
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.
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. Stated geometrically, this says that if a right-angled 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 a2 + b2 = c2 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.
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) by using the Euclidean algorithm. 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. [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.
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,
which is a generalization of an identity that was discovered by Diophantus,
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 the Pell's equation through a series of equations of the form . Unfortunately, Brahmagupta was not able to apply his solution uniformly for all possible values of N, rather he was only able to show that if has an integer solution for k = ±1, ±2, or ±4, then has a solution. The solution of the general Pell's equation would have to wait for Bhaskara II in c. 1150 CE.
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.
So given the lengths p, q, r and s 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 is a special case of this formula and it can be derived by setting one of the sides equal to zero.
Brahmagupta dedicated a substantial portion of his work to geometry. One theorem states that the two lengths of a triangle's base when divided by its altitude then follows,
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.
Thus the lengths of the two segments are .
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.
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].
So, in a "non-unequal" cyclic quadrilateral (that is, an isosceles trapezoid), 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 [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].
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 square-roots from the squares of those two multiplied by ten.
So Brahmagupta uses 3 as a "practical" value of π, and as an "accurate" value of π.
Measurements and constructions
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.
In Chapter 2 of his Brahmasphutasiddhanta, 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 [...]
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 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.
See main article: Brahmagupta's interpolation formula
In 665 Brahmagupta devised and used a special case of the Newton–Stirling interpolation formula of the second-order to interpolate new values of the sine function from other values already tabulated. The formula gives an estimate for the value of a function 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 first-order forward-difference operator, i.e.
It was through the Brahmasphutasiddhanta that the Arabs learned of Indian astronomy. Edward Saxhau stated that "Brahmagupta, it was he who taught Arabs astronomy", The famous Abbasid caliph Al-Mansur (712–775) founded Baghdad, which is situated on the banks of the Tigris, and made it a center of learning. The caliph invited a scholar of Ujjain, by the name of Kankah, in 770 CE. Kankah used the Brahmasphutasiddhanta to explain the Hindu system of arithmetic astronomy. Muhammad al-Fazari translated Brahmugupta's work into Arabic upon the request of the caliph.
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 is maintained in scriptures. He does this by explaining the illumination of the Moon by the Sun.
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] 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. [...]
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.
Some of the important contributions made by Brahmagupta in astronomy are: methods for calculating the position of heavenly bodies over time (ephemerides), their rising and setting, conjunctions, and the calculation of solar and lunar eclipses. Brahmagupta criticized the Puranic view that the Earth was flat or hollow. Instead, he observed that the Earth and heaven were spherical.
- Brahmagupta–Fibonacci identity
- Brahmagupta's formula
- Brahmagupta theorem
- Chakravala method
- List of Indian mathematicians
Citations and footnotes
- Brahmagupta biography
- David Pingree. Census of the Exact Sciences in Sanskrit (CESS). American Philosophical Society. A4, p. 254., Seturo Ikeyama (2003). Brāhmasphuṭasiddhānta (CH. 21) of Brahmagupta with Commentary of Pṛthūdhaka, critically edited with English translation and notes. INSA. p. S2.
- Seturo Ikeyama (2003). Brāhmasphuṭasiddhānta (CH. 21) of Brahmagupta with Commentary of Pṛthūdhaka, critically edited with English translation and notes. INSA. p. S2.
- Shashi S. Sharma. Mathematics & Astronomers of Ancient India. Pitambar Publishing. "He was born in bhillamala. In ancient times it was the seat of power of the Gurjars...Jisnu Gupta.."
- (Plofker 2007, pp. 418–419) The Paitamahasiddhanta also directly inspired another major siddhanta, written by a contemporary of Bhaskara: The Brahmasphutasiddhanta (Corrected Treatise of Brahma) completed by Brahmagupta in 628. This astronomer was born in 598 and apparently worked in Bhillamal (identified with modern Bhinmal in Rajasthan), during the reign (and possibly under the patronage) of King Vyaghramukha.
Although we do not know whether Brahmagupta encountered the work of his contemporary Bhaskara, he was certainly aware of the writings of other members of the tradition of the Aryabhatiya, about which he has nothing good to say. This is almost the first trace we possess of the division of Indian astronomer-mathematicians into rival, sometimes antagonistic "schools." [...] it was in the application of mathematical models to the physical world—in this case, the choices of astronomical parameters and theories—that disagreements arose. [...]
Such critiques of rival works appear occasionally throughout the first ten astronomical chapters of the Brahmasphutasiddhanta, and its eleventh chapter is entirely devoted to them. But they do not enter into the mathematical chapters that Brahmagupta devotes respectively to ganita (chapter 12) and the pulverizer (chapter 18). This division of mathematical subjects reflects a different twofold classification from Bhaskara's "mathematics of fields" and "mathematics of quantities." Instead, the first is concerned with arithmetic operations beginning with addition, proportion, interest, series, formulas for finding lengths, areas, and volumes in geometrical figures, and various procedures with fractions—in short, diverse rules for computing with known quantities. The second, on the other hand, deals with what Brahmagupta calls "the pulverizer, zero, negatives, positives, unknowns, elimination of the middle term, reduction to one [variable], bhavita [the product of two unknowns], and the nature of squares [second-degree indeterminate equations]" - that is, techniques for operating with unknown quantities. This distinction is more explicitly presented in later works as mathematics of the "manifest" and "unmanifest," respectively: i.e., what we will henceforth call "arithmetic" manipulations of known quantities and "algebraic" manipulation of so-called "seeds" or unknown quantities. The former, of course, may include geometric problems and other topics not covered by the modern definition of "arithmetic." (Like Aryabhata, Brahmagupta relegates his sine-table to an astronomical chapter where the computations require it, instead of lumping it in with other "mathematical" topics.)
- Boyer (1991). "The Arabic Hegemony". p. 226. "By 766 we learn that an astronomical-mathematical work, known to the Arabs as the Sindhind, was brought to Baghdad from India. It is generally thought that this was the Brahmasphuta Siddhanta, although it may have been the Surya Siddhanata. A few years later, perhaps about 775, this Siddhanata was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrological Tetrabiblos was translated into Arabic from the Greek."
- (Plofker 2007, pp. 428–434)
- (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, 1,817 AD
- (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: [...]
- (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;
- Boyer (1991). "China and India". p. 220. "However, here again Brahmagupta spoiled matters somewhat by asserting that , and on the touchy matter of , he did not commit himself:"
- (Plofker 2007, p. 426)
- Stillwell, John (2004). pp. 44–46. "In the seventh century CE the Indian mathematician Brahmagupta gave a recurrence relation for generating solutions of , 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." Missing or empty
- Stillwell, John (2004). pp. 72–74. Missing or empty
- (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 straight-forward 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 concrete-number notation that uses names of objects to represent the digits of place-value 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 sine-values 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 sine-value of 225 (although the rest of its sine-table is lost), implying a trigonometric radius of R = 3438 aprox= 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).
- Brahmagupta, and the influence on Arabia. Retrieved 23 December 2007.
- Al Biruni, India translated by Edward sachau.
- (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)
- Teresi, Dick (2002). Lost Discoveries: The Ancient Roots of Modern Science. Simon and Schuster. p. 135. ISBN 0-7432-4379-X.
- Plofker, Kim (2007). "Mathematics in India". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. ISBN 978-0-691-11485-9.
- Boyer, Carl B. (1991). A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7.
- Cooke, Roger (1997). The History of Mathematics: A Brief Course. Wiley-Interscience. ISBN 0-471-18082-3.
- Joseph, George G. (2000). The Crest of the Peacock. Princeton, NJ: Princeton University Press. ISBN 0-691-00659-8.
- Stillwell, John (2004). Mathematics and its History (Second Edition ed.). Springer Science + Business Media Inc. ISBN 0-387-95336-1.
- Brahmagupta's Brahma-sphuta-siddhanta English introduction, Sanskrit text, Sanskrit and Hindi commentaries (PDF)