BRAHMAGUPTA

 

Brahmagupta (c. 598 – 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 was the first to give rules to compute with zero. The texts composed by Brahmagupta were 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.

WORK

Brahmagupta composed the following treatises:

• Brāhmasphuṭasiddhānta, composed in 628 CE.
• Khaṇḍakhādyaka, composed in 665 CE.
• Grahaṇārkajñāna (ascribed in one manuscript)

• Mathematics

  Algebra

  Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphuṭasiddhānta,

  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.

  which is a solution for the equation bx + c = dx + e where rupas refers to the constants c and e. The solution given is equivalent to x = e − c/b − d. 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, diminished that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.

  which are, respectively, solutions for the equation ax² + bx = c equivalent to,

  ${\displaystyle x={\frac {\pm {\sqrt {4ac+b^{2}}}-b}{2a}}}$

  and

  ${\displaystyle x={\frac {\pm {\sqrt {ac+{\tfrac {b^{2}}{4}}}}-{\tfrac {b}{2}}}{a}}}$

  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.

  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:

  The multiplicand is repeated like a string for cattle, as often as there are integrant portions in the multiplier and is repeatedly multiplied by them and the products are added together. It is multiplication. Or the multiplicand is repeated as many times as there are component parts in the multiplier.

  Indian arithmetic was known in Medieval Europe as modus Indorum meaning "method of the Indians". In the Brāhmasphuṭasiddhānta, four methods for multiplication were described, including gomūtrikā, which is said to be close to the present day methods. In the beginning of chapter twelve of his Brāhmasphuṭasiddhānta, entitled "Calculation", 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: 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.

  Series

  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
  .

  Zero

  Brahmagupta's Brahmasphuṭasiddhānta is the first book that provides rules for arithmetic manipulations that apply to zero and to negative numbers. 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 a lack of quantity as was done by Ptolemy and the Romans. In chapter eighteen of his Brāhmasphuṭasiddhānta, Brahmagupta describes operations on negative numbers. He first describes addition and subtraction,

  [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.
  

   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,

  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:

   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 0/0 = 0 and as for the question of a/0 where a ≠ 0 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.

  Diophantine analysis

  Pythagorean triplets

  In chapter twelve of his Brāhmasphuṭasiddhānta, 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.

  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 Nx² + 1 = y² (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,

  ${\displaystyle (x_{1}^{2}-Ny_{1}^{2})(x_{2}^{2}-Ny_{2}^{2})=(x_{1}x_{2}+Ny_{1}y_{2})^{2}-N(x_{1}y_{2}+x_{2}y_{1})^{2}}$

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

  ${\displaystyle (x_{1}^{2}-y_{1}^{2})(x_{2}^{2}-y_{2}^{2})=(x_{1}x_{2}+y_{1}y_{2})^{2}-(x_{1}y_{2}+x_{2}y_{1})^{2}.}$

  Using his identity and the fact that if (x₁, y₁) and (x₂, y₂) are solutions to the equations x² − Ny² = k₁ and x² − Ny² = k₂, respectively, then (x₁x₂ + Ny₁y₂, x₁y₂ + x₂y₁) is a solution to x² − Ny² = k₁k₂, he was able to find integral solutions to Pell's equation through a series of equations of the form x² − Ny² = 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 x² − Ny² = k has an integer solution for k = ±1, ±2, or ±4, then x² − Ny² = 1 has a solution. The solution of the general Pell's equation would have to wait for Bhāskara II in c. 1150 CE.

  Geometry

  Brahmagupta's formula

  

  Diagram for reference

  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 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. 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:

  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 1/2(b ± c² − a²/b).

  He further gives a theorem on rational triangles. A triangle with rational sides a, b, c and rational area is of the form:

  ${\displaystyle a={\frac {1}{2}}\left({\frac {u^{2}}{v}}+v\right),\ \ b={\frac {1}{2}}\left({\frac {u^{2}}{w}}+w\right),\ \ c={\frac {1}{2}}\left({\frac {u^{2}}{v}}-v+{\frac {u^{2}}{w}}-w\right)}$

  for some rational numbers u, v, and w.

  Brahmagupta's theorem
  

  

  Brahmagupta's theorem states that AF = FD.

  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 √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,

  Pi

  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 ${\displaystyle {\sqrt {10}}\approx 3.1622\ldots }$ as an "accurate" value of π. The error in this "accurate" value is less than 1%.

  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.

  Trigonometry
  Sine table

  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.

  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 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:

  ${\displaystyle f(a+xh)\approx f(a)+x{\frac {\Delta f(a)+\Delta f(a-h)}{2}}+x^{2}{\frac {\Delta ^{2}f(a-h)}{2}}.}$

  where Δ is the first-order forward-difference operator, i.e.

  ${\displaystyle \Delta f(a)\ {\stackrel {\mathrm {def} }{=}}\ f(a+h)-f(a).}$

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