ब्रह्मगुप्त BRAHMGUPTA- The gem of the circle of mathematicians

भ्रमगुप्त

Bhramgupta

He was a great Bharatiya mathematician and Astronomer, also called भिल्लमालाआचार्य  .He was born in 598 CE and died in 668 CE at age nearly of 70 years. 

He lived in Bhillamala भीनमाल   in Gurjaradesaगुर्जरदेश (modern Bhinmal in Rajasthan, India) during the reign of the Chavda चावड़ा dynasty ruler, Vyagrahamukhaव्याग्राहामुखा. He was best Known for following:

	Zero Modern number system Brahmagupta's theorem Brahmagupta's identity Brahmagupta's problem Brahmagupta–Fibonacci identity Brahmagupta's interpolation formula Brahmagupta's formula

 the Brāhmasphuṭasiddhānta ब्राह्मस्फुटसिद्धान्त(dated 627) , a theoretical treatise, and the Khaṇḍakhādyakaखण्डखाद्यक या खण्डखाद्यपद्धति ( 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.

Bhillamala was the capital of the 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 Iआर्यभट्ट I, Latadevaलतादेव, Pradyumna प्रद्युम्न, Varahamihiraवराहमिहिर, Simha, Srisena श्रीसेन, Vijayanandin विजयनंदिनी  and Vishnuchandra विष्णुचंद्र. 

In the year 628, at the age of 30, he composed the Brāhmasphuṭasiddhānta ब्रह्मस्फुणसिद्धान्त: which is believed to be a revised version of the received Siddhanta of the Brahmapaksha school of astronomy. He incorporated a great deal of originality into his revision, adding a considerable amount of new material. The book consists of 24 chapters with 1008 verses in the ārya metre. A good deal of it is astronomy, but it also contains key chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain new insights due to Brahmagupta himself.

Later, Brahmagupta moved to Ujjaini, Avanti, which was also a major centre for astronomy in central India. At the age of 67, he composed his next well-known work Khanda-khādyakaखंडा-खद्यक, a practical manual of Indian astronomy in the karana category meant to be used by students.

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

Baskaracharya decribed him as anaka-chakra-chudamani (the gem of the circle of mathematicians)

Algebra

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

which meansa solution for the equation bx + c = dx + e where  constants are 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 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]

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. 

. 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

Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers.

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

12.20. The sum of t

 [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.^[13]

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.he 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].

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-rootwhich means, 

0/0 = 0 

and as for the question of 

a/0

where a ≠ 0 he did not commit himself. 

Pythagorean triplets

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.

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

 Nx² + 1 = y²

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.^[13]

The key to his solution was the identity,^[24]

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

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. 

Brahmagupta's formula

Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals.

 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)

 

Heron's formula 

It is a special case of this formula and it can be derived by setting one of the sides equal to zero.

Triangles

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

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]

π

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, with an error less than 1%.

Trigonometry

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

2.2–5. The sines: The Progenitors, twins; Ursa Major, twins, the Vedas; the gods, fires, six; flavors, dice, the gods; the moon, five, the sky, the moon; the moon, arrows, suns [...]

(Here Brahmagupta uses names of objects to represent the digits of place-value numerals) 

Progentiors --------------------> 14

twins----------------------------->2

Ursa Major---------------------->7

Vedas---------------------------->4

Dice------------------------------>6

...... 

...

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 ${\displaystyle 3270\sin {\frac {\pi n}{48}}}$ for ${\displaystyle n=1,\dots ,24}$)

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 which 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.

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

Astronomy

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.

In chapter seven of his Brāhmasphuṭasiddhānta, entitled Lunar Crescent, Brahmagupta rebuts the idea that the Moon is farther from the Earth than the Sun. He does this by explaining the illumination of the Moon by the Sun.

1. If the moon were above the sun, how would the power of waxing and waning, etc., be produced from calculation of the longitude of the moon? The near half would always be bright.

2. In the same way that the half seen by the sun of a pot standing in sunlight is bright, and the unseen half dark, so is [the illumination] of the moon [if it is] beneath the sun.

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.

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.