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Quick facts for kids Brahmagupta
Hindu astronomer, 19th-century illustration
Born	c. 598 CE Bhillamala, Gurjaradesa, Chavda kingdom (modern day Bhinmal, Rajasthan, India)
Died	c. 668 CE (aged c. 69–70) Ujjain, Chalukya Empire (modern day Madhya Pradesh, India)
Known for	Rules for computing with Zero Modern number system Brahmagupta's theorem Brahmagupta's identity Brahmagupta's problem Brahmagupta–Fibonacci identity Brahmagupta's interpolation formula Brahmagupta's formula
Scientific career
Fields	Astronomy, mathematics
Influenced	Virtually all subsequent mathematics particularly Indian and Islamic mathematics

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 for computing 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.

In 628 CE, Brahmagupta first described gravity as an attractive force, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to describe it.

Life and career
Works
Mathematics
Astronomy
See also

Life and career

Brahmagupta was born in 598 CE according to his own statement. He lived in Bhillamāla in Gurjaradesa (modern Bhinmal in Rajasthan, India) during the reign of the Chavda dynasty ruler, Vyagrahamukha. He was the son of Jishnugupta and was a Hindu by religion, in particular, a Shaivite. He lived and worked there for a good part of his life. Prithudaka Svamin, a later commentator, called him Bhillamalacharya, the teacher from Bhillamala.

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 I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra.

In the year 628, at the age of 30, he composed the Brāhmasphuṭasiddhānta ("improved treatise of Brahma") which is believed to be a revised version of the received Siddhanta of the Brahmapaksha school of astronomy. Scholars state that he incorporated a great deal of originality into his revision, adding a considerable amount of new material. The book consists of 24 chapters with 1008 verses in the 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, a major centre for astronomy in central India. At the age of 67, he composed his next well-known work Khanda-khādyaka, 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.

Works

Brahmagupta composed the following treatises:

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

Mathematics

Algebra

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

which are, respectively, solutions for the equation $ax 2 + bx = c$ equivalent to,

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

and

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.

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.

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", he also details operations on fractions. The reader is expected to know the basic arithmetic operations as far as taking the square root, although he explains how to find the cube and cube-root of an integer and later gives rules facilitating the computation of squares and square roots. He then gives rules for dealing with five types of combinations of fractions: $a / c + b / c$ ; $a / c \times b / d$ ; $a / 1 + b / d$ ; $a / c + b / d \times a / c = a (d + b) / cd$ ; and $a / c - b / d \times a / c = a (d - b) / cd$ .

Squares and Cubes

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

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 $n (n + 1)(2 n + 1) / 6$ 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 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,

But his description of division by zero differs from our modern understanding:

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.

Geometry

Brahmagupta's formula

Diagram for reference

Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. So given the lengths $p$ , $q$ , $r$ and $s$ of a cyclic quadrilateral, the approximate area is $p + r / 2 \cdot q + s / 2$ while, letting $t = p + q + r + s / 2$ , the exact area is

\sqrt (t - p)(t - q)(t - r)(t - s)

Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case. 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. Thus the lengths of the two segments are $1 / 2 (b \pm c 2 - a 2 / b)$ .

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

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.

So, in a "non-unequal" cyclic quadrilateral (that is, an isosceles trapezoid), the length of each diagonal is $\sqrt pr + qs$ .

He continues to give formulas for the lengths and areas of geometric figures, such as the circumradius of an isosceles trapezoid and a scalene quadrilateral, and the lengths of diagonals in a scalene cyclic quadrilateral.

Pi

So Brahmagupta uses 3 as a "practical" value of $π$ , and $\sqrt{10} \approx 3.1622\ldots$ as an "accurate" value of $π$ , with an error 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 traditional die or 6, and so on. This information can be translated into the list of sines, 214, 427, 638, 846, 1051, 1251, 1446, 1635, 1817, 1991, 2156, 2312, 2459, 2594, 2719, 2832, 2933, 3021, 3096, 3159, 3207, 3242, 3263, and 3270, with the radius being 3270 (this numbers represent $3270\sin \frac{\pi n}{48}$ for $n=1,\dots,24$ ).

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. The formula gives an estimate for the value of a function $f$ at a value $a + xh$ of its argument (with $h > 0$ and $-1 \leq x \leq 1$ ) when its value is already known at $a - h$ , $a$ and $a + h$ .

The formula for the estimate is:

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

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

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

Astronomy

Brahmagupta directed a great deal of criticism towards the work of rival astronomers, and his Brāhmasphuṭasiddhānta displays one of the earliest schisms among Indian mathematicians. The division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself. In Brahmagupta's case, the disagreements stemmed largely from the choice of astronomical parameters and theories. 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.

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.

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.