Brahmagupta ( – ) was an Indian mathematician and astronomer who is credited as the first person to understand and formalize the concept of the number zero for nothing in mathematics. 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 Khandakhadyaka ("edible bite", dated 665), a more practical text. He was the first Indian scholar to describe gravity as an attractive force, and used the term "gurutvākarṣaṇam" in Sanskrit to describe it. He is also credited with the first clear description of the quadratic formula (the solution of the quadratic equation) in his main work, the Brāhma-sphuṭa-siddhānta.
Life and career
Brahmagupta, according to his own statement, was born in 598 CE. Born 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,
The historian of science George Sarton called Brahmagupta "one of the greatest scientists of his race and the greatest of his time."
Mathematics
Algebra
Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphuṭasiddhānta,
<blockquote> 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.</blockquote>
which is a solution for the equation where rupas refers to the constants and . The solution given is equivalent to .
He further gave two equivalent solutions to the general quadratic equation
<blockquote>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].<br />
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. 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.</blockquote>
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: ; ; ; ; and .
Squares and cubes
Brahmagupta then goes on to give the sum of the squares and cubes of the first integers.
<blockquote>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].</blockquote>
Here Brahmagupta found the result in terms of the sum of the first integers, rather than in terms of as is the modern practice.
He gives the sum of the squares of the first natural numbers as and the sum of the cubes of the first n natural numbers as .
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,
<blockquote>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 positives, [and that] of two zeros zero.<br />
[...]<br />
18.32. A negative minus zero is negative, a positive [minus zero] is 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. 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:
<blockquote>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.</blockquote>
Or, in other words, if , then a traveller who "leaps" vertically upwards a distance from the top of a mountain of height , and then travels in a straight line to a city at a horizontal distance 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.
<blockquote>The nature of squares:<br />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.<br />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.
Geometry
Brahmagupta's formula
200px|thumb|right|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,
<blockquote>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. 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:
<blockquote>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.
Brahmagupta's theorem
thumb|right|Brahmagupta's theorem states that AF = FD.
Brahmagupta continues,
<blockquote>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].
Trigonometry
Sine table
In Chapter 2 of his Brāhmasphuṭasiddhānta, entitled Planetary True Longitudes, Brahmagupta presents a sine table:
<blockquote>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 [...]</blockquote>
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 <math>3270\sin \frac{\pi n}{48}</math> for <math>n=1,\dots,24</math>).
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 of its argument (with and ) when its value is already known at , and .
The formula for the estimate is:
: <math>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}.</math>
where is the first-order forward-difference operator, i.e.
: <math> \Delta f(a) \ \stackrel{\mathrm{def{=}\ f(a+h) - f(a).</math>
Early concept of gravity
Brahmagupta in 628 described gravity as an attractive force, using the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" to describe it: