Brahmagupta

Indian mathematician

• Born: c. 598
• Birthplace: Bhillamala, Rajputana (now Bhinmal, India)
• Died: c. 660
• Place of death: Possibly Ujjain, Kingdom of Magadha (now in India)

Brahmagupta wrote BrahmasphutŃasiddhānta, a book in verse expounding a complex system of astronomy and containing two chapters on arithmetic, algebra, and geometry. His work on indeterminate equations and the introduction of negative numbers greatly influenced the development of science in both India and Arabia.

Early Life

The Hindu astronomer and mathematician Brahmagupta (brah-mah-GEWP-tah) was born to a man named Jishnugupta from the town of Bhillamala. The suffix -gupta may indicate that the family belonged to the Vaiś;ya caste (composed mostly of farmers and merchants).

In contrast to his predecessor, ĀryabhatŃa the Elder (c. 476-c. 550), who lived in relative obscurity at Kusumapura (modern-day Patna, Bihar), Brahmagupta had the opportunity to live, study, and teach in Ujjain, a town in the state of Gwalior, Central India. Ujjain was then the center of Hindu mathematics and astronomy and had the best observatory in India. At Ujjain, Brahmagupta also had access to the writings of many great scientists who came before him, including Hero of Alexandria (fl. 62-late first century c.e.), Ptolemy (c. 100-178 c.e.), Diophantus (fl. c. 250 c.e.), and ĀryabhatŃa the Elder. Brahmagupta later drew heavily on these sources in his own writings, often correcting their errors. For example, he corrected ĀryabhatŃa's mistake regarding the formulas for the surface areas and volumes of the pyramid and cone. Brahmagupta even borrowed mathematical problems, including one calling for the calculation of the position of a break in a bamboo pole. This problem had first appeared in the Chinese text Jiuzhang shuanshu (c. 50 b.c.e.-100 c.e.; arithmetic in nine sections), the authorship and date of which are uncertain.

Another influence of Ujjain was on Brahmagupta's style of writing. Like other Hindu scientists, including ĀryabhatŃa, he wrote his mathematical texts as poetry . The Indian practice was to clothe all arithmetical problems, especially those in schoolbooks, in poetic garb, fashioning them into puzzles that served as a popular amusement. Brahmagupta wrote that his mathematical problems were undertaken only for pleasure and that a wise man could invent a thousand more or solve those presented by others, thereby eclipsing their brilliance, just as the sun eclipses the other stars in the sky.

At Ujjain, the thirty-year-old Brahmagupta completed his masterwork, the BrahmasphutŃasiddhānta (c. 628; the improved astronomical system of Brahma). The date of this work has been determined by consulting both commentary from later Hindu scholars and, appropriately, astronomical data.

Life's Work

The first ten chapters of Brahmagupta's BrahmasphutŃasiddhānta deal with various astronomical issues, including the mean and true longitudes of the planets, diurnal motion, lunar and solar eclipses, heliacal risings and settings, the lunar crescent and “shadow,” conjunctions of the planets, and their conjunctions with the stars. The following thirteen chapters take up an examination of previous work on astronomy (including ĀryabhatŃa'), additions and problems (and their solutions) supplementing six of the earlier chapters, mathematics, the gnomon, meters, the sphere, instruments, and measurements. The work's twenty-fourth and final chapter summarizes the principles of Brahmagupta's astronomical system in a compendious treatise on astronomical spheres. (Some manuscripts include an additional chapter containing tables.) All but two of the chapters deal with astronomy, but scholars have chosen those two chapters, 12 and 18, which deal with algebra and mathematics, to study most intensely.

Although Brahmagupta studied mathematics only for its applicability to astronomy and considered knowledge of the rules of arithmetic a prerequisite to be a ganaca (a student of astronomy), most scholars in the ages since he lived have studied his mathematics more closely than his astronomy. Of particular interest is his work on indeterminate equations, building on the work of both Diophantus and ĀryabhatŃa. Brahmagupta's work, along with that of Bhāskara II (1114-c. 1185), solved the so-called Pell equation, y² = ax² + 1, where a is a nonsquare integer. Brahmagupta showed that from one solution where x, y, and xy do not equal zero, a general formula indicating an infinite number of solutions could be derived. Brahmagupta also stated that the equation y² = ax² – 1 could not be solved with integral values of x and y unless a was equal to the sum of the squares of any two integers. Brahmagupta's work on these equations, with additions by Bhāskara, is highly regarded because it was not for several centuries that another mathematician, namely Joseph-Louis Lagrange (1736-1813), could completely work out the Pell equation.

Brahmagupta also studied indeterminate equations of the first order, such as this one: Two ascetics live on top of a hill of h units of height, whose base is mh units away from a nearby town. One ascetic descends the hill and walks directly to the town. The other, being a wizard, flies straight up a certain distance, x, then proceeds in a straight line toward the town. If the distance traveled by each ascetic is the same, and h is 12 and mh is 48, find x. The solution comes from the formula x = mh/(m + 2), or in this case, x = 8.

Brahmagupta's work on the geometry of quadrilaterals, which was probably inspired by his studies of Ptolemy and Hero, is also a landmark in the history of Hindu mathematics. Brahmagupta found the formulas, for the first time, for the diagonals (defined as m and n) of a quadrilateral having sides of length a, b, c, and d and opposite angles of A and B, and C and D. He calculated the diagonals thus:

m² = (ab + cd)(ac + bd)/(ad + bc) andn^2M = (ac + bd)(ad + bc)/(ab + cd).

These formulas were later studied by Bhāskara, who, failing to understand that they applied only to quadrilaterals inscribed in a circle, incorrectly pronounced them unsound. Brahmagupta also figured that, if a, b, c, A, B, and C are positive integers such that a^2M + b^2M = c^2M and A^2M + B² = C², then the cyclic quadrilateral having consecutive sides aC, cB, bC, and cA (which came to be called a Brahmagupta trapezium) has rational area and diagonals, and the diagonals are perpendicular to each other. These formulas are most remarkable; nothing like them had previously appeared in Hindu geometry.

Brahmagupta borrowed from Hero of Alexandria the formula for the triangular area, but he brilliantly extended Hero's formula to work with quadrilaterals that can be inscribed within circles. This idea was later built on by the ninth century Hindu mathematician Mahāvīra and was much admired by later commentators. Brahmagupta's other advances in mathematics included proving the Pythagorean theory of the right triangle, deriving formulas for the areas of a square and a triangle inscribed in a circle, and showing that a rectangle whose sides were the radius and semiperimeter of a circle had the same area as that circle.

Although he is now remembered mostly for his advances in mathematics and his influence on the mathematical work of later Hindus such as Mahāvīra and Bhāskara, Brahmagupta considered himself primarily an astronomer. Almost every Hindu commentator on astronomy discusses his work. Indeed, some of his work in astronomy is quite admirable. He provided fairly accurate figures for the circumference of Earth and the length of the calendar year. Brahmagupta gives a figure different from ĀryabhatŃa's for the circumference of Earth: 5,000 yojanas. Assuming that Brahmagupta's yojana was a short league of about 4.5 miles, that would convert his figure to 22,500 miles, which is not too far off the mark. He also tried to correct ĀryabhatŃa's computation for the length of the year, which was 365 days, 15 ghati, 31 pala, and 15 vipala, or 365 days, 6 hours, 12 minutes, and 30 seconds. His own figure was slightly more accurate: 365 days, 15 ghati, 30 pala, 22 vipala, and 30 pratipala (365 days, 6 hours, 12 minutes, and 9.0 seconds).

Much of his astronomy, however, is quite erroneous. Like many Hindu scientists of the time, Brahmagupta was vehemently opposed to ĀryabhatŃa's ideas that Earth revolved around the Sun and spun on its axis. Why then, Brahmagupta asked, do not the lofty bodies fall down to Earth? He also questioned ĀryabhatŃa's theory of an aerial fluid that causes Earth to rotate.

Significance

Although Brahmagupta greatly extended the work of many preceding mathematicians and presented numerous valid theories of his own, it must be acknowledged that he did make some serious scientific errors. In addition to denying ĀryabhatŃa's theories of the place of Earth in the solar system, he gave a faulty formula for the area of an equilateral triangle. In his studies on the circle, he alternately used 3 and the square root of 10 as values for π.

Yet Brahmagupta's importance as a scientist must have been recognized during his lifetime, because he was accused of propagating scientific falsehoods to please the priests and the ignorant commonfolk. The priests were particularly opposed to the ideas that Earth was round and that it rotated around the Sun. Perhaps Brahmagupta had lied to avoid the fate of Socrates (c. 470-399 b.c.e.).

Despite these accusations, at least two of Brahmagupta's algebraic formulations, although originally devised for use in astronomy, became widely used by Hindu traders. Of particular practical use was his rule of three, in which the Argument, the Fruit, and the Requisition are the names of the terms. The first and last terms have to be similar. The Requisition multiplied by the Fruit and divided by the Argument yielded the Produce.

Brahmagupta also introduced the use of negative numbers, which he used to unify three of Diophantus's quadratic equations under a general equation. These negative numbers were especially useful to merchants in representing debts, along with positive numbers, which represented assets. Another advance in mathematics that the merchants must have found helpful was Brahmagupta's work on interest rates.

By 700, Hindu merchants had introduced Brahmagupta's mathematics to the Arabs, with whom they carried on a high volume of trade. In 772, a table of sines from Brahmagupta which, incidentally, was probably based on work by ĀryabhatŃa reached the ՙAbbāsid caliph al-Manṣūr, and it was ordered to be translated into Arabic. The entirety of the BrahmasphutŃasiddhānta was translated into Arabic by 775, around the time works by other Greek and Hindu mathematicians were being translated by Arab scholars. Together, these works would greatly influence the nascent Arabic mathematics, with Brahmagupta's greatest contributions coming in the study of negative numbers and indeterminate equations.

Bibliography

Ball, W. W. Rouse. A Short Account of the History of Mathematics. 4th ed. London: Macmillan, 1908. A thorough overview of the history of mathematics, with a section on Brahmagupta and his work on quadratic equations, right triangles, and algebra, plus scattered information on his later influence on Hindu and Arab mathematicians.

Cajori, Florian. A History of Mathematics. 5th ed. Providence, R.I.: AMS Chelsea, 2000. Gives the solution to Brahmagupta’s broken bamboo problem, plus formulas for Brahmagupta’s work on triangles and quadrilaterals.

Eves, Howard. An Introduction to the History of Mathematics. 6th ed. Philadelphia: Saunders College, 1990. Includes information on Brahmagupta’s studies on indeterminate equations, the Pell equation, cyclic quadrilaterals, and the rule of three, along with a discussion of his place in the history of mathematics. Some problems (with solutions) based on his formula for the cyclic quadrilateral are included.

Joseph, George Gheverghese. The Crest of the Peacock: The Non-European Roots of Mathematics. Rev. ed. Princeton, N.J.: Princeton Unversity Press, 2000. Joseph examines the history of mathematics in cultures throughout the world, including India. Its wide coverage places India within the greater scope of mathematical development. Bibliography and indexes.

Lakshmikantham, V., and S. Leela. The Origin of Mathematics. Lanham, Md.: University Press of America, 2000. Lakshmikantham argues that the importance of the early Indian mathematicians has been underestimated. Bibliography and index.

Prakash, Satya. A Critical Study of Brahmagupta and His Works. New Delhi, India: Indian Institute of Astronomical and Sanskrit Research, 1968. A comprehensive study of Brahmagupta, his works, his sources, and the influence of his work on later writers. Contains an extensive bibliography.

Puttaswamy, T. K. “The Mathematical Accomplishment of Ancient Indian Mathematicians.” In Mathematics Across Cultures: The History of Non-Western Mathematics, edited by Helaine Selin. Boston: Kluwer Academic, 2000. Examines the early Indian mathematicians and their importance.