**Brahmagupta,** whose father was Jisnugupta, wrote important works on mathematics and astronomy. In particular he wrote *Brahmasphutasiddhanta* (The Opening of the Universe), in 628. The work was written in 25 chapters and Brahmagupta tells us in the text that he wrote it at Bhillamala which today is the city of Bhinmal. This was the capital of the lands ruled by the Gurjara dynasty.

Brahmagupta became the head of the astronomical observatory at Ujjain which was the foremost mathematical centre of ancient India at this time. Outstanding mathematicians such as Varahamihira had worked there and built up a strong school of mathematical astronomy.

In addition to the *Brahmasphutasiddhanta* Brahmagupta wrote a second work on mathematics and astronomy which is the *Khandakhadyaka* written in 665 when he was 67 years old. We look below at some of the remarkable ideas which Brahmagupta's two treatises contain. First let us give an overview of their contents.

The *Brahmasphutasiddhanta* contains twenty-five chapters but the first ten of these chapters seem to form what many historians believe was a first version of Brahmagupta's work and some manuscripts exist which contain only these chapters. These ten chapters are arranged in topics which are typical of Indian mathematical astronomy texts of the period. The topics covered are: mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; the moon's shadow; conjunctions of the planets with each other; and conjunctions of the planets with the fixed stars.

The remaining fifteen chapters seem to form a second work which is major addendum to the original treatise. The chapters are: examination of previous treatises on astronomy; on mathematics; additions to chapter 1; additions to chapter 2; additions to chapter 3; additions to chapter 4 and 5; additions to chapter 7; on algebra; on the gnomon; on meters; on the sphere; on instruments; summary of contents; versified tables.

Brahmagupta's understanding of the number systems went far beyond that of others of the period. In the *Brahmasphutasiddhanta* he defined zero as the result of subtracting a number from itself. He gave some properties as follows:-

When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.

He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers):-

A debt minus zero is a debt.

A fortune minus zero is a fortune.

Zero minus zero is a zero.

A debt subtracted from zero is a fortune.

A fortune subtracted from zero is a debt.

The product of zero multiplied by a debt or fortune is zero.

The product of zero multipliedby zero is zero.

The product or quotient of two fortunes is one fortune.

The product or quotient of two debts is one fortune.

The product or quotient of a debt and a fortune is a debt.

The product or quotient of a fortune and a debt is a debt.

Brahmagupta then tried to extend arithmetic to include division by zero:-

Positive or negative numbers when divided by zero is a fraction the zero as denominator.

Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.

Zero divided by zero is zero.

Really Brahmagupta is saying very little when he suggests that *n* divided by zero is *n*/0. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt to extend arithmetic to negative numbers and zero.

We can also describe his methods of multiplication which use the place-value system to its full advantage in almost the same way as it is used today. We give three examples of the methods he presents in the *Brahmasphuta siddhanta* and in doing so we follow Ifrah in [4]. The first method we describe is called "gomutrika" by Brahmagupta. Ifrah translates "gomutrika" to "like the trajectory of a cow's urine". Consider the product of 235 multiplied by 264. We begin by setting out the sum as follows:

2 235

6 235

4 235

----------

Now multiply the 235 of the top row by the 2 in the top position of the left hand column. Begin by 2 × 5 = 10, putting 0 below the 5 of the top row, carrying 1 in the usual way to get

2 235

6 235

4 235

----------

470

Now multiply the 235 of the second row by the 6 in the left hand column writing the number in the line below the 470 but moved one place to the right

2 235

6 235

4 235

----------

470

1410

Now multiply the 235 of the third row by the 4 in the left hand column writing the number in the line below the 1410 but moved one place to the right

2 235

6 235

4 235

----------

470

1410

940

Now add the three numbers below the line

2 235

6 235

4 235

----------

470

1410

940

----------

62040

The variants are first writing the second number on the right but with the order of the digits reversed as follows

235 4

235 6

235 2

----------

940

1410

470

----------

62040

The third variant just writes each number once but otherwise follows the second method

235

----------

940 4

1410 6

470 2

----------

62040

Another arithmetical result presented by Brahmagupta is his algorithm for computing square roots. This algorithm is discussed in [15] where it is shown to be equivalent to the Newton-Raphson iterative formula.

Brahmagupta developed some algebraic notation and presents methods to solve quardatic equations. He presents methods to solve indeterminate equations of the form *ax* + *c* = *by*. Majumdar in [17] writes:-

Brahmagupta perhaps used the method of continued fractions to find the integral solution of an indeterminate equation of the typeax+c=by.

In [17] Majumdar gives the original Sanskrit verses from Brahmagupta's *Brahmasphuta siddhanta* and their English translation with modern interpretation.

Brahmagupta also solves quadratic indeterminate equations of the type *ax*^{2} + *c* = *y*^{2} and *ax*^{2} - *c* = *y*^{2}. For example he solves 8*x*^{2} + 1 = *y*^{2} obtaining the solutions (*x*, *y*) = (1, 3), (6, 17), (35, 99), (204, 577), (1189, 3363), ... For the equation 11*x*^{2} + 1 = *y*^{2} Brahmagupta obtained the solutions (*x*, *y*) = (3, 10), (161/5, 534/5), ... He also solves 61*x*^{2} + 1 = *y*^{2} which is particularly elegant having *x* = 226153980, *y* = 1766319049 as its smallest solution.

A example of the type of problems Brahmagupta poses and solves in the *Brahmasphutasiddhanta* is the following:-

Five hundred drammas were loaned at an unknown rate of interest, The interest on the money for four months was loaned to another at the same rate of interest and amounted in ten mounths to78drammas. Give the rate of interest.

Rules for summing series are also given. Brahmagupta 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}. No proofs are given so we do not know how Brahmagupta discovered these formulae.

In the *Brahmasphutasiddhanta* Brahmagupta gave remarkable formulae for the area of a cyclic quadrilateral and for the lengths of the diagonals in terms of the sides. The only debatable point here is that Brahmagupta does not state that the formulae are only true for cyclic quadrilaterals so some historians claim it to be an error while others claim that he clearly meant the rules to apply only to cyclic quadrilaterals.

Much material in the *Brahmasphutasiddhanta* deals with solar and lunar eclipses, planetary conjunctions and positions of the planets. Brahmagupta believed in a static Earth and he gave the length of the year as 365 days 6 hours 5 minutes 19 seconds in the first work, changing the value to 365 days 6 hours 12 minutes 36 seconds in the second book the *Khandakhadyaka.* This second values is not, of course, an improvement on the first since the true length of the years if less than 365 days 6 hours. One has to wonder whether Brahmagupta's second value for the length of the year is taken from Aryabhata I since the two agree to within 6 seconds, yet are about 24 minutes out.

The *Khandakhadyaka* is in eight chapters again covering topics such as: the longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; and conjunctions of the planets. It contains an appendix which is some versions has only one chapter, in other versions has three.

Of particular interest to mathematics in this second work by Brahmagupta is the interpolation formula he uses to compute values of sines. This is studied in detail in [13] where it is shown to be a particular case up to second order of the more general Newton-Stirling interpolation formula.

**Article by:** *J J O'Connor* and *E F Robertson*

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