Brahmagupta 

Born: 598 in (possibly) Ujjain, India
Died: 670 in India 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 twentyfive 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 placevalue 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 NewtonRaphson
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
type ax + 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 ax2 + c = y2 and ax2  c = y2. For example he solves 8x2 + 1
= y2 obtaining the solutions (x,y) = (1,3), (6,17), (35,99),
(204,577), (1189,3363), ... For the equation 11x2+ 1 = y2 Brahmagupta
obtained the solutions (x,y) = (3,10), (161/5,534/5), ... He also
solves 61x2 + 1 = y2 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 to
78 drammas. 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)(2n+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
NewtonStirling interpolation formula.
Article by: J J O'Connor and E F Robertson Source:
www.history.mcs.standrews.ac.uk/Mathematicians



