Brahmagupta ( ) (598–668)
was a great Indian mathematician and astronomer.Brahmagupta wrote
important works on mathematics and astronomy. In particular he
wrote Brahmasphutasiddhanta (Correctly Established Doctrine of
Brahma), 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.
Life and work
Brahmagupta was born in 598 CE in Bhinmal city in the
state of Rajasthan of northwest India. He likely lived most
of his life in Bhillamala (modern Bhinmal in Rajasthan) in the empire of Harsha
during the reign (and possibly under the patronage) of King
Vyaghramukha. As a result, Brahmagupta is often referred to
as Bhillamalacarya, that is, the teacher from Bhillamala Bhinmal.
He was the
head of the astronomical observatory at Ujjain, and during
his tenure there wrote four texts on mathematics and astronomy: the
Cadamekela in 624, the Brahmasphutasiddhanta in 628, the
Khandakhadyaka in 665, and the Durkeamynarda in
672.The
Brahmasphutasiddhanta (
Corrected
Treatise of Brahma) is arguably his most famous work. The
historian
al-Biruni (c. 1050) in his book
Tariq al-Hind states that the
Abbasid caliph al-Ma'mun had an embassy in India and from India a
book was brought to Baghdad which was translated into Arabic as
Sindhind. It is generally presumed that
Sindhind
is none other than Brahmagupta's
Brahmasphuta-siddhanta.
Although Brahmagupta was familiar with the works of astronomers
following the tradition of
Aryabhatiya,
it is not known if he was familiar with the work of
Bhaskara I, a contemporary. Brahmagupta had a
plethora of criticism directed towards the work of rival
astronomers, and in his
Brahmasphutasiddhanta is found one
of the earliest attested 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.
Mathematics
Brahmagupta's most famous work is his
Brahmasphutasiddhanta. In it he invented many formulas and
mathematical properties. It is composed in elliptic verse, as was
common practice in
Indian
mathematics, and consequently has a poetic ring to it. As no
proofs are given, it is not known how Brahmagupta's mathematics was
derived.
Algebra
Brahmagupta gave the solution of the general
linear equation in chapter eighteen of
Brahmasphutasiddhanta,
18.43 The difference between rupas, when
inverted and divided by the difference 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.
Which is a solution equivalent to x = \tfrac{e-c}{b-d}, where
rupas represents constants. He further gave two equivalent
solutions to the general
quadratic
equation,
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].
18.45. Whatever is the square-root of the
rupas multiplied
by the square [and] increased by the square of half the unknown,
diminish that by half the unknown [and] divide [the remainder] by
its square. [The result is] the unknown.
Which are, respectively, solutions equivalent to,
- x = \frac{\sqrt{4ac+b^2}-b}{2a}
and
- x = \frac{\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.
18.51.
Subtract the colors different from the first
color.
[The remainder] divided by the first [color's
coefficient] is the measure of the first.
[Terms] two by two [are] considered [when reduced to]
similar divisors, [and so on] repeatedly.
If there are many [colors], the pulverizer [is to be
used].
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
In the beginning of chapter twelve of his
Brahmasphutasiddhanta, entitled
Calculation,
Brahmagupta 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, \tfrac{a}{c} + \tfrac{b}{c},
\tfrac{a}{c} \cdot \tfrac{b}{d}, \tfrac{a}{1} + \tfrac{b}{d},
\tfrac{a}{c} + \tfrac{b}{d} \cdot \tfrac{a}{c} =
\tfrac{a(d+b)}{cd}, and \tfrac{a}{c} - \tfrac{b}{d} \cdot
\tfrac{a}{c} = \tfrac{a(d-b)}{cd}.
Series
Brahmagupta then goes on to give the sum of the squares and cubes
of the first
n integers.
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].
It is important to note 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)(2n+1)/6 and the sum of the cubes of the first n natural
numbers as (n(n+1)/2)².
Zero
Brahmagupta made use of an important concept in mathematics, the
number zero. The
Brahmasphutasiddhanta 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 a lack of
quantity as was done by
Ptolemy and the
Romans. In chapter eighteen of his
Brahmasphutasiddhanta, Brahmagupta describes operations on
negative numbers. He first describes addition and subtraction,
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 positive, [and that] of two zeros zero.
[...]
18.32. A negative minus zero is negative, a positive [minus zero]
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.
He goes on to describe multiplication,
18.33.
The product of a negative and a positive is negative,
of two negatives positive, and of positives positive; the product
of zero and a negative, of zero and a positive, or of two zeros is
zero.
But then he spoils the matter some what when he describes division,
18.34.
A positive divided by a positive or a negative divided
by a negative is positive; a zero divided by a zero is zero; a
positive divided by a negative is negative; a negative divided by a
positive is [also] negative.
18.35. A negative or a positive divided by zero has that [zero] as
its divisor, or zero divided by a negative or a positive [has that
negative or positive as its divisor]. The square of a negative or
of a positive is positive; [the square] of zero is zero. That of
which [the square] is the square is [its] square-root.
Here Brahmagupta states that \tfrac{0}{0} = 0 and as for the
question of \tfrac{a}{0} where a \neq 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.
Diophantine analysis
Pythagorean triples
In chapter twelve of his
Brahmasphutasiddhanta,
Brahmagupta finds Pythagorean triples,
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.
or in other words, for a given length
m and an arbitrary
multiplier
x, let a =
mx and
b = m + mx/(x +
2). Then
m,
a, and
b form a
Pythagorean triple.
Pell's equation
Brahmagupta went on to give a recurrence relation for generating
solutions to certain instances of Diophantine equations of the
second degree such as Nx^2 + 1 = y^2 (called
Pell's equation) by using the
Euclidean algorithm. The Euclidean
algorithm was known to him as the "pulverizer" since it breaks
numbers down into ever smaller pieces.
The nature of squares:
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 product of the first [pair], multiplied by the
multiplier, with the product of the last [pair], is the last
computed.
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.
The key to his solution was the identity,
- (x^2_1 - Ny^2_1)(x^2_2 - Ny^2_2) = (x_1 x_2 + Ny_1 y_2)^2 -
N(x_1 y_2 + x_2 y_1)^2
which is a generalization of an identity that was discovered by
Diophantus,
- (x^2_1 - y^2_1)(x^2_2 - y^2_2) = (x_1 x_2 + y_1 y_2)^2 - (x_1
y_2 + x_2 y_1)^2.
Using his identity and the fact that if (x_1, y_1) and (x_2, y_2)
are solutions to the equations x^2 - Ny^2 = k_1 and x^2 - Ny^2 =
k_2, respectively, then (x_1 x_2 + N y_1 y_2, x_1 y_2 + x_2 y_1) is
a solution to x^2 - Ny^2 = k_1 k_2, he was able to find integral
solutions to the Pell's equation through a series of equations of
the form x^2 - Ny^2 = k_i. Unfortunately, Brahmagupta was not able
to apply his solution uniformly for all possible values of
N, rather he was only able to show that if x^2 - Ny^2 = k
has an integral solution for k = \pm 1, \pm 2, \pm 4 then x^2 -
Ny^2 = 1 has a solution. The solution of the general Pell's
equation would have to wait for
Bhaskara
II in c. 1150 CE.
Geometry
Brahmagupta's formula
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,
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.
So given the lengths
p,
q,
r and
s of a cyclic quadrilateral, the approximate area is
(\tfrac{p + r}{2}) (\tfrac{q + s}{2}) while, letting t = \tfrac{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. One theorem states that the two lengths of a triangle's
base when divided by its altitude then follows,
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.
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.
Brahmagupta continues,
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].
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. This leads up to
Brahmagupta's famous theorem,
12.30-31.
Imaging two triangles within [a cyclic quadrilateral]
with unequal sides, the two diagonals are the two
bases.
Their two segments are separately the upper and lower
segments [formed] at the intersection of the
diagonals.
The two [lower segments] of the two diagonals are two
sides in a triangle; the base [of the quadrilateral is the base of
the triangle].
Its perpendicular is the lower portion of the [central]
perpendicular; the upper portion of the [central] perpendicular is
half of the sum of the [sides] perpendiculars diminished by the
lower [portion of the central perpendicular].
Pi
In verse 40, he gives values of
π,
12.40.
The diameter and the square of the radius [each]
multiplied by 3 are [respectively] the practical circumference and
the area [of a circle].
The accurate [values] are the square-roots from the
squares of those two multiplied by ten.
So Brahmagupta uses 3 as a "practical" value of
π, and
\sqrt{10} as an "accurate" value of
π.
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 frustrum 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
In Chapter 2 of his
Brahmasphutasiddhanta, entitled
Planetary True Longitudes, Brahmagupta presents a sine
table:
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 moonl the moon, arrows, suns [...]
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 tradition 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, 1459, 2594, 2719, 2832, 2933, 3021, 3096, 3159, 3207,
3242, 3263, and 3270, with the radius being 3270.
interpolation formula
In 665 Brahmagupta devised and used a special case of the
Newton-Stirling interpolation formula of the second-order to
interpolatenew 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 ≤
x ≤ 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 \left(\frac{\Delta f(a) + \Delta
f(a-h)}{2}\right) + \frac{x^2 \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
It was through the
Brahmasphutasiddhanta that the Arabs
learned of Indian astronomy.
The famous Abbasid
caliph Al-Mansur (712–775) founded
Baghdad, which is situated on the banks of the Tigris, and made it
a center of learning. The caliph invited a scholar of Ujjain by the name
of Kankah in 770 A.D. Kankah used the
Brahmasphutasiddhanta to explain the Hindu system of
arithmetic astronomy.
Muhammad
al-Fazari translated Brahmugupta's work into Arabic upon the
request of the caliph.
In chapter seven of his
Brahmasphutasiddhanta, entitled
Lunar Crescent, Brahmagupta rebuts the idea that the Moon
is farther from the Earth than the Sun, an idea which is maintained
in scriptures. He does this by explaining the illumination of the
Moon by the Sun.
7.1.
If the moon were above the sun, how would the power of
waxing and waning, etc., be produced from calculation of the
[longitude of the] moon? the near half [would be] always
bright.
7.2. In the same way that the half seen by the sun of a pot
standing in sunlight is bright, and the unseen half dark, so is
[the illumination] of the moon [if it is] beneath the sun.
7.3. The brightness is increased in the direction of the sun. At
the end of a bright [i.e. waxing] half-month, the near half is
bright and the far half dark. Hence, the elevation of the horns [of
the crescent can be derived] from calculation. [...]
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.
Some of the important contributions made by Brahmagupta in
astronomy are: methods for calculating the position of heavenly
bodies over time (
ephemerides), their
rising and setting,
conjunction, and the calculation of
solar and lunar
eclipses. Brahmagupta
criticized the
Puranic view that the Earth
was flat or hollow. Instead, he observed that the Earth and heaven
were spherical and that the Earth is moving. In 1030, the
Muslim astronomer Abu al-Rayhan
al-Biruni, in his
Ta'rikh al-Hind, later translated
into
Latin as
Indica, commented on
Brahmagupta's work and wrote that critics argued:
According to al-Biruni, Brahmagupta responded to these criticisms
with the following argument on
gravitation:
About the Earth's gravity he said: "Bodies fall towards the earth
as it is in the nature of the earth to attract bodies, just as it
is in the nature of water to flow."
Citations and footnotes
References
See also
External links