As early Indian astronomers tried to quantify the paths
of the sun, the moon, the planets and the stars on the celestial
sphere with ever more accuracy, or to predict the occurrence
of eclipses, they were naturally led to develop mathematical
tools. Astronomy and mathematics were thus initially regarded
as inseparable, the latter being the maid-servant of the former.
Indeed, about 1400 BCE, the Vedāṅga Jyotiṣa, the first extant
Indian text of astronomy, states in two different versions:
Like the crest on the head of a peacock, like the gem on
the hood of a cobra, jyotiṣa (astronomy) / gaṇita (mathematics)
is the crown of the Vedāṅga śāstras [texts on various branches
of knowledge]. In fact, jyotiṣa initially referred to astronomy
and mathematics combined; only later did it come to mean
astronomy alone (and much later did it include astrology).
First Steps
India’s first urban development, the Indus or Harappan
civilization (2600-1900 BCE), involved a high degree of town
planning. A mere glance at the plan of Mohenjo-daro’s acropolis
(or upper city), Dholavira (in the Rann of Kachchh) or Kalibangan
(Rajasthan), reveals fortifications and streets generally aligned
to the cardinal directions and exhibiting right angles. Specific
proportions in the dimensions of major structures have also
been pointed out. All this implies a sound knowledge of basic
geometric principles and an ability to measure angles, which
the discovery of a few cylindrical compasses made of shell, with
slits cut every 45°, has confirmed. Besides, for trading purposes
the Harappans developed a standardized system of weights
in which, initially, each weight was double the preceding one,
then, 10, 100 or 1,000 times the value of a smaller weight. This
shows that the Harappans could not only multiply a quantity
by such factors, but also had an inclination for a decimal system
of multiples. However, there is no agreement among scholars
regarding the numeral system used by Harappans.
There is no scholarly consensus on the dates of the four
Vedas, India’s most ancient texts, except that they are over 3,000
years old at the very least. We find in them frequent mentions of
numbers by name, in particular multiples of tens, hundreds and
thousands, all the way to a million millions in the Yajur Veda — a
number called parārdha. (By comparison, much later, the Greeks
named numbers only up to 10,000, which was a ‘myriad’; and
only in the 13th century CE would the concept of a ‘million’ be
adopted in Europe.) The Brāhmanas, commentaries on the Vedas,
knew the four arithmetical operations as well as basic fractions.
Early Historical Period
The first Indian texts dealing explicitly with mathematics
are the Śulbasūtras, dated between the 8th and 6th centuries BCE.
They were written in Sanskrit in the highly concise sūtra style
and were, in effect, manuals for the construction of fire altars
(called citis or vedis) intended for specific rituals and made of
bricks. The altars often had five layers of 200 bricks each, the
lowest layer symbolizing the earth, and the highest, heaven; they
were thus symbolic representations of the universe.
Because their total area needed to be carefully defined and
constructed from bricks of specified shapes and size, complex
geometrical calculations followed. The Śulbasūtras, for instance,
are the earliest texts of geometry offering a general statement,
in geometric form, of the so-called Pythagoras theorem (which
was in fact formulated by Euclid around 300 BCE).
They spelt out elaborate geometric methods to construct a
square resulting from the addition or subtraction of two other
squares, or having the same area as a given circle, and vice-versa
— the classic problems of the squaring of a circle or the circling
of a square (which, because of π’s transcendental nature, cannot
have exact geometrical solutions, only approximate ones). All
these procedures were purely geometrical, but led to interesting
corollaries; for instance, √2 was given a rational approximation
which is correct to the fifth decimal
The Śulbasūtras also introduced a system of linear units,
most of them based on dimensions of the human body; they
were later slightly modified and became the traditional units
used across India. The chief units were:
14 aṇus (grain of common millet) = 1 aṅgula (a digit)
12 aṅgulas = 1 prādeśa (the span of a hand, later vitasti)
15 aṅgulas = 1 pada (or big foot)
24 aṅgulas = 1 aratni (or cubit, later also hasta)
30 aṅgulas = 1 prakrama (or step)
120 aṅgulas = 1 puruṣa (or the height of a man with his arm
extended over his head)
A few centuries later, Piṅgala’s Chandasūtras, a text on
Sanskrit prosody, made use of a binary system to classify the
metres of Vedic hymns, whose syllables may be either light
(laghu) or heavy (guru); rules of calculation were worked out
to relate all possible combinations of light and heavy syllables,
expressed in binary notation, to numbers in one-to-one
relationships, which of course worked both ways. In the course
of those calculations, Piṅgala referred to the symbol for śūnya
or zero.
About the same time, Jaina texts indulged in cosmological
speculations involving colossal numbers, and dealt with
geometry, combinations and permutations, fractions, square
and cube powers; they were the first in India to come up with
the notion of an unknown (yāvat-tāvat), and introduced a value
of π equal to √10, which remained popular in India for quite a
few centuries.
With the appearance of the Brāhmī script a few centuries
BCE, we come across India’s first numerals, on Ashoka’s edicts
in particular, but as yet without any decimal positional value.
These numerals will evolve in shape; eventually borrowed by
Arabs scholars, they will be transmitted, with further alterations,
to Europe and become our modern ‘Arabic’ numerals.
The Classical Period
Together with astronomy, Indian mathematics saw its
golden age during India’s classical period, beginning more or
less with the Gupta age, i.e. from about 400 CE.
Shortly before that period, the full-fledged place-value
system of numeral notation — our ‘modern’ way of noting
numbers, unlike non-positional systems such as those depicted
above or Roman numbers — had been worked out, integrating
zero with the nine numerals. It is a pity that we shall never know
who conceived of it. Amongst the earliest known references
to it is a first-century CE work by the Buddhist philosopher
Vasumitra, and it is worked out more explicitly in the Jain
cosmological work Lokavibhāga, written in 458 CE. Soon it was
adopted across India, and later taken to Europe by the Arabs.
This was a major landmark in the world history of science, since
it permitted rapid developments in mathematics.
About 499 CE, living near what is today Patna, Āryabhaṭa I
(born 476 CE) authored the Āryabhaṭīya, the first extant siddhānta
(or treatise) attempting a systematic review of the knowledge of mathematics and astronomy prevailing in his days. The text is so concise (just 121 verses) as to be often obscure, but between the 6th and the 16th century, no fewer than twelve major commentaries were authored to explicate and build upon its contents. It was eventually translated
into Arabic about 800 CE(under the title Zīj al-Ārjabhar), which in turn led to a Latin
translation in the 13th century (in which Āryabhaṭa was called
‘Ardubarius’).
The mathematical content of Āryabhaṭīya ranges from a very
precise table of sines and an equally precise value for π (3.1416,
stated to be ‘approximate’) to the area of a triangle, the sums
of finite arithmetic progressions, algorithms for the extraction
of square and cube roots, and an elaborate algorithm called
kuṭṭaka (‘pulverizing’) to solve indeterminate equations of the
first degree with two unknowns: ax + c = by. By ‘indeterminate’
is meant that solutions should be integers alone, which rules
out direct algebraic methods; such equations came up in
astronomical problems, for example to calculate a whole number
of revolutions of a planet in a given number of years.
It is worth mentioning that despite its great contributions,
the Āryabhaṭīya is not free of errors: its formulas for the volumes
of a pyramid and a sphere were erroneous, and would be later
corrected by Brahmagupta and Bhāskarācārya respectively.
The Classical Period, post-Āryabhaṭa
Born in 598 CE, Brahmagupta
was an imposing figure, with
considerable achievements in
mathematics. In his Brahmasphuta
Siddhānta, he studied cyclic
quadrilaterals (i.e., inscribed
in a circle) and supplied the
formula for their area (a formula
rediscovered in 17th-century
Europe): if ABCD has sides of
lengths a, b, c, and d, and the semi-perimeter is s = (a + b +c +
d)/2, then the area is given by:
Area ABCD = √[(s – a) (s –b) (s – c) (s – a)]
Brahmagupta boldly introduced the notion of negative
numbers and ventured to define the mathematical infinite as
khacheda or ‘that which is divided by kha’, kha being one of
the many names for zero. He discovered the bhāvanā algorithmfor integral solutions
t o s e c o n d - o r d e r
indeterminate equations
(called varga prakriti)
of the type Nx2
+ 1 = y2
.
He was in many ways
one of the founders of
modern algebra, and his
works were translated
into Persian and later
Latin.
D a t e d a r o u n d
the 7th century, the
Bakhshali manuscript,
named after the village
( n o w i n n o r t h e r n
Pakistan) where it was
found in 1881 in the
form of 70 leaves of birch
bark, gives us a rare
insight into extensive
mathematical calculation techniques of the times, involving
in particular fractions, progressions, measures of time, weight
and money.
Other brilliant mathematicians of the siddhāntic era
included Bhāskara I, a contemporary of Brahmagupta, who
did pioneering work in trigonometry (proposing a remarkably
accurate rational approximation for the sine function),
Śrīdhara and Mahāvīra. The last, a Jain scholar who lived in
the 9th century in the court of a Rashtrakuta king (in today’s
Karnataka), authored the first work of mathematics that was not
as part of a text on astronomy. In it, Mahāvīra dealt with finite
series, expansions of fractions, permutations and combinations
(working out, for the first time, some of the standard formulas
in the field), linear equations with two unknowns, quadratic
equations, and a remarkably close approximation for the
circumference of an ellipse, among other important results.Graph showing the high accuracy of Bhāskara I’s rational
approximation for the sine function from 0° to 180° (in blue).
The sine function (in read) had to be shifted upward by 0.05 to
make the two curves distinguishable. (Courtesy: IFIH)
Bhāskara II, often known as Bhāskarācārya, lived in the 12th
century. His Siddhāntaśiromani (literally, the ‘crest jewel of the
siddhāntas’) broke new ground as regards cubic and biquadratic
equations. He built upon Brahmagupta’s work on indeterminate
equations to produce a still more effective algorithm, the
chakravāla (or ‘cyclic method’); with it he showed, for instance,
that the smallest integral solutions to 61x2
+ 1 = y2
are x =
226153980, y = 1766319049 (interestingly, five centuries later,
the French mathematician Fermat offered the same equation as
a challenge to some of his contemporaries). Bhāskarācārya also
grasped the notion of integration as a limit of finite sums: by
slicing a sphere into ever smaller rings, for instance, he was able
to calculate its area and volume. He came close to the modern
notion of derivative by discussing the notion of instant speed
(tātkālika gati) and understood that the derivative of the sine
function is proportional to the cosine.
The first part of Bhāskarācārya’s Siddhāntaśiromani is a
collection of mathematical problems called Līlāvatī, named after
an unknown lady to whom Bhāskara puts problems in an often
poetical language. Līlāvatī became so popular with students of
mathematics across India that four centuries later, Akbar had
it translated into Persian by a court poet.
The Kerala School of Mathematics
Along with astronomy, mathematics underwent a revival
in the Kerala School, which flourished there from the 14th to
the 17th century. Its pioneer, Mādhava (c. 1340–1425), laid
some of the foundations of calculus by working out power
series expansions for the sine and cosine functions (the so-
called Newton series), and by spelling out this fundamental
expansion of π:
This is known as the Gregory–Leibniz series, but ought
one day to be named after Mādhava. He went on to propose a
more rapidly convergent series for π:
which enabled him to calculate π to 11 correct decimals.
Nīlakaṇṭha Somayāji (c. 1444–1545) and Jyeṣṭhadeva (c.
1500–1600) built on such results and considerably enriched what
might be called the Indian foundations of calculus. The latter,
for instance, worked out the binomial expansion:
Features of Indian mathematics
As elsewhere, mathematics in India arose from practical
needs: constructing fire altars according to precise specifications,
tracking the motion of planets, predicting eclipses, etc. But
India’s approach remained essentially pragmatic: rather than
developing an axiomatic method such as that of the Greek
(famously introduced by Euclid for geometry), it focused on
obtaining formulas and algorithms that yielded precise and
reliable results.
Nevertheless, Indian mathematicians did often provide
logically rigorous justifications for their results, especially in the
longer texts. Indeed, Bhāskarācārya states that presenting proofs
(upapattis) is part of the teaching tradition, and Jyeṣṭhadeva
devotes considerable space to them in his Yukti Bhāṣā. The
shorter texts, on the other hand, often dispensed with the
development of proofs. In the same spirit, the celebrated S.
Ramanujan produced many important theorems but did not
take time to supply proofs for them, leaving this for others to do!
Whether those specificities limited the further growth of
Indian mathematics is open to debate. Other factors have been
discussed by historians of science, such as historical disruptions
of centres and networks of learning (especially in north India),
limited royal patronage, or the absence of a conquering
impulse (which, in Europe, did fuel the growth of science and
technology). Be that as it may, India’s contribution in the field
was enormous by any standard. Through the Arabs, many
Indian inputs, from the decimal place-value system of numeral
notation to some of the foundations of algebra and analysis,
travelled on to Europe and provided crucial ingredients to the
development of modern mathematics.
