Mathematics in India

  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.