Indian mathematics
t is without doubt that mathematics today owes a huge debt
to the outstanding contributions made by Indian mathematicians
over many hundreds of years. What is quite surprising is
that there has been a reluctance to recognise this and one
has to conclude that many famous historians of mathematics
found what they expected to find, or perhaps even what they
hoped to find, rather than to realise what was so clear
in front of them.
We shall examine the contributions of Indian mathematics
in this article, but before looking at this contribution
in more detail we should say clearly that the "huge
debt" is the beautiful number system invented by the
Indians on which much of mathematical development has rested.
Laplace put this with great clarity:-
The ingenious method of expressing every possible number
using a set of ten symbols (each symbol having a place value
and an absolute value) emerged in India. The idea seems
so simple nowadays that its significance and profound importance
is no longer appreciated. Its simplicity lies in the way
it facilitated calculation and placed arithmetic foremost
amongst useful inventions. the importance of this invention
is more readily appreciated when one considers that it was
beyond the two greatest men of Antiquity, Archimedes and
Apollonius.
We shall look briefly at the Indian development of the
place-value decimal system of numbers later in this article
and in somewhat more detail in the separate article Indian
numerals. First, however, we go back to the first evidence
of mathematics developing in India.
Histories of Indian mathematics used to begin by describing
the geometry contained in the Sulbasutras but research into
the history of Indian mathematics has shown that the essentials
of this geometry were older being contained in the altar
constructions described in the Vedic mythology text the
Shatapatha Brahmana and the Taittiriya Samhita. Also it
has been shown that the study of mathematical astronomy
in India goes back to at least the third millennium BC and
mathematics and geometry must have existed to support this
study in these ancient times.
The first mathematics which we shall describe in this article
developed in the Indus valley. The earliest known urban
Indian culture was first identified in 1921 at Harappa in
the Punjab and then, one year later, at Mohenjo-daro, near
the Indus River in the Sindh. Both these sites are now in
Pakistan but this is still covered by our term "Indian
mathematics" which, in this article, refers to mathematics
developed in the Indian subcontinent. The Indus civilisation
(or Harappan civilisation as it is sometimes known) was
based in these two cities and also in over a hundred small
towns and villages. It was a civilisation which began around
2500 BC and survived until 1700 BC or later. The people
were literate and used a written script containing around
500 characters which some have claimed to have deciphered
but, being far from clear that this is the case, much research
remains to be done before a full appreciation of the mathematical
achievements of this ancient civilisation can be fully assessed.
We often think of Egyptians and Babylonians as being the
height of civilisation and of mathematical skills around
the period of the Indus civilisation, yet V G Childe in
New Light on the Most Ancient East (1952) wrote:-
India confronts Egypt and Babylonia by the 3rd millennium
with a thoroughly individual and independent civilisation
of her own, technically the peer of the rest. And plainly
it is deeply rooted in Indian soil. The Indus civilisation
represents a very perfect adjustment of human life to a
specific environment. And it has endured; it is already
specifically Indian and forms the basis of modern Indian
culture.
We do know that the Harappans had adopted a uniform system
of weights and measures. An analysis of the weights discovered
suggests that they belong to two series both being decimal
in nature with each decimal number multiplied and divided
by two, giving for the main series ratios of 0.05, 0.1,
0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500. Several
scales for the measurement of length were also discovered
during excavations. One was a decimal scale based on a unit
of measurement of 1.32 inches (3.35 centimetres) which has
been called the "Indus inch". Of course ten units
is then 13.2 inches which is quite believable as the measure
of a "foot". A similar measure based on the length
of a foot is present in other parts of Asia and beyond.
Another scale was discovered when a bronze rod was found
which was marked in lengths of 0.367 inches. It is certainly
surprising the accuracy with which these scales are marked.
Now 100 units of this measure is 36.7 inches which is the
measure of a stride. Measurements of the ruins of the buildings
which have been excavated show that these units of length
were accurately used by the Harappans in construction.
It is unclear exactly what caused the decline in the Harappan
civilisation. Historians have suggested four possible causes:
a change in climatic patterns and a consequent agricultural
crisis; a climatic disaster such flooding or severe drought;
disease spread by epidemic; or the invasion of Indo-Aryans
peoples from the north. The favourite theory used to be
the last of the four, but recent opinions favour one of
the first three. What is certainly true is that eventually
the Indo-Aryans peoples from the north did spread over the
region. This brings us to the earliest literary record of
Indian culture, the Vedas which were composed in Vedic Sanskrit,
between 1500 BC and 800 BC. At first these texts, consisting
of hymns, spells, and ritual observations, were transmitted
orally. Later the texts became written works for use of
those practicing the Vedic religion.
Here is one style of the Brahmi numerals. |
