[Compiled by: Kandiah Thillaivinayagalingam]
The
geometric principles expounded in the Shulba Sutras (800-200 BC) have often
been considered to mark the beginning of mathematics in the Indian
subcontinent.The Sulbasutras are appendices to the Vedas which give rules for
constructing altars.The four major Shulba Sutras, which are mathematically the
most significant, are those composed by Baudhayana, Manava, Apastamba and
Katyayana, about whom very little is known.Baudhayana,800 BC is older than the
other famous vedic mathematicians.He is accredited with calculating the value
of pi before pythagoras, and with discovering what is now known as the
Pythagorean theorem.He describes Pythagoras theorem as: "A rope stretched
along the length of the diagonal produces an area which the vertical and
horizontal sides make together". As you see, it becomes clear that this is
perhaps the most intuitive way of understanding and visualizing Pythagoras
theorem (and geometry in general) and he seems to have simplified the process
of learning by encapsulating the mathematical result in a simple shloka in a
layman’s language.The Manava Sulbasutra is not the oldest,nor is it one of the
most important,Manava would have not have been a mathematician in the sense
that we would understand it today.However his work give π = 25/8 = 3.125.It
would be fair to say that Apastamba's Sulbasutra is the most interesting from a
mathematical point of view and he wrote the Sulbasutra to provide rules for
religious rites and to improve and expand on the rules which had been given by
his predecessors.The mathematics given in his Sulbasutras is there to enable
the accurate construction of altars needed for sacrifices.This work is an
expanded version of that of Baudhayana.The general linear equation was solved
in the Apastamba's Sulbasutra. He also gives a remarkably accurate value for √2
=1 + 1/3 + 1/(3×4) - 1/(3×4×34),which gives an answer correct to five decimal
places [1.4142156861].As well as the problem of squaring the circle,Katyayana
composed one of the later Sulba Sutras, a series of nine texts on the geometry
of altar constructions, dealing with rectangles, right-sided triangles,
rhombuses, etc.However we do not know if these people undertook mathematical
investigations for their own sake, as for example the ancient Greeks did, or
whether they only studied mathematics to solve problems necessary for their
religious rites.The absence of any recorded tradition of geometric knowledge
predating these sutras have led some scholars to suggest a West Asian origin
for the onset of mathematical thinking in India. However, the discovery of the
archaeological remnants of the Indus Valley civilization in parts of Pakistan
and northwestern India over the course of last century has revealed a culture
having a sophisticated understanding of geometry which predated the Sulbasutras
by more than a thousand years.It is difficult to ascertain whether there was
any continuity between the geometry practised by the Indus civilization and
that used by the later Vedic culture;however, it is not impossible that some of
the earlier knowledge persisted among the local population and influenced the
sulbakaras (authors of the Sulbasutras) of the first millennium BC
The
maths used by this early Harrapan civilisation was very much for practical
means, and was primarily concerned with weights, measuring scales and a
surprisingly advanced 'brick technology', (which utilised ratios). The ratio
for brick dimensions 4:2:1 is even today considered optimal for effective
bonding.The well-laid out street plans of the Indus cities and their accurate
orientation along
the cardinal directions have been long been taken as evidence
that the Indus people had at least a working knowledge of geometry also.
Earlier studies have suggested that not only did these people have a practical
grasp of mensuration, but that they also had an understanding of the basic
principles of geometry.Also, many of the weights uncovered have been produced in
definite geometrical shapes (cuboid, barrel, cone, and cylinder to name a few)
which present knowledge of basic geometry, including the circle.Several scales
for the measurement of length were also discovered during excavations.These
scales and instruments for measuring length in different Indus sites indicate
that the culture knew how to make accurate spatial[of or relating to space]
measurements.For example, an ivory scale discovered at Lothal (in the western
coast of India) has 27 uniformly spaced lines over 46 mm, indicating an unit of
length corresponding to 1.70 mm.Other one was a decimal scale found in
Mohenjo-daro,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".Also
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.The sophistication of the metrology practised by the
Indus people is attested by the sets of regularly shaped artifacts of various
sizes that have been identified as constituting a system of standardized
weights.We do know that the Harappans had adopted a uniform system of weights
and measures. An analysis of the weights discovered suggests that they were
based on the binary and decimal systems and in ratios of
1:2:4:8:16:32:64:160:200:320:640:1,600: 3,200: 6,400: 8,000:12,800,from
smallest to largest.A total of 558 weights were excavated from
Mohenjodaro,Harappa,and Chanhu-daro,not including defective weights.The
smallest weight in this series is 0.856 grams,the largest weight,which was
found at the site of Mohenjo-daro weighs 10,865 grams (approximately 25 pounds) and the most common weight is approximately
13.7 grams,which is in the 16th ratio and seems to be one of the units used in
the Indus valley.The first seven Indus weights double in size from 1:2:4:8:16: 32:64.At this point the weight
increments change to a decimal system where the next largest weights have a
ratio of 160, 200, 320, and 640. The next jump goes to 1,600, 3,200, 6,400,
8,000, and 12,800.Further,the existence of a graduated system of accurately
marked weights shows the development of trade and commerce in Harappan society
Indeed, there is a surprising degree of uniformity in the measurement units
used at the widely dispersed centers of the civilization, indicating an
attention towards achieving a standard system of units for measurement.
However, most of the literature available up till now on the subject of Indus
geometry have been primarily concerned with patterns occurring at the
macro-scale (such as building plans, street alignments, etc.).This culture also
produced artistic designs. On carvings there is evidence that these people
could draw concentric and intersecting circles and triangles. The further using
of circles in the Harappan decorative design can be found at the pictures of
bullock carts, the wheels of which had perhaps a metallic band wrapped round
the rim. It clearly points to the knowledge of the ratio of the length of the
circumference of the circle and its diameter, and thus of the value of
π.However,the smaller-scale geometric patterns that are often observed on seals
or on the surface of pottery vessels have not been analysed in great detail
yet. We believe that such designs provide evidence for a much more advanced
understanding of geometry on the part of the Indus people than have hitherto
been suggested in the literature.
Hence,we
have argued that the origin of mathematics, and geometry in particular, in the
Indian subcontinent may actually date back to the third millennium BC and the
Indus Valley civilization, rather than beginning with the Sulbasutras of the
first millennium BC as is conventionally thought. Although the well-planned
cities, standardized system of measurements and evidence of long-range trade
contacts with Mesopotamia and Central Asia attest to the high technological
sophistication of this civilization, the unavailability of written records has
up to now prevented us from acquiring a detailed understanding of the level of
mathematical knowledge attained by the Indus people.By focusing on the geometry
of design motifs observed commonly in the artifacts excavated from various
sites belonging to this culture,we have shown that they suggest a deep
understanding of the properties of circular shapes. In particular, designs
which exhibit space-filling tiling with complicated shapes imply that the Indus
culture may have been adept at accurately estimating the area of shapes
enclosed by circular arcs.This speaks of a fairly sophisticated knowledge of
geometry that may have persisted in the local population long after the decline
of the urban centers associated with the civilization and it may well have
influenced in some part the mathematical thinking of the later Vedic culture of
the first millennium BC
PART :66 WILL FOLLOW
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