### Origins of Tamils?[Where are Tamil people from?] PART :65

[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