Eternal India Encyclopedia

Ancient Concepts, Sciences & Systems

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VEDIC MATHEMATICS

Baudhayana's method of designing spoked wheels by successive circling of concentric squares. Any number of spokes may be placed by dividing one of the outer squares into the desired number of parts before circling them. It is one of the most interesting mathematical examples from antiquity.

series 2, 4, 6, 8, 20, 30, 40, 50, 60, 70, 80, 90, 100. There are numerous examples of seasons and the year being expressed in terms of numbers of days and half days. Interestingly, both the word Aditi (infinite) and Kham (zero) are found in the Vedic literature. There is also evidence of the use of a limited decimal system consisting of multiples of ten. Terms like dasa (10), sata (100), sahasra (1,000), ayuta (10,000) and others are known. Most remarkably the number paradha equal to 10 12 or a trillion is also known. The very first verse of the Atharvaveda refers to trisaptah, or all types of combinations of threes and sevens like: 3+7=10, 3x7=21,3+5+7=15 and many others. But of the famous Indian con- tribution, the use of the zero for the place value notation, there seems to be no signs. Thus even going back to Vedic times, the knowledge of numbers and arithmetic is found to be of no mean order. There are terms describing different aspects of various fire altars that presuppose basic knowledge of geometry also. The spoked wheel described in minute detail in the Rig Veda (1.164. 11-15) shows knowledge of circles, radii and methods of drawing them. This is also confirmed by devices from drawing circles dating back to at least 2500 BC found and described by Mackay. Thus the germs of both arithmetic and geometry taken to such heights in the Sulba works of Baudhay- ana and his schools are already found in the Vedas themselves. THE SULBASUTRAS With the Sulbasutras, or the Sulbas as they are more commonly known, we reach a level of comprehensive mathematical exposition not to be attained until the time of the Greeks. The word Sulba in Sanskrit means a rope or cord, and may be derived from the root sulb or sulv meaning to measure. The Sulbas are concerned primarily with the mathematical details involved in the construction of sacri- ficial altars as prescribed in the Vedas and the Brahmanas. They always appear as appendixes to the Srauta or the ritual part of the Kalpasutras — religious works. Nevertheless, the Sulbas are enormously interesting mathematical works and the earliest com- prehensive treatises of their kind found anywhere. The Sulba of Baudhayana in particular is a work of great perfection. In style and content, the Sulbas may be described as texts of geometric algebra: a problem is stated in geometric terms, but its solution is given in a form that combines geometry and algebra. Two problems dominate the Sulbas: the square on the diagonal theorem and the problem of equivalence of area. The square on the diagonal theorem, which is more often known in its triangular formulation as the ‘Theorem of Pythagoras’ is derived by

BACKGROUND Vedic mathematics as currently used is often an imprecise term, referring to Indian mathematics from the hoariest antiquity to rela- tively modem works that claim the Vedas for their inspiration. But properly speaking, the term can only apply to the mathematics found in the Vedas themselves and to the technical works that followed immediately upon the redaction of the Vedas into the four-fold divi- sion, the form in which they are currently found. These early post- Vedic works, known as the Sulbasutras (or the Sulbas) contain mathematical formulas and geometrical constructions necessary for Vedic rituals. Thus as the American mathematician and historian Seidenberg has noted, the Sulbas preserve the religious origins of mathematics. The present article is concerned primarily with the mathematics of the Sulbas which may be taken as being synony- mous with Vedic mathematics. Vedic astronomy which also pre- supposes considerable knowledge of mathematics is not discussed. ORIGINS The evidence for the existence of mathematics as part of Vedic rituals is both direct and indirect. Until recently, the wholly unten- able belief that the sites of the so-called Indus Valley Civilisation (c.2700-1800 BCE) must have belonged to a pre-Vedic society has been an artificial barrier to a proper appreciation of ancient Indian mathematics. In addition, a strong tendency on the part of nine- teenth century historians to trace all technical knowledge to Py- thagorean Greece has led to irreconcilable contradictions in the history and chronology of ancient mathematics. The very existence of elaborately planned cities like Harappa, Mohenjo-Daro and many other sites going back to the third millennium before Christ is evidence of considerable knowledge of geometry at least two thou- sand years before Pythagoras. But perhaps more interestingly, the so-called Harappan sites from Baluchistan to Lothal in Gujarat to Eastern Uttar Pradesh have yielded Yajnashalas, or sacrificial altars of the kind prescribed in the Vedic literature, and whose method of construction with the necessary mathematical details are found given in the Sulbasutras. Thus, from its very beginnings, mathematics arose from the needs of the religion and ritual, gradu- ally giving rise to secular applications like architecture and town planning. This is precisely how it is described in Indian tradition also. The Vedas being religious literature have little directly to say about mathematics but there is enough even in the ancient Rig Veda suggesting that the mathematical knowledge of its composers was more than rudimentary. In the Rig Veda (ii. 18.4-6) we find the