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October 23, 2007

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Saha had a good sense of humor. But it does appear that BCE-era Indians were pretty good in computational math, including perhaps examples like this. Math achievements of the mid-first-millennium CE Gupta-era were likely built upon this substrate of math acumen, echoed too in the ancient Indian penchant for all manner of classifications of life and mind (6 ways to argue logically, 9 paths to salvation, 24 ways of making love standing up on a wall, etc. -- I'm making up the examples to illustrate the point).

Here is an excerpt from AL Basham's quite excellent editorial compilation, Cultural History of India.

Ancient Hindu mathematics shows an early interest in large numbers expressed in powers of ten, in the nature of numbers and their factors, and in the division of time into smallest units. These large powers occur in the Vedic Samhitas, Brahmanas, and Sutras, in the epics Mahabharata and Ramayana, and in the Lalitavistara (where 10 to power 53 is given). Of particular interest is the Satapatha Brahmana, which lists all the factors of 720 as far as 24, and after stating that 360 nights and days contain 10,800 muhurttas proceeds by four successive multiplications by 15 to reach the ultimate pranas, or breathings. The occurrence of the word rasi (a heap) in the Chandogya Upanishad recalls the use of the same concept by the Ancient Egyptians, and is clearly the humble origin of what was later to become the burden of many a school boy, standing for the unknown quantity x.

Of the greatest importance to the historian of mathematics are the Sulva Sutras, which form part of the Kalpa Sutras and deal with the construction of sacrificial altars used in Vedic ritual. As terminal writings they summarize the knowledge of several preceding centuries and provide an excellent picture of the achievements of Hindu geometry prior to the mathematics of the Jaina sect; furthermore, when temple worship replaced the old rites of the agnicayana, this geometrical tradition lapsed and was subsequently superseded by the growth of analysis for which Hindu mathematicians are justly renowned. From the mass of literature which must have been the prerogative of the priesthood seven Sulva Sutras have survived ... They deal with such matters as the construction of squares and rectangles, the relations of the sides to the diagonals, the construction of equivalent squares and rectangles, the construction of equivalent squares and circles, the construction of triangles equivalent to squares and rectangle, and the construction of squares equal to two or more given squares or equal to the difference between two given squares. In this connection we may note two interesting formulae, those giving the diagonal of a square and the squaring of the circle. Thus, according to Baudhayana and Apastamba, to obtain the dvi-karani or diagonal, "Increase the measure by its third part, and again by the fourth part (of this third part) less the thirty-fourth part of itself (i.e. of the fourth part)." This gives a value of (square root of 2) diverging from modern calculation only in the sixth place of decimals.

Baudhayana says: "If you wish to square a circle, divide its diameter into eight parts; then divide one of these parts into 29 parts and leave out 28 of them; and also the sixth part (of the previous division) less the eighth part of this (last)." A relation between the radius of the circle and the side of the equivalent square if finally obtained in the form [formula I can't write down here].

In the construction of altars (vedi) requiring numbers of bricks of differing sizes in various layers, e.g., in the falcon-shaped fire altar, we see the origin of those indeterminate problems which form a notable part of later Hindu algebra. Bibhutibhusan Dutta, after examining Jaina canonical literature and the commentaries of Kapardisvami and Karavindasvami, inclines also to the view that the irrationality of (square root of two) was understood in the time of the Sulvas.

Most treatments of ancient Hinduism on the Web (and likely also Vedic math) are unfortunately of poor quality -- too many Hindu chauvinists out there. For them, this is above all a matter of voluptuous pride. One of my Indian professors in grad school (US) turned out to be one, devoting his post-tenure life in Electrical Engineering to debunking the "myth" of the Aryan invasion in support for an "out of India" theory.


I had a brilliant physicist uncle (the last person I would have expected) who went the same way as your prof towards the end of his life. He grew roses and delved into the wisdom of the Hindu sages. But from all he said, I always got the feeling that it was he who was interpreting things according his own knowledge of modern physics, rather than what was really present in the texts.

Vedic era computational math probably has more to it than Vedic physics and astrophysics as some claim. But I looked at some of the tutorials. I don't see a whole lot of advantage over conventional computation using multiplication tables - especially in the multiplication of larger numbers.

24 ways to make love standing up? Wow! I know you made that up. But if you have the time, do plow through the pages of the ancient texts. The manual may well be buried somewhere in there. Now that would be a tutorial worth setting up. You could give up your day job without the slightest trepidation.


"Vedic era computational math probably has more to it than Vedic physics and astrophysics as some claim..."

Indeed. And Vedic math is probably not more advantageous on the whole (I'm not certain). I see it as an alternate way to bell the cat, a veritable intellectual feat for its time -- something to make all humans proud.

About the 24 ways, the Gupta-era folks saw the market opportunity long ago. Vatsayana documented many and the sculptors illustrated them on http://www.shunya.net/Pictures/NorthIndia/Khajuraho/Khajuraho.htm>temple exteriors -- I'm 1,500 years too late (I can only take pictures of their tutorials!). Of course, most Indians today are so scandalized by any insinuation of bawdiness in the ancient texts (or on temple walls) that I better keep my day job for now. :-)


Vatsayana I know. 24 ways, I do not. And did you ever get the feeling that Dr. V. had a good laugh after prescribing numerous impossible contortions of carnal delight which he surely knew would leave the average mortal either with severely pulled muscles or a dreadfully red face?

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