Nothing Vedic about ‘Vedic Mathematics’?

Planned city of Mohenjo-Daro

Claims and conflicts

A little more than 40 years ago there appeared a tract entitled Vedic Mathematics attributed to Swami Bharati Krishna Tirtha, the late Pontiff (Shankaracharya) of the Govardhana Peetha of Puri in Orissa. It gave 16 statements in Sanskrit written in the form of sutras (brief assertions or formulas) that were shown to be effective in solving mathematical problems. The book contains numerous computational devices and short cuts reminiscent of but not necessarily equivalent to the well-known Trachtenberg method.

The problems addressed ranged in difficulty from the elementary like the extraction of roots to intermediate at the level of college freshman mathematics; none of them could be considered advanced. At the same time it was clear that the author possessed considerable mathematical skill as well as being an accomplished Vedic scholar. While some of the topics treated are ancient, others including partial fractions and calculus are modern topics that were not known in Vedic times.

The book claimed that the author, the late Swamiji, discovered these sutras in his study of the Atharvaveda. In examining these claims it is important to be aware of the fact that the book Vedic Mathematics is a posthumous publication compiled by his devotees; no manuscript in his own hand has been found. The Atharvaveda is extant and its currently available versions do not contain these sutras. The Swamiji lived during the 1880 – 1964 period, which means he used the same Vedic texts as those available to us today. The Atharvaveda being very ancient, it was available also to ancient Indians including mathematicians.

This raises a question: how did the Swamiji discover the mathematics in these ancient texts (the Vedas) that others have been unable to locate. Indian mathematicians also, going back thousands of years had no inkling of this ‘Vedic Mathematics’ supposedly found in the same texts. The Swamiji’s devotees, however, claim that he was gifted with a divine vision that allowed him discover mathematical truths not apparent to others. This is a matter of faith and cannot be accepted on objective grounds.

(This does not deny the fact that there have been mathematical geniuses like Srinivasa Ramanujam, Leonhardt Euler and Karl Friedrich Gauss and others who saw results beyond the ordinary. But the source of their discoveries was their own genius, not some ancient texts like the Vedas that have been around for thousands of years.)

Fortunately, we have available ample materials from the Vedic times to contrast these claims against what was definitely known in those days. We next take a brief look at a few of these.

Bakshali

Sources

Numerous mathematical terms are found in the Vedas. In the Rigveda (II. 18. 4-6), we find the sequences 2, 4, 6, 8, as well as 20, 30, 40, 50, 60, 70, 80, 90, 100. This suggests that both binary and decimal difference sequences were known. A limited decimal system was also known, consisting of multiples of ten: terms like dasha (10), shata (100), sahasra (1000) and ayuta (10,000) are common.

The very first verse of the Atharvaveda, the claimed source of ‘Vedic Mathematics’, refers to trisapta or different types of combinations of sevens and threes: 3 + 7 = 10; 3 X 7 = 21; and 3 + 5 + 7 = 15. There are references in the Vedas indicating knowledge of geometry also. The spoke-wheel is described, and the Rigveda (I.164. 11-15) shows knowledge of the circle and the radius. All these are quite elementary compared to the examples given in ‘Vedic Mathematics’.

In addition to sacred works like the Vedas, the Brahmanas and the Upanishads, the Vedic literature includes secular works known as the Sutras relating to fields like grammar, etymology, law and mathematics. The Sulbasutras (or the Sulbas) are mathematical texts that describe rules and results of mathematics in the context of then familiar objects like Vedic altars. For this reason, some authorities regard them as ‘religious’ or ‘ritual’ mathematics. This is an oversimplification, for the Sulbas contain a good deal of mathematics that has nothing to do with religion or ritual.

This is now supplemented by the discovery of mathematical terms from the Sulbas found on some Harappan seals deciphered by N. Jha and the present author. This shows that ancient architecture was based on the Sulbas, which happens also to be the traditional Indian view. We have examples of this in Harappan archaeological remains in northwestern India and Pakistan. It is quite clear that Harappans used the mathematics found in the Sulbas in their architecture and town planning. (See figure.)

The Sulbas are best seen as texts on geometric algebra: a problem is stated in geometrical terms, but its solution combines geometry and algebra. The best-known elementary example of it is probably the Pythagorean Theorem, which states that the square on the hypotenuse of a right-angled triangle is equal in area to the sum of the squares on the other two sides. The Sulbas state this theorem in its equivalent rectangular form— in terms of the square on the diagonal and those on the other two sides. An elementary proof is also given.

A particularly interesting example found on the seals is Ö40 = Ö(6² + 2²)— an application of the algebraic version of the Pythagorean Theorem. This is found also in the Manava Sulba (Sulba of Manu), again indicating close connections between the Harappan Civilization and the Vedic literature. Another problem that the Sulbas treat in detail is the conversion of geometric figures from one form into another while preserving the area— like circling a square (approximate). (Pythagorean Theorem is a misnomer; the Sulba author Baudhayana stated it at least 1500 years earlier.)

It leads to important results like determining square roots of numbers and an approximate value for p — the ratio of the circumference to the diameter of a circle. The Manava Sulba gives an approximation that works out to 3.16, with an error in the second decimal place. A similar approximation derived as Ö 10 is found on one of the Harappan seals deciphered by Jha and the author. This apparently was borrowed by Egyptians in the time of Ahmes of Egypt (c. 1550 BC), several centuries after they were written on the seals.

Among the more interesting examples found in the Sulbas of Baudhayana and Apastamba are approximations of the kind:

Ö 2 = 1 + 1/3 + 1/(3.4) – 1/(3.4.34)

Vedic Mathematic

These are related to ‘unit fraction’ approximations found in the records of the Egyptian Middle Kingdom (2050 – 1800 BC) and the Old Babylonian Empire (1900 – 1750 BC). Expressions like dvidha (1/2), tridha (1/3), dashadha (1/10) are found also on some Harappan seals. As the American mathematician A. Siedenberg showed, the mathematics of Egypt and Old Babylon derive from the Sulbasutras. In his words: “The elements of geometry found in Egypt and Babylonia derive from a ritual [mathematical] system of the kind found in the Sulbasutras.”

The discovery of Sulba expressions on Harappan seals by Jha and this author supports Seidenbrg’s conclusion: it shows that trade exchanges between the Harappan Empire and Mesopotamia and Egypt were accompanied by intellectual exchange, especially in mathematics. The same happened three thousand years later when trade and diplomatic exchanges between India and the states of West Asia led to the Arabs borrowing Indian numerals, the decimal system and algebra, all of which eventually made their way to Europe in the 13^th century. The first European to use the decimal system was Leonardo of Pisa, better known as Fibonacci.

Zero and the decimal system

When this is the case, with a good deal of mathematical activity in ancient times, quite remarkable for the age, how are we to explain the complete absence of any ‘Vedic Mathematics’ in explicitly mathematical texts like the Sulbas? After all the Atharvaveda, the claimed source of ‘Vedic Mathematics’ is more ancient than the Sulbasutras and was available to the Sulba authors. Were they so dense as to not see its value?

This is not the only problem. The use of the zero in the number system, arguably India’s (and the World’s) greatest mathematical invention, is not found in the Sulbas. The zero and its use in the modern sense appear for the first time in the Bakshali Manuscript, whose contents may be dated to near the beginning of the Christian era. (It was discovered in 1881 at Bakshali near Peshawar in Pakistan.) Modern research suggests that the Sulbas may date back to before 2000 BC.

This is only the beginning: the next thousand years and more, from the time of the Bakshali Manuscript to Bhaskara II (1114 – 1185) was extremely rich in mathematical discoveries in India, no doubt due to the recently perfected decimal system of computation. (We saw a similar phenomenon after the invention of the computer.) It produced such great mathematicians as Aryabhata, Varahamihira Brahmagupta, Mahavira, Bhaskara and a host of others. Yet none of them is aware of any of the topics found in Bharati Krishna Tirtha’s Vedic Mathematics. They solve problems the hard way, by forging their methods from fundamentals.

All this leaves us in an extraordinary situation. Even ignoring the period from the Sulbas to the Bakshali Manuscript, for two thousand years, from time of the Bakshali Manuscript to the present, a period that saw great advances in Indian mathematics, there is no trace of this ‘Vedic Mathematics’; it appears out of the blue in 1965, supposedly extracted from the ancient Vedic texts that were available to all these people going back to the composers of the Sulbasutras.

Were these great mathematicians from Baudhayana to Bhaskaracharya to our own time so ignorant or so insensitive to the mathematical treasures in the Vedic texts that they spent thousands of years reinventing the mathematical wheels that were already there right before them? The only answer seems to be that this ‘Vedic Mathematics’ is a modern work by a mathematically gifted author (or authors) who happened also to be a Vedic scholar familiar with the style of the ancients. So ‘Vedic Mathematics’ is modern mathematics in ancient style and language.

One can understand the enthusiasm for ‘Vedic Mathematics’ in India, which suffered a long period of European domination. We know that colonial scholars created a highly distorted version of history downgrading Indian achievements, even to the extent of claiming that the Sanskrit language and the Vedas were brought by Eurasian invaders. Similarly they created a myth called the Greek Miracle by which they attributed non-European, particularly Indian achievements to borrowings from the Greeks. (Incidentally, the ancient Greeks didn’t see themselves as Europeans but as Mediterranean people like the Egyptians and the Persians. They regarded Europeans as barbarians,)

The counter to this Eurocentric distortion is not to create another myth like Vedic Mathematics, but a rigorous program of research that brings out the truth about the past. India has made an immense contribution to science, especially mathematics, but it needs to be brought out through research. Fortunately this is happening but more needs to be done. As far as Vedic Mathematics is concerned, there is nothing Vedic about it.