Third Wave West 1: Fertile Crescent Of Mathematics
After the first two waves out of India to Eurasia and Europe, there was a third wave roughly 5000 years ago that carried mathematical ideas to West Asia and Egypt, eventually reaching Greece. This westward movement is recorded both in archaeology and in the mathematical literature of the period.
First of two parts
Pythagoras and Vedic Mathematics
As every schoolchild knows, or should know, the most important theorem in geometry is the Theorem of Pythagoras. Though taught as a theorem in geometry, it is equally a theorem in arithmetic (and algebra). In geometry it is stated in terms of the relationship betweenof the lengths of the hypotenuse and of the other two sides of a right angled triangle (or right triangle). Algebraically, it is expressed as integer solutions for a, b and c in the equation:
a2 + b2 = c2
Integral solutions of this equation are known as Pythagorean triples. Examples of such triples are (3,4,5), (6,8,10), (5,12,13), (12,35,37), (36,77,85), (65,72,97)… It is easy to show that there exist infinitely many such solutions or Pythagorean triples. The equation a2 + b2 = c2 is a special case for n = 2 of the Diophantine equation an + bn = cn. To digress for a moment, the great French mathematician Pierre Fermat in the 1637 stated that though it has infinitely many solutions for n = 2 (Pythagorean triples), for n larger than 2 (like 3, 4, 5, …) it has no solutions. Its proof had defied the efforts of the best mathematicians for over 300 years until it was proved by Andrew Wiles only in 1994.
Even though it is famous as the Pythagorean Theorem, Pythagoras had nothing to do with it. He was less a mathematician than a spiritual master who tried to give special significance to numbers, or what we would now call a. He did, however, found a school that was interested in mathematics. But because of the spirit of the age, Europeans of the late nineteenth and early twentieth century tried to derive everything from what they called Greek Wisdom. Thus was born the myth of Greece as the source of all knowledge. (This still continues through of the late David Pingree and his Kim Plofker who try to trace Indian astronomy and mathematics to Greece, brought to India during Alexander’s invasion.) A typical belief in this can be like what Edith Hamilton wrote in her book The Greek Way:
“Something had arisen in the minds and spirits of the men there… Athens had entered upon her brief and magnificent flowering of genius which so molded the world of the mind and of the spirit that our own mind and spirit are different. …We think and feel differently today because of what a little Greek town did for a century or two, twenty four centuries ago. …In that black and fierce world a little center of white hot spiritual energy was at work.”
This fantastic passage helps one to understand why European scholars tried to trace all knowledge to Greece. To them the whole world was engulfed in a Dark Age, as Europe was until the rediscovery of Greek knowledge. The Aryan invasion, Alexander’s invasion and the like brought all knowledge and civilization to ancient India in accordance with this fantasy. But historians of science, more objective than of poetry punctured this fantasy balloon.
In 1928, the great historian of mathematics Otto Neugebauer discovered that some Pythagorean triples were known to the Babylonians by 1700 BCE or more than a thousand years before Pythagoras. Since they could not have borrowed them from the Greeks of the future, Neugebauer concluded, rather hastily as it turned out, the Greeks must have borrowed the Pythagorean Theorem from the Babylonians. He wrote in 1937: “What was called Pythagorean had better be called Babylonian.”
Indian and Pythagorean geometry: Datta and Seidenberg
Neugebauer’s statement about the indebtedness of Greek geometry to Babylon seemed to settle the problem for good. But the mystery was only beginning: mathematicians soon saw that Greek geometry contains algebraic ideas that could not have come from Babylonian mathematics which is purely arithmetical in character. Babylonians had no Pythagorean Theorem but only some Pythagorean triples that could well have been obtained by trial and error. In short they had no method. To understand this it helps to have an idea of how sciences evolve.
Exact sciences like physics and mathematics have a built in logical sequence that cannot be violated. For example, Einstein’s Relativity Theory could come only after Newtonian Mechanics and Maxwell’s electromagnetic theory. In like manner, calculus could not have come before algebra was firmly established. So it did not take mathematicians long to see that Greek geometry could not have derived from the relatively primitive Babylonian arithmetic that dealt only with numbers.
One of the first to notice this was the eminent American mathematician and historian of science Abraham Seidenberg (1916 – 88), professor of mathematics at the, Berkley. He found that Greek and Babylonian mathematics were totally different in spirit and fundamentally incompatible. After much study he found that in order to understand the origin and evolution of Greek mathematics it was necessary to go to ancient India. After more than fifteen years of that culminated in the monumental 1978 he called The Origin of Mathematics (1972), Seidenberg observed:
“…if one includes the Vedic mathematics, one will get quite a different perspective on ancient mathematics. The Sulvasutras [Vedic mathematics] have geometric algebra. The issue is geometric algebra. …Greece and India have a common heritage [in geometric algebra] that cannot have derived from Old-Babylonia, i.e., Old-Babylonia of about 1700 BCE.”
This means Babylonian mathematics had no geometric algebra and could not therefore be the source. (Old-Babylonia that Seidenberg refers to is the Mesopotamian kingdom Hammurabi and not the Neo-Babylonian Empire of Nebuchadnezzar mentioned in the Old Testament).
Before we proceed further, we need to be very particular about the reference to the ‘Vedic mathematics’ by Seidenberg. THIS IS NOT THE SAME AS IN THE BOOK CALLED VEDIC MATHEMATICS attributed to Swami Bharati Krishna Tirtha. Swamiji’s book is a modern work is used as a pedagogical tool in some schools. Its contents arein the Vedic literature. By Vedic mathematics we mean the ancientSulbasutras.
This point is important because many people are under the mistaken impression that Swamiji’s book is based on the Vedas. To reiterate, the term ‘Vedic mathematics’ here refers to ancient mathematical texts known as the Sulba-sutras (or Sulvasutras as Seidenberg calls them) and to them alone. They are part of the later Vedic literature known as the Sutra literature. We shall see later that they are more or less contemporary with the Harappan civilization. So we may also call them ‘Harappan mathematics’.
The Sutra literature consists of codified knowledge—both sacred and secular— compiled from the Vedic literature and expressed in the form of mnemonic strings or sutras as we do in the computer science today. The two best known Sutra works are the Yogasutra of Patanjali, and Panini’s famous grammar (actually descriptive linguistics) Ashtadhyayi. The Sulbasutras are mathematical appendixes to what are known as Kalpasutras or ritual texts. Though Seidenberg made his comparative study during 1962 – 78, much of his work had been anticipated by the Indian mathematician Bhibhutibhushan Datta nearly thirty years before. (Seidenberg recognized this in his 1978 paper.) Here is how Datta, the greatest modern authority on the Sulbasutras describes them:
“The Sulbas, or as they are commonly known at present amongst oriental scholars, the Sulba-sutras, are manuals for the construction of altars which are necessary in connection with the sacrifices [and other rituals] of the Vedic Hindus. …It was primarily in connection with the construction of sacrificial altars of proper shape and size that the problems of geometry and also of arithmetic and algebra presented themselves and were studied in ancient India…”
So like music, both Indian and Western, mathematics too had religious roots in Vedic ritual. At least eightSulbasutras are known of which the Sulbasutra of Baudhayana is the oldest and the most important. When they became known in the West, to the utter amazement of historians of mathematics, it was found that Baudhayana gives the statement as well as a proof of the so-called Pythagorean Theorem centuries before Pythagoras!
Many textbooks now refer to the theorem as the Baudhayana-Pythagoras theorem. This is a dubious designation since nowhere in the known works of Pythagoras do we find the theorem named after him while we find both the statement (actually several cases) as well as constructions and demonstrations in the Baudhayana Sulbasutra. It is one of the quirks of history.
Baudhayana: Egypt and Old-Babylonia
The next question is— when did Baudhayana live? Western scholars, resting on 1500 BCE for the Aryan invasion and 1200 BCE for the Rig Veda (which in turn assumes the Biblical date of 4004 BCE for the Creation) place Baudhayana in 800 BCE. Since these are no longer acceptable, with the Rig Veda date being pushed back to the early fourth millennium, Baudhayana’s date too calls for a revision. Here, a clue is provided by Baudhayana himself in the following formula for the square root of 2, correct to the fifth decimal place (in modern notation):
These justly famous approximations known as unit fractions are found in Egyptian and Babylonian sources. They go back to before 1700 BCE in Old-Babylonia and before the Egyptian Middle Kingdom (c. 2000 – 1800 BCE). European scholars of the time were not happy with these developments. George Thibaut’s translation of theBaudhayana Sulbasutra (with Pandit Sudhakar Dwivedi) had shown that Greek mathematics had borrowed from India. The eminent German historian of science Moritz Cantor confessed that “he was not enchanted with the idea that in geometry, Pythagoreans were pupils of Indians.” He seems later to have dropped this irrational objection and accepted the evidence. Soon other evidence cropped up: Datta showed that Indian terms are found in the works of the Greek Democritus (c. 440 BCE). An instrument known as the gnomon also links Pythagorean geometry to India.
This was followed the unit fraction and other mathematical results in Babylonian and Egyptian mathematics found in the Sulbasutras. At first Cantor suggested that Indians must have borrowed from the Egyptians. (Many years later, David Pingree and others kept insisting that Indians must have got their astronomy from the Greeks after Alexander. There is no record of any Egyptian Pharaoh invading India. Also, did Alexander bring soldiers or astronomers who within a few months managed to teach astronomy to Indians who didn’t know any Greek?)
Mathematically this again proved impossible: just as it is impossible to derive Greek geometry from Babylonian arithmetic, it is not possible to derive Indian geometry, especially geometric algebra from Old-Babylonian or Egyptian mathematics. Note that in the Sulbasutras, as in Greek mathematics later, algebra and geometry are inseparable. Further, where did unit fractions come from? The sources are nowhere to be found in Egyptian or Old-Babylonian records. They are found only in the Sulbasutras. SoSulbasutras must be older.
And there is more evidence. The approximation for the circumference-to-diameter ratio π = 3.16049 (in modern notation) used by Ahmes of Egypt (c. 1600 BCE) is exactly the same as that given in the Manava Sulbasutra— a relatively late work. And what is more, it is obtained in exactly the same fashion as π = 3.16049 = 4 × (8/9)2. There is still more evidence in the form of ‘Egyptian’ geometric figures found in theSulba texts but this will do for present.
Because of their closeness to Vedic ritual, Seidenberg (and others) have referred to the Sulbasutras as ritual mathematics. In fact, his first paper on the subject was called The ritual origin of geometry (1962). All this means there was a westward movement of mathematical ideas from India to Greece long before Pythagoras (6th century BCE) and also from India to Old-Babylonia and Egypt before 2000 BCE. In the words of Seidenberg:
“The elements of geometry found in Egypt and Old-Babylonia stem from a ritual system of the kind found in the Sulbasutras.”
Since the Egyptian Middle Kingdom existed before 2000 BCE, this takes the Sulbasutras, and Baudhayana, the earliest of them to dates long before 2000 BCE. This also means that the Sulbasutras,and the Sutra literature in general must have been more or less contemporary with the Indus Valley (or the Harappan) civilization of c. 3100 – 1900 BCE.
Vedic mathematics was Harappan mathematics
This means that the mathematics of the Sulbasutras (or real ‘Vedic mathematics’) existed and was available to the builders of the Harappan civilization. This helps solve one of the long standing mysteries that had puzzled mathematicians, engineers and architects: where did the Harappans get the scientific, in particular the mathematical knowledge needed to design and build their highly sophisticated buildings, harbors, streets and water and drainage systems?
Let us take a closer look at this problem, for it lies at the heart of the application of Sulbasutras to practical needs like town planning. Today we use blueprints to design houses and other structures as well as streets and housing communities. This requires knowledge of geometry, especially if one wants a well planned township or a well designed structure. The distinguishing mark of the Harappan civilization is town planning based on geometric design. This is clear from the pictures given. They have no monumental buildings like the Egyptian Pyramids, but do have planned streets, dwellings, water storage and drainages.
All this requires knowledge of geometry. Lothal has dockyards, making it the world’s oldest scientifically designed harbor. No less remarkable is the fact they used standardized burnt bricks of the same dimensions throughout— from Harappa and Mohenjo-Daro in Punjab and Sind to Kalibangan in Rajasthan to Lothal and Dholavira in Gujarat. All this requires knowledge of mathematics— both of geometry and the capacity to do detailed calculations.
We now know that the Sulbasutras or Vedic mathematics existed by the time the cities and harbors of the Harappan civilization were being built. In fact they could not have been built without such mathematics. Their origin may have been in religion and ritual but they soon found secular applications in architecture and town planning. This means Vedic mathematics was also Harappan mathematics.
Then, this mathematics it made its way west to Mesopotamia, Egypt and eventually reaching Pythagorean Greece centuries before Pythagoras. Once there it received further refinement in the form of axiomatic treatment of geometry by Euclid and his successors. Mathematical proofs were known to Indians but the Greeks put it on a systematic foundation based on postulates or axioms. Thus was modern mathematics born.
In summary, the so-called Fertile Crescent may or may not have been the source of agriculture (it was not) but it was certainly a fertile ground for the transmission of mathematical ideas.
(To be continued)
Dr. N.S. Rajaram is a mathematical scientist interested in history and philosophy of science. He is currently working on the book The Genes of Science and the Birth of History on which the article is partly based. He is Contributing Editor of FOLKS.