Abraham Seidenberg

"However, Seidenberg was told by the Indologists that these Sutras, or any Vedic text for that matter, were definitely written later than 1700 BC. But mathematical data cannot be manipulated just like that, and Seidenberg remained convinced of his case: “Whatever the difficulty there may be [concerning chronology], it is small in comparison with the difficulty of deriving the Vedic ritual application of the theorem from Babylonia. (The reverse derivation is easy)… the application involves geometric algebra, and there is no evidence of geometric algebra from Babylonia. And the geometry of Babylonia is already secondary whereas in India it is primary.” [To satisfy the indologists, he said that the Shulba Sutra had conserved an older tradition, and that it is from this one that the Babylonians had learned their mathematics:] “Hence we do not hesitate to place the Vedic (…) rituals, or more exactly, rituals exactly like them, far back of 1700 BC. (…) elements of geometry found in Egypt and Babylonia stem from a ritual system of the kind described in the Sulvasutras.”"

"By examining the evidence in the Shatapatha Brahmana, we now know that Indian geometry predates Greek geometry by centuries. For example, the earth was represented by a circular altar and the heavens were represented by a square altar and the ritual consisted of converting the circle into a square of an identical area. There we see the beginnings of geometry! Two aspects of the 'Pythagoras' theorem are described in the Vedic literature. One aspect is purely algebraic that presents numbers a, b, c for which the sum of the squares of the first two equals the square of the third. The second is the geometric, according to which the sum of the areas of two square areas of different size is equal to another square. The Babylonians knew the algebraic aspect of this theorem as early as 1700 BCE, but they did not seem to know the geometric aspect. The Shatapatha Brahmana, which precedes the age of Pythagoras, knows both aspects. Therefore, the Indians could not have learnt it from the Old-Babylonians or the Greeks, who claim to have rediscovered the result only with Pythagoras. India is thus the cradle of the knowledge of geometry and mathematics."

"Its mathematics was very much like what we see in the Sulvasutras [szulbasu utras]. In the first place, it was associated with ritual. Second, there was no dichotomy between number and magnitude … In geometry it knew the Theorem of Pythagoras and how to convert a rectangle into a square. It knew the isosceles trapezoid and how to compute its area … [and] some number theory centered on the existence of Pythagorean triplets … [and how] to compute a square root. …The arithmetical tendencies here encountered [ie in the SZulbasuutras] were expanded and in connection with observations on the rectangle led to Babylonian mathematics. A contrary tendency, namely, a concern for exactness of thought … together with a recognition that arithmetic methods are not exact, led to Pythagorean mathematics."

"‘Greece and India have a common heritage that cannot have derived from Old-Babylonia, i.e., the Old-Babylonia of about 1700 B.C.’ (Seidenberg 1978)"

"‘A comparison of Pythagorean and Vedic mathematics together with some chronological consideration showed that the current view [of Greek influence on Vedic thought] is not tenable. A common source for the Pythagorean and Vedic mathematics is to be sought either in the Vedic mathematics [i.e. the Sulbasutras] or in an older mathematics very much like it.’ (Seidenberg 1978)"

"I think its mathematics [of the common source] was very much like what we see in the Sulbasutras. (Seidenberg 1978)"

"I cannot help thinking that it shows battle weariness rather than a considered opinion."

"Yet long before 1937 people had suggested a non-Greek origin to Greek mathematics; and long before 1943 people had pointed out that sacred books of the East contain the ‘Pythagorean numbers’ [and indeed the ‘theorem’]. Such numbers are mentioned in the Sulvasutras; ancient Indian works on altar constructions. […] Neugebauer does not mention the Sulvasutras in his book Vorlesungen über Geschichte der Antiken Mathematischen Wissenschaften, nor does B. L. van der Waerden them in his book Science Awakening. Why this omission?"

"[…] nor did he [Thibaut] formulate the obvious conclusion, namely, that the Greeks were not the inventors of plane geometry, rather it was the Indians. At least this was the message that the Greek scholars saw in Thibaut’s paper. And they didn’t like it."

"If the Indians invented plane geometry, what was to become of Greek ‘genius’ or of the Greek ‘miracle’?"

"‘a number of points in Greek geometry are illuminated by the hypothesis that it started from a tradition of peg-and-cord constructions of the kind we find with the Vedic ritualists’ [45]."

"As N.S. Rajaram has rightly observed, Seidenberg traces Babylonian mathematics and astronomy to Indian models. He suggests the Kassite dynasty (18th-16th century) as the channel of transmission, as the Kassite language has an Indo-Aryan substrate. This is eminently reasonable. Thus, Babylonian astronomy divided the ecliptic in 18, yet by the first millennium it had adopted a division in 12, the same as existed in Vedic culture, where a nightly division into 28 lunar houses was complemented by a daily division of the ecliptic in 12 half-seasons (Madhu, Madhava etc.), and where the rishi Dirghatamas introduced the first-ever division of the circle into 12 and 360. Till today, the division into 360 is explained in textbooks as a Babylonian invention, but the earliest mention is Indian."

"Seidenberg (1983), a historian of science, described the reaction to Thibaut's claim: "Thibaut himself never belabored or elaborated these views, nor did he formulate the obvious conclusion, namely, that it was not the Greeks who invented plane geometry, it was the Indians. At least this was the message that the Greek scholars saw in Thibaut's paper. And they didn't like it" (103). Eventually, Thibaut was pressured into proposing a date for the sutras that would disarm the Greek scholars, but he would not forgo the possibility that the two different peoples could at least have developed the same knowledge independently. The date he offered was the fourth or third century B.C.E. Seidenberg (1978) comments: "A terrible statement! I cannot help thinking that it shows battle- weariness rather than a considered opinion. . . . Anyway, the damage had been done and the Sulvasutras have never taken the position in the history of mathematics that they deserve" (306)"

"Seidenberg (1983), "regard[s] it as certain that knowledge of Pythagoras' Theorem was known to the Satapatha Brahmana, which mentions calculations connected with the purusa bird altar, and to the Taittiriya Samhita, which showed similar geometrical awareness" (106). Since these texts are generally dated to around 1000-800 B.C.E., "Greek geometry did not somehow make its way into Vedic geometry, as Greek geometry is only supposed to have started about 600 B.C." (108). Scholars no longer consider Vedic geometry to have been borrowed from the Greeks, so Seidenberg's more controversial claim in the modern context is his rejection of the possibility that the algebra either of the Indians or the Greeks was derived from Babylonia, since the former were aware of aspects of the theorem to which the Babylonians make no reference. He also rejects the possibility that these aspects could have been discovered by the Indians after receiving the basic theorem from Babylonia and then transforming it, and concludes that either "Old Babylonia got the theorem of Pythagoras from India or that Old Babylonia and India got it from a third source" (Seidenberg 1983, 121)."

"Seidenberg's next steps involve a series of assumptions that need to be laid out. He first states that since the Babylonian sources are dated to 1700 B.C.E., the mathematical knowledge in the Sulvasutras must predate that. His next statement is significant: "Now the Sanskrit scholars do not give a date for the geometric rituals in question as early as 1700 B.C. Therefore I postulate a pre-Babylonian (i.e., pre-1700 B.C.) source for the kind of geometric rituals we see preserved in the Sulvasutras, or at least for the mathematics involved in these rituals" (1983, 121). Seidenberg assigns a date of 2200 B.C.E. for the common source, but upon being told by his Indological colleagues that the Aryans were not even in India in 1700 B.C.E. (1978, 324), let alone 2200 B.C.E., he is forced to postulate that the Aryans and Greeks inherited their mathematics from the joint Indo- European period, and the Indo-Aryans brought the knowledge into the subcontinent with them."