Did Indian learn trigonometry from Greeks? Responses to the Aryan race conjecture in the African context, and the relevance to Indology

Recently, I presented my talk on “Pre-colonial appropriations of Indian ganita: epistemic issues”. This was at a round table at IIAS Shimla which replaced the now-postponed conference on Indology. My talk was primarily about the inferior math we teach in school today based on the European misunderstanding of the Indian ganita which Europe imported.

Shimla Indology lecture

But as a sidelight, I took up a novel aspect of the Aryan race conjecture. Indologists have so far talked about the Aryan conjecture solely in the Indian context. However, I pointed out the need to link this discussion also to the Aryan race model as it applies to the African context. In particular, to the issue of the Aryan model vs Ancient model as in Martin Bernal’s Black Athena, vol. 1: The fabrication of ancient Greece 1785-1985. (The date of 1785 alludes to William Jones whose philological researches started these wild speculations on race.)

The fabrication of ancient Greece has a direct bearing on the history of Indian math. But first let us understand how racists did it.

Racist history

Bernal’s key point was that after 1785 racist historians systematically rewrote history to appropriate all achievements of Black Egyptians to White Greeks. This aligned with George James’ Stolen Legacy: Greek philosophy is stolen Egyptian philosophy. But instead of philosophy, Bernal applied it, for example, to architecture where the evidence of Greeks copying Egyptians is not easily contested: the so-called Greek architecture of columns is manifestly copied from Egypt and Iran (Persepolis).

Bernal made only scattered remarks on math and science, perhaps out of deference to his father J. D. Bernal, who wrote his famous (but now hopelessly dated) volumes on the history of science. However, after going through my PHISPC volume Cultural Foundations of Mathematics, Bernal (Jr) strongly encouraged me to look at the related issues of concern to the history of math where undue credit has been given to Greeks (as explained in an earlier blog “Greediots and Pythagoras”, which also provides the relevant background to this post).

One point in my above book relates closely to Afrocentrist concerns about undue credit to Greeks in the history of math.

Thus, my point (later summarised e.g. in Is Science Western in Origin?) was that the church falsified history even before racist historians. This process of falsifying history went virulent during the Crusades against Muslims. (Bernal agreed with me here.) The Toledo mass translations of Arabic texts into Latin, beginning 1125, involved learning from the books of the religious enemy. The church, which had earlier consistently burnt heretical books, needed to justify learning from the books of the religious enemy. It provided this justification through the coarse falsehood that all scientific knowledge in Arabic books came from the sole “friends of Christians”, the early Greeks. As such, it claimed that knowledge in Arabic books as a Christian inheritance: and that Arabs contributed nothing to it. Later racist historians modified the church thesis by insisting that the authors of Greek books, even in Africa, were white-skinned, hence claimed it as part of White achievements. The racist historian Florian Cajori is an example of how religious chauvinism was absorbed into racist chauvinism. No evidence exists, and none was needed!

Egyptian and Persian texts were translated into Greek, by Alexander and the Ptolemy dynasty, but any material coming from these texts was all attributed by racist historians to Greeks. Western historians against Afrocentrism, such as Lefkowitz, falsely state that there is no evidence for such translation. As I pointed out in my UNISA lectures, Zoroastrians have been complaining about the burning and Greek translation of their texts for over 2000 years. Western historians rightly assume that their parochial readers would be unfamiliar with those texts. Obviously, also, for the Greediotic brain it is equally easy to imagine (when required) that skin color relates to the language of the text: thus, any Indian author writing in English, such as this one, must be white-skinned! There are no early original Greek sources available, but even if they were a claim of any Greek originality (e.g. on Sphere and Cylinder, attributed to Archimedes), would need proof, since this is also found in the Ahmes papyrus from a thousand years earlier, as pointed out by Diop. Lefkowitz has only some utterly foolish comments to offer claiming that Archimedes compared the area of a cylinder to the volume of a sphere. That is the typical standard of racist historians.

Relevance to Indology

Anyway, the fact is (1) that the Abbasid khilafat in Baghdad made huge investments in knowledge (e.g. Bayt al Hikma), so that, following the knowledge gradient, numerous Arabic texts were translated FROM Arabic into Byzantine Greek (then Constantinople was a tributary of Baghdad). The fact also is that (2) much Indian knowledge travelled to Baghdad, as is well known and as repeated and explained during my talk (e.g. al Khwarizmi’s Hisab al Hind). As stated in the abstract, a striking example of both (1) and (2) is the case of the Panchatantra which was translated from Sanskrit to Farsi to Arabic and then to Byzantine Greek to other European languages as Aesop’s fables. Knowledge of Indian math could similarly have got into late Arabic and Byzantine Greek texts.

So, the question that arises, and was raised in Cultural Foundations of Mathematics, was this: could Indian knowledge have been mis-attributed to Greeks in the process of appropriating Arabic texts to Greeks? Specifically, on the strength of this appropriation, people like Pingree and his students have been clamouring that trigonometry was transmitted from Greeks (“Ptolemy”) to Indians. My question challenged this claim (and Pingree ducked the challenge in 2004 when, on a trip to the US, I directly challenged him to publicly debate the claim).

My counter-points to that claim are the following.

(0) Non-existence of primary Greek sources. There are no primary sources for claims about Greek achievement in math. (This was admitted by the famous historian of Greek math, David Fowler: “We possess no original versions of any Greek mathematical text, and most texts survive only in the form of Byzantine minuscule from the mid-ninth century AD onwards”. That is all the primary sources for Greek math are from another land, in another language, and from another time, thousand years or more later. For the source of the quote, see my lecture 3 on “Not out of Greece” at the University of South Africa, posted online at http://ckraju.net/unisa.)

That is, the earliest available sources for Greek math date to at least a century AFTER the known arrival of Indian math among Arabs, and some are from many centuries later as in the case of “Ptolemy’s” Almagest etc. As such, it is easily possible that Indian math and astronomy went into Arabic or Byzantine Greek manuscripts, and some of it was later indiscriminately attributed with Crusading and racist fervour by credulous or dishonest Western historians to the Greeks.

(1) Epistemic test. Then, of course, there is the epistemic test that (leave alone early Greeks) even later-day Europeans did not understand trigonometry properly as the very term “trigonometry” shows. The Jesuit general Clavius did publish in 1607 in his own name trigonometric values stolen from India. Though these values had the same ten decimal place precision, as found in India, neither Clavius nor any other European was then able to use them to correctly calculate the size of the earth (as the students of my history and philosophy of science course do, and as explained in my Class IX school text on Rajju Ganita). As explained in my talk, it was because of its mathematical backwardness that Europe hence had a navigational problem with longitude, which persisted until (at least) the mid-18th c. (The relation of earth size to longitude determination was already mentioned in my abstract by quoting Brahmagupta’s statement that ignorance of earth’s radius makes longitude calculations futile. The matter is discussed at length in Cultural Foundations of Mathematics. That is, the epistemic test and the European longitude problem show the persistent European lack of understanding of trigonometry until the 17th c.

But dishonest historians like Pingree (and his students) keep insisting that trigonometry was transmitted to India from the early Greeks. They want people to believe the quaint story that early Greeks had this knowledge which then suddenly vanished from Europe exactly like fairy godmother appears and disappears! This is an obvious case of anachronistic attribution of knowledge to early Greeks.

In my presentation, I also quickly made several other points, related to my general theory that there has been a whole lot of fraud in the Western history of science, so that there is an urgent need to correct and decolonise the history and philosophy of science (as I have been trying to do).

(a) Non-textual evidence: numerals. Because textual evidence in the early Greek case is not at all reliable (see below) we need to examine the non-textual evidence. Greeks and Romans were backward even in arithmetic, and lacked knowledge of fractions, and therefore could not have done any trigonometry. There are other pointers to Greek and Roman arithmetic backwardness: e.g. the largest number they had was the myriad (10000) pitifully small compared to the number of 1053
named as tallakshana by the Buddha. (No name for it even today in English.)

(b) Non-textual evidence: calendar. For what purpose did Greeks do trigonometry? Indians did it for astronomy and navigation. Forget about navigation, the Greek ignorance of astronomy is further corroborated by the highly defective Greek calendar. The Greek calends were a butt of jokes for Romans, as even the OED accepts, though the Romans themselves had a highly defective calendar.

(c) Social conditions. The social conditions of early Greeks (e.g. the death-sentence on Socrates on the charge of impiety by doing astronomy) demonstrates that the early Greeks were a superstitious lot, who punished astronomy with death. How, then, could astronomy (or any science) flourish among Greeks under these circumstances? Likewise, it is strikingly strange that a Euclid should have written a text so well suited to the political requirements of the Crusading church, 1500 years later, that the church adopted the book as a text for centuries. (But Greediots are gullible people with no brains, which is OK, except that they demand the same from others!)

(d) Accretion in scientific texts. Later-day (Arabic or even later Byzantine Greek) sources of Greek math are likely to be accretive. As I teach in my decolonoised HPS course, scientific texts tend to be accretive since frequently updated with the latest knowledge. The Almagest is such an accretive text, e.g. its star list is headed by the present-day pole star which was not even remotely near the pole in the time of its purported Greek author Claudius Ptolemy (2nd c. CE). As such, attributing authorship of, or all knowledge in, these late texts to early Greeks (such as Ptolemy and Archimedes) is grossly anachronistic. (But Greediots are bent on self-glorification.)

(e) Scriptural significance given to isolated passages. Since early Western historians were Christian priests, or trained by them, they assigned scriptural significance to isolated passages in these late texts. For example, the supposed “evidence” for “Euclid” is one such single passage in a text by Proclus. As in scriptural analysis, reliance on individual passages conveniently ignores the context; it invites us to overlook that this passage contradicts the rest of Proclus’ prologue which speaks of the religious significance of geometry. (Theology, of course, can digest all contradictions by demanding faith.)

(f) Interpolation, either innocent or deliberate. The more natural approach is to regard such misfit passages as later-day interpolations. The late texts, which are the source of Greek history of science, come to us from the hands of dishonest Christian priests who could easily have interpolated remarks or passages. It has always been the official tradition of church history (since Eusebius and Orosius) to write falsehoods to glorify itself and denigrate others. There are well documented cases of forgeries by Christian priests, such as the fake “Testimonium Flavium”, or the spurious “Award of Constantine” on which the Vatican is founded. They introduced extensive forgeries even into the Bible (“gospel truth”) as the scientist Isaac Newton pointed out in his suppressed seven-volume History of the Church, suppressed to this day. Of course, some interpolations could also be non-malicious errors, as in the wrong claim that Euclid was from Megara in the first English translation of the Elements from Byzantine Greek in the 16th c. In short, isolated passage in late sources of Greek math are not reliable for they may involve interpolations, whether deliberate or introduced innocently.

In view of all the above arguments, in my talk I repeated the point made in Cultural Foundations of Mathematics that the Almagest (of Egyptian origin) a later version of which was mis-attributed to a non-existent early Greek called Claudius Ptolemy could have accreted from later-day Indian math and astronomy texts. In fact, I argued that the version available to us did so accrete, for example because it speaks of the difficulty of multiplication exactly as do Arabic zijes of the 9th c. And it starts with a paraphrase of a long-drawn controversy in Indian tradition over Aryabhata’s statement that the earth rotates.

Someone in the audience (a Greediot?) understood only the point about the lateness of sources for claims about Greek math. Looks like the rest of my arguments went above his head, or he just ignored them since he had no answer to them. He objected that physical sources for Aryabhata are similarly late. This is an objection raised even by school children (while teaching them Rajju Ganita) hence here is my response in detail.

First, I have no objection to valid inference: only to wild speculation as used in the Greek case. To reiterate, one needs to separate valid inference from wild speculation.

Thus, the physical Indian sources of math come from the same land, and in the same language, only from a different time, unlike the Greek sources which are not only from a different time, but also from a different land, and in a different language. Apart from wild racist assumptions, by what process exactly do we know what percent of the text, if any, was actually the work of Greeks? How much was of Egyptian origin, and how much due to accretion from Indian texts? Obviously, Western historians never explained. They never will be able to.

Second, the objections (a), (b), (c), (d) obviously do NOT apply to Indian sources of math. E.g. there is evidence for sophisticated arithmetic and a sophisticated calendar right from Vedic and Buddhist times (the Buddha has a name, tallakshana, for 1053). The social conditions in India were against superstition, as I have repeatedly pointed out (e.g. see this magazine article on scientific temper in ancient India, and this extract describing Indians against superstition), though, of course, all sorts of false and derogatory stories have been spread about India. This situation is contrary to Greeks where even Socrates was killed for doing astronomy, and disbelieving the divinity of the sun and the moon, a charge he denied. But Greediots are stuck to their myths and just neglect any contrary textual or non-textual argument, and they justify this neglect by refusing to provide space for counter-views in the journals of the history of science which they control, like Pingree did.

Further, Indian sources are free from the objection of accretion and anachronistic attribution, which applies to Greek sources. Thus, for example, the tradition in India was that commentaries reproduced the original text in full. As such, by examining Nilkantha’s आर्यभटीयभाष्यwe can clearly separate Aryabhata’s contribution from that of Nilakantha a thousand years later. No anachronistic attribution here.

Because there are so many different commentaries from different times and places, all of which verbatim reproduce the original text on which they comment, we can also rule out accretion. But Greediots don’t seem understand this argument. They are bent upon glorifying Greeks by falsely and anachronistically attributing knowledge to mere names like Archimedes.

Also, in the Indian case, there are so many commentaries on Aryabhatiya from so many different places in India. This is quite unlike the Greek case, where there is just one (out-of-context) passage in one manuscript used as evidence for Euclid. That is the Indian sources are (probabilistically) independent, so that any later-day interpolations can easily be identified, and can also be easily excluded from a critical edition. In short, anachronistic attributions, accretion, and interpolations, can be ruled out in the Indian case of Aryabhata, say, but not in the case of Greek “sources”.

Again, because of the existence of both commentaries and objections (raised by opponents of Aryabhata, such as Varahamihira, and critics such as Brahmagupta) and responses to them, there is a continuity in Indian sources, which is completely absent in the Greek case, where all we have is a single late text. We are asked to rely on wild (and sometimes demonstrably dishonest) speculations (such as those of Heiberg) based on a single late text such as the Archimedes palimpsest.

Anyway, the simple upshot is that contrary to what has been stated by dishonest Western historians like Pingree, trigonometry (and aspects of Indian astronomical controversies) were accreted into the Almagest, and the early Greeks were innocent of both trigonometry and the crime of astronomy. This is one of the reasons Western historians have failed to engage seriously with my book Cultural Foundations of Mathematics on the origin of calculus and trigonometry in India. They have no answer, and instead have the cheek to ask me to engage with their later texts, and not tell a different story.

The time has come to puncture the bloated Western self-image, based on falsehoods. By beginning to tell our own stories, we are nearing the end of false Greek glorification since the Crusades and by racist historians who never dare take our objections into account.

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