Note: My book The Imperishable Seed: How Hindu Mathematics Changed the World and Why This History Was Erased, is available at
https://garudabooks.com/the-imperishable-seed
and
https://www.amazon.in/IMPERISHABLE-SEED-Mathematics-Changed-History/dp/B0BG7RRLVR
Introduction
Most people assume that the ‘Pythagoras Theorem’ was discovered and proved by Pythagoras (why else would it be called the ‘Pythagoras Theorem’?). Only a handful of people are aware that it was known in the Mesopotamian region much before Pythagoras. An even smaller number has heard of the Sulbasutras, which are conservatively dated to 800 BCE, and which contain the first known statement of the Pythagoras Theorem. What is more, its geometrical constructions even imply a knowledge of the proof of the ‘Pythagoras Theorem’. And even among this group, only a small fraction is aware of how it was systematically attempted by Western historians of the 19th and 20th centuries to expunge the Sulbasutras from mainstream historical narratives of mathematics. This article discusses in detail this attempted erasure of the Sulbasutras by providing a systematic historiography of the ‘Pythagoras Theorem’ and the Sulbasutras.
Did Pythagoras discover and/or prove the ‘Pythagoras Theorem’? The Available Evidence
There is no evidence that Pythagoras discovered the result known after his name, and there is no evidence that Pythagoras proved the result known after his name.
The legend of the ‘Pythagoras Theorem’ arose more than 500 years after Pythagoras (572-497 BCE, as stated in (Heath 1921: 67)). The earliest writers who impute this result to Pythagoras are Vitruvius (1st cent. BCE), Plutarch (born 46 CE), Athenaeus (200 CE) and Diogenes Laertius (200 CE). Vitruvius claimed that Pythagoras ordered the sacrifice of a 100 cattle (hecatomb) to celebrate the supposed discovery. This is inconsistent with the fact that killing of animals was strictly forbidden by Pythagoras, a point which was also pointed out by Cicero. To accommodate this inconsistency, ‘oxen’ were replaced by ‘oxen made out of wheat’ in later accounts.
In the words of Thomas Heath, one of the most well known scholars of Greek mathematics:
‘Though this is the proposition universally associated by tradition with the name of Pythagoras, no really trustworthy evidence exists that it was actually discovered by him.’ (Heath 1921: 144)
The fifth century mathematician Proclus writes as follows:
If we listen to those who like to record antiquities, we shall find them attributing this theorem to Pythagoras and saying that he sacrificed an ox on its discovery. For my part, though I marvel at those who first noted the truth of this theorem, I admire more the author of the Elements for the very lucid proof by which he made it fast. (Morrow 1970: 337).
Evidently Proclus has no evidence to link Pythagoras to the ‘Pythagoras Theorem’ as well and refuses to commit himself to any statement that claims so.
Hermann Hankel (1839-1873), a noted mathematician and also known for his writings on the history of mathematics, mentions the same points in the following passage in his book Zur Geschichte der Mathematik in Alterthum und Mittelalter (Hankel 1874: 97-98)
Der erste Schriftsteller, der diesen Satz als von Pythagoras gefunden bezeichnet, ist Vitruv, schwerlich ein zuverslässiger Zeuge; von da an verbreitete sich diese Angabe allgemeiner, aber immer in Verbindung mit jener bekannten Hekatombe, welche Pythagoras vor Freude über die Entdeckung jenes Lehrsatzes dargebracht haben soll – eine Anekdote, welche die Glaubwürdigkeit der ganzen Nachricht stark beeinträchtigt. Denn dies Opfer verträgt sich nicht mit dem strengen Verbote alles blutigen Opfers, welches uns aus den pythagorischen Ritualgesetz die Schriftsteller derselben Zeit, ja oft dieselben überliefert haben, die anderswo von der Hekatombe erzählen. Schon Cicero nahm daher an jener Anekdote Anstoss und in der spätesten Tradition der Neupythagoriker wird das blutige Opfer durch das eines ‘aus Mehl geformten Ochsen’ ersetzt.
Proklus, ein einsichtiger Schriftsteller, drückt sich auffalend unbestimmt so aus: ‘Wenn wir die, welche alte Geschichten erzählen wollen, hören, so finden wir, dass sie dieses Theorem auf Pythagoras zurückführen.’ Auch ihm war, wie hieraus hervorgeht, keine sichere Quelle bekannt.
Fig. 1. From Hermann Hankel, Zur Geschichte der Mathematik in Alterthum und Mittelalter, 1874.
A further perusal of Hankel’s and Heath’s work uncovers further confusion and question marks in associating Pythagoras with the ‘Pythagoras Theorem’ which, however, for the sake of brevity, will not be discussed in this article.
In the absence of any concrete evidence, the one thing that can link Pythagoras with the ‘Pythagoras Theorem’ is faith or belief. As Heath says, ‘I like to believe that tradition is right and that it [the ‘Pythagoras Theorem’] was really his’ (emphasis added).
Fig. 2. From Thomas Heath, A History of Greek Mathematics, Vol. I, 1921.
The Sulbasutras
In 1875, the first translations of parts of the Sulbasutras became available into English. These were carried out by Thibaut and published in the Journal of the Asiatic Society of Bengal (Thibaut 1875). These translations provided the first ever exposure of the Sulbasutras to the Western world and shattered the myth that the Indians had no geometry of their own. The Sulbasutras are ancient texts that contain geometric rules for creating altars for Vedic rituals. They can rightfully be called ‘Vedic mathematics’, as they are part of the Vedas. The name Sulbasutras comes from Sulba, which means rope, as a rope was used to measure the dimensions of altars. The Sulbasutras are further named after their respective composers: thus there are the Baudhayana, Apastamba, Katyayana, and the Manava Sulbasutras. The Sulbasutras being extremely ancient texts, their dates of composition are not known for certain. Conservative estimates by Western Indologists put them at around 800 BCE.
The Sulbasutras describe different rules for creating altars. There are are rules for constructing altars of different shapes but of the same area, altars of the same shape but of different areas, altars whose area is sum of the areas of two different altars, etc. These rules in turn are based on a number of geometric results explicitly stated in the Sulbasutras. These results include the first ever general statements of the so-called ‘Pythagoras Theorem’ in its full generality, and not just some special cases with integers. For example, in the Baudhayana Sulbasutra:
दीर्घचतुरस्रस्याक्ष्णया रज्जुः पार्श्वमानी तिर्यङ्मानी च यत्पृथग्भूते कुरुतस्तदुभयं करोति ॥
and a similar statement almost identically expressed in the Apastamba Sulbasutra:
दीर्घस्याक्ष्णयारज्जुः पार्श्वमानी तिर्यङ्मानी च यत्पृथग्भूते कुरुतस्तदुभयं करोति ॥
The figure shows a snapshot from the translation by Thibaut’s translation:
Fig. 3. From Thibaut, G. ‘On the Sulvasutras’, 1875.
(Note: the word oblong means a rectangle)
As mentioned, Thibaut’s translation was published in the year 1875. It is interesting what Hankel would have said had he lived to see Thibaut’s translation of the Sulbasutras. His writings show that he was ready to treat Indian mathematics in its own right without invoking the Greek connection. Unfortunately he died in the year 1873, two years before Thibaut’s English translation was published, and his book was printed posthumously by his father.
Pythagoras and his Potential Sources
Even if we do assume that Pythagoras knew the result today known after his name, what were his potential sources?
While attempting to answer this question, it is helpful to remember that Pythagoras was known more for philosophy than mathematics, and even his mathematical ideas are accompanied by a philosophical or mystical background. Among the various ideas that he is supposed to have believed in, the one about which there is the least doubt is the idea of transmigration of souls. Pythagoras is also said to have remembered his previous lives (Zeller 1892: 449, Barnes 1987: 28-35). Further, the rationale behind the transmigration of souls is exactly the same as in Hindu thought (as the snapshots from (Zeller 1892) show). Added to this are claims by ancient Greek historians that Pythagoras had visited India in his search for knowledge, which is plausible considering the similarity between Pythagoras’s view of transmigration of soul and Hindu thought, and the fact that the Hindu civilization extended over a much greater area westwards than today. In any case, interaction between Pythagoras and Hindu Brahmins well versed in the Vedas and Hindu philosophy cannot be ruled out. And the very same Brahmins were well versed in the Vedas, part of which were the Sulbasutras, and among one of whose statements was the so-called ‘Pythagoras Theorem’ and which was used for constructing Vedic altars.
Fig. 4. From Eduard Zeller, Die Philosophie der Griechen in ihrer geschichtlichen Entwicklung, 1892.
In 1884 the German Indologist Leopold von Schroeder wrote an article wherein he discussed these very points in thorough detail, i.e., how Pythagoras could have received the so-called ‘Pythagoras Theorem’ from Hindu sources such as the Sulbasutras. Apart from the points mentioned above relating to the philosophy of transmigration of souls, Schroeder made another very interesting observation that has largely remain unnoticed. This relates to the system of SAnkhya (साङ्ख्य) philosophy, enunciated by the sage Kapila to his mother Devahuti, for the purpose of liberating the Jiva (roughly translated as the individual soul) from the bonds of worldly existence. The sAnkhya philosophy is one of the central pillars of Hindu thought and is described, among other places, in book 3 of the Srimadbhagavatam, and in chapter 13 of the Bhagavadgita. The word sAnkhya comes from the Sanskrit word sankhyA (सङ्ख्या), which means number, and this is so because the various levels of existence in nature are enumerated in this philosophy. According to the Practical Sanskrit English Dictionary by V. M. Apte and shown in the figure:
This philosophy is so called because it ‘enumerates’ twentyfive Tattvas or true principles; and its chief object is to effect the final emancipation of the twenty-fifth Tattva i.e. the Purusha or soul, from the bonds of this worldly existence - the fetters of phenomenal creation - by conveying a correct knowledge of the twenty-four other Tattvas and by properly discriminating the soul from them. (Apte 1957: 1664)
Schroeder mentions this very same relation between sAnkhya (the philosophy) and sankhyA (number) and argues that it is in this relation where the source of Pythagoras’ belief in the mystical quality of numbers is to be found: it is basically a corrupted, distorted view of sAnkhya philosophy. So while in sAnkhya philosophy, numbers are used to enumerate the philosophical principles, Pythagoras may have confounded the numbers with the principles themselves and thus ascribed them mystical qualities.
Fig. 5. From V. S. Apte, The Practical Sanskrit-English Dictionary, Vol. II 1957.
Although the Sulbasutras contain the first ever known statement of the result known as the ‘Pythagoras Theorem’, Egypt and Babylonia are often claimed as the sources for the ‘Pythagoras Theorem’, and the Sulbasutras receive only a passing mention in most accounts, if at all. Thus it is often argued that Pythagoras got the ‘Pythagoras Theorem’ from Babylonia (modern day Iraq), as clay tablets found in the region indicate that the result was known there. But we have seen the closeness of Pythagoras’ philosophy with Hindu philosophy and the connection between sAnkhya philosophy and Pythagoras’ interest in numbers, all of which carry signs that his philosophy was borrowed from Hindu sources. If he got his philosophy from Hindu sources and if the same sources contain exactly the mathematical result that he is famous for, then, assuming he indeed knew the result known after his name, the possibility that he got both his philosophy and mathematics from the same source becomes a strong one indeed. This natural inference was not considered by historians before Babylonian mathematics became known, although they had no problems in tracing the origins of the ‘Pythagoras Theorem’ to Babylonian sources once the latter became known.
It is not the aim of this article to claim that Pythagoras indeed got the ‘Pythagoras Theorem’ from Hindu sources (assuming that he knew it in the first place), but to show the asymmetric attitudes of Western historians with regard to Indian and non-Indian mathematical sources, especially in the context of the Sulbasutras, ‘Pythagoras Theorem’, and Babylonian/Mesopotamian mathematics, and how these relate to the creation of Western accounts of the history of mathematics in which mathematics is essentially a Western innovation.
Proof of the ‘Pythagoras Theorem’ implied in the Sulbasutra Constructions:
Before proceeding, it is necessary to make an important and interesting observation. And that is, not only do the Sulbasutras contain the first known statement of the ‘Pythagoras Theorem’ in its full generality, but also that the constructions given in the Sulbasutras imply a knowledge of why the sutra is true. In other words, we are here talking about a proof of the ‘Pythagoras Theorem’ implicit in the Sulbasutra constructions. This is to be found in the sutra that deals with creating a square altar whose area is equal to the sum of the areas of two other square altars of known sides. The sutra is given, for example, in the Baudhayana Sulbasutra I.50:
नानाचतुरस्रे समस्यन्कनीयसः करण्या वर्षीयसो वृध्रमुल्लिखेत् वृध्रस्याक्ष्णयारज्जुः समस्तयोः पार्श्वमानी भवति।
The same statement expressed in slightly different words is also found in the Apastamba Sulbasutra II.4 and Katyayana Sulbasutra II.22.
These sutras says that to create a square with an area equal to the sum of the areas of two other squares of known areas, a length equal to the side of the smaller square should be marked on one of the sides of the bigger square. Then, the square formed by the diagonal of the resulting rectangle has an area which equals the sum of the areas of the two given squares. That the above construction implies a knowledge of why the ‘Pythagoras Theorem’ can be seen from the following animation.
Fig. 6. Animation of the proof of the ‘Pythagoras Theorem’ implied in the Sulbasutra constructions.
The above observation was made by Bibhutibhushan Datta (Datta 1932) and by the German mathematician Conrad Müller (Müller 1929). A snapshot from Müller's article is shown in the following figure:
Fig. 7. From Conrad Müller, Die Mathematik der Sulvasutra, eine Studie zur Geschichte indischer Mathematik, 1929.
The following figure shows another snapshot from Müller’s article, where he mentions that the same construction is found also in Apastamba Sulbasutra II.4 and Katyayana Sulbasutra II.22:
Fig. 8. From Conrad Müller, Die Mathematik der Sulvasutra, eine Studie zur Geschichte indischer Mathematik, 1929.
Other constructions in the Sulbasutras imply a knowledge of other types of proofs. More details can be found in (Datta 1932) and (Dutta 2022).
Interestingly, the very same proof of the ‘Pythagoras Theorem’ as shown in the above animation is discussed in (Stillwell 2010: 10) and (Magnus 1974: 159) (without reference to the Sulbasutras, however).
Fig. 9. From John Stillwell, Mathematics and Its History, 2010.
The Erasure of the Sulbasutras from History
In spite of the lack of evidence of Pythagoras having anything to do with the ‘Pythagoras Theorem’, and the earliest known statement of the result to be found in the Sulbasutras, along with possible knowledge of why it holds true, how is that the Sulbasutras have been practically erased from mainstream historical accounts? How is it that most people have never even heard about the Sulbasutras? And by the same token, in spite of the absence of evidence that Pythagoras discovered or proved the ‘Pythagoras Theorem’, how does Pythagoras remains strongly associated with the ‘Pythagoras Theorem’ or, in slightly more informed circles, Babylonia becomes the source of the ‘Pythagoras Theorem’ whereas the Sulbasutras are as well as nonexistent?
Two episodes can be recognized as having been greatly influential in the erasure of the Sulbasutras from historical accounts and from the mainstream consciousness. The first is the attempt to show the Sulbasutras as having been derived from Greek geometry, which was carried out by influential historians of mathematics such as Moritz Cantor; and the second is Otto Neugebauer’s work on Babylonian mathematics which further served to push the Sulbasutras out of mainstream historical accounts and concentrating the focus almost entirely on Babylonian mathematics.
As we have seen, Hankel had written a major work on the history of mathematics which was published posthumously in 1874. At the time he was not aware of the Sulbasutras and therefore these do not find mention in his book. But he nonetheless mentions the total absence of evidence to link the ‘Pythagoras Theorem’ to Pythagoras, and is at the same time aware about the works of other Hindu mathematicians such as Brahmagupta and Bhaskara I and II, which he duly mentions and is of the opinion that Indian mathematics developed on its own, independent of Greek influence. At one point he also points out a proof of the ‘Pythagoras Theorem’ given by Bhaskara II in the 12th century which, as he also points out, appeared in Europe only much later. All these points are missing or are glossed over in Cantor’s works. Understanding the chronology of events in the context of the writings of Cantor and Neugebauer will provide interesting insights into how Pythagoras still is associated with the ‘Pythagoras Theorem’, and how the Sulbasutras got pushed out of the picture.
Introduction of the Sulbasutras in the Western World (1875)
Pre-1875: Prior to 1875, the Western world was not acquainted with the existence of the Sulbastras. Hindu mathematics was restricted to the mathematics in the Hindu civilization starting from Aryabhata (5th cent. CE) till Bhaskara II (12th cent. CE). Pythagoras is believed to have visited Egypt and Babylonia in the search for knowledge. While accounts such as those by Apuleius mention that Pythagoras went also to India; see (Butler 1909: 184), these are not emphasized by European historians of this era. As regards the ‘Pythagoras Theorem’, it was associated exclusively to Pythagoras. Pythagoras’ religious beliefs, such as reincarnation and the cycle of birth and death, were considered to be irrelevant as far as his supposed mathematical output was concerned.
1875: The English translation of the Sulbasutras carried out by G. Thibaut appeared in the Journal of the Royal Asiatic Society of Bengal, which introduced them to the Western world for the first time. Thibaut refrained from assigning any dates to the Sulbasutras, but he made it clear that they were summaries of things that were known and practiced much before the Sulbasutras were formulated. This made it clear that the Indians had a geometry of their own. As he says:
[…] whatever is closely connected with the ancient Indian religion must be considered as having sprung up among the Indian themselves, unless positive evidence of the strongest kind point to a contrary conclusion.
Thibaut was moreover confident that the Sulbasutras showed signs of having what he termed as ‘a purely Indian origin’.
Episode 1: Cantor’s Cover Up (1875-1920)
Now we come to the first important episode responsible for erasing the Sulbasutras out of historical accounts.
In the final quarter of the 19th century, the history of mathematics came to be dominated by the writings of Moritz Cantor. His major work, Vorlesungen über die Geschichte der Mathematik (Lectures on the History of Mathematics), along with his various published articles, played a huge role in shaping how the history of the development of mathematics across the ages was perceived. Writing in the late 19th century, when, as discussed by Francois Charette, the ‘history of science and mathematics’ was shaped into a colonial enterprise by Western scholars, (Charette 2012: 292), establishing Western superiority over ‘non-Western’ peoples became a major academic and political priority. As a result, Cantor consciously or unconsciously wove his writings with the thread of his times, as will be discussed here in the context of the Sulbasutras.
With the accomplishment of the imperialist enterprise and the general confidence that space, people and nature could be successfully dominated, Western Europeans acquired the ultimate certainty of their superiority over the rest of the world. It is no wonder, then, that the Romantic and Orientalist enthusiasm, omnipresent in the first half of the century, was quickly annihilated. Dismissing previous attempts to proclaim the originality of ‘Oriental’ science and consolidating the integrity of ‘Western’ science was, indeed, a major characteristic of scholarship in the history of science during the last quarter of the nineteenth century. The views became increasingly and consensually Helleno- and Eurocentrist, not in the ingenuous and instinctive manner of previous generations, but in systematic and dogmatic ways. (Charette 2012: 292)
1877-1904: After the publication of the English translation of the Sulbasutras, the Western world was forced to consider the existence of an original Indian geometry independent from Greece and modify its view of the history of mathematics accordingly. As has been mentioned, in the last quarter of the 19th century the most influential figure in the history of mathematics was Moritz Cantor (1829-1920). Cantor took it upon himself to fit the Sulbasutras in the general historical framework of mathematics. Although Thibaut had not given any dates to the Sulbasutras, and was confident that the Sulbasutras had a ‘purely Indian origin’, this did not stop Cantor from creating a chronology of his own for the Sulbasutras, based on some comments by Albrecht Weber relating to the word ayana and extrapolating it to Baudhayana and Katyayana. This chronology then made it possible to assign a Greek origin to the Sulbasutras, which Cantor accordingly did in an article published in 1877 with the title Gräko-indische Studien, Historisch-literarische Abteilung der Zeitschrift für Mathematik und Physik (Cantor 1877). In this article, Cantor hinted a date as late as 140 CE to the Sulbasutras, from which it became very easy for him to ‘conclude’ that the Sulbasutras came from Greek geometry. (Incidentally, he also stated this to be the date of Panini, which is off by several centuries, as conservative estimates put Panini’s date to about 500 BCE.) In this article, Cantor discusses the Sulbasutras and points out apparent similarities between Heron’s geometry and the treatment of the ‘Pythagoras Theorem’ in the Sulbasutras, again concluding that the Sulbasutras were derived from Greek geometry. As Seidenberg summarizes:
Cantor concludes that Indian geometry and Greek geometry, especially of Heron, are related; and the only question is, Who borrowed from whom? He expresses the opinion that the Indians were, in geometry, the pupils of the Greeks.
Speaking of the Hindu mathematicians such as Aryabhata (5th cent. CE), Brahmagupta (7th cent. CE) and Bhaskara (12th cent. CE) who come later, he writes (slightly paraphrased for clarity):
Es ist kaum denkbar, dass diese Schriftsteller [Aryabhata, Brahmagupta, Bhaskaracharya etc.], mit den griechischen Quellen, welche diesem zu Gebote standen, unbekannt gewesen sein sollen; es ist noch wenig denkbar, dass sie, mit denselben bekannt, sie gar nicht auf sich wirken liessen. Es ist somit auch wieder eine zum Voraus zu vermutende, nur auf Bestätigung angewiesene Folgerung, dass griechische Mathematik ihre Spuren in jene indische Werke hineingetragen habe.
It is hardly conceivable that these writers [Aryabhata, Brahmagupta, Bhaskaracharya etc.], should have been unacquainted with the Greek sources, which were available to Varahamihira [the earliest of these writers]; it is even less conceivable that they [the Indian writers], acquainted with the same, did not let themselves be influenced by them [the Greek sources]. Thus, it is again a conclusion to be assumed in advance, only waiting for a confirmation, that Greek mathematics had brought its traces into those Indian works.
One encounters an interesting sentence in the above passage: A conclusion to be assumed in advance, only waiting for a confirmation. Needless to say, his chronology for the Sulbasutras appears to be motivated by a similar assumption. As I have written in my book The Imperishable Seed in the context of similar assumptions by James Mill:
Because the Greeks were ‘ingenious and inventive’, they must have developed mathematics, and that is what we must assume even in the absence of evidence. And because the Hindus were backward and primitive, they could not have created mathematics, and that is what we must assume even in the presence of evidence. (Kamble 2022).
So here we see the first sign of faulty research apropos the Sulbasutras: claiming a Greek origin for the Sulbasutras based on their proposed dates, with the proposed dates chosen specifically for the purpose of validating the claim of Greek origin.
Meanwhile, what are Cantor’s views regarding Pythagoras and the ‘Pythagoras Theorem’? Judging from the 1880 version of his influential book Vorlesungen über die Geschichte der Mathematik (Cantor 1880), it would appear that the Sulbasutras made no difference to his view on Pythagoras’ role in the ‘Pythagoras Theorem’. Namely, that it was exclusively Pythagoras who discovered and proved the theorem without external influence. After all, with 140 CE as their proposed date, the Sulbasutras are as good as irrelevant to the discourse. He also acknowledges that Pythagoras learnt some mathematics from Egyptian and Babylonian sources, but what he learnt from there has nothing to do with the ‘Pythagoras Theorem’. Let us have a look at some his statements in this regard:
Wir zweifeln daher keinen Augenblick, dass der Aufenthalt des Pythagoras in Aegypten, dass der Unterricht, welchen er bei den dortigen Priestern genoss, zu den Dingen gehört, die landläufige Wahrheit waren. (Cantor 1880: 127)
Therefore, we do not doubt for a moment that Pythagoras’ stay in Egypt, that the lessons he received from the priests there, were among the things that were common truth.
and:
Als sicher gestellt erscheint uns damit so viel, dass Pythagoras in Babylon hätte gewesen sein können. (Cantor 1880: 128)
So much appears certain to us, that Pythagoras could have been in Babylon.
However, as mentioned, when it came to the ‘Pythagoras Theorem’, he believed that it was discovered and proved by Pythagoras himself, and was not influenced by Egyptian or Babylonian sources, as this passage shows:
Einige von den Dingen, welche ganz besonders der Geschichte der Mathematik angehören, werden wir allerdings nicht verzichten Pythagoras selbst zuzuschreiben. Dazu gehört der pythagoräische Lehrsatz, den wir unter allen Umständen ihm erhalten wissen wollen. (Cantor 1880: 129)
Translation: Some of the things, which belong particularly to the history of mathematics, we will not refrain from attributing to Pythagoras himself. Among them is the Pythagorean theorem, which we want to preserve under all circumstances. (Cantor 1880: 129)
At one place he mentions in passing that the Egyptians and Babylonians perhaps knew about the 3-4-5 triangle, and makes no hypothesis about where Pythagoras may learnt about the 3-4-5 triangle.
Pythagoras bemerkte, meinen wir, dass 9+16=25. Als er diese unter allen Umständen interesante Bemerkung machte, kannte er bereits, gleichviel aus welcher Quelle, die Erfahrungstatsache, dass ein rechter Winkel durch Annahme der Maasszahlen 3, 4, 5 für die Längen der beiden Schenkel und für die Entfernung der Endpunkte derselben construirt wird. Wir haben darauf hingewiesen, dass die Aegypter und die Babylonier vielleicht die gleiche Kenntniss besassen, dass die Chinesen ihrer sicherlich teilhaftig waren.
Pythagoras remarked, we think, that 9+16=25. When he made this under all circumstances interesting remark, he already knew, no matter from which source, the experienced fact that a right angle is constructed by assuming the measure numbers 3, 4, 5 for the lengths of the two legs and for the distance of the end points of the same. We have pointed out that the Egyptians and the Babylonians perhaps possessed the same knowledge that the Chinese certainly had (Cantor 1880: 153)
Indeed, at page 56 when he discusses Egyptian geometry, he says allerdings noch ohne jede Begründung, that the Egyptians knew the 3-4-5 triangle (Cantor 1880: 56). And in fact nowhere in the book does he hint that Pythagoras may have known the 3-4-5 triangle from Egypt.
Further he says that Pythagoras got some knowledge from Egypt and Babylon, but is careful not to imply the ‘Pythagoras Theorem’ as being part of that acquired knowledge:
Pythagoras hat, so suchten wir zu erweisen, sicherlich in Aegypten, vielleicht in den Euphratländern mathematisches Wissen sich angeeignet. Ersteres geht wie aus den ausdrücklichen Ueberlieferungen, so auch aus dem ägyptischen Gepräge mancher geometrischer Entwicklungen, Letzteres aus den babylonisch anmutenden Zahlendifteleien der Pythagoräer hervor. Die Summe des geometrischen Wissens, welches von Pythagoras und seiner Schule den Griechen vor dem Jahre 400 zugänglich gemacht wurde, ist eine nicht ganz geringfügige. […] Sie umfasste den pythagoräischen Lehrsatz und den goldenen Schnitt.
Pythagoras, so we tried to prove, certainly acquired mathematical knowledge in Egypt, perhaps in the Euphrates countries. The former can be seen from the explicit traditions, as well as from the Egyptian character of some geometrical developments, the latter from the Babylonian seeming number differences of the Pythagoreans. The sum of geometrical knowledge, which was made accessible by Pythagoras and his school to the Greeks before the year 400, is not quite a minor one [...] It included the Pythagorean theorem […]. (Cantor 1880: 158)
It is clear that according to Cantor the ‘Pythagoras Theorem’ has nothing to do with this stay in Egypt and Babylon.
In this book he again hints at the Sulbasutras as being from 140 CE, as in his 1877 paper. Considering Thibaut’s comments on the originality of the Sulbasutras and his belief that they summarize knowledge that was known and practiced since much earlier ages, Cantor’s views appear to be a case of backfitting the dates to match the conclusion he wanted to reach.
1884 (Leopold von Schroeder’s article Pythagoras und die Inder): In 1884, the German Indologist Leopold von Schroeder published an article called Pythagoras und die Inder (‘Pythagoras and the Indians’) (Schroeder 1884). In this article, he gave his supposed dates for the Sulbasutras as around 800 BCE and categorically dismissed Cantor’s views pertaining to the Greek origins of the geometry of the Sulbasutras. He further presented compelling reasons as to why Pythagoras must have come to know of the ‘Pythagoras Theorem’ from Indian sources such as the Sulbasutras, some of which we have discussed previously.
1894: Cantor’s second edition of Vorlesungen über Geschichte der Mathematik
In 1894 Cantor came up with another edition of Vorlesungen über Geschichte der Mathematik, in which, in spite of Schroeder’s 1884 article, Cantor persists with his belief that the Sulbasutras were derived from Greek geometry and were formulated in 140 CE. As regards the ‘Pythagoras Theorem’, he attributes it exclusively to Pythagoras, with Egypt or Babylonia playing no role in it. In this edition he further repeats verbatim all the passages from the 1880 edition which we have quoted above (that Pythagoras stayed in Egypt and Babylonia, that the ‘Pythagoras Theorem’ is to be attributed exclusively to Pythagoras, and how Pythagoras must have proved the ‘Pythagoras Theorem’ starting from the 3-4-5 triangle, only making a passing comment that Egyptians may have known about the 3-4-5 triangle, but without mention that Pythagoras could have learnt about this triangle from Egypt).
The highlighted text above is meant to serve as a reminder to the contrast with which the Sulbasutras were treated compared with Babylonian mathematics, which we will discuss later. For example, in spite of the evidence presented in Schroeder’s article, the Sulbasutras were not considered as the origin of the ‘Pythagoras Theorem’, whereas this hesitation is absent in the case of Babyonian mathematics. Also, at this the stage, the discovery of the ‘Pythagoras Theorem’ is by Pythagoras alone, independent of Egypt or Babylon, and in spite of the Sulbasutras, which has been dismissed on the basis of faulty dating. This changes, however, when the Sulbasutras cannot be denied to predate Pythagoras, and Cantor is now glad to invoke Egyptian origins to the ‘Pythagoras Theorem’, as we will see.
In the 1894 edition of Vorlesungen, Cantor does not mention Schroeder’s article at all. In fact, in an article published in 1905 titled Über die älteste indische Mathematik (to be discussed below), he writes that Schroeder was friendly enough to send him a copy of his article, but that he (Cantor) made no mention of Schroeder’s article in the 1894 edition of his Vorlesungen über die Geschichte der Mathematik because the conclusion arrived at by Schroeder, namely, that Pythagoras was to be considered as a student of the Indians appeared to him ‘very unlikely’ (Cantor 1905).
Hence, even after the publication of Schroeder’s article, Cantor continued referring the ‘Pythagoras Theorem’ exclusively to Pythagoras.
Thibaut’s 1899 article on Indian mathematics and astronomy
Such was Cantor’s influence that in an article by G. Thibaut (the person who had introduced the Sulbasutras to the West through his English translation in 1875) on Indian mathematics and astronomy published in 1899 titled Astronomie Astrologie und Mathematik (Thibaut 1899), Thibaut gave in to Cantor’s authority and reputation by repeating the same cliches which Cantor used in Vorlesungen regarding Hindu mathematics. Cantor had claimed that the later mathematicians such as Brahmagupta and Bhaskara had access to Greek sources and thus their work is derived from Greek sources, and Thibaut repeats this. Very carefully he asserts that the Sulbasutras were original and not influenced by Greek mathematics, and also reiterates their age as predating Pythagoras, but the writing is peppered with disparaging remarks, which appears to be in deference to Cantor’s authority and to toe Cantor’s line as far as possible. The whole tone is of a contemptuous nature, and the historical significance of the Sulbasutras is reduced to as consisting of a ‘limited amount of practical knowledge’ about which there is nothing remarkable. He was however careful enough to maintain the originality of the Sulbasutras (in defiance to Cantor’s views), but not their novelty or historical significance. Thibaut was clearly overawed by Cantor’s authority, which is reflected in his disparaging remarks towards Hindu mathematics. In Seidenberg’s words: ‘I cannot help thinking that it shows battle weariness rather than a considered opinion.’ Similarly, another mathematician and historian of mathematics, Max Simon wrote that Thibaut was a victim to Cantor’s authority (Simon 1909: 139). Seeing the change of opinions as exhibited by Thibaut, starting from a keen, appreciative assessment of the Sulbasutras in 1875, to reducing them to a ‘limited amount of practical knowledge’ in his 1899 article, arising from his reverence for Cantor’s authority, Max Simon wrote:
ich wunderte mich, wie befangen sich dieser hervorragende Kenner des indischen Wissens auf dem Gebiet der exakten Wissenschaften der Autorität Cantors gegenüber zeigte.
I was amazed at how this outstanding connoisseur of Indian knowledge in the field of exact sciences showed himself to be so captive to Cantor’s authority.
1901: The German Indologist Albert Bürk published a German translation of the Apastamba Sulbasutra. Bürk placed the age of the Apastamba Sulbasutras at 5th or 4th century BCE (Heath 1921: 145).
1905: Cantor’s article Über die älteste indische Mathematik
Following Bürk’s translation of the Apastamba Sulbasutra, Cantor published an article called Über die älteste indische Mathematik (On the Oldest Indian Mathematics) in 1905. In this article, Cantor claims to be convinced by Bürk’s assessment of the date of the Apastamba Sulbasutra. Perhaps there was an increasing acceptance of the fact in the last 25 years or so since Thibaut’s English translation of the Sulbasutras that, whatever the age of the Sulbasutras, they clearly predated Pythagoras, and because of which Cantor probably realized that the dates assigned by him earlier to the Sulbasutras were clearly untenable.
Prior to Bürk’s translation, Cantor had maintained that Pythagoras had started off with the observation that in a right-angled triangle with sides 3, 4 and 5, the relation 3²+4²=5² holds. From this, he (Pythagoras) had gone on to prove the general result for any right-angled triangle. As to from where Pythagoras came to know about this triangle, he made no mention. But he now emphasizes, after Bürks translation appears, that Egypt had the 3-4-5 triangle and that Pythagoras got it from there, and which he use to prove the general result. So, while till now, based on faulty dating, he does not consider the Sulbasutras as a possible origin of the ‘Pythagoras Theorem’, he now invokes Egypt only after the Sulbasutras becomes a distinct possibility as an origin of the ‘Pythagoras Theorem’. Before this, Cantor was happy to claim that the ‘Pythagoras Theorem’ was exclusively due to Pythagoras, which he proved using the 3-4-5 triangle.
As he says in the article:
Für Ägypten genügte nach meiner Auffassung die eine Erfahrungstatsache, daß es ein rechtwinkliges Dreieck 4, 3, 5 gebe. Dann nahm Pythagoras die Sache geometrisch in die Hand und bewies den nach ihm benannten Lehrsatz, indem er in altgriechischer Weise von Einzelfall zu Einzelfall, zuletzt zum ganz allgemeinen Satze aufstieg.
For Egypt, in my opinion, the one fact of experience was sufficient that there was a right-angled triangle 4, 3, 5. Then Pythagoras took the matter geometrically in hand and proved the theorem named after him by ascending in ancient Greek manner from individual case to individual case, finally to the quite general theorem. (Cantor 1905: 70).
To repeat: while earlier Cantor does not credit Egypt for the 3-4-5 triangle (as reflected in the phrases gleichviel aus welcher Quelle and vielleicht in the texts quoted earlier), he now maintains that Pythagoras got 3-4-5 from Egypt and proved the general case.
Why the Sulbasutras could not be a possible source for Pythagoras is justified on rather peculiar grounds which, for the sake of brevity, we will summarize as follows. The argument is that the Sulbasutras list out more triples than Pythagoras could have known, hence could not be the source of the ‘Pythagoras Theorem’! The existence of extra triples in Sulbasutras should make one take Sulbasutras even more seriously, as the absence of these triples in Pythagoras’ mathematics can very well indicate an incomplete understanding of the original sources. (Of course considering the inappropriateness of associating Pythagoras with the ‘Pythagoras Theorem’ it makes even less sense to talk about this triple formula as coming from Pythagoras. This just goes to show how an entire edifice can be created from an initial false assumption) And one would also have thought that the explicit mention of the ‘Pythagoras Theorem’ in the Sulbasutras along similarities in Pythagoras’ religious beliefs with Hindu beliefs are very important points, leading to suggest Sulbasutras as a strong potential source for Pythagoras.
This is not all. Still attached to his Heron hypothesis, Cantor attempts to save it by even insinuating that parts of the Sulbasutras are later day interpolations. He ends his article with the following words:
[…] sollte es […] darauf hinauslaufen, daß einzelne Teile der Sulbasutras verhältnissmäßig moderne Einschiebsel sind?
[…] should it […] boil down to the fact that individual parts of the Sulbasutras are relatively modern interpolations?
As Max Simon writes in his Geschichte der Mathematik im Altertum (Simon 1909: 138-139):
Aber statt dass nun Cantor die Selbstständigkeit oder wenigstens die relative Selbstständigkeit der Inder, d.h. die Unabhaängigkeit ihrer Geometrie von den Griechen zugegeben, druckt er sich äusserst gewunden aus, ja selbst seine Heron-Hypothese gab er nicht auf, indem er sie hinter der zweifelnden Frage am Schluss versteckt, ob nicht am Ende in den Sulbasutras verhältnissmässig moderne Einschiebsel seien.
But instead of Cantor admitting the independence or at least the relative independence of the Indians, i.e. the independence of their geometry from the Greeks, he expresses himself extremely tortuous, even he did not give up his Heron hypothesis by hiding it behind the doubtful question at the end, whether there are not at the end in the Sulbasutras relatively modern insertions.
Incidentally, Burks translation does not say anything substantially different from Thibaut’s translation, so one wonders what led Cantor to his conclusions of Greek origin of the Sulbasutras after Thibaut’s translation of 1875. It is also remarkable that even after Burk’s translation and his proposed dates of the Apastamba Sulbasutras, the latter hardly featured in any account of mathematics, in spite of the fact that the only explicit general statement of ‘Pythagoras Theorem’ was to be found in the Sulbasutras. In fact, T. H. Heath writes in 1921 the following of the Sulbasutras (Heath 1921: 145):
True, the discovery is also claimed for India. The work relied on is the Apastamba-Sulba-Sutra, the date of which is put at least as early as the fifth or fourth century B.C., while it is remarked that the matter of it must have been much older than the book itself ... There is a proposition stating the theorem of Eucl. I. 47 as a fact in general terms, but without proof […] Certain considerations suggest doubts as to whether the proposition had been established by any proof applicable to all cases. Thus Apastamba mentions only seven rational right angle triangles; he had no general rule such as that attributed to Pythagoras for forming any number of rational right-angle triangles; [his words imply] that he knew no other such triangles. […] the theorem is enunciated and used as if it were of general application; there is, however, no sign of any general proof; there is nothing in fact to show that the assumption of its universal truth was founded on anything better than an imperfect induction from a certain number of cases, discovered empirically, of triangles with sides in the ratios of whole numbers in which the property (1) that the square on the longest side is equal to the sum of the squares on the other two was found to be always accompanied by the property (2) that the latter two sides include a right angle. (emphasis added.)
In the above passage, while Heath claims that there is no evidence that the composers of the Sulbasutras had a proof of the ‘Pythagoras Theorem’, it is noteworthy that there is no evidence that Pythagoras discovered, or knew, the theorem known after him either. In spite of this, there is no hesitation on Heath’s part to ascribe the ‘Pythagoras Theorem’ to Pythagoras.
Episode 2: Neugebauer’s Fixation with Babylonia
1928-1945: Rise of the Theory of Babylonian Origins
Even though the Sulbasutras were the oldest source known till the 1920s to contain an explicit statement of the ‘Pythagoras Theorem’, and the oldest source known even today to contain such a statement, various reasons were suggested by prominent historians of the era as to why the ‘theorem’ could not have originated from the Sulbasutras, as we have seen in the previous sections. This was intensified with the discovery of Babylonian mathematics. Starting from the 1920s onwards, Babylonan mathematics started getting more and more prominence due to Otto Neugebauer’s research on the subject. In contrast to the Sulbasutras, Babylonian mathematics was immediately feted as the origin of the ‘Pythagoras Theorem’ and as the precursor of ‘Greek geometry’ as soon as signs of the ‘Pythagoras Theorem’ started becoming known in Babylonian mathematics. In other words, Babylonian mathematics was posited as the forerunner of Greek mathematics, and the ‘Pythagoras Theorem’ and other aspects of Babylonian mathematics were supposed to have reached Greece and were transformed by the ‘Greek genius’, as if by the Midas touch, into the ‘Greek mathematics’ as known today, viz., as given in accounts of Heath, etc.
In this section we will examine the factors that led to this preferential treatment accorded to Babylonian mathematics as compared to the Sulbasutras.
Before proceeding,it may be relevant to recount a few points as to what is meant by ‘Babylonian mathematics’. ‘Babylonian mathematics’ refers to the mathematics known and practiced in the region earlier called Mesopotamia, which today corresponds to modern day Iraq. Evidence for mathematical knowledge in the region comes from about 400 clay tablets, discovered in the 1850s, written in the cuneiform script and which contain specific mathematical examples with steps for solutions:
… The region known as Mesopotamia was the home of many civilizations. … the region was invaded and conquered many times, and the successive dynasties spoke and wrote in many different languages. The convention of referring to all the mathematical texts that come from this area between 2500 and 300 BCE as ‘Babylonian’ gives undue credit to a single one of the many dynasties that that ruled over this region. The cuneiform script is used for writing several different languages. The tablets themselves date to the period from 2000 to about 300 BCE. (Cooke 2005: 35)
Starting from the late 1920s, Otto Neugebauer (1899-1990), an Austrian-American historian of mathematics and astronomy, involved himself with the analysis and translation of mathematical inscriptions on Babylonian clay tablets. Lasting till the 1940s, his work has been greatly responsible for the huge importance accorded to Babylonian mathematics when it comes to the history of the ‘Pythagoras theorem’. It carries on in the spirit of Cantor’s work, in that it attempts to further relegate the Sulbasutras to a footnote in mathematical history and as a derivative of something that came earlier.
In this section we take a closer look at Neugebauer’s works to understand how a preferential treatment was accorded to Babylonian mathematics in comparison to the Sulbasutras, while the Sulbasutras were ignored, even before Neugebauer started his work on Babylonian mathematics.
In the following we will be following the same chronological approach that we did while discussing Cantor’s works.
1928: Zur Geschichte des Pythagoräischen Lehrsatzes
In 1928 Neugebauer published one of his first articles relating to Babylonian mathematics called Zur Geschichte des Pythagoräischen Lehrsatzes published in Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (Neugebauer 1928). In this article he described a mathematical problem along with its solution sketched out on a Babylonian clay tablet. The problem consists in finding the length of the diagonal of a rectangle with sides of length 10 and 40. In the solution given on the tablet, the length of the diagonal is stated to be 40(1+(1/2)(10/40)). This gives an approximate value of root(10²+40²), and will be known to readers familiar with the binomial expansion:
√(1+x) ≈ 1+x/2
From the solution as stated on the tablet, Neugebauer concludes that this is a sign that the Babylonians knew the ‘Pythagoras Theorem’; that is, that the Babylonians knew that the above is an approximation to the square root and hence that they knew the ‘Pythagoras Theorem’. However, the formula given could very well be an empirically arrived at approximation, and nothing in the statement of the solution indicates that this served as an approximation to √(10²+40²). This we know only in hindsight. (The approximation can also be arrived at geometrically, as shown in (Fowler and Robson 1998), but it still does not indicate that the above approximation was derived from there.) The above exhibits Neugebauer’s eagerness to make assumptions most favorable to Babylonian mathematics.
Later on, indications surfaced in the 1930s that the Babylonians were probably indeed aware of the so-called ‘Pythagoras Theorem’ for a general right angled triangle, as discussed by Hoyrup (Hoyrup 2018). These consist of clay tablets involving problems for finding a side of a right angle triangle, given two its other sides, and where the square root is explicitly mentioned. But these surfaced later and Neugebauer could not have known these at this time.
Another thing to be noted is that in the above article, Neugebauer takes the dates of the Sulbasutras as between 800 BCE and 200 CE. From this, he concludes that the Sulbasutras are also derived from Babylonian mathematics, just like Cantor had concluded on the basis of his assumed dates that the Sulbasutras were derived from Greek geometry. As Neugebauer writes:
Die Schwierigkeiten, die einer direkten griechischen Entlehnung aus Indien entgegenstehen, fallen bei der Annahme von Babylon als gemeinsames Ursprungsgebiet ohne weiteres weg.
The difficulties which arise from assuming a direct borrowing by the Greeks from India fall away on the assumption of a common origin in Babylonia.
Much like Cantor, the supposed chronology of the Sulbasutras is also invoked by Neugebauer to conclude that they are derived from something older, this time from Babylonian mathematics. However, the supposed dates of the Sulbasutras as between 800-200 BCE are conservative conjectures by Western Indologists, and may not indicate the real antiquity of the Sulbasutras. Also, it is not clear from the above quoted text what ‘the difficulties which arise from a direct borrowing by the Greeks from India’ are, as they have not been explicitly mentioned, just as it was not clear what exactly Cantor’s objections had been for considering a direct borrowing from India.
1930s-1940s Mathematical Cuneiform Texts
Neugebauer’s work on Babylonian mathematics lasted till the 1940s. His work involving the deciphering and translating of the Babylonian clay tablets was published in German in the 1930s as Mathematische Keilschrift-Texte, Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik (Neugebauer 1935-1937). In 1945 he once again compiled his findings on Babylonian mathematics in ‘Mathematical Cuneiform Texts’, this time in English, where the Babylonian tablets were discussed in detail (Neugebauer 1945).
1945 The square root of 2 in Babylonian mathematics and the Sulbasutras. There are two things in the (Neugebauer 1945) which are of interest in the context of the Sulbasutras and the ‘Pythagoras Theorem’. The first relates to the value of the square root of two as given in the Sulbasutras and the Babylonian clay tablets (Neugebauer 1945: 43). On a Babylonian clay tablet, known as YBC 7289, a square is depicted accompanied by certain numbers in the Babylonian notation. Near a side of the square the number 30 is etched (which can also mean 30/60, or ½, depending on the context), which stands for a square of side 30 (or ½). On one of the diagonals the following numbers are etched in Babylonian notation: 1, 24, 51, 10. In decimal notation this is interpreted as (1+(24/60)+(51/60²)+(10/60³)) which equals 1.414212962962963 and equals the square root of two up to 6 decimal places. It is assumed that the clay tablet is an instruction to multiply the side of the square with this number to get the diagonal of the square. The extremely accurate value of the square root of 2 is taken to imply that the Babylonians knew the diagonal equals the square root of two and hence the ‘Pythagoras Theorem’ in the case of an isosceles right angle triangle.
Fig. 10. Clay Tablet YBC 7289. Links: https://en.wikipedia.org/wiki/YBC_7289#/media/File:YBC-7289-OBV-labeled.jpg and https://maa.org/press/periodicals/convergence/the-best-known-old-babylonian-tablet
Now, an approximation for the square root of two is given also in the Sulbasutras of Baudhayana, Apastamba and Katyayana. For example, the Baudhayana Sulbasutra I.61 states the following sutra:
प्रमाणं तृतीयेन वर्धयेत्तच्च चतुर्थेनात्मचतुस्त्रिँशोनेन। सविशेषः।
which translates to the following approximation:
√2 ≈ 1 + 1/3 + 1/(3.4) – 1/(3.4.34)
and which equals 1.4142156862745097, accurate to five decimal places. The same approximation is stated also in the Apastamba and Katyayana Sulbasutras. Although the Babylonian value is slightly more accurate, the point here is not the accuracy of the approximation, as the composers of the Sulbasutras knew that the above approximation exceeds the actual value. This is made clear as shown by the word savisheshah in the above sutra. Also, there was no harm if the value slightly exceeded the exact value, as the offerings made to the Devas in the Vedic rituals could exceed the prescribed value; however, offerings less than the prescribed amount were strictly prohibited. This is the importance of the term savisheshah: The above value indeed exceeds the actual value of √2, and that by about 2x10⁻⁶. Further, as pointed by Henderson, it is possible to fathom from other sutras in the Sulbasutras how the above approximation was arrived at, and it is even possible to extend the procedure to achieve more and more accurate approximations to root2 (Henderson 2000). All these are signs of the originality of the Sulbasutras and their independence from Babylonian mathematics.
However, this does not stop Neugebauer from claiming that the Sulbasutra value for the square root of two is derived from Babylonian sources. His claim is based (1) on the dates ascribed to the Sulbasutras and (2) on the observation that the first three terms in the Sulba approximation, which add up to 1+(25/60), and which equal 1.4166666666666667 in decimal notation, is also found elsewhere in Babylonian mathematics. Based on this, he makes the claim that the correction to 1+(25/60), that is -1/(3.4.34) (the fourth term in the formula) is also derived from Babylonian mathematics. In Neugebauer’s words:
The possibility seems to me not excluded that that both the main term and the subtractive correction are ultimately based on the two Babylonian approximations. (Neugebauer 1957: 35)
The dismissive treatment given to the Sulbasutras becomes evident from this example. It may be remembered that before Babylonian mathematics came into prominence, the Sulbasutras were hardly given much attention and the focus had been to show them as being derivatives of Greek geometry, and later when it could not be denied that the Sulbasutras predate Greek geometry, the possibility of the Sulbasutras being the source of the ‘Pythagoras Theorem’ was hardly ever considered.
We give here another example of the highly dismissive treatment the Sulbasutras received in comparison to Babylonian mathematics. Even before the YBC 7289 was deciphered and much before Babylonian mathematics came into prominence, the same Sulba approximation for the square root of two was discussed by Heath in 1921 in Greek Mathematics in the following words:
[…] the truth of the theorem was recognized in the case of the isosceles right-angled triangle; there is even a construction for root2, or the length of the diagonal of a square with side unity, which is constructed as (1+1/3+1/(3.4)-1/(3.4.34)) of the side: the length taken is of course an approximation to √2 […]; but the author does not say anything which suggests any knowledge on his part that the approximation value is not exact. Having drawn by means of the approximate value of the diagonal an inaccurate square, he proceeds to use it to construct a square with area equal to three times the original square, or, in other words, to construct root3, which is therefore only approximately found. (Heath 1921: 146)
One wonders if Heath was expecting that they should find √3 exactly. If so, he had rather high standards, considering that √3 contains an infinite number of digits after the decimal. But what is more concerning is that Heath insinuates that they did not know that the value they found was an approximation. As we saw in the context of savisheshah, this is a patently false statement: the term savisheshah means that the stated value is an overestimate. Clearly Heath’s focus clearly was on trying to disprove or deny the possibility of Indian influence on Greek mathematics, especially on the mathematics attributed to Pythagoras and his supposed ‘theorem’.
In contrast, while writing about YBC 7289, Neugebauer implies that the Babylonians knew that the length of the diagonal they have stated is an approximation to the square root of 2, and also uses it an evidence to show that they knew the ‘Pythagoras Theorem’. Earlier in the 1928 article on the 10-30 right triangle, he had ascribed to the Babylonians a full awareness that the formula used by them was an approximation for the square root of the sum of the squares of the sides, again implying that they knew the ‘Pythagoras Theorem’. In fact Neugebauer has no hesitation in attributing Babylonia as the source of Greek mathematics, as is obvious from the following quote ((Seidenberg 1961) and (Neugebauer 1945: 41)):
What is called Pythagorean in the Greek tradition had better be called Babylonian….
1940s: Plimpton 322:
The Plimpton 322 tablet is the most well known artifact of Babylonian mathematics. It is a cuneiform clay tablet dated to about 1800 BCE. In the words of the historian Eleanor Robson:
‘Plimpton 322 […] is a clay tablet, measuring some 12.7 × 8.8 cm as it is preserved, ruled into four columns. It was excavated illegally sometime during the 1920s, along with many thousands of other cuneiform tablets, not from Babylon but from the ancient city of Larsa (modern Tell Senkereh […]) […] The tablet had originally been acquired by the New York publisher George Arthur Plimpton for his private collection of historical mathematical artifacts, which was bequeathed to Columbia shortly before his death in 1936.’ (Robson 2001)
As mentioned in the above passage, the tablet contains four columns, which consist of numbers etched in Babylonian notation. Column 4 just contains the serial numbers 1, 2, 3 etc. Columns 2 and 3
contain the lengths of the hypotenuses and widths [shorter side] respectively of 15 right-angled triangles. (Robson 2001)
While column 1 apparently contain certain ratios of the sides of the triangles; there is some uncertainty associated with it. The tablet shot to fame after Neugebauer introduced them in his books ‘Mathematical Cuneiform Texts’ and ‘Exact Sciences in Antiquity’ ((Neugebauer 1945) and (Neugebauer 1957)). Neugebauer not only saw it as definitive proof that the Babylonians knew the ‘Pythagoras Theorem’, but also that they knew the general formula for generating Pythagorean triples. Later historians such as Bruin and Robson showed the last claim to be a flawed one and an effort to foist modern day mathematical knowledge on to the Babylonians (Robson 2001). In 1937 Neugebauer had quoted:
What is called Pythagorean in the Greek tradition had better be called Babylonian...
After Plimpton he goes on to say,
This is fully confirmed by the texts discussed here. (mentioned also in Seidenberg 1961).
And thus it was, starting with the works of Moritz Cantor and culminating with the works of Otto Neugebauer, that the Sulbasutras, introduced to the Western world in 1875, were by 1945 removed for good from any serious discussion on the history of mathematics.
‘Why this Omission?’
Even before Babylonian mathematics came into prominence, the Sulbasutras were known to contain the ‘Pythagoras Theorem’, ‘Pythagorean triples’, and a host of other geometric knowledge believed to have arisen in Greece. Why then were they not considered as a possible source of early Greek geometry and the ‘Pythagoras Theorem’ before Babylonian mathematics was known? Why were efforts made to date them later and show them as being derived from Greek geometry, and later, when it became clear that they were much older than Greek geometry, why were efforts made to downplay their importance? And why, later, with the discovery of Babylonian mathematics, was there no hesitation on the part of Western scholars to attribute early Greek geometry to the Babylonians? Indeed, we may join hands with Seidenberg and pose the question: ‘Why this omission’?:
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? (Seidenberg 1961).[emphasis added]
A part of the answer was given by Seidenberg himself in 1978:
‘[…] 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.’ (Seidenberg 1978) [emphasis added].
If the Indians invented plane geometry, what was to become of Greek ‘genius’ or of the Greek ‘miracle’? (Seidenberg 1978) [emphasis added]
Indeed the most obvious aspect in this omission has to do with the narrative of Greek mathematics. I have discussed in my book The Imperishable Seed: How Hindu Mathematics Changed the World and Why This History Was Erased about the reasons behind the general tendency of Western scholars throughout the ages to downplay the role of Hindu mathematics. Clearly some of the reasons mentioned there have an overlap in the specific case of the Sulbasutras as well. The narrative of ‘Greek mathematics’ was such a crucial part of Western identity that questioning it was almost tantamount to blasphemy. Added to that were the colonial and racial attitudes of the time, and the writings of Cantor and Heath should come as no surprise.
But that is not the whole answer, however, because while any mention of the Sulbasutras as being a possible influence on Greek geometry was scrupulously avoided before anything about Babylonian mathematics was known, and wherever possible the Sulbasutras being derived from Greek geometry was treated as a near certainty by assuming dates which would serve this purpose, there was no hesitation in crediting Babylonia as the source of Greek geometry (and also of the Sulbasutras, as Neugebauer did) once Babylonian mathematics came into prominence. Thus, while there is an overlap in the specific case of the Sulbasutras when discussing about the reasons why Western scholars and historians downplayed the role of Hindu mathematics in general, there are some specific differences as well which merit a closer look. To correctly identify the reasons, it is necessary to go beyond the narrow confines of mathematics and to understand the role of Mesopotamia and Babylon in the wider context of the Western historical narrative.
The most important aspect regarding Mesopotamia and Babylonia, and one which receives far less attention that it should, is the Biblical connection. The name ‘Mesopotamia’ comes from the Greek and literally means ‘the land between two rivers’; the two rivers in this case being the Euphrates and the Tigris, both of which are mentioned in the Bible:
In the Bible, the Tigris was called the Hiddekel, the Hebrew pronunciation of the river’s authentic name, while the Euphrates was simply called the Prat. The book of Genesis describes them as two of the four rivers that flowed out of Eden and watered its famous garden. Biblical tradition thus connects Mesopotamian geography with the beginnings of the human race. (Bertman 2003: 2)
In addition, the cities discovered in the Mesopotamian region, such as Babylon, were also mentioned in the Bible. When archaeological excavations began in this region in the mid-nineteenth century, they were at least partly motivated to validate the truth of Biblical stories. Given the Biblical connection, the excavations sought to establish the ‘roots of the Western civilization’ and locate that region as part of the Western heritage. As described by Robson:
In the 1840s rival British and French teams began to uncover and document the remains of vast stone palaces near Mosul, now in northern Iraq but then part of the Ottoman Empire whose capital was Istanbul. The adventurers quickly identified the ruins they were digging as the ancient Assyrian city of Nineveh, known to them through the stories of the Old Testament and Classical authors. Thus they claimed it as part of their own, European heritage, and were little interested in its place in Middle Eastern history and tradition per se. Thus unwittingly the tone was set for interpreting ancient Assyrian – and, later, Babylonian and Sumerian – remains. […] The finds represented the ‘cradle of civilization,’ mankind’s first tottering steps toward European sophistication. [And] they were potential witnesses to events described in the Old Testament, appearing at a crucial juncture in Western European intellectual history […]. (Robson 2007: 59).
In other words, driven on by Biblical folklore, positioning Babylonia as the origin of Western civilization, both in a Biblical and secular sense, had started already in the mid-nineteenth century. Exactly the same point is made by Bahrani:
The acquisition of monuments and works of art that were shipped to London, Paris and Berlin in the mid-nineteenth century was thus not seen solely, or even primarily, as the appropriation of historical artifacts of Iraq but as the remains of a mythical pre-European past. Mesopotamian cultural remains unearthed in the first days of archaeological exploration then served to illustrate how the modern West had evolved from this stage of the evolution, and that Biblical accounts were true, thus that the Judeo-Christian God was the true God. (Bahrani 1999: 166)
Since human civilisation was thought to originate in Mesopotamia, and this civilisation was transferred from the East to the West, the two justifications for the archaeological expeditions were repeatedly stated as being the search for the ‘roots’ of Western culture and to locate the places referred to in the Old Testament. (Bahrani 1999: 166)
These very points were emphasized by the Deutsche Orient-Gesellschaft in the following essay which was printed in the Orientalistische Literatur-Zeitung of 1898 (Orientalistische Literatur-Zeitung 1898):
Die Kultur des alten Morgenlandes, die Kultur von Ninive und Babylon gewinnt an und für sich wie durch ihre Beziehungen zu der biblischen, ägyptischen und altgriechischen Welt von Jahr zu Jahr ein höheres Interesse. Die ihr gewidmete Forschung hat für die Geschichte der Menscheit so überaus wichtige Thatsachen ergeben, dass ein in Religion und Staat, Kunst und Litteratur reich gegliedertes Leben der Völker am Euphrat und Tigris an der Hand zuverlässiger Urkunden bis in ein hohes Altertum zurück verfolgt werden kann, das noch bis vor Kurzem für die menschliche Erkenntnis völlig unerreichbar schien. Das Studium der in Babylonien und Assyrien ausgegrabenen Kunst- und Litteratur Denkmäler hat unser Wissen von dem Werdegang der Menschheit um die Kenntnis vieler Jahrhunderte, man darf sagen – mehrerer Jahrtausende bereichert und einen Einblick in jene Urzustände eröffnet, in denen die Wurzeln unserer Kultur, der Zeitrechnung und Himmelskunde, des Mass- und Gewichtswesens, sowie wichtige Teile der im alten Testamente niedergelegten religiösen Vorstellungen ruhen.
The culture of the ancient Orient, the culture of Nineveh and Babylon, gains in and of itself, through its relationships with the biblical, Egyptian, and ancient Greek worlds, a growing interest from year to year. The research dedicated to it has revealed, for the history of mankind, such exceedingly important facts that a richly structured life of the peoples along the Euphrates and Tigris in religion and state, art and literature, can be traced back to a distant antiquity through reliable documents, an antiquity that until recently seemed completely unreachable for human knowledge. The study of the artistic and literary monuments excavated in Babylonia and Assyria has enriched our knowledge of the development of humanity by the understanding of many centuries, one might say - several millennia, and has provided insight into those primal states where the roots of our culture, the reckoning of time and astronomy, the systems of measurement and weight, as well as important aspects of the religious beliefs laid down in the Old Testament, reside.
Fig. 11. From Orientalistische Literatur-Zeitung 1898.
From the above we can see that the excavations in Mesopotamia were to a large extent motivated by a need to locate it within the Western historical narrative (as evinced by the phrase ‘roots of our culture’). As Bahrani writes further:
Archaeology, like other human sciences such as anthropology and history, allowed a European mapping of the subjugated terrain of the Other. While ethnography portrayed the colonised native as a savage requiring Western education and whose culture needed modernisation, archaeology and its practices provided a way of charting the past of colonised lands. (Bahrani 1999: 160)
And further:
This obsessive desire to disassociate the past of the region from its present and to present it instead as a primitive stage in the evolution of mankind facilitated the concept of ‘Mesopotamia’ as the rightful domain of the West […] (Bahrani 1999: 166)
Hegel’s interpretation of history is indispensable to understand the source of terms such as ‘cradle of civilization’, ‘first infantile steps towards civilization’ and other such terms which are used to describe Mesopotamia. According to Hegel, civilization evolves in a linear fashion and culminates in a final point of perfection (telos). And this telos is the ‘perfection’ of the West, according to him. The earlier imperfect stages of civilization can be found, apart from civilizations of the past, also in the present in other non-Western civilizations or cultures, which, therefore deserve to be colonized in order to be led to the perfection that has been attained by the West. Hegel’s view of history can be compared with the excavations in Mesopotamia through the following words of Bahrani’s:
The image of Mesopotamia, upon which we still depend, was necessary for a march of progress from East to West, a concept of world cultural development that is explicitly Eurocentric and imperialist. (Bahrani 1999: 172)
The creation of a historical narrative in which space and time became transcendental horizons for the Being—Mesopotamia, was part of the larger discursive project through which Europe attempted its mastery of the colonised. The narrative of the progress of civilisation was an invention of European imperialism, a way of constructing history in its own image and claiming precedence for Western culture. (Bahrani 1999: 171)
Incidentally, it may be pointed out that the same was done with the fabrication of the ‘Aryan race’: the Hindu civilization as a corrupted version of the perfection of the West and claim it as a Western heritage.
And this is where Mesopotamia comes in:
Mesopotamia as primitive stage in the evolution of mankind (Bahrani 1999: 166)
Even the choosing of the name ‘Mesopotamia’ is not without a purpose. ‘Mesopotamia’ comes from Greek, and means ‘between two rivers’ (meso=between and potamus=river, just like hippopotamus means ‘horse of the river’; hippo=horse), the two rivers being Tigris and Euphrates. The name was used by Greek travelers in early times. It was revived once the excavations in the mid-nineteenth century began, and was used to locate the region as part of the Western historical narrative and erase any connection with its present:
Within this disciplinary organisation the term that came to be the acceptable name for Iraq in the Pre-Islamic period was ‘Mesopotamia.’ This revival of a name applied to the region in the European Classical tradition came to underscore the Babylonian/Assyrian position within the Western historical narrative of civilisation as the remoter, malformed, or partially formed, roots of European culture which has its telos in the flowering of Western culture and, ultimately, the autonomous modern Western man. Thus the term Mesopotamia refers to an atemporal rather than a geographical entity, which is, in the words of the renowned Mesopotamian scholar, A. Leo Oppenheim (1964), a ‘Dead Civilisation.’ This civilisation had to be entirely dissociated, by name, from the local inhabitants and contemporary culture in order to facilitate the portrayal of the history of human civilisation as a single evolutionary process with its natural and ideal outcome in the modern West. (Bahrani 1999: 165)
In the above discussion we see how 20th century scholars tried to portray Babylon and Mesopotamia as the ‘cradle of civilization’, where supposedly the first signs of culture and civilization arose which were ‘[...]“passed” as “a torch of civilization” to the Graeco-Roman world’ (Bahrani 1999: 166). Subsumed into the Western framework, Babylonia becomes part of the West where the first tentative beginnings of civilization arise, culminating in the Western civilization of the modern world.
Exactly the same mechanism is at play in mathematics. Following the same narrative as above, where Babylon represents the first steps towards civilization, it, unsurprisingly, also represents the first steps into the realm of mathematics, which found its perfection in Greek and eventually European mathematics:
For many people, the attraction of Plimpton 322 has been exactly its status as a ‘first infantile step’ on the way to modern Western-style mathematics. (Robson 2001).
This then explains why it is acceptable, and even desirable, to attribute mathematical knowledge to the Babylonians, rather than to the Indians. By doing so, the origins of mathematics get transferred to an extinct civilization (Mesopotamia) with Biblical connections and which could be portrayed as handing over the reins to the Greeks and quietly exiting the picture, without compromising the narrative of Greek mathematics at the same time. Thus the history of mathematics becomes solely Western, and unwanted, foreign elements such as Indian mathematical sources are kept out of the way. This is important because the Hindu civilization is a living tradition and is fundamentally different to the West. So, while Pythagoras no longer enjoys the credit of having discovered the ‘Pythagoras Theorem’, the latter is now credited to another civilization, and that civilization is reduced to play the role of being a precursor to ‘Greek mathematics’, and portrayed as taking the ‘first infantile steps on the way to modern Western-style mathematics’ (Robson 2007: 60). The aim is not to elevate Babylonian mathematics, but to show it as a crude beginning, which needed the mature, rational acumen of the Greeks to be brought to perfection:
Strictly speaking, Old Babylonian (and, in general, Mesopotamian) mathematics might therefore better be characterized as computation; instead of ‘mathematicians’ we should speak of ‘calculators’ and ‘teachers of calculation’; supra-utilitarian activities represent ‘pure calculation’ rather than ‘pure mathematics.’ The ultimate interest in finding a number is of course also a characteristic of most present-day applications of mathematics; but it remains a feature which distinguishes both the Mesopotamian and the contemporary calculating orientation from the investigation of the properties of mathematical objects which (since the Greeks) constitutes our ideal type of mathematics proper. (Hoyrup 2018)
Another important aspect in the above is the role of the ‘dead civilization’ in narrative building. Appropriating a dead civilization into the dominant narrative is easier, since, being dead, it is not a threat to the dominant civilization, and can be twisted and interpreted at will. Attributing to India rather than extinct civilizations like Egypt or Babylon would destroy the narrative, as India is an ancient yet living civilization, and has no connection to Biblical events. It thus cannot be dissociated from the present and cannot be integrated into a European framework the way Mesopotamia was. (The way this was attempted to be done for India was via the Aryan race theory: see (Malhotra 2011) and (Kamble 2022).) That is why it is acceptable to invoke Babylonia, a dead civilization, rather than India, a living civilization.
The Situation Today
One expects that, given the availability of information in today’s world, there would be increasing awareness of the fallacy of naming Baudhayanasutra I.50 as the ‘Pythagoras Theorem’, and of the attempts to erase the Sulbasutras from mainstream history, and to reduce them to a mere footnote. Thus it is with surprise that one notes the continued usage of the term ‘Pythagoras Theorem’ in modern popular narratives, implicitly implying that the result was discovered and / or proved by Pythagoras.
In spite of the total absence of any sort of evidence to link Pythagoras to ‘Pythagoras Theorem’, there is no shortage of articles and media in the pop-science world and even the academic world supporting and propagating this myth. For example, the figure shows tweets from two prominent Twitter handles on popular science called ‘Fermat’s Library’ and ‘Physics in History’:
Fig. 12. Links: https://twitter.com/fermatslibrary/status/1687462427162460160 and https://twitter.com/PhysInHistory/status/1670083879372488705
‘Physics in History’ even claims that ‘this theorem [was] proposed by ancient Greek mathematician Pythagoras’.
Similarly, the Glenn Research Center website of NASA claims that starting from the 3-4-5 triangle known in Egypt, ‘Pythagoras generalized the result to any right triangle’ (similar to what Cantor claimed). Link: https://www.grc.nasa.gov/www/k-12/rocket/pythag.html
This is just a small selection and serves to show how the myth of the ‘Pythagoras Theorem’ is propagated by the mainstream media and communication channels.
A Proper Appraisal of the Sulbasutras
Due to the repeated attempts to downplay the importance of the Sulbasutras on the basis of their supposed dates as documented in this article, and thereby chronologically giving a priority to Babylonian mathematical sources, the Sulbasutras usually receive a passing mention in most modern accounts of the history of mathematics. However, there are two aspects which make this chronological approach problematic. The first is that the Sulbasutras were repeating things that were already known and were in practice in the Hindu civilization centuries before they were formally put down in the form of sutras. Secondly, the date of 800 BCE as the time of composition of the Sulbasutras is an estimate by Western Indologists is itself something that deserves a closer scrutiny. They are in all probability much older, with scholars such as Feuerstein placing them contemporary to Babylonia. At any rate, considering the eager attempts of Indologists in the late nineteenth century to assign a much later date to the Sulbasutras, the estimate of 800 BCE is also to be treated with suspicion.
With the above information, instead of treating Babylonia as the cradle of (Western) civilization with all its Biblical connections, several other scenarios become possible. For example, there could have been a common cultural space that connected the regions of India, Greece and Mesopotamia. This is clearly seen from the similarities between ancient Greek and Hindu philosophy (McEvilly 2001), a point which I have examined in detail in my book. This brings us to the point I have made earlier: that considering the strong Hindu influence on early Greek philosophical ideas, a strong influence of Hindu philosophical ideas on early Greek mathematical ideas cannot be ruled out either.
One scholar who carefully and meticulously considered these possibilities was Abraham Seidenberg. He made a careful analysis of Babylonian mathematics and the mathematics of the Sulbasutras. I have examined his work in detail in my book The Imperishable Seed: How Hindu Mathematics Changed the World and Why This History Was Erased, hence I will not repeat them here. Here I will content myself with quoting the following statements:
‘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)
When we consider along these lines, instead of the narrow confines of Mesopotamia as the cradle of (Western) civilization ending in the telos of the West, it becomes clear that the Sulbasutras have played a much more important role in the history of mathematics than hitherto supposed.
Acknowledgements: I am thankful to Prof. M. D. Srinivas for useful inputs and discussions.
My book The Imperishable Seed: How Hindu Mathematics Changed the World and Why This History Was Erased, is available at https://garudabooks.com/the-imperishable-seed and https://www.amazon.in/IMPERISHABLE-SEED-Mathematics-Changed-History/dp/B0BG7RRLVR
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