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Reading Wikipedia's articles on the flat earth, spherical earth, and history of geodesy makes it clear that virtually every society recognizing the spheroidal shape of the earth today owes the observation to the ancient Greeks. But one possible exception is India. It's not clear to me, after reading the articles and the sources I could reach online, whether India came to this conclusion via cultural exchange with the Hellenistic world like others did or if their own astronomers reached it independently.

I suspect that the reason it wasn't clear to me is that historians aren't certain themselves. If that's the case, a good answer will lay out the evidence pro and con and explain the general views of historians.

Mr. Bultitude
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    India was engaged in intensive cultural exchange with the Hellenistic world after the Alexandrian conquests, there is no mention of spherical Earth in Indian sources prior to c. 300 BC. – Conifold Apr 16 '18 at 20:50
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    @Conifold If you have citations, that comment could be an answer – Carl Witthoft Apr 17 '18 at 12:17
  • It's not that hard. First contact with the upper New Guneans revealed they knew the earth was round. The shape of the earth is directly observable and European scientists had simply overlooked the fact and mispredicted what indigenous people would conclude. The earth is round, and this can be seen from the shape of the shadow cast on the moon during a partial lunar eclipse. – Joshua May 30 '23 at 19:32

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(I started writing this answer from memory, but on a second look at some sources it turns out to answer the question for the statement of a rotating earth rather than that of a spherical earth... but I'll leave it here; hope the other question is still interesting to you.)

The most notable statement of a spherical rotating earth, in Indian astronomy, is by Āryabhaṭa. In his Āryabhaṭīya (499 CE), he explains the familiar observation of the night sky, of the stars collectively appearing to go around the earth, as being because of the earth's rotation. (He gives the analogy of being in a boat, where things on the shore appear to be moving.)

Āryabhaṭa was always universally revered in the Indian astronomical tradition, and had his followers (especially in Kerala) until pretty much modern times (in the 18th century, the French astronomer Guillaume Le Gentil was in India during a lunar eclipse, and found that the prediction of its length by local astronomers based on Āryabhaṭa's method was more accurate than the European one), but this particular statement about the earth's motion was rejected by many later Indian astronomers (even his own commentators kind of explained it away), because of various physical objections. For example:

And if the earth turns, then how do birds reach their nests again after leaving them? Arrows released up toward the sky should fall down in the west if there is an eastward rotation of the earth, and clouds should move toward the west. Now, if you argue that these effects do not happen because the motion is very slow, then how can the earth complete a full rotation in one day?

In Indian astronomy by this time there were many Greek sources known (starting with the influence of the Indo-Greek kingdoms after the breakup of Alexander's empire); consider works (either available or mentioned) like Pauliśa-siddhānta, Romaka-siddhānta, Yavana-jātaka, etc. Astronomers Varāhamihira and Brahmagupta, both later than Āryabhaṭa, mention these Greek sources and also praise them: “Though the Yavanas [Greeks] are barbarians (mleccha), this science (astronomy) is good among them, and they are revered (pūjyante) like sages — a scholar (div) in these sciences even more so.” However when Brahmagupta criticizes Āryabhaṭa's spinning-earth theory, he does not attribute it to a Greek source.

(I suppose that eventually in a few centuries the rotating earth was widely accepted, or maybe it was not, but in any case it doesn't matter much for the calculations.)

So to summarize, we see that neither Āryabhaṭa nor his commentators nor other astronomers attributed the theory of a rotating earth to Greek astronomy even though they were aware of it, nor was the theory widespread the way (some) ideas from Greek astronomy were, so it seems quite probable that the idea originated in India independently.

Sources: see Kim Plofker's Mathematics in India, the course “Mathematics in India” by M.D.Srinivas, M.S.Sriram and K.Ramasubramanian (I can't find the link to the lecture notes and associated writing, but the videos are here), etc.

(As I said, I realize this is not exactly the question you were asking. The idea of a spherical earth itself (even if possibly fixed instead of rotating) was strongly defended by both Āryabhaṭa's commentators and by other astronomers, and is older than Āryabhaṭa... it was universally preferred in Indian astronomy over any kind of flat earth model.)

ShreevatsaR
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    Thanks! There's definitely at least a partial answer in here to my question, so I've given it an upvote. – Mr. Bultitude Apr 18 '18 at 01:29
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    Since the idea was common at the time of Āryabhaṭa lack of specific attribution hardly indicates independent origin, at best it indicates that it spread prior to Ptolemy's Almagest. And lack of credible mentions before Hellenistic times confirms it. I understand the popular feelings behind "Indians were no worse than Greeks", but there is simply no documentary evidence for it before the Alexandrian conquests. – Conifold Apr 19 '18 at 21:44
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    @Conifold My answer was about the earth's rotation. The idea was neither common at the time of Aryabhata, nor later, nor is it found in Ptolemy's model AFAICT. I know the question was about the spherical earth (about which I don't know, as I said at the top, middle and bottom of the answer---it may well be from Greek sources), but as far as the rotation is concerned, I find the evidence persuasive that Aryabhata came up with it himself. – ShreevatsaR Apr 19 '18 at 22:11
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    Rotating Earth is discussed in Almagest, and Ptolemy gives the standard fly-off arguments against it, the idea goes back to at least Hellensitic times as well. So what makes you think Āryabhaṭa came up with it? He simply accepted a known idea while others rejected it. And it is not even clear how he dealt with the counterarguments, if at all. – Conifold Apr 19 '18 at 22:29
  • @Conifold From what I know of the tradition (and Bhaskara's work, etc.), if it was a known idea with known counterarguments, they would be discussed and addressed by his commentators (Bhaskara and others). Similarly, when the counterarguments were presented (by Lalla in this case) they too would be accompanied by names if they were known. As these are all absent in this particular case, it does not seem likely that they were known to the astronomical tradition at the time. – ShreevatsaR Apr 19 '18 at 23:22
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    Let's say you are right, even though Almagest was already known in India Āryabhaṭa thought of the idea on his own, but either did not think of the objections or had no reply to them. Just because A thought X and X turned out to be right (centuries later for reasons A knew nothing about) does not mean that A knew or "realized" X. – Conifold Apr 19 '18 at 23:39
  • @Conifold Well if your baseline belief (prior) is that nothing came from India by default unless there's indisputable documentary evidence, then nothing will convince you. :-) To me the question is about the preponderance of evidence, comparison with similar cases etc. To paraphrase what you said I understand the popular feelings behind “everything came from the West”, but sometimes others can have simple ideas too. As for “knew” or “realized”, that's philosophy and surely the ancient Greeks didn't “know”/“realize” it either, by those standards. – ShreevatsaR Apr 20 '18 at 00:39
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    To the contrary, there are so many more worthwhile things that came from ancient India (like the Vedic tradition) that I do not understand the inferiority complex of web-wide obsession with "proving" that Indians "independently knew" that Earth was spherical and rotated. And given the context of the times rotating Earth was not even a good conjecture for Āryabhaṭa to make, which is why so many of his colleagues (rightly) rejected it. – Conifold Apr 20 '18 at 00:54
  • I did not claim that “the Indians” “knew” anything. In fact, I pointed out that the Indians mostly did not “know” a rotating earth, because even when Aryabhata postulated it, it was nearly universally rejected. So why assume that? All I'm saying that when he makes this statement it seems likely to be his own. Frankly I think it's meaningless to speak of what “the Indians” (or even “the Greeks”) knew; there were specific people with specific ideas. Even what Indian astronomers widely assumed (spherical earth, etc) was not necessarily so in wider society. (See: their comments about eclipses.) – ShreevatsaR Apr 20 '18 at 01:13
  • @Conifold I'd encourage you to write your own answer. – Mr. Bultitude Apr 22 '18 at 15:48
  • @conifold: Even in the modern era you get coincidences such as Liebniz and Newton independently coming up with the calculus round about the same time; there are plenty of other examples; so it's quite feasible that India & Greece came up with a spherical earth independently. – Mozibur Ullah Apr 28 '18 at 02:38
  • @Conifold Reading this conversation again, I'm actually surprised by the suggestion that the “Almagest was already known in India” at the time of Āryabhaṭa. How do you say that? Everything I can find suggests the opposite, and that the Greek sources/influences on the Indian astronomical tradition seem to be pre-Ptolemaic ones that have not survived today. The first proper introduction of Ptolemy's Almagest and Euclid's Elements seems to have been first via the Islamic tradition, and more completely when Jayasiṃha commissioned their translations into Sanskrit during his reign (1700–1743). – ShreevatsaR Apr 22 '19 at 00:12
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    Aryabhata’s computational schemes suggest familiarity with the equant, and hence with Almagest, see Duke, The Equant in India. – Conifold Apr 22 '19 at 04:58
  • @Conifold See pages 6–7 of that very paper: “it is impossible not to notice that all the sophisticated elements of mathematical astronomy included in Ptolemy’s Almagest are completely missing in any known Indian astronomy until late in the first millennium. […] Presumably they would have used Ptolemy’s more accurate tabular interpolation scheme if they had known about it.” and “an essentially universally accepted (except among some Indian scholars, see ref. 20) view that the astronomy we find in the Indian texts is pre-Ptolemaic.” So what you propose is against the near-universal consensus. – ShreevatsaR Apr 22 '19 at 05:29
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    I guess you skipped pp.5 and 8: the "consensus" is dated, and was based on lack of evidence which is no longer lacking:"In either case, however, there must be a significantly different relationship between the Indian models and ancient Greek astronomy than the one universally accepted until now." See also Duke's review of post-2000 evidence in The Circulation of Astronomical Knowledge in the Ancient World, esp. pp.564, 569 and 574. – Conifold Apr 22 '19 at 08:11
  • @Conifold On the equant, Duke considers (p5) two cases [I've renumbered]: (1) independently discovered in India, (2) Greco-Roman in origin. After rejecting (1)—which is what is elaborated in the other paper—further considers cases (2.1) it is pre-Ptolemy (2.2) post-Ptolemy. His preferred is (2.1) (abstract and bottom of p7), but for (2.2) he points out (p8) two cases: (2.2.1) (simplest) something post-Ptolemy came in later, (2.2.2) (more radical) the Greek material in India is post-Ptolemy. What you're suggesting is a hyper-radical (2.2.3) that the Almagest itself was known (impossible; p6-7). – ShreevatsaR Apr 22 '19 at 19:23
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    I can't follow this, but there is no plausible alternative explanation either for the simplified equant, or for the 4-step method in India. Whether Ptolemy took the equant from some hypothetical earlier source that made its way to India (and forgot to mention it, like many other things), or it was taken from a dumbed down version of the Almagest itself, does not seem to make much of a difference. – Conifold Apr 22 '19 at 19:58
  • @Conifold If you notice, what you claimed (and what we're discussing) is that the Almagest itself as a text in its entirety was known in India at the time of Āryabhaṭa -- not whether something specific like the equant or 4-step method came into Indian astronomy from Greco-Roman sources (on which I'm inclined to agree with you). Now after reading the sources you linked, it seems even more clear that no historian on any side thinks "Aryabhata’s computational schemes suggest familiarity with [...] Almagest"… and in fact that would be incredible, given the list on pp 6-7 of the paper you showed. – ShreevatsaR Apr 22 '19 at 20:33
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    What appears above is "at best it indicates that it spread prior to Ptolemy's Almagest" and nothing about Almagest "in its entirety". Why would one need "its entirety" for rotating Earth? That was definitely known before Ptolemy, unlike the equant, and is more likely to be mentioned by any prior source or dumbed down summary than an obscure computational correction. – Conifold Apr 22 '19 at 21:02
  • @Conifold Firstly, our recent comments above (i.e. starting at “Reading this conversation again”, not the one from 2018) is entirely about your claim that “Almagest was already known in India”. Do you at least agree, after reading the sources you gave, that that is not the case — that in fact much of the Almagest was not known in India? Else it is hard to see anything productive out of discussing the original matter. – ShreevatsaR Apr 23 '19 at 02:09
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    Perhaps I should have written "material from Almagest was known in India", but then this is a comment thread, and I am not sure where you are getting your "entirelys" and "entireties" from, the evidence is too spotty for any sweeping generalizations. What is your substantive claim? That Almagest/prior sources/summaries that reached India talked of the equant, but not of the rotating Earth? I do not find that plausible. And I agree that this discussion is unproductive and it is time to end it. – Conifold Apr 23 '19 at 20:05