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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.

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    $\begingroup$ 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. $\endgroup$ – Conifold Apr 16 '18 at 20:50
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    $\begingroup$ @Conifold If you have citations, that comment could be an answer $\endgroup$ – Carl Witthoft Apr 17 '18 at 12:17
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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.)

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    $\begingroup$ Thanks! There's definitely at least a partial answer in here to my question, so I've given it an upvote. $\endgroup$ – Mr. Bultitude Apr 18 '18 at 1:29
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    $\begingroup$ 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. $\endgroup$ – Conifold Apr 19 '18 at 21:44
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    $\begingroup$ @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. $\endgroup$ – ShreevatsaR Apr 19 '18 at 22:11
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    $\begingroup$ 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. $\endgroup$ – Conifold Apr 19 '18 at 22:29
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    $\begingroup$ 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. $\endgroup$ – Conifold Apr 23 at 20:05

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