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Archimedes's work The Sand Reckoner (also known by its original Greek title Ψαμμίτης or by the Latin translation thereof, Arenarius) sets out to prove that the number of grains of sand in the Universe is finite by providing an upper bound. From my point of view, this text is remarkable because not only it introduces a kind of scientific notation for numbers but also for the idea of putting an upper bound on the order of magnitude of something we cannot directly count (Archimedes arrives at an upper bound of $10^{63}$), and for the introduction of truly large numbers: Archimedes considers numbers like $10^{8\cdot 10^8}$ and $10^{8\cdot 10^{16}}$ (according to Wikipedia — I didn't check) which I presume were far beyond any finite quantity explicitly described by anyone thitherto.

Of course, now, truly huge numbers have been constructed and used for various reasons in mathematics: some have even proposed the word "Googology" for this art(?).

So my question would be about the history of "Googology" and what the main steps after Archimedes were in the description of large numbers. Specifically: when was the first time in history that a number greater than $10^{8\cdot 10^{16}}$ was explicitly mentioned (that is sufficiently precisely described so that it can be compared)? But generally speaking, any comments on the history of large numbers interest me.

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    $\begingroup$ I've been collecting appearances of large numbers in the literature for many years, and one thing that I've found quite interesting is that I have not found found ANYONE who came close to Archimedes until after 1900. Unfortunately, my extensive literature collection on this topic is not easily available to me right now (I moved a couple of months ago and my stuff on this particular topic has still not been unpacked). However, I do recall that the "number curiosity" $9^{9^9},$ which is approximately $10^{3.7\times 10^8},$ began appearing in popular math articles and puzzle books by the 1890s. $\endgroup$ – Dave L Renfro Mar 15 '16 at 21:39
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    $\begingroup$ The 1910 (first) edition of Hardy's Orders of Infinity is an early appearance of some large numbers, which can be found at the end of the book. The largest seems to be $10^{10^{10^{10.3}}},$ which can be found on p. 62. Incidentally, as a possibly interesting historical footnote that I don't recall anyone ever mentioning before, note that $10^{10^{100}}$ (now called a googleplex, but not then) appears on p. 61. $\endgroup$ – Dave L Renfro Mar 15 '16 at 21:54
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    $\begingroup$ Just to clarify one small point: Archimedes did not calculate "the number of grains of sand in the Universe", but rather the number of grains of sand that would fill the entire space of the universe if it were full of sand. $\endgroup$ – fdb Mar 15 '16 at 23:27
  • $\begingroup$ @DaveLRenfro Thank you, that makes Archimedes's work seem all the more remarkable. You should copy your comment as an answer so I can approve it, because it's unlikely I'll get a more precise answer than this. $\endgroup$ – Gro-Tsen Mar 16 '16 at 13:44
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(comments assembled to form an answer, as OP suggested)

I've been collecting appearances of large numbers in the literature for many years, and one thing that I've found quite interesting is that I have not found found ANYONE who came close to Archimedes until after 1900. Unfortunately, my extensive literature collection on this topic is not easily available to me right now (I moved a couple of months ago and my stuff on this particular topic has still not been unpacked). However, I do recall that the "number curiosity" $9^{9^9},$ which is approximately $10^{3.7 \times 10^{8}},$ began appearing in popular math articles and puzzle books by the 1890s.

The 1910 (first) edition of Hardy's Orders of Infinity is an early appearance of some large numbers, which can be found at the end of the book. The largest seems to be $10^{10^{10^{10.3}}}$ which can be found on p. 62. Incidentally, as a possibly interesting historical footnote that I don't recall anyone ever mentioning before, note that $10^{10^{100}}$ (now called a googleplex, but not then) appears on p. 61.

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The Central Asian mathematician al-Biruni, writing in the year AD 1000, calculated the solution to the “chess problem”: Take a chessboard with 64 squares, put one grain on the first square, double this on the second square, double it again on the third, and so on. How many grains are there on the whole chessboard? His answer is (in decimal notation) 18,446,744,073,709,551,615, or (in sexagesimal notation) : 30. 30. 27. 9. 5. 3. 50. 40. 31. 0. 15. This is of course much less than the figures discussed by Archimedes, but still quite substantial.

What is perhaps interesting is that neither Archimedes nor al-Biruni was actually interested in grains of sand. Both of them are concerned only with proving that it is possible to express very large numbers in some sort of notation.

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I have heard the rumor that Carl Friedrich Gauss (1777 - 1855) called $9^{9^{9^9}}$ "the measurable infinity" (die messbare Unendlichkeit).

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