Talk:YBC 7289

Some suggestions
Since the prose is so tight, especially after the good-article review, I'm reluctant to do even minor surgery, but I do have some suggestions. Will Orrick (talk) 16:56, 5 March 2019 (UTC)
 * 1) The sentence in the lead starting "It is believed to be the work of a student…" might mislead some readers into thinking that the computation of &radic;2 is due to the student, especially since it follows immediately on "'the greatest known computational accuracy ... in the ancient world'".  Later in the article it is made clear that "[t]he student would likely have copied the sexagesimal value of the square root of 2 from another tablet," but it might be good to hint at that earlier.  Presumably the student's task was to compute the diagonal of the square by multiplying the side by a constant taken from a standard table.  It might be good to add to the article that such a table is actually known: according to Fowler and Robson (page 370) the 10th line of the tablet YBC 7243 reads "1;24 51 10, the diagonal of a square."
 * 2) The quotation from Fowler and Robson used to support the alternative interpretation (namely that the tablet shows a computation of the reciprocal of &radic;2) is taken out of context.  As the article stands, it makes it look like Fowler and Robson's article is agreeing with the alternative interpretation, whereas the surrounding text makes it clear that this is an interpretation that one of the authors (Fowler) once found appealing, but is now questioning under the influence of the other author (Robson).  The passage from which this quotation is taken concludes with "a lot more detail … puts this simple and attractive interpretation into question, just as it may be passing into general circulation."  I have not read Fowler and Robson's article from start to finish, but it looks to me like they don't return to this point in the remainder of the article, and it's hard to tell what they really believe.  My guess is that they think 30 was chosen as the side length because it is a nice round number in the sexagesimal system and makes computations easy, not because of any connection with reciprocals, but I'm not sure about this.
 * 3) The article seems incomplete since it doesn't even mention the back of the tablet, although an image reverse side is included.  My understanding is that scholars believe the computation on the reverse has been partially erased.  I don't know whether anyone has investigated what the calculation on the reverse might have been.
 * 4) I suspect that many readers will visit the Wikipedia article because they have read somewhere that YBC 7289 demonstrates Old Babylonian knowledge of the Pythagorean theorem and they want to know what that's all about. Such a reader may be left dissatisfied since we don't mention that point at all.  This is a tricky issue because it leads to the speculative side of the story, whereas the article as it stands mostly sticks to the facts.  My own view&mdash;and I think the experts mostly agree&mdash;is that evidence for Old Babylonian knowledge of the Pythagorean theorem is overwhelming, and that YBC 7289 is hardly the main piece of evidence.  I fear that less careful treatments of YBC 7289 often leave readers with the impression that the tablet contains not just an accurate approximation of &radic;2, but also some kind of proof of the Pythagorean theorem.  One can ask why YBC 7289 is so famous and is shown in all the histories, whereas YBC 7243 (which arguably contains more information&mdash;see above) remains relatively obscure.  Undoubtedly this is because of the diagram, which is vaguely suggestive of well-known visual proofs of the Pythagorean theorem, but is, in fact, at best a degenerate case of those diagrams.  It is also precisely the diagram described in Plato's Meno dialog, but there is no evidence that I know of that the goal on this particular tablet was to reason in the way suggested by that dialog.  Perhaps then it is best to stick just with the facts.  Nevertheless, YBC 7289 does demonstrate that the Old Babylonians were willing to resort to approximation of the diagonal when they had to, and didn't restrict themselves to exact Pythagorean triples.  There are other tablets, such as VAT 6598, that do the same.  (See J. Høyrup, Pythagorean ‘Rule’ and ‘Theorem’&mdash;Mirror of the Relation Between Babylonian and Greek Mathematics.)
 * Good Article reviewer here. You do raise some good points. I don't think any of these issues are fatal for a GA, although they would probably be good to note in case you all want to get this to Featured Status.

-John M Wolfson (talk) 18:50, 5 March 2019 (UTC)
 * 1) I would simply substitute "the tablet" for "it" in that case.
 * 2) I haven't read the Fowler and Robson article, but my suggestion would be simply to add something to the effect of "though Fowler and Robson have some doubt on the interpretation" at the end of the relevant sentence.
 * 3) I agree that the back would be a nice addition to the article, although the article as it stands implies that the tablet's claim of significance is the calculation of $\sqrt{2}$ on the front, so mentions of the back should be relatively brief.
 * 4) That might be a tricky issue. I would stick to cited sources to avoid OR, and mention the Pythagorean theorem iff you find reliable sources that relate it to this tablet specifically. Otherwise I'd suggest putting it as a "See Also" item if there's a separate article for it.
 * I agree, substitute "the tablet" for "it" to make the antecedent clearer.
 * No strong opinion here.
 * I didn't mention the reverse because I didn't know of sources for its content, but I think the Yale library web site says "partially erased" so we can at least say that much.
 * I agree that the connection to the Pythagorean theorem is tenuous, and that the evidence of this tablet that the Babylonians were willing to approximate is interesting. We would need sources for both, of course. One reason that the connection is tenuous is that it's not clear from this tablet itself whether the Babylonians understood the number in question to be the square root of two (the formula you would get by applying Pythagoras) or whether it's just a number they computed as the diagonal of a square, and that happens to equal the square root of two. We could at least add a mention of Plimpton 322, our only other article on a specific mathematical tablet, with a mention that it has been used (disputedly) as evidence for Babylonian knowledge of the Pythagorean theorem.
 * —David Eppstein (talk) 21:01, 5 March 2019 (UTC)
 * Thanks John Wolfson and David Eppstein for your replies. My thoughts:
 * I will replace the pronoun in the lead as you both suggested, and agree that that should fix the issue. I will try to fit in a mention of YBC 7243 somewhere if I can do so without damaging the flow of the article.
 * The only source Fowler and Robson cite that mentions the alternative interpretation is Friberg's unpublished synopsis of Neugebauer and Sachs's Mathematical Cuneiform Texts. Unless other sources favoring the alternative interpretation turn up, we are in the awkward position that the only published source for the alternative interpretation is a skeptical one.  Of course it is perfectly correct that 30 could be 1/2 and therefore that the length of the diagonal could be &radic;2/2, and it is not a problem to mention that.  The speculative part is that this calculation was done because of the Old Babylonian interest in reciprocal pairs.  So far we have one published source mentioning that possibility, but that source says the idea is unlikely to be correct.  Yet the quotation we have chosen to use from that source gives the reader the opposite impression.
 * In Robson's contribution on Mesopotamian mathematics to Katz's The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook contains this about YBC 7289: "The previously unpublished reverse shows the much-erased remains of a solution to an analogous problem about the diagonal of a 3-4-5 rectangle. The calculations below it may belong to another problem entirely."
 * It seems very unlikely that the Old Babylonians were unaware that the area of the square on the diagonal of the unit square is 2 given the numerous known examples of calculations of squares on diagonals of rectangles using the Pythagorean rule. Høyrup's article that I link to in my previous comment contains some of these, but the more recent books of Friberg and Robson contain many more.  I'm also unsure why it is so often stated (outside of the professional literature) that there is a dispute concerning the connection of Plimpton 322 and the Pythagorean theorem.  Robson's 2001 article surveys much of the prior research on Plimpton 322, and the only authors she cites that claim the tablet doesn't relate to the Pythagorean theorem are Schmidt and Voils/Buck.  But Voils's article was never published and Buck presents Voils's hypothesis as one of several, never fully endorsing it.  Robson herself comes down against Voils's interpretation and says that Schmidt's explanation for the column headings containing "width" and "diagonal" is "unconvincing".  I have not been able to access Schmidt's article, so I don't know what it says, but all of the more recent papers I have read on the subject connect the numbers on Plimpton 322 to the width and diagonal of a rectangle (or right triangle).
 * —Will Orrick (talk) 12:58, 7 March 2019 (UTC)
 * I've made changes addressing points 1–3, except for mentioning YBC 7243. I've not dealt with point 4, as I have not yet turned up much in the reliable sources at my disposal.  Friberg does list both YBC 7243 and YBC 7289 in Appendix 8 of his book A Remarkable Collection of Babylonian Mathematical Texts on a list of 18 tablets exhibiting Old Babylonian knowledge of the Pythagorean relation, but he does not elaborate on his reasons for including them.  Høyrup's earlier list of eight such tablets does not include these two, an omission that must have been deliberate.  Friberg mentions another such list made by Damerow, but I have not been able to find it; it seems to be in German.  Both Friberg's list and Høyrup's list include Plimpton 322, by the way.


 * I mentioned Meno in an earlier comment, and have since stumbled across the paper Socrates in Babylon by Peter Damerow, which argues that a series of problems stated on BM 15285 are meant to stimulate the same line of reasoning as in Socrates' dialog. This series of problems starts with the problem of finding the area of a square inscribed in a square; subsequent problems rephrase the question or add additional lines to the figure apparently to encourage the conclusion that the inscribed square has half the area of the outer square.  Although Damerow does connect BM 15285 with YBC 7289, the former asks only about areas, not lengths.  BM 15285 does, however, strongly indicate that the Babylonians were familiar with the fact that the square constructed on the diagonal of a square has twice the area.


 * This is not the place for a discussion of Plimpton 322, but I did want to mention that I now have a copy of Schmidt's paper, mentioned above, and now think either that I misunderstood Robson, or Robson misunderstood Schmidt: as far as I can tell, Schmidt does subscribe to the traditional interpretation of Plimpton 322 as a table of exact side lengths of right triangles.
 * —Will Orrick (talk) 19:12, 22 March 2019 (UTC)

Interpreting 42:25:35
The articles currently states: "The second of the two numbers is 42;25,35 = 30547/720 ≈ 42.426. This number is the result of multiplying 30 by the given approximation to the square root of two, and approximates the length of the diagonal of a square of side length 30."

I think this misses the meaning of 42;25,35. Note that 42/60 + 25/60^2 + 35/60^3 = 0.7071064815, to be compared with 1/√2 = 0.7071067812. Thus the number represent the edge length of the right triangle when the hypotenuse is set to 1. This seems to be the obvious explanation of those digits. Dbkaplan1958 (talk) 20:18, 18 June 2024 (UTC)


 * I suspect that what appears to be an obvious difference of interpretation, when these numbers are written as decimal with a decimal point, is not so obvious in the notation used on this tablet, which used place-value notation but did not (at least not here) use any marker to indicate where the sexagesimal point would go. So 30 is more or less indistinguishable from 1/2. Regardless of whether the Babylonians might have thought of those as different numbers, they would not have written them as different numbers.
 * So your variation amounts to: the current text states that the length of the diagonal of a square of side length 1/2 is (numerical value for 1/sqrt 2). You think it should instead be interpreted as saying that the edge length of a square of diagonal 1 is (same numerical value). 1 is simpler than 1/2 (or 30). So far so good.
 * But you are missing something important: the tablet does not just show the numbers; it is a diagram, showing a square, with one side labeled by 1/2 (or 30), and with the diagonal labeled by 1/sqrt2 (or 30/sqrt 2). Those labels, placed in that way, make sense for the interpretation given in our article, but not for your interpretation. It would be a simpler way of deriving the same numerical value, but not one that corresponds to the diagram and the placement of labels in the diagram. —David Eppstein (talk) 21:04, 18 June 2024 (UTC)


 * Further to David's point, according to Neugebauer the number 1; 24, 51 , 10 was probably copied from a reference table of technical constants, such as YBC 7243, where it is described as the scaling factor that converts the side of a square into ‘the diagonal of a square’. So while it is mathematically true that 42;25,34 is the side of a square with diagonal one, this number has no contemporary meaning whereas 1; 24, 51, 10 does. Daniel.mansfield (talk) 02:57, 20 June 2024 (UTC)