|WikiProject Mathematics||(Rated C-class, Mid-importance)|
|WikiProject Ancient Near East||(Rated C-class)|
- 1 Image of the tablet
- 2 Importance
- 3 Arc Sine in the Second Millenia BCE ??
- 4 The interpretation of Donald L. Voils (1975 et seq.)
- 5 Mathematical underpinnings and reconciliation of interpretations
- 6 Attempt at a scholarly apparatus
- 7 References
- 8 Exhibit calls for clearer article lead and more focused, balanced sections on varied interpretations of: 1) apparent errors, 2) method of calculation, & 3) function
- 9 Real disagreement?
- 10 Something wrong with this maths
- 11 Recent scholarship
Image of the tablet
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It would be nice to add a picture of the tablet to the article, something like those available at  or . The tablet itself is of course in the public domain, but the photographs may not be. We could probably add one under fair use, given the historical importance of the topic, but I was wondering if we could argue that the photograph is in the public domain, according to Bridgeman Art Library v. Corel Corp. ? The tablet is a 3D object, but it is photographed as a flat object, with the goal of reproducing it accurately. Any opinion ? Schutz 11:20, 20 April 2006 (UTC)
- Bridgeman Art Library v. Corel Corp. fits perfectly for these images. └ VodkaJazz / talk ┐ 13:49, 11 November 2006 (UTC)
- Tablets are 3d. So no Bridgeman Art Library v. Corel Corp. probably does not apply.Geni 03:35, 2 December 2007 (UTC)
I'm bumping the importance in the math rating header to mid. This tablet is the basis of claims that the Babylonians knew the Pythagorean theorem prior to the Pythagoreans themselves and prior to the Sulba Sutras; in terms of the history of mathematics it's quite an important document, although its impact on modern mathematics per se is minimal. —David Eppstein 15:49, 20 May 2007 (UTC)
- I cannot see there would be any doubt whatsoever that the Babylonians (indeed pre Babylonians according to Robson's thesis) understood Pythagoras theorem long before Pythagoras. How else do you explain both the Plimpton tablet and the one which has a 1,1,sqrt(2) triangle inscribed with a very good approximation of sqrt(2) in sexagesimal notation. Just to blow your mind a bit further calculate [(col 1)/(1+(col 1)] + .25 for each of the triples (see below) and write your answers in sexagesimal notation. Then compare with arc sine [(col 2)/(col 3)] from the same row in the table. It should be noted that Robson's excellent thesis concerning reciprocal pairs - despite her dismissal of the 'generating' theory - results in the generating tuple:
- [(x-1/x)/2 ; 1; (x+1/x)/2]
- and that will certainly churn out any 'Pythagorean' triple you care to name. Again it's difficult to understand why there could be any doubt that the numbers in column 2 and column3 are indeed 2 sides of 'Pythagorean' triples. And more intriguingly Pythagorean triples which populate the trigonometrically significant range 30 to 45 degrees - what school pupil doesn't have a 45,45,90 and 30,60,90 square in his/her geometry set ?!
Clarification: Plimpton 322 has been regarded as a basis of claims that the Babylonians had early familiarity with at least a Pythagorean rule. The thrust of Robson (2001) is to knock Plimpton 322 off this particular pedestal. However, perhaps a firmer, and certainly an independent, basis for the claim is provided by Db2-146 = IM67118, a tablet from Eshnunna from about -1775, as discussed, for example, by Høyrup (2002) (Høyrup is one of the principal authors cited in Robson (2001) for naive geometry, but, of course this book postdates that article).
Arc Sine in the Second Millenia BCE ??
|Col 2||Col 3||Calculate||Calculate||Col 1||1 + Col 1||Divide:||Angle x|
|Opp||Hyp||Adj||Angle x||= tan^2(x)||= sec^2(x)||= sin^2(x)||=60*[sin^2(x)+.25]|
Is it beyond the bounds of credibility to imagine that the Babylonians may have in some way noted:
30 degrees sin^2(30) = 1/4 = 0:15 sexagesimal
45 degrees sin^2(45) = 1/2 = 0:30 sexagesimal
60 degrees sin^2(60) = 3/4 = 0:45 sexagesimal
and concluded that if '0:15' is added to the squared sine, it's equal to the angle? And indeed a remarkably linear relationship does exist for angle vs sin^2(angle) in the range 30 to 60 degrees.
It's an accepted part of Maths history that the Babylonians divided the circle into 360 equal parts so we need to ask the question how exactly did they achieve that division and how accurate were they? (presumably they would have had a more sophisticated method than merely marking 360 approximately even divisions around a circle circumference). Neil Parker (talk) 08:18, 2 September 2009 (UTC)
The interpretation of Donald L. Voils (1975 et seq.)
The interpretation of Donald L. Voils (b. 1934) is worthy of note as being historically intermediate between that of Bruins (1949) and Robson (2001) (at current writing (September, 2010), the main article Plimpton 322 references, but otherwise does not mention, Bruins (1949), notwithstanding that Robson graciously acknowledges it as containing the thesis in Robson (2001) in a different guise). Robson expresses interest in, but ignorance of, Voil's interpretation based on passages in Buck (1980) and indeed one passage in particular resonates with the account of Robson's own thesis as described in the main article Plimpton 322.
Voils, then at the University of Wisconsin at Oshkosh, spoke at the April, 1975 meeting of the Wisconsin Section of the Mathematical Association of America (MAA) on the question Is the Plimpton 322 a Cuneiform tablet dealing with Pythagorean triples?, as reported in the issue of the American Mathematical Monthly for December that year (p. 1043). We may follow Robson (2001) in picking up the story from Buck (1980), p. 344, recalling that Buck was on the faculty at the University of Wisconsin at Madison:
- Voils adds to this suggestion of Bruins the observation that the numbers A are exactly the results obtained at the end of the second step in the solution algorithm, (d/2)2, applied to an igi-igibi problem whose solution is x and xR. Furthermore, the numbers B and C can be used to produce other problems of the same type but having the same intermediate results in the solution algorithm. Thus Voils proposes that the Plimpton tablet has nothing to do with Pythagorean triplets or trigonometry but, instead, is a pedagogical tool intended to help a mathematics teacher of the period make up a large number of igi-igibi quadratic equation exercises having known solutions and intermediate solution steps that are easily checked .
Unfortunately, Buck's reference , apparently an item by Voils slated to appear in Historia Mathematica, was never published. Cooke (2005), pp. 163-164, in an extensive discussion of Plimpton 322, gives a sympathetic account of Voil's interpretation, but again based only on Buck (1980). Voils recalls the submission, written after taking a class in the history of Babylonian mathematics at the University of Wisconsin at Madison, was rejected on some technical ground and is now uncertain whether any copy survives, as he changed interests into computer science at about the same time. The class was taught by William D. Stahlman (1923-1975), who had taken his doctorate at Brown University under Otto Neugebauer.
This quotation from Buck (1980) also serves to remind us that, while this paper does discuss a trigonometic interpretation of Plimpton 322, as noted in the main article Plimpton 322, it was by no means confined to it, nor did it endorse it. Rather, in the light of Robson (2001), Buck's contribution seems to show an uncanny prescience of the limitations of the detective genre (Buck (1980), p. 345):
- Unlike Doyle's stories, this has no final resolution. Any of these reconstructions, if correct, throws light upon the degree of sophistication of the Babylonian mathematician and breathes life into what was otherwise dull arithmetic.
Mathematical underpinnings and reconciliation of interpretations
It is a usual and customary part of the scholarly apparatus in the discussion of multiple interpretations to consider their underpinnings and possible reconciliation qua interpretations. Robson (2001) points the way and sets the standard in volunteering that the thesis being advanced already appeared in Bruins (1949) in a different guise. What is meant here by in a different guise, is that Robson recognises a broad ressemblence between the two interpretations (even if Bruins (1949) might not achieve the same elect state of perfection and grace accorded Robson (2001) in the main article Plimpton 322). But naturally there is no suggestion, nor should readers of Robson (2001) infer, that, say, Bruins thought the same way as Robson or would agree with this assessment. Examination of mathematical underpinnings and possible reconciliation tells us only how interpretations stand one to another, but is neutral on what is being interpreted.
Plimpton 322 has often been taken as the basis of claims that the Babylonians had some early acquaintance with a Pythagorean or diagonal rule, in keeping with the thesis ascribed to Neugebauer in the main article Plimpton 322. On the other hand, the thesis attributed there to Robson, but advanced previously in Bruins (1949) in a different guise, in recontextualizing Plimpton 322 within the corpus of Babylonian mathematics, removes it from this supporting role. It might be helpful to indicate (as the main article Plimpton 322 does not) that the claim to early acquaintance has a firmer, and certainly an independent, foundation in Db2-146 = IM67118, a tablet from Eshnunna from about -1775, as discussed, for example, by Høyrup (2002). The tablet works a computation of the sides of a rectangle given its diagonal and area. The working prefigures a dissection of a square on the diagonal into a ring of four congruent right triangles surrounding a square of side the difference between the sides of the proposed rectangle. The general form of this dissection yields the Pythagorean rule on rearrangement of the pieces, although the working on the tablet skirts this observation (compare also the illustrated discussion in Friberg (2007), pp. 205-207). But, for good measure, the tablet also runs a check on the working by applying the Pythagorean rule to the sides to get back to the prescribed diagonal. (An updated listing of Babylonian appearances of the Pythagorean rule is given in Friberg (2007), pp. 449--451, building on an earlier listing by Peter Damerow as well as Høyrup (2002).)
The theses attributed to Neugebauer and to Robson are linked mathematically by two standard, age old tricks, taking the semi-sum (average) and semi-difference of two quantities coupled with difference of squares (notice that this internal link immediately gives a problem with Wikipedia's policy on references and sources, as this entry is currently flagged as open to challenge and removal). For, suppose that l, s and d stand in the Pythagorean relation l2 + s2 = d2, so that l2 = d2 - s2. Application of difference of squares then yields l2 = (d + s)(d -s). Thus, taking x = (d + s)/l, we also have 1/x = (d - s)/l, and can then recover d and s from x and 1/x by the trick of taking the semi-sum and semi-difference: x + 1/x = 2d/l, x - 1/x = 2s/l. Consequently, we have solved the quadratic equation x - 1/x = c, where c = 2s/l, and for that matter also the quadratic equation x + 1/x = k, where k = 2d/l. The algebra is reversible, so starting from solutions to these quadratics, we can recover three quantities standing in the Pythagorean relation.
This mathematical exercise only tells us how the two theses are related (as the main article Plimpton 322 does not), not what skills the Babylonians possessed, still less what the purpose of the tablet might have been. However, as it happens, it is a commonplace of accounts of Babylonian mathematics that it exhibits a propensity to work with the semi-sum and semi-difference of a pair of quantities, as noted, for example, in Cooke (2005). But the mathematically careful account there misses the trick with the difference of squares, so fails to see that whenever solutions of certain quadratics are present so, too, are Pythagorean triads, and vice versa, although such fraility is itself a corrective in historical analysis. Nevertheless, turing back to Bruins (1958), we find the acknowledged progenitor of the thesis in Robson (2001) in a different guise reprising much of this mathematical exercise, with the claim that the approach was used by the Babylonians (Bruins (1958) is not cited in Robson (2001): it is a minor publication easily overlooked on account of its location; but that it appears in a popular journal makes it more accessible to a general reader in the tradition of Robson (2002)):
- We begin by remarking that if we put one of the sides I of a right-angled triangle equal to unity, the Pythagorean relation between the remaining sides d and b is d2 - b2 = (d + b)(d - b) = 1. If therefore we set d + b = λ, then d - b = 1/λ. Now a reciprocal value can be calculated in Babylonian Mathematics only for numbers containing no prime factors other than 2, 3 and 5, i.e. for so-called regular numbers. Extensive tables of such reciprocals were calculated by the Babylonian mathematicians, and therefore by reference to such a table the numbers d = ½(λ+1/λ), b = ½(λ - 1/λ), l = 1, satisfying the Pythagorean relation, could be simply computed. Are there any indications that the Babylonians used this relation? Yes, there are.
Attempt at a scholarly apparatus
Exchanges with David Eppstein
Plimpton 322 It is difficult to "source" mathematically elementary observations about right triangles and it would be embarrassing to describe a subject which has been worked over for thousands of years as "original research"; it is just mathematics. In contrast, it is natural and proper to source interpretations, as is done in the article, as they are proposed by individuals. If anything, it is surprising that the mathematical reconciliation, being completely trivial, had not already been included in such a "definitive" article. —Preceding unsigned comment added by 126.96.36.199 (talk) 04:47, 11 September 2010 (UTC)
The calculations themselves may be trivial, but by putting those calculations in that context as if to lead to a conclusion about what Plimpton 322 was used for, you are committing original research by synthesis. —David Eppstein (talk) 04:56, 11 September 2010 (UTC)
However, routine calculations are allowed and what is given is entirely routine. You seem to be misreading the text. No comment is made about what Plimpton 322 was used for, although comment is made about how the Pythagorean rule can and was used (that can be sourced, for example, in the writings of Jens Høyrup. Rather, without giving weight to any interpretation, the remarks show how they are related mathematically. It was puzzling how such an elementary observation had been left out of an otherwise "definitive" article. Let me restore the comments in good faith, since otherwise readers who are not so mathematically deft are deprived of pertinent information. Of course, you are free to edit the section so as to give only mathematical trivialities that say absolutely nothing about the use of Plimpton 322. —Preceding unsigned comment added by 188.8.131.52 (talk) 05:10, 11 September 2010 (UTC) I have now qualified the section heading to emphasise that only the mathematics of two contending interpretations is being reconciled (as you might expect to have been done already in a "definitive" article when mathematically speaking the points are so trivial). You are clearly anxious about the making of inferences about how Plimpton 322 was used. Can you say how clarifying the very simple mathematics in the two interpretations has bearing on that? —Preceding unsigned comment added by 184.108.40.206 (talk) 05:25, 11 September 2010 (UTC)
What is your point in adding that passage to the article? It's not just a calculation — if I wrote 1+1=2 at the end of an article on Fibonacci, it would be a true statement of mathematics, but it would not lead anywhere. I am similarly having a difficult time seeing how what you wrote in Plimpton 322 connects to anything in the article, but if it does connect, it is (I assume) in order to make some particular point about the Babylonians' ability to solve quadratic equations or generate Pythagorean triangles. That point, whatever it is that you are trying to make, needs a source. It is not good enough to say that the mathematics in what you wrote is true, and that any conclusion is in the mind of the reader. Either you are adding pointless irrelevant calculation to the end of the article, or you are committing original research by synthesis. Either way, it doesn't belong. —David Eppstein (talk) 05:40, 11 September 2010 (UTC)
I believe that you are a very distinguished computer scientist, so your comment is bewildering. One interpretation of Plimpton 322 is in terms of Pythagorean triples, another is that it is an exercise set for the solution of a certain quadratic. Non-mathematical readers might not notice that the mathematics of these two interpretations is closely related, indeed that you can use the Pythagorean triples to solve just such quadratics, not just the one mentioned. So, the mathematically trivial computation is closely tied to the existing text and designed to assist those readers. You are reading into this a suggestion of what the Babylonians could do, but it is not there nor does it need to be there, although just such issues have been discussed (as I say, for instance, by Jens Høyrup). Moreover, what you are also throwing out, is the very simple observation that certain right triangles, such as the 3-4-5 triangle, have all their sides determined as segments of grid lines in a square grid. So, in fact, you really do not need to know all that much, other than to count. So, I submit that the section is pertinent to the existing text, helpful to readers, but not original research, whether by synthesis or in some other way. If indeed I am right in thinking you are a computer scientist, I should be surprised if you did not want to help non-mathematical readers see how the mathematics of the two interpretations is related. You do agree that the mathematics is related as stated and also that Pythagorean triples can be generated in the square grid without knowledge of the Pythagorean Theorem or number theory? —Preceding unsigned comment added by 220.127.116.11 (talk) 06:03, 11 September 2010 (UTC)
I think this section needs to be removed. There are several reasons I can see:
It is original research by synthesis given that there are no sources given. The reason given: "Just in case the solution algorithm for the quadratic equation might seem divorced from Pythagorean triples" is not valid. There are already connections to quadratic equations given right above it. Contrary to what is claimed, this will not help any non-mathematical reader in any way shape or form. My experience developing and teaching liberal arts mathematics courses tells me that even the average freshman at a university (so reasonably well-educated) is going to take one glance at the writing and ignore it. The reference to folding squares/triangles is questionable given that they wrote on clay tablets.
I agree with Dave Eppstein, this needs to be removed. --AnnekeBart (talk) 14:11, 11 September 2010 (UTC)
Clearly original research by synthesis resonates in the Wikipedia community. But surely it is to stop synthesis that is tendentious. AnnekeBart helps out by supplying an instance: yes, indeed, quadratic equations are mentioned immediately beforehand, but the elementary calculations connecting them with Pythagorean triples are not. So, that is why the material is inserted, "just in case". As it happens, one of the leading authors in the history of mathematics in the USA has just written in privately to say the reasoning is excellent and he regrets having missed it, simple though it is. Why was it not already in a "definitive" article? The reference to folding right triangles side to side is to help visualise the significance of the half angles. But writing on clay tablets has nothing to do with it - yet another synthesis gone wrong. I agree that, if that is the level of the readership, then very little, not just the inserted section, is going to register - eyes are likely to glaze quickly encountering the elaborate account of the vs in the algorithm for solving the quadratic. Against that, my guess is that college freshmen, like the leading historian, might rather say, "Pyth to solve quadratics. That's cute".
So, let me try to say yet again what this section is intended to do. The Neugebauer thesis draws on Pythagorean triples. The Robson thesis draws on solutions to quadratic equations. Already here then in the article are suggestions of Babylonian skills. But what sort of mathematical threshold do these skills represent? The talk of number theory for the triples might seem to make it less plausible even if the triples themselves are fairly concrete. But, no, this need not be the case, because right triangles with commensurable sides can be identified in playing on the square grid. Again, the talk of solutions of quadratics, with numerous equations for the algorithm, might remind readers of why they were never any good at mathematics and, indeed, where they lost the plot. But, no, this too need not be a challenge, because a computational trick with Pythagorean triples, little more than difference of two squares, brings out the solution. So, the section supports the existing content of the article by indicating the skills threshold that might be required for one or other of these two interpretations. Moreover, it reveals that they are not exactly exclusive. However, it does not come with any tendentious suggestion as to the use of Plimpton 322 or the skills achieved by the Babylonians. Why deprive readers of this support?
Mathematics as elementary as this cannot be said to be original research. Just imagine trying to publish this in order to generate a source. But I suspect that even if there were a published source to quote at this juncture, that does not seem to be really what is troubling David Eppstein or AnnekeBart. I am afraid that they come over as strangely hostile to the idea of noting for readers how the theses of Neugebauer and Robson are linked, so not chalk and cheese, as might appear from the article. To that extent, it is the article that is tendentious. I agree that the section is in the nature of a footnote or an aside. I should hope that Wikipedia was sufficiently versatile to handle this. But Wikipedia does already have options for leaving material in, while cautioning that original research might be present.
On the other hand, there certainly are cognate sources. There is a large body of problems in old Babylonian mathematics. For instance, BM13901 looks at the problem of two squares for which the sum of areas is known along with either the sum or the difference of the sides, giving the same haunting, suggestive mix of the Pythagorean rule and quadratic equations. One researcher who has written extensively about this material and who is widely quoted is Jens Høyrup. I refrained from putting any of this sourced comparative material in because it might upset the focus of the article on Plimpton 322, although it might be helpful to put Plimpton 322 back in a context from which it has become somewhat detached by being such a centre of attention. In that wider context, the comingling of the Pythagorean rule and quadratics is familiar, at least in our latter day understanding of the subject.
As I say, I was startled and amazed not to find the inserted section already present in a "definitive" article, and now I am bewildered that there is this insistence that supportive information of a non-tendentious nature be deleted. —Preceding unsigned comment added by 18.104.22.168 (talk) 21:40, 11 September 2010 (UTC)
If the section is badly written, why not say that at the start? Unhelpful to any reader? You seem to be changing your tune the more your objections are answered.
Here, in contrast, is the message from the author of one of the leading histories of mathematics published in the USA: Your reasoning here is excellent. I feel I ought to have noticed this connection before, but somehow I missed it. Thus, it appears that even if Plimpton 322 is about problems in algebra or Diophantine equations specifically, the connection with Pythagorean triples is quite immediate. And, of course, the argument that shows how to generate all primitive Pythagorean triples in the form (m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2 works off the same idea of factoring the difference of two squares.
Are you not rather undercutting the spirit of Wikipedia here? —Preceding unsigned comment added by 22.214.171.124 (talk) 00:04, 12 September 2010 (UTC)
Reviewing the discussion, and acknowledging the proper place of sources in Wikipedia, it occurs to me that it might be worth reminding ourselves of the abstract for one of the key sources for the Wikipedia article, Robson's contribution to Historia Mathematica in 2001. In view of Robson's final sentence, maybe I was wrong not to have included mention of some of those other texts, such as BM13901: Ancient mathematical texts and artefacts, if we are to understand them fully, must be viewed in the light of their mathematico-historical context, and not treated as artificial, self-contained creations in the style of detective stories. I take as a dramatic case study the famous cuneiform tablet Plimpton 322. I show that the popular view of it as some sort of trigonometric table cannot be correct, given what is now known of the concept of angle in the Old Babylonian period. Neither is the equally widespread theory of generating functions likely to be correct. I provide supporting evidence in a strong theoretical framework for an alternative interpretation, first published half a century ago in a different guise. I recast it using regular reciprocal pairs, Høyrup’s analysis of contemporaneous “na¨ıve geometry,” and a new reading of the table’s headings. In contextualising Plimpton 322 (and perhaps thereby knocking it off its pedestal), I argue that cuneiform culture produced many dozens, if not hundreds, of other mathematical texts which are equally worthy of the modern mathematical community’s attention
- Richard, Thanks very much for entering the discussion and deleting the section on mathematical reconciliation. I was puzzled that a section like that was not there already, and now mystified that anyone should want to excise it altogether. In contrast, the author of one of the leading histories of mathematics in the USA writes privately:
- Your reasoning here is excellent. I feel I ought to have noticed this connection before, but somehow I missed it. Thus, it appears that even if Plimpton 322 is about problems in algebra or Diophantine equations specifically, the connection with Pythagorean triples is quite immediate. And, of course, the argument that shows how to generate all primitive Pythagorean triples in the form (m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2 works off the same idea of factoring the difference of two squares.
- Now, you are an advocate of contacting academics, so just possibly you might want to ask around here among your academic contacts. At the moment, your policy is to guarantee that people go on missing a connection that, once seen, they feel they ought to have noticed. I know that rules are rules, but, with all due respect, might I suggest that you are cutting against the spirit of Wikipedia? —Preceding unsigned comment added by 126.96.36.199 (talk) 23:52, 11 September 2010 (UTC)
The spirit of wikipedia holds Reliable sources as one of its core principles, this means that every item in wikipedia can be traced back to a verifiable source so people can check the accuracy. Further original research consisting of unsourced results is prohibited. Your addition falls foul of these two principles.--Salix (talk): 06:30, 12 September 2010 (UTC) Further, you edits are asserting that the Babylonian knew of this connection between Pythagorean triples and quadratic equations and used a technique based around grids to do this. Yet there is no evidence for such. If we compare your addition to the work of Robinson who has based her theory on extensive research of the other writings of the Babylonian we see two very different level of scholarship. Maintaining high levels of scholarship is the reason behind the policies on original research.
Richard, Might I just possibly correct you in some places there, apart from the obvious lapses in attention, such as "Robinson" for "Robson". There was no assertion in the excised section that the Babylonians had any particular skills, and indeed when I became aware through exchanges with David Eppstein in which he shifted his ground, I redrafted the text to make that absolutely specific so there could be no doubt. Rather, both the theses of Neugebauer and of Robson presume that the Babylonians had certain skills. All I was doing was pointing to the mathematical level of these skills and a link between them. It just so happens that certain right triangles do have all their sides determined as integral grid-line segments, and they turn out to be the Pythagorean triangles. So, if you were playing on the grid, you might notice that, without having to know the Pythagorean rule or number theory. In a sense, you have them for free. So, the simple point here is that the mathematical level might not be very high, not as high as might be supposed.
Just a comment on mathematical level; nothing about the Babylonians, you understand. It is just a property of right triangles that depends on tangents of half-angles. It is true that Wikipedia does not have that property on in its articles, such as Right_Triangle, so perhaps the way to build the scaffolding of verification and sources is to edit it in.
Again you are completely mistaken about any assertion that the Babylonians knew a link between Pythagorean triples and quadratic equations. I made no such comment. Instead there was a simple mathematical observation that the age-old trick of completing the square does unlock a link of this sort for those who know it. It would seem from the writings of Jens Hoeyrup, which Robson draws on extensively, that this trick, and so this comingling, might not actually have been unknown to some of the Babylonians. But what I wrote was not in any way in competition with Robson's research, so your comparison is otiose as well as gratuitous, I regret to say. Rather the leading historian in the USA has the right reading: completely granting Robson's thesis, nevertheless the connection with Pythagorean triples is immediate. But notice significantly that the leading historian says he had missed this himself, even while feeling he should have spotted it. That is why I did not say, and would never say, the Babylonians or anyone else knew this or that: even the obvious can be hidden in plain sight.
Now, you have exposed yourself brilliantly as having totally misread what was written, even in the face of an explicit disclaimer. But it is some consolation that you are so concerned to maintain high levels of scholarship. Perhaps you might make a start yourself by reading more carefully, instead of jumping to unwarranted conclusions.
You are, in effect making totally false accusations about me in public on this page. I have never written anything of the sort you attribute to me.
- Bruins, Evert M. (1949), "On Plimpton 322, Pythagorean numbers in Babylonian mathematics", Koninklijke Nederlandse Akademie van Wetenschappen Proceedings 52: 629–632 .
- Bruins, Evert M. (1951), "Pythagorean triads in Babylonian mathematics: The errors on Plimpton 322", Sumer 11: 117–121 .
- Bruins, Evert M. (1958), "Pythagoreans triads in Babylonian mathematics", Mathematical Gazette, 41: 25-28, http://www.jstor.org/stable/3611533
- Buck, R. Creighton (1980), "Sherlock Holmes in Babylon", American Mathematical Monthly (Mathematical Association of America) 87 (5): 335–345, doi:10.2307/2321200, http://jstor.org/stable/2321200
- Cooke, Roger L. (2005), The History of Mathematics: A Brief Course, 2nd ed., Wiley, pp. 159-164, ISBN 0-471-44459-6.
- Conway, John H.; Guy, Richard K. (1996), The Book of Numbers, Copernicus, pp. 172–176, ISBN 0-387-97993-X
- Friberg, Jöran (2007), A Remarkable Collection of Babylonian Mathematical Texts (Sources and studies in the history of mathematics and physical sciences; Volume 1 of Manuscripts in the Schøyen Collection: Cuneiform texts; Volume 1 of Manuscripts in the Schøyen Collection, Springer. ISBN 0-387-34543-4, ISBN 978-0-387-34543-7
- Høyrup, Jens (2002), Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and its Kin (Sources and studies in the history of mathematics and physical sciences Sources and Studies in the History of Mathematics and The Graduate Texts in Mathematics), Springer, pp. 257-261, ISBN 0-387-95303-5, ISBN 978-0-387-95303-8
- Knorr, Wilbur R. (1998), ""Rational Diameters" and the discovery of incommensurability", American Mathematical Monthly (Mathematical Association of America) 105 (5): 421-429, http://www.jstor.org/stable/3109803
- Neugebauer, O.; Sachs, A. J. (1945), Mathematical Cuneiform Texts, American Oriental Series, 29, New Haven: American Oriental Society and the American Schools of Oriental Research .
- Neugebauer, Otto (1969) , The Exact Sciences in Antiquity (2 ed.), Dover Publications, ISBN 978-048622332-2
- Robson, Eleanor (2001), "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322", Historia Math. 28 (3): 167–206, doi:10.1006/hmat.2001.2317, MR1849797 .
- Robson, Eleanor (2002), "Words and pictures: new light on Plimpton 322", American Mathematical Monthly (Mathematical Association of America) 109 (2): 105–120, doi:10.2307/2695324, MR1903149, http://mathdl.maa.org/images/upload_library/22/Ford/Robson105-120.pdf .
- Voils, Donald L. (1975), "Is the Plimpton 322 a Cuneiform tablet dealing with Pythagorean triples?". Talk listed by title only in Notice of April Meeting of the Wisconsin Section, American Mathematical Monthly (Mathematical Association of America) 82 (10): 1043, http://www.jstor.org/stable/2318291, allusion in Buck (1980).
Exhibit calls for clearer article lead and more focused, balanced sections on varied interpretations of: 1) apparent errors, 2) method of calculation, & 3) function
I've added references to current exhibit at NYU and NY Times review of same. This article needs work to satisfy the lay public's newly aroused curiosities. This article need not be written in academic style -- this is a subject requiring little more than a few reminders of high school math and elementary number theory and one that if well written could easily engage high school students.
I've also linked to specific pages of Neugebauer & Sachs for those who want to read the original 1945 source. Will do same for his Exact Sciences in Antiquity, also avail. on Google Books, at least in relevant parts of ch. 2. A couple of External links added for inspiration re: clarity is certainly possible.
Lead needs serious work:
- Pythagorean triples can be simply defined in context, keeping wikilink for those who want more: two simple examples below.
- Otto Neugebauer needs to be given priority in lead and discussion, commensurate with his status as dean of historians of ancient mathematics, his prior status in time re: this artifact, and for NPOV: currently, the vast majority of the cites are to Robson alone. Jury is still out. I've added an External reference to a recent paper that may help restore some sense of balance to these active, competing interpretations and unresolved controversies.
- Interpretation falls naturally in two: mechanics of calculations and function of tablet -- these separate questions could be split for clarity. Perhaps even a third category, prior to above, for varying interpretations of apparent errors.
- NYU exhibit page on Plimpton 322 is esp. clear on two-pronged controversies over method & function:
The most renowned of all mathematical cuneiform tablets since it was published in 1945, Plimpton 322 reveals that the Babylonians discovered a method of finding Pythagorean triples, that is, sets of three whole numbers such that the square of one of them is the sum of the squares of the other two. By Pythagoras' Theorem, a triangle whose three sides are proportional to a Pythagorean triple is a right-angled triangle. Right-angled triangles with sides proportional to the simplest Pythagorean triples turn up frequently in Babylonian problem texts; but if this tablet had not come to light, we would have had no reason to suspect that a general method capable of generating an unlimited number of distinct Pythagorean triples was known a millennium and a half before Euclid.
Plimpton 322 has excited much debate centering on two questions. First, what was the method by which the numbers in the table were calculated? And secondly, what were the purpose and the intellectual context of the tablet? At present there is no agreement among scholars about whether this was a document connected with scribal education, like the majority of Old Babylonian mathematical tablets, or part of a research project.
- Columbia makes the 2 key interpreters the primary carriers of an interesting storyline in a few sentences:
Plimpton 322 is known throughout the world to those interested in the history of mathematics as a result of the interest that Otto Neugebauer, chair of Brown University's History of Mathematics Department, took in the tablet. In the early 1940s, he and his assistant Abraham Sachs interpreted it as containing what is known in mathematics as Pythagorean triples, integer solutions of the equation a2 + b2 equals c2, a thousand years before the age of Pythagoras.
Recently, Dr. Eleanor Robson, an authority on Mesopotamian mathematics at the University of Cambridge, has made the case for a more mundane solution, arguing that the tablet was created as a teacher's aid, designed for generating problems involving right triangles and reciprocal pairs. Mr. Plimpton, who collected our tools of learning on a broad scale, would have been delighted with this interpretation, showing the work of an excellent teacher, not a lone genius a thousand years ahead of his time.
Given the current interest in this artifact, can't we do better for the lay public? If no objections, I'll revise the lead to more closely match the professionally conceived summaries cited above.
I respectfully challenge the more mathematically qualified to split the interpretation section into two or three manageable pieces and to distribute citations to varied sources more equitably. -- Paulscrawl (talk) 22:27, 28 November 2010 (UTC)
Robson's main article on the subject is written very polemically, yet buried in it is a conclusion that is much less at odds with the traditional interpretation than one would expect from the way the argument is framed:
[...] the question “how was the tablet calculated?” does not have to have the same answer as the question “what problems does the tablet set?” The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems.
(Robson, "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322", Historia Math. 28 (3), p. 202).
By the way, Robson takes Voils to task for ignoring the table headings (which make it obvious that the scribe saw the relation to what is now called Pythagoras's theorem, or something very close thereto). Robson's main real contribution seems to have been to make a case for how the table was most likely derived: Babylonian algebra does not equal our high school algebra, even when their methods are mathematically equivalent from our perspective. Van der Waerden also posited what amounts to the derivation Robson believes to have been more likely, but did not emphasise its difference from the form chosen by Neugebauer and others.
I am touching on these issues very briefly at User:Garald/Number_theory, which is intended to become a replacement to the main Number Theory page (once it's finished); we should probably go into greater length about this here. Garald (talk) 14:13, 8 December 2010 (UTC)
From the article heading:
"Although the table has been interpreted by leading mathematicians as a listing of Pythagorean triples, more recently published theories give it a different function."
This isn't really an "although". The list is (from our perspective) a listing of Pythagorean triples (that is, integers a, b, c such that ). Moreover, the headings (as Robson herself says) suggests that these were indeed thought of as numbers corresponding to lengths of a right triangle. This says nothing by itself about the *function*. Robson's statements about the function of the table are in logical contraposition, not to the interpretation of the table as a list of Pythagorean triples, but to its interpretation as a trigonometric table. Garald (talk) 09:10, 6 October 2011 (UTC)
On this note - we should transcribe not just the numbers, but the headings of the columns. They make it clear that these are lengths of sides of triangles; Robson herself makes that they make the extreme position taking by Voils untenable. Garald (talk) 11:50, 20 October 2011 (UTC)
... and we should point out that the exposition in van der Waerden's Science Awakening is actually very close to what Robson takes to have been the reasoning behind the method used to construct the table. Van der Waerden (a) was fully familiar with Babylonian methods, and thus had a good feel for what would have been natural to a very able scribe; (b) was a mathematician, and thus may not have been inclined to see much of a difference between two interpretations (Neugebauer's and his own) that are after all mathematically equivalent. Garald (talk) 11:54, 20 October 2011 (UTC)
Something wrong with this maths
The article says: If p and q are two coprime numbers, then form a Pythagorean triple, and all Pythagorean triples can be formed in this way.
Take the triangle with sides (12, 16, 20), which is both a Pythagorean Triple and is of the form where p = 4 and q = 2 which are not coprime. Therefore, this particular Pythagorean Triple cannot be formed with p, q both coprime and therefore the claim that all Pythagorean triples can be formed in this way is not true. Cottonshirtτ 11:30, 15 September 2012 (UTC)
- The formula is valid for all primitive triples, ones that are not integer multiples of some smaller triple. Your example is a multiple of the (3,4,5) triple. I added a clarification.
Robson's work now being over a dozen years old, it would be nice if those who have contributed to this article would consider adding references to more recent scholarship. The only more recent paper I know of is
Britton, John P., Proust, Christine, and Shnider, Steve, Plimpton 322: a review and a different perspective, Arch. Hist. Exact Sci. 65 (2011), no. 5, 519–566.
While it comes down on the "reciprocal pair" side of how the tablet was constructed (with refinements to Robson's argument), it argues against the "school text" interpretation of why it was written. I do not feel sufficiently qualified to be bold, and would thus prefer it if someone else took on the task, however. Michael Kinyon (talk) 15:12, 26 March 2014 (UTC)
I would strongly support rewriting this page to take this new article into account. At any rate, Robson's article was written so polemically that some of her conclusions got obscured. This is a tablet in which reciprocal pairs are the *method*, and constructing Pythagorean triples is the *problem* (and she herself seems to agree with this, if you read her closely). I tried to go into this in the notes to the Number Theory article. Garald (talk) 22:29, 27 April 2014 (UTC)