This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 |
The editors who have taken the lead in blocking any mention of Mansfield and Wildberger's rational trig paper in a leading journal were editing this page before that article is published and I think it is reasonable to conclude that they don't agree with its findings. Some new editors have come along and expressed surprise that M and W don't get a mention but haven't got involved. I did get involved and opened a Request for Comment which was subsequently closed - I was one against several - and when another more experienced editor Prokaryotes tried to open another RFC this was denied. It's quite possible that other potential editors who have heard of M and W's thesis will come to the wiki article and be surprised they are not mentioned and turn to the talk page. If any should read this and think M and W should be mentioned please state so here and add this page to your watchlist, in future there may be a majority for mentioning them. Nine-and-fifty swans ( talk) 16:34, 27 October 2017 (UTC)
@
Nine-and-fifty swans: So go ahead! I'm one of those who object to the way some people try to protect articles from even mentioning certain interesting things, citing various Wikipedia policies.
Eric Kvaalen (
talk) 11:00, 15 February 2018 (UTC)
Eric, I am glad you agree with me, but ther e is no point in making changes, they would only be reverted. In fact quite a few people take the same few as us but nobody stays around to make a voting bloc. Still the good thing is that this page is a perfect example of one of the glaring defects of wikipedia and can be used to demonstrate this. Nine-and-fifty swans ( talk) 15:58, 7 March 2018 (UTC)
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:
Unfortunately, Buck's reference [7], 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):
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)):
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. Cottonshirt τ 11:30, 15 September 2012 (UTC)
Superposition exists if a positive integer right triangle, PIRT, is specified in an elementary fashion, (short side)^2+(long side)^2=(diagonal)^2. And use of this form ensures that the function (d/l)^2 will always lie between 1 and 2, the first two regular sexagesimal numbers. Thomas L. O"Donnell ( talk) 02:17, 7 September 2018 (UTC)
I've just read the 2002 paper by Eleanor Robson, after reading this Wikipedia article. There are a couple things that don't make sense to me. Supposedly this is a list of "teacher's notes" for solving , with the second and third columns being and respectively. But that's usually not the case. Take a look at the first row. We should have and rather than just 119 and 169. Obviously if you subtract 119 from 169 you don't get the reciprocal of their sum. So how does this help the teacher in any way?
(Row 11 does work if you interpret 45 as 45/60 and 75 as 75/60. But that's the only row that can be made to work by "shifting the sexigesimal point".)
Secondly, if this is just a series of exercises, then why are the angles of the triangles spaced out by about 1°? There are many other examples that could have been used, based on similarly sized sexigesimal numbers, but they weren't used.
And if it's a list of problems, why doesn't it give the answer (x)?
Have any of these questions been raised by the proponents of other theories, such as the trigonometry theory?
Eric Kvaalen ( talk) 17:34, 25 March 2018 (UTC)
For Row15 we get a sample of what the Teacher is teaching. p/q=7/2. And we need to realize that this p/q is in the complementary range, 45deg to 60deg, 2.4142<(p/q)<3.7321, and therefore we want the alternate set (which would be in the 45deg to 30deg range). Using p/q=7/2, our "reduced triple" alternate set becomes:
a=1 b=((p/q)-(q/p))/2 c=((p/q)+(q/p))/2 with p/q=7/2 a=1 b=((7/2)-(2/7))/2 c=((7/2)+(2/7))/2 and solving: a=1 b=45/28 c=53/28 and multiplying all sides of this "reduced triple" by 28: a=28 b=45 c=53, a primitive PIRT Multiply by 2 and get: a=56 b=90 c=106 not a primitive
So this is the set to use when the PIRT is primitive and a is even, b is odd. And the Teacher would have pointed out that (b) should be a "regular" number, and it is.
Hope this helps. — Preceding unsigned comment added by 336sunny ( talk • contribs) 17:56, 3 July 2018 (UTC)
The derivation for Row15 looks wrong because a p of (7) is not regular. But it's right because our test fundamentally is that "the long side must be regular". And the long side is now (p^2-q^2) which is (7^2-2^2)=(49-4)=45 which is indeed regular. Which brings up a very important point: The use of the short (s), and long (l) designations for the sides of the right triangle put it in superposition. Each one of two solutions for p&q pairs and for reduced triples gives exactly the same result. The orientation of the right triangle remains undefined until (a) and (b) labels are attached. Thomas L. O"Donnell ( talk) 14:14, 9 August 2018 (UTC)
1. And if it's a list of problems, why doesn't it give the answer (x)?
The answer is the beginning, x=p/q, the starting point. To work backwards, the complete reduced-triple equation must be used. It is at /info/en/?search=Number_theory#Dawn_of_arithmetic
The equation is (s/l)^2 + (l/l)^2 = (d/l)^2. (The left-hand column of P322). Dividing Row11 thru by 15, and dividing Row15 thru by 2 will get them back to primitive PIRTs. Substitute for (s/l) and solve for (x) in the primitive reduced-triple gets p/q for each of the 15 rows. (The resulting quadratic is solved by completing the square).
2. X numerators and denominators that are regular numbers up to 125.
Between 1 and 125 there are only 11 odd and 26 even regular numbers. How about that?
3. why are the angles of the triangles spaced out by about 1°?
Would you believe happenstance? With requirements of p and q beween 1 and 125 (nice regular numbers) and long sides a regular number and 15 PIRTS total, what fell out, put in (d/l)^2 decending order, is what we got. (With a few extras).
4. Who uses PIRT?
Everyone serious about Akkadian mathematics. Thomas L. O"Donnell ( talk) 05:10, 11 September 2018 (UTC)
5.Plimpton 322 tablet gives all of them except one, namely X=125/64. This should go between rows 11 and 12.
That PIRT is s=11529, i=16000, d=19721. Note 5 of the article says "See also Joyce, David E. (1995), Plimpton 322". Dr. Joyce in his paper asks the same question. The answer will be posted on my (Talk) page.
Thomas L. O"Donnell ( talk) 15:48, 24 September 2018 (UTC)
That would seem very true for an edit of the article. TALK sounds like a discussion that might include scholarly speculation. The idea of superposition came from just such a discussion. David, aren't you being too restrictive? Thomas L. O"Donnell ( talk) 11:02, 11 September 2018 (UTC)
@
Will Orrick: Thank you for your response to me, and for the extensive work you have recently done on the article!
Eric Kvaalen (
talk) 08:27, 6 February 2019 (UTC)
The article finishes its description of the content of P322 with the sentence
The sixty sexagesimal entries are exact, no truncations or rounding off
This is a peculiar description, since the numbers are rational. The use the of the words 'sexagesimal' and 'exact' is similar to the phrase 'exact sexagesimal trigonometry', from the title of Mansfield and Wildberger's controversial paper. 'Truncation' and 'rounding' are particular bugbears of Wildberger in his ideas on the non-existence of real numbers. I would suggest removal, not sure if anything is needed in its place; perhaps just a comment that the numbers are rational? — Preceding unsigned comment added by 129.127.37.121 ( talk) 01:06, 16 July 2018 (UTC)
The entries should be regarded as exact values, that is rational numbers, rather than as truncated or rounded sexagesimal expansions approximating irrational real numbers.
"Either convention may be appropriate for use in Wikipedia articles depending on the article context." MOS:ERA The context isn't Christian. Rupert Loup ( talk) 00:47, 7 November 2019 (UTC)
Do not change the established era style in an article unless there are reasons specific to its content. Seek consensus on the talk page first, applying Wikipedia:Manual of Style § Retaining existing styles. Open the discussion under a subhead that uses the word "era". Briefly state why the style is inappropriate for the article in question.You edit warred the change, You opened the thread, There's no consensus to make the change. Meters ( talk) 05:55, 9 November 2019 (UTC)
The article is too technical, too lengthly-wordy-opinionated, relies too much on a handful of sources, and still omits roughly the past two decades of studies. For starters, a specific paper is continuously removed from mention, even though it is frequently discussed and won The Best Writing on Mathematics 2018 by the European Mathematical Society. Someone even moved the more recent talk page entries to the archive, why? prokaryotes ( talk) 10:20, 22 April 2019 (UTC)
>When they find that the M and W paper was indeed touted as a breakthrough in 2017, they probably are going to wonder what the breakthrough was, and what the difference is between the new hypothesis and the similar earlier ones. I, and I think many other editors of this page, share that puzzlement. I would ask those editors who feel it is important to mention M and W in Wikipedia to suggest how this should be explained to readers.< why explain? Better for readers to be puzzled by lack of explanation than lack of mention. For my own part I find M and W's exposition of the tablet so beautifully simple that whether it is historically accurate is neither here nor there. — Preceding unsigned comment added by 90.154.71.175 ( talk) 05:59, 9 October 2019 (UTC)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 |
The editors who have taken the lead in blocking any mention of Mansfield and Wildberger's rational trig paper in a leading journal were editing this page before that article is published and I think it is reasonable to conclude that they don't agree with its findings. Some new editors have come along and expressed surprise that M and W don't get a mention but haven't got involved. I did get involved and opened a Request for Comment which was subsequently closed - I was one against several - and when another more experienced editor Prokaryotes tried to open another RFC this was denied. It's quite possible that other potential editors who have heard of M and W's thesis will come to the wiki article and be surprised they are not mentioned and turn to the talk page. If any should read this and think M and W should be mentioned please state so here and add this page to your watchlist, in future there may be a majority for mentioning them. Nine-and-fifty swans ( talk) 16:34, 27 October 2017 (UTC)
@
Nine-and-fifty swans: So go ahead! I'm one of those who object to the way some people try to protect articles from even mentioning certain interesting things, citing various Wikipedia policies.
Eric Kvaalen (
talk) 11:00, 15 February 2018 (UTC)
Eric, I am glad you agree with me, but ther e is no point in making changes, they would only be reverted. In fact quite a few people take the same few as us but nobody stays around to make a voting bloc. Still the good thing is that this page is a perfect example of one of the glaring defects of wikipedia and can be used to demonstrate this. Nine-and-fifty swans ( talk) 15:58, 7 March 2018 (UTC)
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:
Unfortunately, Buck's reference [7], 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):
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)):
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. Cottonshirt τ 11:30, 15 September 2012 (UTC)
Superposition exists if a positive integer right triangle, PIRT, is specified in an elementary fashion, (short side)^2+(long side)^2=(diagonal)^2. And use of this form ensures that the function (d/l)^2 will always lie between 1 and 2, the first two regular sexagesimal numbers. Thomas L. O"Donnell ( talk) 02:17, 7 September 2018 (UTC)
I've just read the 2002 paper by Eleanor Robson, after reading this Wikipedia article. There are a couple things that don't make sense to me. Supposedly this is a list of "teacher's notes" for solving , with the second and third columns being and respectively. But that's usually not the case. Take a look at the first row. We should have and rather than just 119 and 169. Obviously if you subtract 119 from 169 you don't get the reciprocal of their sum. So how does this help the teacher in any way?
(Row 11 does work if you interpret 45 as 45/60 and 75 as 75/60. But that's the only row that can be made to work by "shifting the sexigesimal point".)
Secondly, if this is just a series of exercises, then why are the angles of the triangles spaced out by about 1°? There are many other examples that could have been used, based on similarly sized sexigesimal numbers, but they weren't used.
And if it's a list of problems, why doesn't it give the answer (x)?
Have any of these questions been raised by the proponents of other theories, such as the trigonometry theory?
Eric Kvaalen ( talk) 17:34, 25 March 2018 (UTC)
For Row15 we get a sample of what the Teacher is teaching. p/q=7/2. And we need to realize that this p/q is in the complementary range, 45deg to 60deg, 2.4142<(p/q)<3.7321, and therefore we want the alternate set (which would be in the 45deg to 30deg range). Using p/q=7/2, our "reduced triple" alternate set becomes:
a=1 b=((p/q)-(q/p))/2 c=((p/q)+(q/p))/2 with p/q=7/2 a=1 b=((7/2)-(2/7))/2 c=((7/2)+(2/7))/2 and solving: a=1 b=45/28 c=53/28 and multiplying all sides of this "reduced triple" by 28: a=28 b=45 c=53, a primitive PIRT Multiply by 2 and get: a=56 b=90 c=106 not a primitive
So this is the set to use when the PIRT is primitive and a is even, b is odd. And the Teacher would have pointed out that (b) should be a "regular" number, and it is.
Hope this helps. — Preceding unsigned comment added by 336sunny ( talk • contribs) 17:56, 3 July 2018 (UTC)
The derivation for Row15 looks wrong because a p of (7) is not regular. But it's right because our test fundamentally is that "the long side must be regular". And the long side is now (p^2-q^2) which is (7^2-2^2)=(49-4)=45 which is indeed regular. Which brings up a very important point: The use of the short (s), and long (l) designations for the sides of the right triangle put it in superposition. Each one of two solutions for p&q pairs and for reduced triples gives exactly the same result. The orientation of the right triangle remains undefined until (a) and (b) labels are attached. Thomas L. O"Donnell ( talk) 14:14, 9 August 2018 (UTC)
1. And if it's a list of problems, why doesn't it give the answer (x)?
The answer is the beginning, x=p/q, the starting point. To work backwards, the complete reduced-triple equation must be used. It is at /info/en/?search=Number_theory#Dawn_of_arithmetic
The equation is (s/l)^2 + (l/l)^2 = (d/l)^2. (The left-hand column of P322). Dividing Row11 thru by 15, and dividing Row15 thru by 2 will get them back to primitive PIRTs. Substitute for (s/l) and solve for (x) in the primitive reduced-triple gets p/q for each of the 15 rows. (The resulting quadratic is solved by completing the square).
2. X numerators and denominators that are regular numbers up to 125.
Between 1 and 125 there are only 11 odd and 26 even regular numbers. How about that?
3. why are the angles of the triangles spaced out by about 1°?
Would you believe happenstance? With requirements of p and q beween 1 and 125 (nice regular numbers) and long sides a regular number and 15 PIRTS total, what fell out, put in (d/l)^2 decending order, is what we got. (With a few extras).
4. Who uses PIRT?
Everyone serious about Akkadian mathematics. Thomas L. O"Donnell ( talk) 05:10, 11 September 2018 (UTC)
5.Plimpton 322 tablet gives all of them except one, namely X=125/64. This should go between rows 11 and 12.
That PIRT is s=11529, i=16000, d=19721. Note 5 of the article says "See also Joyce, David E. (1995), Plimpton 322". Dr. Joyce in his paper asks the same question. The answer will be posted on my (Talk) page.
Thomas L. O"Donnell ( talk) 15:48, 24 September 2018 (UTC)
That would seem very true for an edit of the article. TALK sounds like a discussion that might include scholarly speculation. The idea of superposition came from just such a discussion. David, aren't you being too restrictive? Thomas L. O"Donnell ( talk) 11:02, 11 September 2018 (UTC)
@
Will Orrick: Thank you for your response to me, and for the extensive work you have recently done on the article!
Eric Kvaalen (
talk) 08:27, 6 February 2019 (UTC)
The article finishes its description of the content of P322 with the sentence
The sixty sexagesimal entries are exact, no truncations or rounding off
This is a peculiar description, since the numbers are rational. The use the of the words 'sexagesimal' and 'exact' is similar to the phrase 'exact sexagesimal trigonometry', from the title of Mansfield and Wildberger's controversial paper. 'Truncation' and 'rounding' are particular bugbears of Wildberger in his ideas on the non-existence of real numbers. I would suggest removal, not sure if anything is needed in its place; perhaps just a comment that the numbers are rational? — Preceding unsigned comment added by 129.127.37.121 ( talk) 01:06, 16 July 2018 (UTC)
The entries should be regarded as exact values, that is rational numbers, rather than as truncated or rounded sexagesimal expansions approximating irrational real numbers.
"Either convention may be appropriate for use in Wikipedia articles depending on the article context." MOS:ERA The context isn't Christian. Rupert Loup ( talk) 00:47, 7 November 2019 (UTC)
Do not change the established era style in an article unless there are reasons specific to its content. Seek consensus on the talk page first, applying Wikipedia:Manual of Style § Retaining existing styles. Open the discussion under a subhead that uses the word "era". Briefly state why the style is inappropriate for the article in question.You edit warred the change, You opened the thread, There's no consensus to make the change. Meters ( talk) 05:55, 9 November 2019 (UTC)
The article is too technical, too lengthly-wordy-opinionated, relies too much on a handful of sources, and still omits roughly the past two decades of studies. For starters, a specific paper is continuously removed from mention, even though it is frequently discussed and won The Best Writing on Mathematics 2018 by the European Mathematical Society. Someone even moved the more recent talk page entries to the archive, why? prokaryotes ( talk) 10:20, 22 April 2019 (UTC)
>When they find that the M and W paper was indeed touted as a breakthrough in 2017, they probably are going to wonder what the breakthrough was, and what the difference is between the new hypothesis and the similar earlier ones. I, and I think many other editors of this page, share that puzzlement. I would ask those editors who feel it is important to mention M and W in Wikipedia to suggest how this should be explained to readers.< why explain? Better for readers to be puzzled by lack of explanation than lack of mention. For my own part I find M and W's exposition of the tablet so beautifully simple that whether it is historically accurate is neither here nor there. — Preceding unsigned comment added by 90.154.71.175 ( talk) 05:59, 9 October 2019 (UTC)