Talk:Babylonian mathematics

Untitled

Part of this article seems to copied from http://www-history.mcs.st-andrews.ac.uk/HistTopics/Babylonian_mathematics.html

Babylonian multiplication formulae

How do we know that the Babylonians used

$ab={\frac {(a+b)^{2}-a^{2}-b^{2}}{2}}$

$ab={\frac {(a+b)^{2}-(a-b)^{2}}{4}}$

rather than

$ab={\frac {a^{2}+b^{2}-(a-b)^{2}}{2}}$?

If one is calculating from a table of squares of the numbers from 1 to $N,$ the last formula works for the product of any $a$ and $b$ that are both in the table, i.e. in the range $1..N.$ The two earlier formulae require that the sum, $a+b$, also be in the range $1..N.$ What evidence do we have that the Babylonians used the first two formulae, but not the third? Alma Teao Wilson (talk) 21:26, 16 October 2008 (UTC)

Is there any historical evidence that the Babylonians used any of these formulas for multiplication? I could not find any. Or did the authors presumed that they used these formulas because of the table of squares? Goutliebsf (talk) 01:50, 29 June 2021 (UTC)

Critique on the use of algebra for explanations

Using algebra to explain how babylonians did math suggests they set up equalities and derived new equations from old using an algebraic system This seems unlikely; the likely truth is that they assembled formula that were useful for solving particular problems either by insight or experiment. You need to include actual text or literal translations of their method.Mrdthree 23:58, 6 April 2006 (UTC)

Possible copyright violation.

I've removed some text some of which seems to have been copied from [1], as a potential copyright violation. Paul August ☎ 15:21, 21 June 2006 (UTC)

I've restored the text on the Chaldeans, most of which came from the Hipparchus article (although I'm not sure about the original sources). The Old Babylonian section however would need to be re-written without directly copying from the History of Mathematics archive. Jagged 18:16, 21 June 2006 (UTC)

I have trimmed, re-written, wikified and re-ordered the Old Babylonian Mathematics section. I think it is now sufficiently different from the source to be no longer copyvio, so I have put the re-written version back into the article. Gandalf61 13:20, 22 June 2006 (UTC)

Well done. Paul August ☎ 20:19, 22 June 2006 (UTC)

Name of the article

The current name (Assyro-Babylonian mathematics) was changed to without discussion, let ous establish the articles name based on WP:NAME.

• Find sources: Google (books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL hits:7 ; books: 32; scholar: 2
• Find sources: Google (books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL hits:12,500 ;books: 742; scholar: 642

Clearly the most recognizable name is Babylonian mathematics and not Assyro-Babylonian matematics. So what's the reason for the name change on the article? The TriZ (talk) 02:51, 26 November 2008 (UTC)

I'll change back the article to its original name. Feel free to reach consensus next time you move a page. The TriZ (talk) 03:03, 28 November 2008 (UTC)

Shoddy wording

In the opening paragraph: "...accurate to nearly six decimal places." Does this mean it was 'accurate to five decimal places'?--FimusTauri (talk) 12:37, 28 January 2009 (UTC)

Fixed it myself.--FimusTauri (talk) 11:30, 5 March 2009 (UTC)

Divisors

In the text, the divisors of 60 are given, but with 1 and 60 left out. —Preceding unsigned comment added by 86.137.170.8 (talk) 12:26, 28 July 2009 (UTC) 1 and 60 have now been added to the list. —Preceding unsigned comment added by 86.184.201.103 (talk) 09:16, 19 April 2010 (UTC)

Pioneers

The Babylonians are said to be pioneers in using the base 60. The article on Babylonian numerals correctly points out that it was inherited from the Sumerians. —Preceding unsigned comment added by 78.105.36.65 (talk) 14:45, 10 October 2009 (UTC)

Angles

The article starts out by crediting Babylonian use of base 60 with such things as dividing a circle into 360 degrees, and then later states that ancient Babylonian mathematicians had no concept of the measures of angles, but only used computations related to the sides of triangles, which seems to be a contradiction. Did they use triangle measure in their astronomy too? Or was it simply that they found computations based upon triangle sides easier for terrestrial use? — Preceding unsigned comment added by 166.70.15.233 (talk) 14:37, 14 August 2011 (UTC)

There is no direct evidence for the Babylonian use of angles in trigonometry yet. All the calculations seem to be purely arithmetic. They were very interested in calendars, and divided the zodiac (on which they based their calendar) into 36 parts. It seems to have been the Greeks who subdivided these 36 parts for a year to 360 degrees in a circle. The Greeks recognized the value of the sexagesimal system over the available alternatives and continued to use it. — Preceding unsigned comment added by 68.232.114.38 (talk) 04:23, 10 May 2018 (UTC)

Plimpton

"Though the table was formerly popularly interpreted by leading mathematicians as a listing of Pythagorean triples and trigonometric functions, in 2002 the Mathematical Association of America published Robson's research and (in 2003) awarded her with the Lester R. Ford Award for a modern day interpretation formally rejecting prior mathematical misconceptions."

This goes too far in several ways.

• Publishing historical research does not mean enshrining it as the ultimate truth. (The same goes for scientific research, for that matter, but in the case of history, this has to be emphasised particularly strongly.)
• For that matter, MAA is mostly an association of mathematics educators, not the main association of professional mathematicians (AMS). (This is of course by no means the main issue.)
• Two issues are mixed here. It may be more or less clear by now that interpreting Plimpton 322 as a trigonometric table may be an anachronism. At the same time, Plimpton 322 is a table of Pythagorean triples (in the sense of valid integer or rational lengths of a triangle), and the language of the headings suggests that it was also conceived as such.
• Robson's papers on the subject are written in a polemical style (as she herself says in one of them). This is common practice in parts of history (perhaps especially ancient history?) -- in order to make some headway, you have to be polemical. It is (a) not common practice among mathematicians writing about their subject, (b) not something that, by itself, makes her work a definitive rebuttal of previous work.

The following paragraph in her work is key:

[...] 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.

(E. Robson, "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322", Historia Math. 28 (3), p. 202). We would be much better off by putting this forward (as a view with strong support) than by being led off by an arguably inaccurate and overly enthusiastic interpretation of the polemics that precede it. Garald (talk) 08:57, 6 October 2011 (UTC)

Regarding the statement "The triples are too many and too large to have been obtained by brute force." I am no polished mathematician, but in my youth managed to develop a crude algorithm for generating Pythagorean triplets (PTs) in a few hours simply by pondering and playing with a hand written table of squares. This was pre-internet, with no reference to other mathematicians or mathematical works. So it shouldn't seem incredible that the Plimpton 322 tablet had either "too many" or "too large" examples. The size of algorithm-generated PTs are arbitrarily chosen by the size of the starting values used. See "Pythagorean Triplets" here on Wikipedia for more detail. — Preceding unsigned comment added by Sarookha (talk • contribs) 15:26, 26 November 2012 (UTC)

what the heck are you even talking about — Preceding unsigned comment added by 94.93.92.143 (talk) 15:02, 20 October 2015 (UTC)

The "spreadsheet" (for that is what it is) of Plimpton 322 is without question trigonometric. It is a list of fifteen Pythagorean Triple triangles, in roughly one-degree increments (bracketed between 45˚ to 30˚) which are all exactly solvable by sexagesimal notation (no approximations). (But a question: is it a characteristic of Pythagorean Triples that they are all exactly solvable by sexagesimal calculation? Or just the triangles on this list?) The value of the long side of the triangle is omitted, so hypothetically this is an exercise tablet given to students: which challenges them to calculate the value of the missing long side. A ‘giveaway’ is the item on line 11, which is a 3,4,5 triangle, but expressed as a 15x multiple (45,60,75) so it is more “complicated” for the hypothetical student to figure out. Alternatively, the tablet is broken on the left, so there could have been a column which listed the long side of the triangle (or rectangle).

The figures in column one are the square of the tangent of the short leg. So if you calculate the square root of the number in column one and divide it into the corresponding number in column two, you will get the length of the long side of the triangle. If you calculate the arctangent of the square root of the number in column one, you will get the value of the angle opposite the short side of the triangle. Pleroma (talk) 11:56, 31 August 2017 (UTC)

We (Wikipedia that is) typically add comments at the end (unlike other places on the web) and use indentation to clarify the different voices in a discussion. I have moved your comments accordingly and hope that you do not mind.

Your edit summary stating that this is fact and interpretation comes second was disturbing on several levels, but I'll limit my discussion. Plimpton 322 is a clay tablet with numbers on it, arranged in columns, and that is the only fact that there is. It did not come with a user's manual to tell us what these numbers mean or how they are to be used! For all we know for certain, these could just be the schedule of stops on the Babylonian underground . We look at these numbers and try to determine a pattern that makes sense to us–and that is an interpretation of the data we are looking at. There is more than one interpretation of the data, this is far from settled science, scholars are still positing their theories on what this all means. If this interpretation is to be used (and I am not objecting to that) it needs to be given a citation for readers to know the source, and should not be presented as fact! Especially not in the middle of a paragraph that says that such interpretations could be wildly off the mark. I reverted the inclusion so that someone who felt that this was necessary–there is a question here about an undue amount of detail in a general article on Babylonian mathematics–could do it properly.--Bill Cherowitzo (talk) 16:55, 31 August 2017 (UTC)

But certainly one of the analyses of the tablet must be trigonometric. If you take a number in the left-hand column - ignoring the (1) for the moment, and find its square root, and then divide this square root into the number in the same row's next column, the result is a whole number, and its value is the long leg of a Pythagorean Triple triangle, whose hypotenuse is the number in the third column of that same row. The arctan of that square root is (by our present reckoning) an angle. And the arrangement of the 15 rows is in decreasing values of that angle, bracketed by 45˚ and 30˚. This is so consistent that it should be taken seriously.Pleroma (talk) 23:24, 31 August 2017 (UTC)

Let's see. The first thing you do is to ignore the (1) because if you didn't the following computation would not work out the way you think it should. If one of my students did this I would congratulate them for being so imaginative and then flunk them for fudging the data. Now maybe this is the correct thing to do, I really don't know, but I do know that we are not going to find out if it is by staring at this piece of clay in isolation. I understand that there have been some very recent advances in translating these clay tablets, and perhaps this may shed some light on Plimpton 322 in due course. Meanwhile, every interpretation we refer to in our article needs to be supported by a citation.--Bill Cherowitzo (talk) 03:41, 1 September 2017 (UTC)

Don't flunk me yet! The 1 is bracketed (in the original cuneiform, then interpreted by Neugebauer? or Buck?) and that presumably means it is optional, to be included or not. If we do include it (I said "for the moment" in the paragraph above), and take the square root of that number, and divide the number in the third row (the hypotenuse) by this square root, we also get the long leg of the triangle. In the case of the first row: 1.9834028 is the number in the first column. Its square root is 1.408333. Divide that into 169 (the number in the third column) and you get exactly 120, the third side of the triangle (the long leg). This is true for all fifteen rows. Pleroma (talk) 10:21, 1 September 2017 (UTC)

You may be temporarily reprieved from the wrath of my red pen, but it still awaits in the wings. You present a compelling (as in "I can't think of a better one") argument, but not a conclusive (as in "all the evidence strongly supports this as the correct interpretation") one. There are two issues which send up red flags in my mind. The first is that the left side of the tablet is missing and anyone claiming to know what the numbers in the first column really are is making an assumption (sorry, I was being a little snippy about the 1 above, Neugebauer hypothesized that they might have been there, but there is no evidence for that assumption). And then there are the errors. An error in this context seems to be a value that does not fit the assumed computation! Most people assert that these are transcription errors, but there is a small possibility that maybe the assumption is incorrect. How does your assertion that the argument given is true for all fifteen rows relate to these errors. Does the calculation give the value in the tablet, or do you have to "fix" the clearly incorrect value to get it right! --Bill Cherowitzo (talk) 21:12, 1 September 2017 (UTC)

Calculus

Looks like they knew Calculus too: http://www.abc.net.au/news/2016-01-29/ancient-babylonian-text-earliest-use-of-calculus-for-astronomy/7121548 Yadojado (talk) 22:56, 28 January 2016 (UTC)

Dewi7 (talk) 16:22, 24 July 2021 (UTC)==Subject. translation paper with "Old Babylonian triangle coefficients list"==

Can you help me out on a reference to a cuneiform translation paper that I lost? About ten years ago, there was a translation paper that was purportedly on a tablet with a table with "Old Babylonian triangle coefficients list" or in different terms, Babylonian gradient coefficients. Maybe my memory is wrong, but the gist of the table is carried in the "Wedge" (triangle) coefficients numbers below. I have these numbers in a local spreadsheet too.I wrote something up on my webpage. Maybe, these were really rectangle coefficients? Possibly, this is a list of ukallu (modern 1/slope) for triangles or rectangles, called sa.gal in text, Sumerian for “constant_coefficient its.” Is there a bibliography of "Old Babylonian coefficients lists" Thank you. Note. On related triangle papers. articles at academia.edu, D.A.R. DeSegnac reports the triangles and other figures (trapezoid ladders) on MAH 16055 ..., and Erm 15189 ...7/24/2021 Dewi7 (talk) 16:07, 24 July 2021 (UTC)

normalized_adjacent_side in base_60, gradient fraction in rational numbers,normalized_opposite_side (1), degrees, cotangent ratio decimal tangent 1; 1 1 45.0 1.0 1. 0;30 2 1 26.56 2. 0.5 0;20 3 1 18.43 3. 0.333 0;15 4 1 14.04 4. 0.25 0;12 5 1 11.309 5. 0.2 0;10 6 1 9.462 6. 0.1666 0;09 7 1 8.130 7. 0.142 0;08 8 1 7.125 8. 0.125 0;07 9 1 6.340 9. 0.111 0;06 10 1 5.710 10. 0.1 0;09 1/7 1 8.130 7. 0.142 0;10 1/6 1 9.462 6. 0.1666 0;12 1/5 1 11.309 5. 0.2 0;15 1/4 1 14.036 4. 0.25 0;20 1/3 1 18.435 3. 0.333 0;30 1/2 1 26.565 2. 0.5 0;40 2/3 1 33.690 1.5 0.666 Note. Using modern concepts here for clarity (to my engr brain). Babylonians did not use modern angles and trig definitions in this manner. Note. MAH 16055 ca. 1900 – 1600 BC A HARMONY OF NUMBERS D.A.R. DeSegnac - A draft for an essay Note. On related triangle papers. articles at academia.edu, D.A.R. DeSegnac reports the triangles and other figures on MAH 16055 ..., and Erm 15189 ...7/24/2021 Dewi7 (talk) 16:07, 24 July 2021 (UTC) Dewi7 (talk) 16:22, 24 July 2021 (UTC)

Revert

An editor has reverted the following edit with reason UNDUE:

According to Dr.Daniel Mansfield, a researcher at UNSW Sydney, speaking about the "surveyor's tablet" Si.427 and its relationship with Plimpton 322, "The rectangles are perfect". The surveyor achieved this by using Pythagorean triples. {{reftalk}

This seems like overkill for two sentences in a section of the article body directly related to the edit, anyone care to comment? Selfstudier (talk) 17:01, 9 August 2021 (UTC)