They used base 60.
There are no angles indicated on the tablet.
(I still wonder what the table header says; seems like it could shed some light on the purpose of the table.)
A propos not of this tablet: the Babylonians knew how to approximate irrational square roots, if necessary.
They mean exact in the sense that the numbers have a finite representation in base sixty. I’m a mathematician, and I believe you are or have been a math teacher, and I’m pretty sure we can agree that that is a fairly non-standard definition of exact.
In this article by Eleanor Robson there is a transcription she made (either from a photograph or from the actual tablet— it does not say), and it is stated that
The text was written in Akkadian, probably between 1822-1784 BC
The columns are labelled, from left to right:
So, perhaps not crystal-clear to the uninitiated, but at least one can read the text, and the reference to sides and diagonals proves that triangles are involved (which was obvious to begin with, from the numbers)
Yes, but it doesn’t mean that what they said is “pure bullshit.” It simply means they are using a non-standard meaning of the word. A better criticism is to say that what they said tortures the meaning of the word “exact.”
One of the fairly standard usages of “pure bullshit” is “tortures the meaning of X”.
IMO “This trig table has more exact numbers” doesn’t mean “This trig table has numbers of greater precision or accuracy”. It means “This trig table has a greater quantity of entries that are rational = exactly representable as a fraction.”
IMO the word “exact” is fine. The confusion arises from whether “more” means greater exactness or greater quantity. I sure read it the first way the first few times I saw the phrase. But the second way make a lot more sense and isn’t crazy.
As Chronos says, if the tangents, secants, etc. are rational then the angles aren’t.
I’m going to suggest that for ancient construction purposes, they *probably *built nearly everything using integer quantities of their standard length unit (cubit, etc?). These integer measurements were applied to rise & run or to North/South & East/West. Then they simply let the slope angle fall wherever it did. They cared about the length of the 3rd side of the triangle so they could make parts of the correct length or estimate materials requirements. So it was convenient for them that that value also came out integral in their units.
What they did not need was computing or measuring angles. So the fact their trig tables resulted in irrational angles was immaterial.
We do care about rational angles, but we build lots of deliberate arbitrary angles and use tools like transits to measure angles and then convert freely between angles & lengths. We also have the precision to measure and make a part 100.123 meters long if we need to. Unlike them, we’re not afraid of a few extra sigfigs on the right of the [del]decimal[/del] er [del]sexigecimal[/del] … er … radix point.
Or so it seems to me.
This was impressive math earlier in history than previously expected. So an impressive feat of current anthropology. But not anything of interest to current mathematicians. Math historians, sure. But not mathematicians. Or engineers.
Sure, a table like this would certainly be useful for many practical purposes. That doesn’t really make it a trig table, though, and it’s definitely not a better trig table than what we have now.
Are the entries in this table sorted by one of the ratios (equivalent to sorting by angle), or are they sorted by the length of one of the sides?
It is sorted by the first column, which is equivalent to by angle. The next thing to look at is whether there is an obvious pattern to the entries; Robson claims there isn’t one, the recent article mentioned has an elaborate theory, and there seems to be various speculation floating about. Might be fun to think about, but lack of familiarity with all the other tablets could be a handicap.
I agree with LSLGuy.
Angles are not used at all, only ratios of lengths.
Each line of the table contains rational numbers, and the table systematically covers a wide range of triangles of different proportions. In base 60 there are far more triples than in base 10.
When you do a calculation using a line of the table, you will get out a rational number as a result. No precision is lost when multiplying, as with modern trig tables, which are calculated to a certain number of significant digits.
To solve a real world problem, you pick the two closest relevant lines of the table, do the calculation with each (getting two exact rational numbers), and then interpolate. The Babylonians knew about interpolation. You end up with a pretty accurate result.
Whether rational trigonometry is a better approach than the one we use is another question. One of the authors of this paper is Prof. Norman Wildberger, Associate Professor of Mathematics at the University of New South Wales. He has written a book, and also has many YouTube videos and articles, proposing that rational trigonometry is a better system. Whether he’s right or not is a question for mathematicians, but he is certainly a reputable academic.
Here is a review of Wildberger’s book Divine Proportions by Prof. William Barker (Professor of Mathematics at Bowdoin College). His conclusion is: “Divine Proportions is unquestionably a valuable addition to the mathematics literature. It carefully develops a thought provoking, clever, and useful alternate approach to trigonometry and Euclidean geometry. It would not be surprising if some of its methods ultimately seep into the standard development of these subjects. However, unless there is an unexpected shift in the accepted views of the foundations of mathematics, there is not a strong case for rational trigonometry to replace the classical theory.”
Not needed for whatever problems and applications are represented by the particular tablet in question. I wonder if there is evidence of it elsewhere?
They were not afraid of mathematical precision, but precision engineering and construction was somewhat limited 4000 years ago compared to today. π approximated as 25/8 was considered close enough.
How is “rational” trigonometry in any way different from the trigonometry we all know? All I see from your link is more of an emphasis on some of the (nowadays) lesser-known trigonometric functions like haversine.
Since I blurted that out, I should point out that if you simply list all primitive Pythgorean triples (a,b,c) where b is of the form 2[sup]x[/sup]3[sup]y[/sup]5[sup]z[/sup] and not too big, then sort them by decreasing value of the first column (c/b)[sup]2[/sup] from 2 on down, then you can essentially reconstruct the table. The triple (175,288,337) should be next?
This was obvious when the tablet was first discovered. It still does not tell us for sure how the table was originally produced, whether or not it is a fragment of a larger table with more rows and/or columns, for what purpose it was used, and what were the precise algorithms.
Unfortunately it’s beyond my education, but the linked Wikipedia article seems to make a big deal of it supposedly eliminates the need to use infinite series.
It’s equivalent. Rational trigonometry just uses a different approach to derive the same information. Some people think this approach has pedagogical value, others just like it for aesthetic reasons, because it eliminates a lot of messy numerical solutions in favor of algebraic ones.
What trig table do we have now ?
I think that Mansfield is over emphasising accuracy.
There’s a mistaken conflation between the perfectness of the table, and the way that the sides have finite representations in base 60, and accuracy.
So you go to build one of these perfect triangles? Well all measurements have an error margin, every time you make a 65 ’ long edge, you make it with some inaccuracy.
Mansfield is putting pure maths ahead of the real world.
And for that reason there is no great signifance in a “trig table” of right angle triangles with sides of finite represenatation … Such high accuracy of the values in the table ? Wow they are perfect values, you are going to build a perfect triangle ? no because each real world measurement has an error. Every single one, (except “countable” things… but thats not measurement, thats counting.). And how to get of very high (but not perfect) accuracy for angles not in a table ?
So do we have “better trig tables”. Yes and no, depending how pedantic you want to be. We actually use , by way of calculator or computer, the first 10 (or something suitable to the number of decimal digits) values from the also perfectly accurate series expansion of the function. Because then stored values of 10 (or whatever) coefficients is enough to store a table that would be otherwise extremely huge (and take way more time to access. )
That is one of the reasons some Oriental historians dispute that the table was a general-purpose trigonometric table, and suggest it was used in certain algebra problems instead.
60 like 12 for dozen - is useful due to the number of different factors. I assume they mean this helps simply fractions.
Oops. to clarify - a dozen can be divided in 2,3,4,6 shares…
Sixty, even more. 2,3,4,5,6,10,20,30.
Before decimal calculations were commonly known and taught in school, this simple feature was pretty useful for everyday activities.
(Same reason, IIRC, why a pound used to be 12x20 pence)
Don’t forget 12 and 15.
One question upthread was about the Babylonian 360° circle. Presumably they had six groups of 60° to correspond to the six sides of an inscribed hexagon. Did they have six names for these six parts?
(I’ve asked this question before: Left/Right/Up/Down, N/E/S/W are among the ways to label four sides or four directions, but is there a “nice” way to label six sides? [Starting with 8 directions N/NE/E/SE, etc. and discarding two is NOT a “nice” way to label six sides.])
Now *that * is an interesting question and one that never occurred to me. :smack: Can you link to your earlier musings on this? I’d like to learn more.
Any answer to which consists IMO of four parts:
In terms of dependencies I think 1 & 2 can be answered in either order. Maaybe 2 is largely a supercase of 1 and ought to go first. 3 depends on 1 & 2 and 4 depends on 3.