Plimpton 322 clay tablet

Originally posted on the RetroMat website as part of a page on programming the Sinclair Spectrum to generate Pythagorean Triples.

Source: wikipedia

Plimpton 322 is a Babylonian clay tablet containing a list of triangles composed of Pythagorean Triples, and their associated properties, in the form of a table with 4 columns and 15 rows ordered in descending sequence of (short side/long side) squared. 

These triples can be generated by using a method, such as the one we know as Euclid's Formula, and the following analysis refers to using that formula. Presumably the Babylonians used a similar method.

There are 15 triples implied on the tablet (it lists the short sides and hypotenuses, but not the longer sides). Of these, 13 are primitive. Eight have c > 1000 and the largest has a = 12709 and c = 18541. These are large triangles! (To generate the largest, m=125 and n=54). 

(I wonder what method they used to determine that these are co-prime and if they knew about what we now call Euclid's Algorithm? (Not to be confused with the above similarly-named formula!))

The tablet is broken and it is possible that there are missing columns at the left. Angles aren't used, but can be determined. If we define the angle as being next to the long side and hypotenuse, then angles* range from ~45 degrees at the top of the table to ~32 degrees at the bottom. The difference between the angles associated with successive rows is just under one degree on average, but actually varies from less than half a degree to nearly two and there seems to be no pattern to this.

Why were those particular triples chosen? I found that by increasing the maximum value for m to the 125 needed for the largest triangle in the tablet, the formula generates 441 triples in this range, providing many more suitable candidates for a more evenly-spaced table.

Why this range of angles? Perhaps it is one of a set - four such tablets could cover angles from 0 to 45 degrees (albeit with 60 rather than 45 entries!). For 45 to 90 degrees, the triples could be reused, switching the short and longer sides.

Was the tablet intended to be used as a trigonometric table, as we understand that term, with angles being present originally in a column now missing? (Presumably, angles would have had to be obtained by direct measurement of suitably scaled-down triangles). If so, it predates those of the Ancient Greeks, who are usually regarded as having developed the first ones, by 1500+ years! However, such a table would have been of limited practical use:

• the angles, if present, would not have been 'tidy' integers or rational numbers 
• there is an uneven difference in angle between successive entries
• the number of entries (15) also counts against (would have expected 14 for the range given).

Perhaps instead they were searching for triples that could produce (near) whole integer angles and this was a work in progress, a prototype of trig. tables to come?

Regardless, what I find truly incredible, perhaps even more than the information it contains, is that in spite of its great antiquity, it is in a form that we recognise today - an ordered numbered table consisting of rows and columns!

* angles of the triangles associated with the triples: 44.76, 44.25, 43.79, 43.27, 42.07, 41.54, 40.32, 39.77, 38.72, 37.44, 36.87, 34.98, 33.86, 33.26, 31.89.