This tablet is a round school tablet of unknown provenance from the
Old Babylonian period. It has a picture of a square with both diagonals
drawn in. On one side of the square is written the number `30`,
along one of the diagonals is the number `1,24,51,10` and below
it is `42,25,35`.

It is easy to see that `30` times `1,24,51,10` is `42,25,35`
(or, recalling that the reciprocal of `30` is `2`, that
`42,25,35`
times `2` is `1,24,51,10`). From the positioning of the
numbers,
the natural interpretation to make is that a square with side of length
`30` (or 1/2) has diagonal of length `42,25,35`. This
means
that the number `1,24,51,10` must be the 'coefficient of the
diagonal
of a square' and, indeed we do have an Old Babylonian coefficient list
that has this number (see MCT text Ue (YBC 7243)).

We know that the ratio of the side to diagonal in a square is 1 to
the
square root of 2. Since root(2) is irrational, it cannot be expressed
as
a finite sexagesimal number, so `1,24,51,10` can only be
approximate.
In fact, the square of `1,24,51,10` is `1,59,59,59,38,1,40`,
a remarkably good approximation to `2`

Why choose a side of `30` in a school exercise on diagonals?
Because it is the only choice that makes the diagonal of the square (`42,25,35`)
equal to the reciprocal of the coefficient (`1,24,51,10`) That
is,
`1,24,51,10` is root(2) and `42,25,35` is 1/root(2).
Given
the importance the Babylonians attached to reciprocals, this can hardly
be a coincidence. It is a nice exercise in algebra to see why you want
a square of side 1/2 rather than 1.

For a recent analysis of how Mesopotamian scribes might have
determined
the approximation used, see Fowler, D.H. and Robson, E.R. (1998).
'Square
root approximations in Old Babylonian mathematics: YBC 7289 in
context.'
*Historia Mathematica* 25, 366-378.

Some excellent large-size photographs of the tablet
along with some commentary and analysis are available on Bill
Casselman's website. I encourage you to visit there.

I thank Professor W.W. Hallo for permission to use the image of the tablet and Professor A. Aaboe for permission to reproduce his copy.

Go to Mesopotamian
Mathematics.

Last modified: 18 September 2006

Duncan J. Melvilledmelville@stlawu.edu