The Stephenson Abacus™, A base-60 multiplication example.

by Steve Stephenson
Revised April 24, 2008 

In Historia Mathematica 29 (2002), 193–198, A Note on Old Babylonian Computational Techniques,
Jens Høyrup wrote,

Analysis of the errors in two Old Babylonian “algebraic” problems shows
(1) that the computations were performed on a device where additive contributions were no longer
identifiable once they had entered the computation;
(2) that this device must have been some kind of counting board or abacus where numbers were
represented as collections of calculi;
(3) that units and tens were represented in distinct ways, perhaps by means of different calculi.

It has been known for more than a century that Babylonian calculators made use of
tables of multiplication, reciprocals, squares, and cubes. It is also an old insight that such
tables alone could not do the job—for instance, a multiplication like that of 2 24 and 2 36
(performed in the textVAT 7532, obv. 15, ed. [Neugebauer 1935, I, 294]) would by necessity
require the addition of more partial products than could be kept track of mentally, even if
simplified by means of clever factorizations. It has therefore been a recurrent guess that the
Babylonians might have used for this purpose some kind of abacus—Kurt Vogel [1959, 24]
also pointed to the possibility that the creation of the sexagesimal place value system might
have been inspired by the use of a counting board.

Below are snapshots of performing the multiplication of 2 24 and 2 36 on The Stephenson Abacus. It's such a straightforward process that the question arises: Could it be that the Old Babylonians used a counting board very similar to The Stephenson Abacus?

Note that the multiplication makes no use of any memorized or written multiplication tables; relying instead only on addition of partial products that are produced by duplicating tokens, and by doubling and halving tokens.

© 2006-2008 Stephen Kent Stephenson. Some Rights Reserved. Creative Commons License
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