The Stephenson Abacus™,
A base-60 multiplication example.
Mathematica 29 (2002), 193–198, A
Note on Old Babylonian
Revised April 24, 2008 (copyright)
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