of the beads in step with unit increasing values of twelfths. Likewise, if each of the beads in the lower slot is assumed to have a value of 1/6 of an uncia, there is again an irregular series of values available to the user, no possible value of 1/12 and an extraneous value of 13/12. It is only by employing a value of 1/12 for each of the beads in the lower slot that all values of twelfths from 1/12 to 11/12 can be represented and in a logical ternary, binary, binary progression for the slots from bottom to top. This can be best appreciated by reference to the figure below. Alternative usages of the beads in the lower slot It can be argued that the beads in this first column could have been used as originally believed and widely stated, i.e. as ½, ¼ and ⅓ and ⅔, completely independently of each other. However this is more difficult to support in the case where this first column is a single slot with the three inscribed symbols. To complete the known possibilities, in one example found by this author, the first and second columns were transposed. It would not be unremarkable if the makers of these instruments produced output with minor differences, since the vast number of variations in modern calculators provide a compelling example. What can be deduced from these Roman abacuses, is the undeniable proof that Romans were using a device that exhibited a decimal, place-value system, and the inferred knowledge of a zero value as represented by a column with no beads in a counted position. Furthermore, the biquinary nature of the integer portion allowed for direct transcription from and to the written Roman numerals. No matter what the true usage was, what cannot be denied by the very format of the abacus is that if not yet proven, these instruments provide very strong arguments in favour of far greater facility with practical mathematics known and practised by the Romans in this authors view. The reconstruction of a Roman hand abacus in the Cabinet, supports this. The replica Roman hand abacus at, shown alone here, plus the description of an Roman abacus on page 23 of provides further evidence of such devices. References Further reading Abacus Ancient Roman technology Roman mathematics