Roman Hand Abacus
The Roman hand abacus predated the Chinese and Japanese versions as described in the Wikipedia Abacus page. Other examples include this at the the London Science Museum, and probably the best example at the Online Museum.
This latter version clearly shows the seven leftmost columns for integer calculations in a decimal place value layout, from ones, then tens, hundreds and all the way up to millions. Each bead in the lower slot counted as one of that power while the bead in the top slot denotes five of that power. This clearly shows that the Romans were conversant with a decimally based place value system. The separation into values of one in the bottom slots and five times that value in the upper slots allowed for very easy translation of values from the Roman written numbers with their I and V for 1 and 5, X and L for 10 and 50, C and D for 100 and 500 and so on for the other powers, onto the abacus itself. Calculations could then be performed on the abacus and the result easily translated back to the Roman written format. It is unfortunate that they did not make the mental leap to a decimally based written place value format.
The second column from the right has 5 beads in the lower slot and 1 in the top. This column was a duodecimal based form of fractions, where each bead in the lower section was worth 1/12 and the bead in the top had a value of 6/12. This fitted with the usual Roman usage of duodecimal fractions for which they had names for 11/12, 10/12 down to 1/12 (and also a name for 1/8 which was one and a half twelfths), and duodecimal subdivisions of 1/12 down to 1/2304 (1/16 x 1/144).
The first column was used for fractions of 1/12. In some versions there was a single slot with three markings and in others three separate slots with the same markings, one to each slot. The top slot was for 1/2 of 1/12 and the middle slot for 1/4 of 1/12, while most current documents state the lower slot to be for thirds of 1/12. This is supported by the arrangement of beads with one bead in the top and middle slots each, and two beads in the lower slot.
There is however an alternative suggestion that seems to provide greater sense and regularity which claims the beads in the lower slot represent twelfths of a twelfth (1/12 x 1/12). This is supported by two strong pieces of evidence, that of the logical progression of values allowed only if the value of the beads in the lower slot are twelfths of a twelfth, and by the evidence from the tables of Gottfried Friedlein that states the symbol that resembles a digit 2 on the abacus denotes a value of 1/72 or 1/6 x 1/12. This can only be true if each bead has a value of 1/12 of 1/12 and the two together sum to 2/12 for a result of 2/12 or 1/6 x 1/12. This is described in full detail in the Wikipedia entry for the Roman Abacus.