Roman abacus

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A reconstruction of a Roman Abacus, made by the RGZ Museum in Mainz, 1977. The original is bronze and is held by the Bibliothèque nationale de France, in Paris.

The Romans developed the Roman abacus, an advancement over the previous Babylonian abacus. It was the first portable calculating device for engineers, merchants and presumably tax collectors. It greatly reduced the time needed to perform the basic operations of Roman arithmetic using Roman numerals.

1 Layout
2 Inference of Zero and Negative Numbers
3 Other facts
4 References

[edit] Layout

The Late Roman abacus shown here as a reconstruction contains seven longer and seven shorter grooves used for whole number counting, the former having up to four beads in each, and the latter having just one. The rightmost two grooves were for fractional counting. The abacus was made of a metal plate where the beads ran in slots. The size was such that it could fit in a modern shirt pocket.

 | |   | |   | |   | |   | |   | |   | |   | |
 | |   | |   | |   | |   | |   | |   | |   | |
 |O|   |O|   |O|   |O|   |O|   |O|   |O|   |O|

 MM    CM    XM     M     C     X     I     0    ~3
 ---   ---   ---   ---   ---   ---   ---   ---   ---
 | |   | |   | |   | |   | |   | |   | |   | |   | |
 | |   | |   | |   | |   | |   | |   | |   | |   | | )
 |O|   |O|   |O|   |O|   |O|   |O|   |O|   |O|   | |
 |O|   |O|   |O|   |O|   |O|   |O|   |O|   |O|   | |
 |O|   |O|   |O|   |O|   |O|   |O|   |O|   |O|   | |
 |O|   |O|   |O|   |O|   |O|   |O|   |O|   |O|   |O| 2
                                           |0|   |O|

The diagram is based on the Roman abacus at the London Science Museum.

The lower groove marked I indicates units, X tens, and so on up to millions. The beads in the upper shorter grooves denote fives—five units, five tens, etc., essentially in a bi-quinary coded decimal system.

Computations are made by means of beads which would probably have been slid up and down the grooves to indicate the value of each column.

The upper slots contained a single bead while the lower slots contained four beads, the only exceptions being the two rightmost columns, marked 0 and ~3.

The longer slot with five beads below the 0 position allowed for the counting of 1/12th of a whole unit, making the abacus useful for Roman measures and Roman currency. Many measures were aggregated by twelfths. Thus the Roman pound ('libra'), consisted of 12 ounces (unciae) (1 uncia = 28 grams). A measure of volume, congius, consisted of 12 heminae (1 hemina = 0.273 litres). The Roman foot (pes), was 12 inches (unciae) (1 uncia = 2.43 cm). The actus, the standard furrow length when plowing, was 120 pedes. There were however other measures in common use - for example the sextarius was two heminae.

The as, the principal copper coin in Roman currency, was also divided into 12 unciae. Again, the abacus was ideally suited for counting currency.

It is most likely that the rightmost slot or slots were used to enumerate fractions of an uncia and these were from top to bottom, 1/2 s , 1/4 s and 1/12 s of an uncia. The upper character in this slot (or the top slot where the righmost column is three separate slots) is the character most closely resembling that used to denote a Semuncia or 1/24. The name Semuncia denotes 1/2 of an uncia or 1/24 of the base unit, the As. Likewise the next character is that used to indicate a Sicilius or 1/48 th of an As which is 1/4 of an uncia. These two characters are to be found in the table of Roman Fractions on P75 of 'Numbers' by Graham Flegg. Finally, the last or lower character is most similar but not identical to the character in Flegg's table to denote 1/144 of an As, the Dimidio Sextula which is the same as 1/12 of an uncia.

The reconstruction of a Roman abacus in the Cabinet des Médailles, Bibliothèque nationale, supports this. The replica Roman hand abacus at Landesinstitut für Lehrerbildung und Schulentwicklung, shown alone here Replica Roman Abacus, provides even more evidence.

[edit] Inference of Zero and Negative Numbers

When using a counting board or abacus the rows or columns often represent nothing, or zero. Since the Romans used Roman numerals to record results, and since Roman numerals were all positive, there was no need for a zero notation. But the Romans clearly knew the concept of zero occurring in any place value, row or column.

It may be also possible to infer that they were familiar with the concept of a negative number as Roman merchants needed to understand and manipulate liabilities against assets and loans versus investments.

[edit] Other facts

The Roman abacus predates all records and specimens of the Suan Pan by centuries;
The Roman abacus has the refinements attributed to the modern Japanese Soroban; i.e. one bead above and four beads below the bar; and
The Roman abacus incorporates mixed-base arithmetic (in the two rightmost columns), another original enhancement by the Romans that is not present in any other abacus.