Roman abacus
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The Romans developed the Roman abacus, an advancement over previous Greek counting boards. 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.
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 groves 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.
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MM CM XM M C X I 0 ~3
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|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
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The diagram is based on the Roman abacus at the London Science Museum.
The leftmost seven rows are used in whole item arithmetic and the abacus can count up to ten million. The two rightmost slots where used to count fractions - 1/12ths and 1/3rds.
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.
The rightmost position, the ~3, with only two beads, allowed the counting of 1/3rds of a whole unit. The use of this position is not clear but could be used to count partially full containers.
I do not agree with the statement above regarding the rightmost column of beads for the following reasons.
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. This is supported by a combination of factors as follows. 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. Although these characters are not identical, it is clear that a wide variation did occur over the decades and centuries this system was in use. According to 'Ifrah' this lower slot is used for 1/3 of an uncia and is quoted by him as 2/72nds or duae sextula. However why denote a value as 2/72nds when the Romans had a symbol and a value for 1/36 = 2/72, the duella. Furthermore, why introduce another non-essential value when that value of 1/3 of an uncia can be represented as 1/4 + 1/12 of an uncia with one bead up in the lower slot and the single bead up in the middle slot? Introducing 2/72nds breaks the simple progression by 1/144ths up to 11/144ths, then next to 1/12 or 12/144ths by moving all beads in the first column down and moving one bead in the uncia column up?
Further evidence of this conjecture is provided by the format of the reconstruction of a Roman abacus in the Cabinet des Médailles, Bibliothèque nationale, Paris shown above. This is supported even more firmly by the replica Roman hand abacus at Landesinstitut für Lehrerbildung und Schulentwicklung shown alone here Replica Roman Abacus
As is obvious, the lower right hand slot has 2 beads, while the upper two slots have only 1 bead. This makes sense only if the lower slot is used for counting 1/12 s of an uncia. Starting with all beads in the down position, the lower two beads can be used to count 1 then 2 twelfths of an uncia. Adding one more twelfth is accomplished by sliding the lower two beads down and moving the single 1/4 bead up. The two lower beads are then moved up one at a time to add first 1 more then another twelfth to give 5/12 of an uncia. Adding one more twelfth is done by moving the lower three beads down and moving the single top bead up, denotin 6/12 or 1/2 of an uncia. This same process is extended to progress up to 11/12 when all four beads in the three slots are in the top position. Adding one more 1/12 of an uncia is acheived by moving all four beads down to their bottom positions and moving one bead up in the uncia column. This second column has two slots with five beads in the lower and one bead in the upper section to allow for all valuse fro 0 to 11 uncia where the upper bead has a value of six uncia.
Since the second column from the right is used to enumerate unciae as most will agree the symbol in that column is that for unciae, and since I have shown that the first column is for 1/2, 1/4 and 1/12 of an uncia, then this first column is in effect counting units of 1/144. Thus the Roman hand abacus was even more sophisticated that generally believed. With graduations of 12ths and 144ths, the Romans could express a very wide range of fractional values including 1/2s, 1/4s, 1/8ths (1/8 = 1/12 + 1/24), 1/3rds, 1/6ths, 1/12ths, 1/18ths, 1/24ths, 1/36ths, 1/72nds 1/144ths and multiples of these. I am sure that many Roman merchants, builders and craftsmen had more than a passing knowledge of such aritmetical calculations, otherwise there would have been no need for such a sophisticated device. The fact that this form of abacus predated Chinese models also indicates the level of Roman mathematical knowledge that allowed them to build those long lasting constructions that we are familiar with 2000 years later.
Ray Greaves 2006
:Flegg, Graham, "Numbers, Their History and Meaning" ISBN 0140225641 :Ifrah, Georges, "The Universal History of Numbers" ISBN 186046324X
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.
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.