Roman abacus

The Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of 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.

As Karl Menninger says on page 315 of his book [Menninger, Karl, 1992. Number Words and Number Symbols: A Cultural History of Numbers, German to English translation, M.I.T., 1969, Dover Publications.] , "For more extensive and complicated calculations, such as those involved in Roman land surveys, there was, in addition to the hand abacus, a true reckoning board with unattached counters or pebbles. The Etruscan cameo and the Greek predecessors, such as the Salamis Tablet and the Darius Vase, give us a good idea of what it must have been like, although no actual specimens of the true Roman counting board are known to be extant. But language, the most reliable and conservative guardian of a past culture, has come to our rescue once more. Above all, it has preserved the fact of the "unattached" counters so faithfully that we can discern this more clearly than if we possessed an actual counting board. What the Greeks called "psephoi", the Romans called "calculi". The Latin word "calx" means 'pebble' or 'gravel stone'; "calculi" are thus little stones (used as counters)."

Both the Roman abacus and the Chinese suanpan have been used since ancient times. The Roman abacus' appearance, with one bead above and four below the bar, is systematically similar to the modern Japanese Soroban, although the physical construction of the soroban is clearly derived from that of the suanpan.

Layout

The Late Roman hand 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.

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O| |O| |O| |O| |O| |O| |O| |O
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
O| |O
"The diagram is based on the Roman hand 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 place value 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.Fact|date=October 2007 These latter two slots are apparently for mixed-base math, a development unique to the Roman hand abacus.cite web|title=The Roman Hand-Abacus | author=Stephenson, Steve | url=http://www.ee.ryerson.ca/%7Eelf/abacus/roman-hand-abacus.html | accessdate=2007-07-04]

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 first column was arranged either as a single slot with three different symbols or as three separate slots with one, one and two beads or counters respectively and a distinct symbol for each slot. 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 Graham Flegg's book [Flegg, Graham, 1984. Numbers, Their History and Meaning, Penguin Books] . 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.

This is however even more strongly supported by Gottfried Friedlein [Friedlein, Gottfried, Die Zahlzeichen und das elementare rechnen der Griechen und Römer und des Christlichen Abendlandes vom 7. bis 13. Jahrhundert (Erlangen, 1869)] in the table at the end of the book which summarizes the use of a very extensive set of alternative formats for different values including that of fractions. In the entry in this table numbered 14 referring back to (Zu) 48, he lists different symbols for the semuncia (¹/₂₄), the sicilicus (¹/₄₈), the sextula (¹/₇₂), the dimidia sextula (¹/₁₄₄), and the scriptulum (¹/₂₈₈). Of prime importance, he specifically notes the formats of the semuncia, sicilicus and sextula as used on the Roman bronze abacus, "auf dem chernan abacus". The semuncia is the symbol resmbling a capital "S", but he also includes the symbol that resembles a numeral three with horizontal line at the top, the whole rotated 180 degrees. It is these two symbols that appear on samples of abacus in different museums. The symbol for the sicilicus is that found on the abacus and resembles a large right single quotation mark spanning the entire line hight. The most important symbol is that for the sextula, which resembles very closely a cursive digit 2. Now, as stated by Friedlein, this symbol indicates the value of ¹/₇₂ of an "as".

If this refers to each of the two beads in this slot as ¹/₇₂th of an "as", then together they sum to ²/₇₂ths or ¹/₃₆th of an "as". Thus this slot can only represent ¹/₆th or ²/₆th (¹/₃rd) of an "uncia", contradicting the stated usage of representing ¹/₃rd and ²/₃rds of an "uncia".

If this symbol refers to the total value of the slot (i.e ¹/₇₂th of an "as"), then each of the two counters can only have a value of half this or ¹/₁₄₄th of an "as" or ¹/₁₂th of an "uncia". This then suggests that these two counters did in fact count twelfths of an "uncia" and not thirds of an "uncia".

A further argument which suggests the lower slot represents twelfths rather than thirds of an uncia is best described by the figure below. The diagram below assumes for ease that we are talking about fractions of an uncia as a unit value equal to one (1). If the beads in the lower slot of column I represent thirds, then the beads in the three slots for fractions of ¹/₁₂ of an uncia cannot show all values from ¹/₁₂ of an uncia to ¹¹/₁₂ of an uncia. In particular, it would not be possible to represent ¹/₁₂, ²/₁₂ and ⁵/₁₂. Furthermore this arrangement would allow for seemingly unnecessary values of ¹³/₁₂, ¹⁴/₁₂ and ¹⁷/₁₂. Even more significant is the fact that there would not be a rational progression of arrangements of the beads in step with increasing values of twelfths. It is only by employing a value of ¹/₁₂ for the beads in the lower slot that all values of twelfths from ¹/₁₂ to ¹¹/₁₂ can be represented and in a logical trinary, binary, binary progression for the slots from bottom to top. This can be best appreciated by reference to the figure below.

Alternate Usages of Column One

Penultimately 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, it must be noted that in one example found by the author, the first and second columns were transposed. It is left to the reader to ponder upon this and put forward their own interpretations of the use of these devices. No matter what the true usage was, what cannot be denied is that these instruments provide very strong arguments in favour of far greater facility with practical mathematics known and practised by the Romans.

The reconstruction of a Roman hand abacus in the Cabinet des Médailles, Bibliothèque nationale, supports this. The replica Roman hand abacus at [http://www.hh.schule.de/metalltechnik-didaktik/users/luetjens/abakus/rom-abakus-en.htm Landesinstitut für Lehrerbildung und Schulentwicklung] , shown alone here [http://www.hh.schule.de/metalltechnik-didaktik/users/luetjens/abakus/ab96.jpgReplica Roman Hand Abacus] , provides even more evidence.

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.

References

Additional sources

*Flegg, Graham, "Numbers, Their History and Meaning" ISBN 0-14-022564-1
*Ifrah, Georges, "The Universal History of Numbers" ISBN 1-86046-324-X