Talk:Roman abacus

General comments
The stuff about the Roman empire being based on trad does not need to be in this article, or it needs to be drawn in to the article. Also, the bottom bullet points need to be NPOV'd Reid 01:24, Sep 1, 2004 (UTC)


 * Article on Roman mercantilism has been created. --Denise Norris 19:12, Sep 1, 2004 (UTC)

How many grooves?
Am I mis-counting, or is that actually eight long and eight short grooves? PhilHibbs 13:52, 1 Sep 2004 (UTC)


 * Now Addressed in Article --Denise Norris 19:11, Sep 1, 2004 (UTC)

Mixed-base arithmetic
Specifically how was mixed-base arithmetic used? And why doesn't the article say? Michael Hardy 16:46, 1 Sep 2004 (UTC)


 * Now Addressed in Article. --Denise Norris 19:10, Sep 1, 2004 (UTC)


 * Thank you. It's quite complicated. The whole article, like others about the Romans in wikipedia seems to be aimed at the Romans as a 'nation of shopkeepers,' to coin a phrase (quotes: "the abacus was ideally suited for counting currency"; "Roman merchants needed to understand and manipulate liabilities against assets and loans versus investments"), when Roman engineering achievements demonstrate that they understood algebra as well as the Greeks. In fact it is hard to imagine how they could have build great arched aqueducts and the Dome of the Pantheon without a form of pre-calculus, and the more complex features of the abacus were surely helpful in the calculations. 173.174.85.204 (talk) 17:22, 3 March 2017 (UTC) Eric

Paris library
The caption notes that the image is from the Paris library. There are quite a few libraries in Paris, to say the least. And is it on a wall in a library or in a book in a library? I do not think it is necessary to state in the caption where the image was taken, but if there is a statement about origin it should not be so vague. AlainV 17:03, 1 Sep 2004 (UTC)


 * Updated Article --Denise Norris 21:05, Sep 1, 2004 (UTC)

Fractional Columns
I have been trying to determine how the fractional columns are applied in the Roman Abacus, in part to determine what fractional representations where used (with the view that the Roman number system is essentially a transcription of the Abacus).

There are a number of other illustrations or pictures of abaci or replicas online. But the most interesting reference was to Frontinus on the Water Supply of Rome, which includes a number of fractional numbers, with symbology which can be related to the abacus (I dont have access to pre-typeset versions of this document, so I am relying on the referenced version and a couple of other online versions).

Fractional numbers are shown in Frontinus as: 'XVII S ==- £ 3 VI' (my spacing). The 'XVII' is the integer part, 'S' represents 'semis' (1/2) as necessary, '==-' (dots or dashes) mark uncia (1/12) upto 5/12th. The '£' is interpreted as half-uncia (1/24th), and appears to be the same symbol as written '~3' in the Wiki article. The '3' is actually ')-bar' (reverse-C with horizontal bar through centre), indicates scripulum (1/288), and the 'VI' is the number of scripula (upto XI, 'I' may be omitted).

Onto the Abacus...

Firstly, the Uncia column, labelled O-bar here, is fairly straight forward. Other illustrations show 5 lower and 1 upper bead, thus representing 1/12ths -- upper bead = 6/12 = 1/2, representing by 'S' (semis) in writing, lower beads 0-5/12, represented by dashes. This is essentially the same as the integer multiple columns to the left.

The three short fraction columns, labelled '~3' (cf. '£'), ')' and '2', appear as seperate columns with their own beads in some instances, and as a single column with three labels (as per the diagram) in others. I strongly suspect that the correct interpretation is as seperate columns, with single-columns variations being split into three domains for operation (after all, you cannot unsplit the first case).

The top column, labelled '~3' or '£' (or capital-sigma), contains a single bead, which maps to a representation of half-uncia, or 'semuncia'. This corresponds to the notation in Frontinus.

The middle column, labelled ')', holding two beads would represent 'sextulae', 1/6-uncia. These are not represented independantly in Frontinus.

Finally, the lower column, labelled '2', is still uncertain. If this represents scripula (1/288, 1/24 uncia) then there should be three beads (four scripula to the sextulae). The only clear examples show 2 beads -- but it is not impossible that one has been lost... The symbol '2' could be related to the ')-bar' used in Frontinus for scripula: ')' + '-' becomes ')' + '_' becomes '2'.

So, it does appear to be possible to reconcile the all the Abacus fields with the fractional numbers represented in Frontinus. Does this make sense?

--Sawatts 23:13, 7 Dec 2004 (UTC)

Historic and Copyright problems
There are some obvious historical problems with this article. It is by no means clear to me that Roman trade was the engine of the Empire, or even significant to the population. There is no evidence of trasference to China and it is not clear that the Roman counting board predates the Chinese abacus. With all due respect to the Romans, their counting board is a counting board and so similar to the Salamis counting board for instance, as well as other ad hoc counting devices going back to Sumer. It is not an abacus which has beads on wires or strings (as opposed to groves). On top of which this article looks awfully similar to some others on the internet. Which came first? Anyone know if this is someone else's work or not? Lao Wai 14:21, 7 August 2005 (UTC)

Fractional Columns Proposed Use
The main description has been updated with a shortened version of my submission. I include the whole argument in order to show the reasoning behind my proposition. Only this way is it clear why I posit the value of the lower section of the right hand column.

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. This is supported even more firmly by the replica Roman hand abacus at Jörn's Online Museum 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, denoting 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 values from 0 to 11 uncia where the upper bead has a value of six uncia.

The following example shows the unciae column with one uncia and the first column from 0/144 to 11/144 of an As or alternatively 0/12 to 11/12 of an uncia. Above each version of column 1 is the fractional number of an uncia in 12ths and on the right end the reduced fraction that each separate groove represents and below each column the reduced fraction of an uncia represented. The last line shows the total value of the two columns.

 | | | |  |O| 0    1     2     3     4     5     6     7     8     9    10    11         ---   ---   ---   ---   ---   ---   ---   ---   ---   ---   ---   ---   0   ~3| |   | |   | |   | |   | |   | |   |O|   |O|   |O|   |O|   |O|   |O| ---   |O|   |O|   |O|   |O|   |O|   |O|   | |   | |   | |   | |   | |   | |    1/2 |O| | |  )| |   | |   | |   |O|   |O|   |O|   | |   | |   | |   |O|   |O|   |O|   | |    |O|   |O|   |O|   | |   | |   | |   |O|   |O|   |O|   | |   | |   | |    1/4  |O|     |O|    | |   |O|   |O|   | |   |O|   |O|   | |   |O|   |O|   | |   |O|   |O|    |O| 2  |O|   | |   |O|   |O|   | |   |O|   |O|   | |   |O|   |O|   | |   |O|    1/12  |0|    |O|   |O|   | |   |O|   |O|   | |   |O|   |O|   | |   |O|   |O|   | |          0   1/12   1/6   1/4   1/3  5/12   1/2  7/12   2/3   3/4   5/6  11/12   Last column only        1/12 13/144  7/72  5/48  1/9 17/144  1/8 19/144 5/36  7/48  11/72 23/144  Both columns 

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 than 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, 1/9ths, 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 arithmetical 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.

Note that this determination differs from that of Sawatt above. This can possibly be due to different variations of abacii. The one referred to by me above was reconstructed from a relief on a column in Rome. It is obvious that the three slots in column one could only allow 1 bead in the top two slots and 2 in the bottom slot as is as shown on the relief. Other makers of such devices could have produced variations. I still feel it is more likely that the lower slot in the abacus at Replica Roman Abacus does not represent 1/3 unciae as this restricts the use of the device since 1 and 2 thirds unciae can be represented by 1/4 plus 1/12 and 2/12 unciae respectively.

Ray Greaves 2006

Similar to Soroban?... errr, not really
The construction of the Soroban has rods and beads, making it physically similar to the Chinese suanpan.

The Roman abacus has fewer rows than either the suanpan or soroban, and it is in grooves.

How "similar" can the two be? Not very. —Preceding unsigned comment added by Facial (talk • contribs) 22:11, 9 December 2007 (UTC)

There is a significant conceptual similarity, a similarity of ideas, an operational or functional similarity, call it what you like. Then, there is a slight insignificant difference in the way the two abaci realize this functionality.--Niels Ø (noe) (talk) 08:09, 10 December 2007 (UTC)

Have redirected roman arithmetic here
I have redirected Roman arithmetic here as there seems to be no evidence that they ever did arithmetic with Roman numerals. Possibly this article could have a little bit more about roman arithmetic but I can't think of much to say except that the evidence is they either used their fingers or an abacus. I can see no reason they would even do mental arithmetic with Roman numerals, Japanese schools teach using the soroban and people are that transfers well to them using a mental soroban, very often they're even faster and more accurate doing it mentally. Dmcq (talk) 19:31, 21 October 2011 (UTC)

Survey of Evidence
The article, as written, cites a number of theories and reconstructions, but there is no survey of the archaeological or literary evidence for the form of the abacus. On what do the reconstructors base their work? Are metal abaci found in the archaeological record? Where and from when? There needs to be some mention of *how* any of this information is more than mere conjecture.

Removed original research
I have removed the following section which seems to be original research. If citations can be provided, it can be reinstated, though without the repeated references to the author himself and his education. Mnudelman (talk) 14:15, 10 April 2015 (UTC)

Why was this removed from the main page? I have already attested that the main image is my creation and NOT a copy from elsewhere ( who else would include that picture that is evidence of my assertions?). Also, the person who removed it (Mnudelman) is no longer known on Wikipedia.

Secondly, he asked for citations, but not for which points. The work is entirely my own independent logic based on references given in the text.

Finally he make a disparaging remark about Repeated references to myself, where are these repetitions? Ruthe (talk) 00:54, 26 April 2016 (UTC)

There is however a third possibility.

If this symbol refers to the total value of the slot (i.e. 1/72 of an as), then each of the two counters can only have a value of half this or 1/144 of an as or 1/12 of an uncia. This then suggests that these two counters did in fact count twelfths of an uncia and not thirds of an uncia. Likewise, for the top and upper middle, the symbols for the semuncia and sicilicus could also indicate the value of the slot itself and since there is only one bead in each, would be the value of the bead also. This would allow the symbols for all three of these slots to represent the slot value without involving any contradictions.

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 one is using 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, it is logically impossible for there to be a rational progression of arrangements 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.



I have supplied a copyright for the image!!!Ruthe (talk) 00:05, 8 November 2016 (UTC)

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.

Tone of Lead Section
This lead section may not follow Wikipedia's Manual of Style. Parts are written with lots of opinion, i.e. "But language, the most reliable and conservative guardian of a past culture, has come to our rescue once more." The information is there, but it's just not presented properly. Bass77 (talk) 00:26, 15 February 2017 (UTC)

P.S.  Parts are also written using we/you rather than in the third person.