page_title: Roman abacus
section_title: Symbols and usage
hierarchical_section_title: Roman abacus / Symbols and usage
context_page_description: The Ancient Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of earlier abacuses like those used by the Greeks and Babylonians. 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 arithmetic using Roman numerals.
  As Karl Menninger says on page 315 of his book, "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.
context_section_description: 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 rightmost 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 sicilicus or 1/48 of an As, which is 1/4 of an uncia. These two characters are to be found in the table of Roman fractions on page 75 of Graham Flegg's book. 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 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 resembling 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 height.
  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. However, he stated specifically in the penultimate sentence of section 32 on page 23, the two beads in the bottom slot each have a value of ¹/₇₂. This would allow this slot to represent only ¹/₇₂ (i.e. ¹/₆ × ¹/₁₂ with one bead) or ¹/₃₆ (i.e. ²/₆ × ¹/₁₂ = ¹/₃ × ¹/₁₂ with two beads) of an uncia respectively. This contradicts all existing documents that state this lower slot was used to count thirds of an uncia (i.e. ¹/₃ and ²/₃ × ¹/₁₂ of an As.
  This results in two opposing interpretations of this slot, that of Friedlein and that of many other experts such as Ifrah, and Menninger who propose the one and two thirds usage.
  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 above. 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/
caption: Alternative usages of the beads in the lower slot
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