Since the ancient Greeks used letters for their numbers, could they have used a counting system like an abacus that relies on a base?
Did the Greeks use the letters to do the calculations or only as names for numbers? Were the Greeks really baseless? What is meant by an abacus?
The letter-based numbering system may have started fairly late. Certainly by the time the Greeks were applying the names of their letters as the names of their numbers, they were already performing what we would call arithmetic operations. Using fingers and toes for counting must have come automatically to early humans, so even if the Greeks didn't think of arithmetic in terms of a base 5 or 10, they still would have practiced it, and the Greek word used for counting in Homer (pempathai) has the number 5 as its stem, suggesting that hands (and base 5) were indeed used in reckoning.
From Marks in the Dirt to the Abacus
Tallying (by making lines standing for individual animals) in the dirt and sand was an early form of counting that supplemented fingers and would presumably have been used when a shepherd or other animal tender had more than 10 animals to watch.
Roman numerals may have been a way of more permanently recording tallies.
At some point, making marks in the dirt with a pointed stick gave way to using stones. This could be thought of as a stage along the path to the abacus.
We have (Roman) archaeological evidence of manufactured stones. It is assumed that they -- or their equivalents found in nature -- were used for calculations.
- Each stone represented 1 sheep (or cow, or vote, or, in commerce, whatever a merchant wished to trade).
- The stones could be moved along finger-drawn rows in sand. The stones were also used on wooden counting boards and moved along rows that were either carved into the surface or painted on.
- Stones could be added or taken away as desired because it was not a closed system. This procedure would have been somewhat like using an open-ended bottom half of a modern abacus (which has stones threaded through and moved across a bamboo or metal pole divided into a top and bottom section in set ratios) except that the counting boards are thought to have been used to keep track of sums rather than to perform the actual calculations.
If the counting boards were not used for calculating, the calculations would instead have been done in the head.
The word "abacus" comes from the Greek, and there is a fragment from Lysias that specifically mentions the word abacus. There are other passages in ancient Greek literature that describe specific stone counting procedures.
Sources: "Herodotos and the Abacus," Mabel Lang, Hesperia,Vol. 26, No. 3 (Jul., 1957), pp. 271-288
The "Algorists vs. the Abacists: An Ancient Controversy on the Use of Calculators," Barbara E. Reynolds, The College Mathematics Journal Vol. 24, No. 3 (May, 1993), pp. 218-223
(www.ee.ryerson.ca:8080/~elf/abacus/history.html)The Abacus A Brief History