It’s clear that historically most people did calculation using some form of counting board or frame, and then wrote their finished calculations down using some symbols (roughly) directly representing the state of the counting board. The evolution of tally marks into unique symbols for each numeral (or the adoption of alphabetic letters for numbers) was a more recent development, often initially somewhat incoherent, and probably at first just used for recording calculations still done by abax. But the precise instructions/methods and most of the materials for doing the counting have by now been lost, since the former were mostly transmitted as an oral culture (much like early music, etc.) and the latter were often made of wood, cloth, hides, or lines in the dirt, etc. or confused by archaeologists for game boards and pieces.

Anyway, I’m curious to hear how people set up counting boards, ideally with some pictures of what is involved. There are many possibilities, including using two-sided counters or multiple types of counters, arranging various kinds of grids and assigning the sections of the grids multiple meanings, putting multiple counting boards side by side, using counting boards for accumulation with other scratch work done using pen+paper arithmetic to the side, etc.

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This is all apropos of an interesting site I came across by Steve Stephenson, a retired engineer (and more recently a retired secondary school math teacher) who spent a decade or two researching ancient counting boards and speculatively inventing possible algorithms for them.

His specific speculations are vaguely plausible but probably not quite historically accurate (again, there’s not a whole lot of direct evidence from, say, 500 BCE, or even 1000 CE), but I thought the system he worked up was pretty interesting.

His page is here http://ethw.org/Ancient_Computers

Looking at the Salamis Tablet and the Roman hand abacus and various Babylonian cuneiform tablets with sexagesimal numbers written on them, he speculates (a bit fancifully perhaps) that the basic form of the counting board went unchanged for ~2 millennia, and was very similar to the form of the Salamis tablet, originally designed for sexagesimal computations and later repurposed for decimal computations by the Greeks and for duodecimal fractions by the Romans. (Stephenson speculates that the Romans would use a {2:2:3} base for duodecimal fractions, but I personally recommend {4:3})

Salamis Tablet:

Here’s an example demonstrating his way of writing 1946

_{d}on such a table:

He proposes using the left column for negative counters, and the right for positive, allowing a significant reduction in the number of counters needed to represent any given number, and suggests that this is the origin of the IV = 4, IX = 9 conventions in writing Roman numerals, and certain unusually notated cuneiform numbers.

E.g. he proposes {A, B, C, D} in the above represent 19, {E, F, G, H} represent 18, and {I, J} might represent 18;20 (i.e. 20 – 1 – 0;40)

He then suggests that the extra grid lines at the top could be used to store an exponent, so that the full number would be “floating point”, think of scientific notation.

For solving multiplication and division problems, he proposes putting three such two-column grids side by side.

He made a series of YouTube videos explaining his proposals:

https://www.youtube.com/view_play_list? ... C6BA8D6F44

Anyway, I wouldn’t suggest that Stephenson’s speculation is necessarily historically accurate, but I think it’s a pretty nice setup for practical arithmetic, well worth trying, especially by anyone trying to come to grips with sexagesimal or similar bases.