The Roman Abacus

If you're like me and enjoyed history at school, you may also remember being taught about the Romans in Maths. If your lessons were anything like mine, you may have been taught Roman numerals with the caveat that they are inferior to the modern Hindu-Arabic numerals that we use today and that to get around this deficiency they had to use an abacus. After this your maths teacher might quickly move on to something far more useful like Algebra!

This disparaging assessment of an empire that undertook some of the greatest civil engineering projects in history, not to mention the administration required to maintain said empire, doesn't seem very fair, to say the least!

Now the purpose of this post is not to get into the details of Roman numerals (that will surely come in another post) but this mysterious abacus that often gets mentioned. Perhaps you will be familiar with the Victorian abacus, often sold as a plaything for toddlers and never, or rarely, used to teach arithmetic these days. As this is the most common reference that most people have of anything called an abacus, this is often what is depicted.

A standard 10 x 10 bead abacus. Photo: Classroom Adventures

But this is not what a Roman abacus looked like!

Let's start at the beginning

According to writers like Ifrah, Menninger and Flegg, the earliest indications of an abacus is from the 3rd millennium BC in the Middle East (I'm assuming from written sources, like cuneiform tablets) but the earliest actual abacus that has ever been found is the Salamis tablet (found on the Island of Salamis) which dates to around the 3rd century BC.

Old black and white photograph of the Salamis Tablet with engravings highlighted.

Now before you think about walking off with this in your pocket... it's huge! And heavy! A solid slab of marble measuring approximately 150 x 75 cm and 4.5 cm thick, this thing was not designed to be moved. Presumably similar in nature to the medieval counting board of the Exchequer, this was something for calculating sums of importance and therefore needed to be large enough for enough eyes to be able to scrutinise said calculations.

But how is a slab of stone an abacus, it doesn't have any beads?

There is argument as to the origin of the word abacus. Some say that it stems from Ancient Greek, some say it comes from the Hebrew word for dust. It is very easy to imagine calculations being reckoned between traders by drawing in dust or sand. However, the slab at the top of a pillar in a Greek temple, before the lintel is called an abacus. Perhaps they are both true, we may never know.

Example of an abacus in Ancient Greek Architecture. Photo: Archaeotravel.eu

What was likely used, prior to the beads that we use today, were pebbles, and in fact the Latin for pebble is "calculi", which is where we get the word "calculate" from. And the Roman way of saying "to calculate" was "ponere calculi", or literally "to place pebbles". One problem with the ancients using pebbles for counting pieces is that they are non-descript. There is no way you would ever know that you had found a pebble that had been used on an abacus. Another possibility are finds of carved discs of bone or wood, but archaeologists aren't always able to work out if these are for games or counting. It is possible that these early abaci were used for both. I can certainly picture officials unwinding after some hard maths with a game on a ready made board. In fact, a known game from Ancient Greece, Pente Grammai, matches the top section of the Salamis Tablet perfectly.

To date, no Roman abacus in the form of a slab or board has ever been found. There are only three known depictions of a calculator in antiquity; the Darius Vase, the Etruscan Cameo and a 1st century gravestone now held in the Museo Capitolino, Rome, and these are only illustrative. It stands to reason that the Romans must have had an abacus like the Greeks but any attempt to reconstruct it would be completely speculative. There are some clever people out there who have made some guesses, but that is something we may visit another day.

From left to right: detail from the Darius Vase, the Etruscan Cameo, funerary stele held in the Capitoline Museum in Rome.

Okay let's see them...

What has survived is, quite literally, a handful of examples of a portable abacus that seems to have been invented by the Romans. This new abacus, or at least the examples that survive, was made of a small bronze sheet with slots cut out and loose-fitting rivets hammered into place. These rivets can slide in the slots to act as the counters. The surviving examples are all predominantly base 10 (we'll come onto that in a moment), with the caveat that no instruction manual has come down to us, meaning that their use is based on a mixture of intelligent guesswork, deduction and touch of speculation!

Confirmed Roman abacus finds. From left to right: Paris find (provenance unknown), Rome find (provenance unknown), Aosta find (found in a cemetery at St. Martin-de-Corléans)

Other useful examples of Roman abacai. From left to right: drawing from Marci Velseri's Letters (late 16th century), 18th century copy held in the British Museum, former exhibit from IBM's European Museum of Computing (now missing)

Now I did say that I wasn't going to delve into Roman numerals here, but what I will say is that everything that you were taught about Roman numerals at school was probably wrong, at least from a Roman perspective. When taught them at school we are taught relatively modern conventions, which no Roman from the 1st century would recognise. One major difference is that Romans didn't use M to represent 1,000, they used ↀ (among other things) and to add another loop would increase the value to the next power of ten: ↂ = 10,000, ↈ = 100,000. This symbol appears on the abacus finds and you can see another good example on the Columna Rostrata in Rome:

Engraving from the Columna Rostrata in Rome. Note the many numerals for 100,000! Contrary to popular belief Roman numerals went beyond M/1,000. Photo: Livius.org

All of the finds follow mostly the same format. All except the Aosta find (now illegible) have very similar numerals meaning that we can decipher most of it. Whilst the Paris and Rome finds are clearly missing rivets, the Aosta find confirms the number (and this ties in with the other examples). Also comparing with extant versions of the abacus, such as the Japanese soroban, we can get a pretty good idea of how it may have worked.

The Japanese Soroban is configured in the same way as the surviving Roman abacai, with 1 bead at the top and four beads at the bottom. Learning to use the soroban gives a great insight into how the Roman abacus may have been used. Photo: Classroom Adventures

So how does it work then?

The conclusion forms two parts to the Roman abacus. Part 1 is for integers (whole numbers) going from millions in the left-hand most column going down to ones in the third column from the right, and is set to work in base 10 arithmetic. Part 2 is not fully understood in its entirety (despite what some might say) but the consensus is that it is for fractions.

A replica of the Paris find. Photo: Classroom Adventures

The integer side appears to be pretty straight forward when compared with the soroban. We know that Roman numerals worked in base 10, but they also had symbols for the mid points, V = 5, L = 50, D = 500 etc. Therefore, we can easily deduce that the rivets in the lower slots have a value of 1 and the rivets in the upper slots have a value of 5.

Here is an example:

That's a big number!

As you can see, rivets away from the central line with the numerals are not counted, only the ones touching are active. If we look at the ones/units column we can see that the 5 bead is active and the 4 one beads are active. (1 x 5) + (4 x 1) = 9. Obviously, each column only needs to go up to 9, because 1 more than that and then we get ten, which is 1 rivet on the next column.

If you have a go with the soroban then you will learn pretty quickly that number-bonds (or complementary numbers) for 10 and 5 are essential for efficiency. If you look at the ones column again, all of the rivets are active. If you want to add 2 to this sum you clearly don't have any more rivets to add, and so (by using number bonds) you can complete this sum by:

"add 10, minus 8"

If you do that then you will be left with the correct new number "91" (or 2,615,391). If you are not familiar with number bonds, then this works because 8 & 2 are number bonds of 10, and adding 10 then subtracting 8 still leaves you with +2.

If you would like to explore this in more detail then I would recommend learning the soroban, there are a variety of lessons in different formats: online, books etc.

What about the Fractions?

What is universally agreed upon is that the second column from the right (the one to the left of the green line) is for twelfths. Romans had a word for twelfths "unciae". Unciae meant a measure of one 1/12, for example the thumb (called pollex in Latin) was sometimes called unciae because 12 thumb widths roughly equated to a foot (you are now itching to see for yourself aren't you). Base 12 is really handy for divisions and you can represent most common fractions in twelfths (which you cannot do with base ten): 1/2 = 6/12, 1/3 = 4/12, 1/4 = 3/12 etc.

Having an extra rivet in the bottom slot means that you can represent 11/12, and one more twelfth would equal a whole 1.

In this example we have set the integers to 0 and set the unciae to 7.

The last column is where the main disagreement is. Most resources will explain how this works with three separate slots as with Velseri's depiction:

Marci Velseri's depiction of a Roman handheld abacus (late 16th - early 17th century)

With the 3-slot model the top represents a half an unciae (a 24th), the middle represents a quarter of an unciae (a 48th) and the bottom represents either thirds, sixths or twelfths of an unciae depending on who you read (Menninger, Ifrah, Flegg etc.). The issue I have with all of this is that the explanation is made via (as far as I am aware) no actual archaeological example. All of the actual finds have a single slot and therefore we do not know if the 3-slot version was contemporary with the single slot version or was a revised idea post the fall of the Western Roman Empire. The Reading Ancient Schoolroom have made a suggestion that you could split the 4 beads into 2 groups of two, but this has no easy definite visual like with the rest of the abacus where the rivets are either on or off.

So, there you have it!

Well, at least as much of it as there is to have! What we know and have deduced about the Roman handheld abacus gives us enough to have a play and excitingly there is still room for discovery.

If you would like to have a play yourself, either for curiosity sake, or perhaps you would like to see if you can come up with a great theory on how to use the final fractions column then you can buy your own kit here: Roman Abacus Kit | History Adventures

Our Roman abacus kits are available in 2 models: Paris and Velseri so that you can choose which type you prefer.

Happy counting!