What can go wrong during scaling? Anything. As in any stage. With this pretty unhelpful opening...
One of the recent weekends, when reading up on Inca history, I came across something which, to my shame, I wasn't familiar with. And that made me go “wow”. And – boom, the rest of the day was gone. And then one more day the following weekend. That something was called “counting board”. I started looking at what else was used for counting and when, and as a result connected some dots that I didn´t expect to connect. So, this post is about counting and simple math.
Use what you have
There is an obvious reason why majority of cultures rely on a decimal system. Using 10 fingers for counting is called finger-counting (surprise, surprise) or dactylonomy to use a smart name. According to Wikipedia it was used as early as ancient Egypt – but I suspect some version was used much earlier. I simply can’t believe that very early on someone didn’t figure out that you can match objects to fingers.
Other body-based systems were used as well. For example, pre-Columbian Maya used base-20 counting systems by adding (you guessed it) finders on their feet.
Fingers can actually take you a surprisingly long way past counting 10 things. Here´s an example. And for those interested, here’s a tutorial of how it’s done. Check out the same guy’s channel for more complex operations, like multiplication or dates.
This system, called Chisanbop, was created only in 1940s. I am guessing in the ancient days the finger-based apparatus was more limited and eventually run out of power. What do you do next? You look for things around you. You find stones, sticks, and other things you can count with. Then someone (actually, many someones in different locations and times) came up with a simple but powerful idea.
The rolling stones
“The counting board is the precursor of the abacus, and the earliest known form of a counting device (excluding fingers and other very simple methods). Counting boards were made of stone or wood, and the counting was done on the board with beads, pebbles etc”, Wikipedia counting board
The first example was found on the Greek island Salamis in 1846. It is appropriately named the Salamis Tablet and dates to 300 BC, making it the oldest counting board discovered so far. This video shows how it was used.
But it wasn´t the only type in use. Babylon used conceptually the same devices in between 300 to 500 BC, as did Romans and many others. The Chinese started to use counting boards around the 4th century BC, with sticks made of wood or bones, which were placed vertically or horizontally in various combinations. I don’t speak Chinese, but at first glance you can see traces of those sticks in modern Chinese numerals.
Here´s an example of representing a number on a Roman counting board
The Romans came up with the first found portable calculating device for engineers, merchants, and presumably tax collectors. This one is 11.5 x 7.2 cm and dates to c. 300 AD
In the 1st century BC, Romans also used the wax abacus, a board covered with a thin layer of black wax on which columns and figures were inscribed using a stylus.
The ancient Egyptians also used something like a counting board, although no physical evidence exists (or I didn’t try hard enough to find it). The counting frame was mainly a flat surface on which pebbles were moved from right to left to perform basic counting operations. Egyptians were also using a “sand abacus”, a table or tablet on which calculations were written in sand or dust, and then easily erased.
The Incas
Now to the one that started this whole theme for me – the Inca counting board called yupana. Since Incas had no writing system, no records of how to use yupana are left, so some people argue that yupana was a board game. But to me it looks a lot like a counting board. And the fact that the name yupana, is derived from Quechua yupay (count) seems like a bit of a giveaway as well.
There are several types and I looked at two. The first one is very similar to all above, except it has 5, 3, 2 and 1 grovels on the rows. One interesting fact to point out is that 1, 2, 3, 5 is a Fibonacci sequence base.
The other one is rather sophisticated and a lot less straight forward than the ones above. It has pockets of different shapes and several levels. Below you can see how it looked and what values were assigned to each compartment (the values are a theory that can’t be proven due to the fact that no living Incas or written instructions are left). The bottom level is Fibonacci based as well (1,2,3), while the two higher levels break away from that sequence and need multiplying by 6 and 12 respectively.
Now, Incas were different not only in doing different shapes of the counting boards. While having no writing they did develop a way of storing information called quipu. One way to think of it is that yupana is a short-term memory, a device used during a calculation (RAM in computer terms or a piece of paper), while quipu is a long-term storage (disks or books). In short, it’s a very clever information storing system made of strings of various lengths with different knots on them. These book equivalents stored both numeric and word-based information. Check out Quipu in Wikipedia.
They obviously were not very useful for doing calculations – tying and untying all these knots would probably confuse you more rather than help in doing a calculation. So, the calculations were done in yutanas and stored in quipu. Think of it as “annual company accounts are prepared in yutanu and then reported accounts are stored in the filing cabinet in the form of quipu”. Maybe with a copy sent to the tax office as well?
Who said “glyphs”?
I spent quite a bit of time trying to find counting boards used by the Maya civilisation before giving up. They didn’t seem to have them. But they did have numbers of course - a unique vigesimal (base-20) numeral system.
The Maya was the only civilisation in the Americas, and only one of five cultures in the world to develop a proper writing system. They had paper and paper books. Which probably explains why my time searching for “maya counting board” was wasted. They used paper and wrote on all kinds of other surfaces. How very modern of them and can I have that hour of searching back please.
Here are Maya numerals in the Dresden Codex (Maya paper book) with explanation and examples of math operations
By the way one more thing I learnt is that “glyphs” is not a name for Maya writing, but short for “hieroglyphs”.
Turning it inside out
It only makes sense that at some point someone (again, probably several someones in different parts of the world) said: “Carrying all these stones or sticks is cumbersome, losing them is annoying, why don’t we attach them to the board?”. Enters abacus, called by the Latin word abacus originating from Greek άβακασ – board or table, so the inheriting relationship is undeniable.
Mesopotamia or Sumerian civilization, one of the oldest civilizations in human history, used the first abacus to count between 2700bc to 2300 BC.
In the 2nd century BCE, the Chinese chipped in with Suan-pan. Unlike the simple counting board, suanpan techniques have been developed to do multiplication, division, addition, subtraction, square root and cube root operations at high speed. I am not sure at which point and in which exact sequence they came in, but it surely moved things along.
A few derivatives were developed. One of them I remember having at home as a child. Very easy to use and adds and subtracts rather efficiently. While adding you move the beads on the rod to the same side of the frame (up/down or right/left). You start with the rod that has lower decimal position value and once it fills up you use a bead on the higher value rod and reset the lower value one.
Pre-Columbian Americans (namely Aztecs) had something called Nepohualtzintzin, which is similar, but is more rooted in a human body. The section with four beads is equivalent to the model of the four fingers of the hand, with the fifth, the thumb, representing the full hand and being in turn represented in the three-bead section, where each bead has the value of five (=fingers=full hand). That gets you to 20, the maximum in a 20-based numeric system. The Nepohualtzintzin has 13 columns, lines or vertical axes, each representing the main joints of the body: two ankles, two knees, two groin, two wrists, two elbows, two shoulders and neck.
What does Chancellor of the Exchequer do?
Spoiler: he (ac)counts. But let’s come back to it.
Those who used counting boards in their shop or office would probably misplace it every now and then, wouldn’t they? So why not fix it to some surface, say, hm, I don’t know, a table? They did and it stuck. And then someone must have asked – why don’t we do a whole counting table instead of a small tablet? Everyone else said, yeah, let’s - and they did.
When that came to England, the French were the higher class. In Old French they had a word coming from medieval Latin “scaccarium” and meaning ‘chessboard” – “eschequier”. So, it evolved to The Exchequer, first a table, then, in medieval England, an office responsible for the collection and management of the royal revenue. Another version says that the name came from the chequered tablecloth on which accounts were kept by means of counters. To save mental cycles, the Exchequer (the office) was named after a table used to perform calculations for taxes and goods. 10 feet by 5 feet with a raised edge or "lip" on all sides of about the height of four fingers to ensure that nothing fell off it. Counters representing various values were placed on the tabletop. The top had cloth with stripes of about the breadth of a human hand in a chequer-pattern. The spaces represented pounds, shillings and pence. Wikipedia article here.
Mechanising pebbles
In 1600s people came up with mechanical calculators, a couple of which are shown below. Basically, this is taking columns and playing with what´s in them. In one case columns are replaced by wheels, in another with linear moving parts. To me, fundamentally, it´s building on the same idea, while nudging it forward.
Which was then leapfrogged to an arithmometer, an invention of the 1800s. The first one was called Thomas Arithmometer, below. It´s really fascinating to watch this thing in action here while recognising familiar features – vertical sliders that represent digits and are used to input the value of the operands (instead of pebbles), digits rotating in a window building on wheels from before from the columns or rows from before that. A pretty clever addition is that the whole panel with inputs could move from left to right shifting decimal positions.
How far have we evolved
Turns out – not very far on the fundamental level. The same concepts and methods are used today. The easiest way to showcase that is to show a couple of counting aids for kids. Same pebbles and same grooves, albeit in a different composition and with a different purpose. And, of course, the abacus is the same.
But the thing that got me when I realised it is the relation of counting boards to the math methods, which we all use - long addition, subtraction, multiplication, division. Take a look at this school worksheet I downloaded from Internet to recognise familiar checkerboard pattern of the counting boards with decimal places arranged in exactly the same way. Is doing it with a pen on paper all that different from the Romans writing in wax, the Egyptians on sand or moving pebbles on a checkerboard surface?
On second thought, I shouldn’t be surprised by this. It is inevitable because it’s dictated by the fundamental rules of math and its reliance on decimal positions. But I still find it mystical and beautiful that we not only rely on the same rules as the people in the ancient world, but even use the same shape of things to apply these rules in practice.
You could be sitting next to an ancient Greek or an ancient Chinese person jointly doing a calculation using the same tools. No language barrier.
Ain’t that cool?
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