Associated Topics || Dr. Math Home || Search Dr. Math

### Roman Counting Instruments

```
Date: 04/24/98 at 09:17:58
From: Helen Griffith
Subject: Roman Numerals/Counting
I noticed your letter of 10/19/96 from Doctor Mason about the Romans
contribution to Maths.

http://mathforum.org/dr.math/problems/lovejoy.10.18.96.html

I am researching a schools programme for Channel 4 Schools about Maths
from History, and one of the programmes is on the Romans. I too have
found that they did not contribute a great deal to mathematics, but
that they were great builders and surveyors. However, I have not come
across the board with grooves and holes that you described in your
answer and wondered if you could tell me a little bit more about it -
how it was designed and used, and what it looked like.

Many thanks,

Helen Griffith
Television Junction
```

```
Date: 04/25/98 at 10:46:22
From: Doctor Jen
Subject: Re: Roman Numerals/Counting

It's been said that the Romans' greatest contribution to mathematics
was the killing of Archimedes - but that's neither here nor there.

Counting boards were forerunners of the abacus we know today. No
examples of Roman counting boards survive today (although Greek
boards, which were similar, survive), but we know they existed from
pictorial evidence, although this is scarce. There is also linguistic
evidence which shows that the Romans used pebbles on a board for
calculations; the Roman expression for "to calculate" is "calculus
ponere" - literally, "to place pebbles". When a Roman wished to
settle accounts with someone, he would use the expression
"vocare aliquem ad calculos" - "to call them to the pebbles."

The earliest counting boards had columns marked on them; naturally,
grooves were more convenient, since the pebbles would then be
prevented from rolling off. The grooves were known as "alveoli" and
the pebbles as "calculi" (from which we get our word "calculate").
At the top of each groove was marked the number represented by each
counter in that column.

Thus the number 5316 is represented as shown below:

(|)  C   X   I
| o | o | o | o |
| o | o |   | o |
| o | o |   | o |
| o |   |   | o |
| o |   |   | o |
|   |   |   | o |

Note that (|) is an early form of M and seems to have been the form
used on counting aids at this time.

Addition is carried out by putting counters representing the number to
be added onto the board, and then carrying groups of 10 counters into
the next column on the left. Similarly for subtraction, but not for
operations such as multiplication.

At some time (I have not been able to find an indication of when;
records seem to be somewhat hazy) the Romans introduced a horizontal
line crossing the decimal columns. Counters above the line count as 5
corresponding units.

On this type of board, 5316 is given as:

|   |   |   |   |
| o |   |   | o |
(|)  C   X   I
|   | o | o | o |
|   | o |   |   |
|   | o |   |   |

This type of board had an advantage in that fewer counters were
required.

In addition to counting boards, the Romans also used hand abaci, which
developed from the counting board, and were smaller and more portable.
Two examples of these survive, and according to my references, one of
these is in Paris, and the other is in Rome.

On both types of large counting board, you could represent two or more
numbers by placing more counters on the board. On a hand abacus more
counters could not be added, as the counters were fixed in their
grooves and were moved against the dividing bar as required. The
grooves below the dividing bar contain four counters; those above
contain one counter, which again counts as 5 corresponding units.

On an abacus of this type, only counters hard up against the dividing
bar are in the number.

Now 5316 looks like this. It is essentially the same, but with the
surplus counters present.

|   | o | o |   |
| o |   |   | o |
(|)  C   X   I
|   | o | o | o |
| o | o |   |   |
| o | o | o | o |
| o |   | o | o |
| o | o | o | o |

Abaci and counting boards were not restricted to the four columns
given above. The hand abacus in Paris apparently has eight. These are
_
|X|   (((|)))  ((|))   (|)   C    X   I   O   @   )   2

10^6    10^5    10^4  10^3  100   10  1

The symbols O, @, ) and 2 require clarification. They represent
unciae. Note that 12 unciae make 1. So:

O is 1 uncia: 1/12
@ is 1/2 uncia [@ would be something like an L ]
) is 1/4 uncia [ ) represents a backwards C ]
2 is 1/3 uncia

Roman abaci survived in Europe after the collapse of the empire, most
likely in Christian monastic communities, but there is no further
record of calculation on them until the tenth century AD, by which
time they had departed significantly from their original form.

I would think that any reasonable text about the history of numbers
One author I have found particularly helpful on the subject is Graham
Flegg, but I am not sure his books are available outside the UK
(unless you are in the UK, in which case you are all right).

Anyway, hope this helped some!

-Doctor Jen,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```

```Date: 08/20/2002 at 08:08:43
From: Steve Stephenson
Subject: Roman Hand Abacus layout

Hi,

I just came home from a vacation in London where, in the Science Museum,
I saw a replica of a Roman Hand Abacus.

Here's the London Science Museum's Roman Hand Abacus layout, where the
~3 was actually a symbol that looked like a 3 that was flattened on
the top then flipped top to bottom and right to left, or rotated 180
degrees:

| |    | |    | |    | |    | |    | |    | |    | |
| |    | |    | |    | |    | |    | |    | |    | |
|*|    |*|    |*|    |*|    |*|    |*|    |*|    |*|
_____
| X | (((I))) ((I))   (I)     C      X      I      0     ~3

| |    | |    | |    | |    | |    | |    | |    | |    | |
| |    | |    | |    | |    | |    | |    | |    | |    | | )
|*|    |*|    |*|    |*|    |*|    |*|    |*|    |*|    | |
|*|    |*|    |*|    |*|    |*|    |*|    |*|    |*|    | |
|*|    |*|    |*|    |*|    |*|    |*|    |*|    |*|    | |
|*|    |*|    |*|    |*|    |*|    |*|    |*|    |*|    |*| 2
|*|    |*|

The abacus was made of a metal plate where the beads ran in slots. The
size was such that the abacus could fit in a modern shirt pocket.

Note the longer slots below the 0 and ~3 positions, the 5 beads in the
lower slot of the 0 position, the 2 beads in the lower slot of the ~3
position, and the absence of an upper slot in the ~3 position. I
wonder what the ')' and '2' symbols along the right side of the ~3
slot meant?

Obviously the units in the 0 position were 1/12 of the I position,
and the units in the ~3 position were 1/3 of the 0 position. So the
upside down reversed 3 character seems appropriate to represent
1/3; or, more likely, our symbol for 3 came from the Roman symbol for 1/3.

Regards,
-Steve Stephenson
```
Associated Topics:
Elementary Math History/Biography
Middle School History/Biography

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search