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.

Any help you could give would be much appreciated.

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 
should be able to give you pictures, references and more information. 
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