It largely supercedes other material on this site. It determines:

- that the Romans used The Salamis Tablet structure as their line abacus and used it for both duodecimal and decimal calculations;
- that the Babylonians designed The Salamis Tablet abacus structure to accomodate their cuneform sexagesimal numbers;
- how the pebbles were manipulated to do all four arithmetic operations in all three bases (10, 12, 60); and
- that the methods of use are very rapid, the total number of pebbles needed very small, and the line abacus very portable.

by

B.S., M.Eng.(Elect.), M.Ed.

High School Math Teacher

sks23cu AT gmail.com

© 2005-2008 by Stephen K. Stephenson. Some Rights Reserved

- Overview
Structure of Base 60 Counting Boards

Operations on Base 60 Counting Boards

Representing Digits as Sums of Five, Ten, Three, or Six Complements

How did the Babylonians actually do their base-60 numeric
calculations? This paper builds on a previous one,
*Ancient
Scientific Calculators*
(Stephenson), that
demonstrated methods to use copies of The
Salamis Tablet counting board for base 10 calculations with
numbers formed in what we call scientific notation.
This paper describes and demonstrates extending those methods to do
base-60 calculations as well. As a tutorial
example, the length of a diagonal of a square whose sides are
½ units long is calculated, duplicating the
results found on Yale
tablet YBC 7289.

Further extensions into other multiple base counting boards are also possible, but not explored here. These could have aided calculations in various metrological systems.

Figure 1 shows two possibilities for structuring a base-60 counting board following the format of The Salamis Tablet. Both structures assume a pebble on a space between lines is worth ½ a pebble on the line above it. In support of that assumption, the Sumerians had words for numbers that were 5 times multiples of 10 (Ifrah, p.94) and Greek and Roman written numerals have symbols for them (Ifrah, pp.182-200). The Salamis Tablet itself has inscriptions of Greek Attic (or Acrophonic) Numerals that include these 5 times multiples of 10 numbers. The Roman Hand Abacus and its more modern copy, the Japanese Soroban, have at each digit position four one-count beads and one five-count bead. A five-count bead, or pebble, can also be thought of as a ½-count of the next higher digit, and it is that concept that we carry over to the base-60 counting board. The practical impact of using ½-count pebbles is to significantly reduce the number of pebbles needed both to represent a number on the counting board, thereby freeing up space on the board, and those needed to carry around to do calculations.

From the structure of their cuneiform numerals, the
Babylonians obviously chose the second base-60 counting
board structure. Numbers are entered on the board in the form
±*a*×60^{±}* ^{b}*
where (0,0;0,0,0,1) ≤

To add or subtract numbers their exponent parts must be equal,
so one or both may have to be multiplied by appropriate
powers of 60. This is accomplished by moving the pebble representation
of the *a* part of a number up or down
the table by full sexagesimal digits and adding the number of digits
moved to the exponent part, *b*; an extremely
similar process to adding numbers in our scientific notation.

Positive numbers have their pebbles added to the right of the
median (vertical line), negative numbers have
their pebbles added to the left of the median. Subtraction is
accomplished by adding opposites: *x* - *y* = *x* + (-*y*). The
Chinese documented the use of "false" number rods (negative numbers)
about 200 B.C.E. (Ifrah,
p.287, & Burton,
p.236); and the Babylonians actually represented negative numbers
before
1300 B.C.E. (Burton, p.66).

Multiplication and division are accomplished by adding or
subtracting partial products until the multiplier
or dividend is exhausted, forming the product or quotient in the
process. See *Ancient
Scientific Calculators* (Stephenson)
for an example of multiplication on a base-10
counting board. The partial products are formed mostly by doubling or
halving powers of 60 times the multiplicand
or divisor. Since doubling or halving each pebble is pretty simple, no
multiplication table is really needed; but
sometimes other multiples of the four pebble locations are useful (even
they are simple; see Figure 2). That
the ancients used doubling and halving is well documented in the case
of the Egyptians
(Burton, pp.35-40), so the
technique was probably known to other ancients, including the
Babylonians.

The
MacTutor History of Mathematics site has an An
overview of Babylonian Mathematics page (retrieved 7/7/2005)
that states:

Perhaps the most amazing aspect of the Babylonian's calculating skills was their construction of tables to aid calculation. Two tablets ... give squares of the numbers up to 59 and cubes of the numbers up to 32. ... The Babylonians used the ... formula

ab= [(a+b)^{2}- (a-b)^{2}]/4 which shows that a table of squares is all that is necessary to multiply numbers, simply taking the difference of the two squares that were looked up in the table then taking a quarter of the answer.

To use this formula, and a table of squares of numbers from (;1) to (;59), to multiply two five digit sexagesimal numbers we would proceed as follows:

(;

a_{1},a_{2},a_{3},a_{4},a_{5}) (;b_{1},b_{2},b_{3},b_{4},b_{5})= (;

a_{1}) (;b_{1},b_{2},b_{3},b_{4},b_{5}) + (;0,a_{2}) (;b_{1},b_{2},b_{3},b_{4},b_{5}) + (;0,0,a_{3}) (;b_{1},b_{2},b_{3},b_{4},b_{5}) + (;0,0,0,a_{4}) (;b_{1},b_{2},b_{3},b_{4},b_{5}) + (;0,0,0,0,a_{5}) (;b_{1},b_{2},b_{3},b_{4},b_{5})= (;

a_{1}) (;b_{1}) + (;a_{1}) (;0,b_{2}) + (;a_{1}) (;0,0,b_{3}) + (;a_{1}) (;0,0,0,b_{4}) + (;a_{1}) (;0,0,0,0,b_{5})

+ (;0,a_{2}) (;b_{1}) + (;0,a_{2}) (;0,b_{2}) + (;0,a_{2}) (;0,0,b_{3}) + (;0,a_{2}) (;0,0,0,b_{4}) + (;0,a_{2}) (;0,0,0,0,b_{5})

+ (;0,0,a_{3}) (;b_{1}) + (;0,0,a_{3}) (;0,b_{2}) + (;0,0,a_{3}) (;0,0,b_{3}) + (;0,0,a_{3}) (;0,0,0,b_{4}) + (;0,0,a_{3}) (;0,0,0,0,b_{5})

+ (;0,0,0,a_{4}) (;b_{1})+(;0,0,0,a_{4}) (;0,b_{2})+(;0,0,0,a_{4}) (;0,0,b_{3})+(;0,0,0,a_{4}) (;0,0,0,b_{4})+(;0,0,0,a_{4}) (;0,0,0,0,b_{5})

+ (;0,0,0,0,a_{5}) (;b_{1})+(;0,0,0,0,a_{5}) (;0,b_{2})+(;0,0,0,0,a_{5}) (;0,0,b_{3})+(;0,0,0,0,a_{5}) (;0,0,0,b_{4})+(;0,0,0,0,a_{5}) (;0,0,0,0,b_{5})= (1;) [(;

a_{1}) (;b_{1})] + (;1) [(;a_{1}) (;b_{2}) + (;a_{2}) (;b_{1})] + (;0,1) [(;a_{1}) (;b_{3}) + (;a_{2}) (;b_{2}) + (;a_{3}) (;b_{1})] + (;0,0,1) [(;a_{1}) (;b_{4}) + (;a_{2}) (;b_{3}) + (;a_{3}) (;b_{2}) + (;a_{4}) (;b_{1})] + (;0,0,0,1) [(;a_{1}) (;b_{5}) + (;a_{2}) (;b_{4}) + (;a_{3}) (;b_{3}) + (;a_{4}) (;b_{2}) + (;a_{5}) (;b_{1})] + (;0,0,0,0,1) [(;a_{2}) (;b_{5}) + (;a_{3}) (;b_{4}) + (;a_{4}) (;b_{3}) + (;a_{5}) (;b_{2})] + (;0,0,0,0,0,1) [(;a_{3}) (;b_{5}) + (;a_{4}) (;b_{4}) + (;a_{5}) (;b_{3})] + (;0,0,0,0,0,0,1) [(;a_{4}) (;b_{5}) + (;a_{5}) (;b_{4})] + (;0,0,0,0,0,0,0,1) [(;a_{5}) (;b_{5})]

If we only wanted the five most significant figures in the
result, we could throw away the terms in red. Then
we have nineteen products to calculate using the formula *a*_{i}*b*_{j}
= [(*a*_{i} + *b*_{j})^{2}
- (*a*_{i}
- *b*_{j})^{2}]/4.
In (*a*_{i} + *b*_{j})^{2}
the sum will be of the form (1;*c*_{ij})
half the time, so that (*a*_{i}
+ *b*_{j})^{2}
= [(1;) + (;*c*_{ij})]^{2}
= (1;) + 2(;*c*_{ij})
+ (;*c*_{ij})^{2}.
On average, the calculation of each product would then require:
two additions, two table lookup, half a doubling, two subtractions, and
two halvings. Combining the nineteen products
will require another 18 additions, being careful to add into the proper
place value. In all, there are 56 additions,
38 table lookups, 8 doublings, 38 subtractions, and 38 halvings; a
total of 178 operations!

How would you keep track of all this in cuneiform? How many
errors would you make? How would you find them?
Is this a simpler process than using counting boards to multiply, like
the example in *Ancient
Scientific Calculators* (Stephenson)?

Whenever pebble counts are reduced on lines and spaces between
lines, digits should be represented as a sum
of their five, ten, three, or six complements, as in Figure 3.* **Ancient
Scientific Calculators* (Stephenson)
includes an analysis of the striking impact
this has on reducing pebble count and increasing pebble efficiency. The
board space freed by using these representations
allow adding a second number onto a board without immediately combining
it with the number already there. The new
number can then be double checked before combining with the old. This
reduces errors considerably.

Subtractive numeral notation was used by the Sumerians before 2500 B.C.E. (Ifrah, p.89), Babylonians (Burton, p.22), Etruscans (Ifrah, p.190, 38 = XXXIIX), and Romans (e.g., 4 = IV), with many other cultures using "back-counting" in their number names (Menninger, p.74).

The official methods to use the Japanese Soroban make extensive use of five and ten complements in order to "mechanize" operations "to minimize … mental labor … and let the result form … mechanically and naturally on the board" (Kojima, p.42-44).

The same thing should happen when using the operations described for a Salamis Tablet counting board.

To demonstrate the use of base-60 counting boards, let's
calculate the length of a diagonal of a square whose
sides are ½ units long (Yale
tablet
YBC 7289). If the length of the diagonal is *d*,
then we know, as did the Babylonians,
that *d*^{2} = (;30)^{2}+(;30)^{2}
= 2(;30)(;30) = (;60)(;30)
= (1;)(;30) = (;30). If we now think of *d*
as the side of another square, that square must have
an area of *A* := (;30). Squares
are rectangles whose width and length are equal. If we look at
rectangles that have area *A* and find one whose
width and length are equal, then we have found *d*.
We start with a guess for *d*, *g*_{0} = (1;).
The two dimensions of the rectangle must then be *g*_{0} and
*A*/*g*_{0}. Our
next
and subsequent guesses for *d*
will be the average of the previous two dimensions, *g*_{i}=
(*g _{i}*

- Burton, D. M.
- 1999
*The History of Mathematics: An introduction*. Fourth Edition. New York: McGraw-Hill. - Ifrah, G. [Bellos, D., Harding, E.F., Wood, S. & Monk, I., French Translators].
- 2000
*The Universal History of Numbers: From prehistory to the invention of the computer*. New York: John Wiley & Sons, Inc. - Kojima, T.
- 1954
*The Japanese Abacus: Its Use and Theory*. Tokyo, Japan: Charles E. Tuttle Co. - Menninger, K.
- 1969
*Number Words and Number Symbols: A Cultural History of Numbers*. Cambridge, Massachusetts: M.I.T. Press. - Stephenson, S. K.
- 2005 "Ancient Scientific Calculators"

©
2006-2008 Stephen Kent Stephenson. Some
Rights
Reserved.

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.

sks23cu AT gmail.com

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.

sks23cu AT gmail.com