by Steve Stephenson
Revised July 17, 2010 (copyright)
1 Basics 2 Addition & Subtraction + Roman Numeral Calc 3 Multiplication 4 Division 5 Base 60 Numbers 6 Base 60 Multiplication 7 Base 60 Division 8 YBC 7289 root(2) calc (3 parts) Conclusions: 9.1 The Salamis Tablet 9.2 Conjectures and Challenge to Historians 10 Roman Calcs (6 parts) With proof Romans used The Salamis Tablet. If them, probably the Greeks and Babylonians. |
The Stephenson Abacus™ is a counting-board type abacus with a particular structure and a set of methods to perform addition, subtraction, multiplication, and division on decimal (base-10) or sexagesimal (base-60) place-value numbers in the form ± c × b^{ ± p}.
Without violating the methods of operation, the parameter ranges for base-10 arithmetic are: 0.0000000001 ≤ c ≤ 0.9999999999, b = 10, and 0 ≤ p ≤ 9,999. For base-60 arithmetic they are: (0;0,0,0,0,1) ≤ c ≤ (;59,59,59,59,59), b = 60, and 0 ≤ p ≤ (59,59;).
The Stephenson Abacus™ can be used to do any modern engineering calculation without knowing or referring to addition or multiplication tables. If ancient peoples used The Stephenson Abacus™, it would have been their supercomputer.
The Stephenson Abacus™ may today be an inexpensive and useful tool to:As a quick example, here's how the Romans could have found the number of years between 1997 and 2000:
Figures 1 and 2 below show the structure of The Stephenson Abacus™. The two semicircles indicate that a number is composed of two parts, its coefficient and its exponent; like our scientific and engineering notations. The two ends of both arcs point to the positive and negative parts of the coefficient and exponent.
A base-60 number uses two horizontal lines and the space above them for each digit, so the coefficient lines are exactly enough to contain a number of the form h.hhhh, like a human hand's five digits: a thumb and four fingers. The exponent part has space for two digits; one not being enough. Perhaps the ancient Sumerians and Babylonians, and the scholars of ancient Greece, et al, used a counting-board similar to The Stephenson Abacus™ to do calculations with their base-60 place-value numbers.
A base-10 number uses one horizontal line and the space above
it for
each digit. Perhaps the ancient Romans, Greeks, Egyptians, and
Akkadians used counting-boards similar to The
Stephenson Abacus™ for their base-10 numeric
calculations; borrowing the basic structure from the base-60 board.
Numbers are represented with tokens placed on, and between, the
horizontal lines. The earliest tokens were pebbles.
"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.'" In the middle ages flat coin-like
tokens called
jetons
were used. Pennies make very good tokens.
Figure 4 shows the process of counting from 0 to 8 and demonstrates
converting 3 to
5 – 2, clearing positive-negative
pairs of tokens
(–1 + 1 = 0), and
promoting two 5 tokens to one 10 token. New tokens are entered
next to the bar, checked for correct entry, and then combined with
existing tokens and positioned away from the bar to make room for
the next entry.
Figures 5 - 8 show the steps in a series of additions of positive and negative numbers, i.e.:
+95,641 |
–380,276 |
+90,375 |
–6,813 |
–201,073 |
Notice in Figures 6 and 7 that intermediate results don't have to be immediately readable on a digit by digit basis. To make digits readable, simply add zero pairs of tokens, e.g., –1 +1 or –5 +5, in appropriate places. An example of how to make a result readable on a digit by digit basis can be seen on page 7 here, as can a base 10 multiplication example (which includes addition and subtraction).
A multiplication in base-60 is shown here and a division here.
This Excel model can be used to add, subtract, multiply, or divide in either base-10 or base-60. Or a paper model can be used with pennies as tokens (For each abacus print 3 copies, then cut, tape, and annotate to match the figures. Do it three times to make a set of three for multiplication and division).
Here are some historical conjectures related to The Stephenson Abacus™. And here are some references with comments.