### August 13, 2013: While this site is interesting, my latest works are listed here: http://bit.ly/sks23cuMyACPubs The Stephenson Abacus™, An Introduction

by Steve Stephenson

Ancient Computers, a paper of mine posted 7/15/2010 on IEEE Global History Network.
It largely supercedes other material on this site. It determines:
1. that the Romans used The Salamis Tablet structure as their line abacus and used it for both duodecimal and decimal calculations;
2. that the Babylonians designed The Salamis Tablet abacus structure to accomodate their cuneform sexagesimal numbers;
3. how the pebbles were manipulated to do all four arithmetic operations in all three bases (10, 12, 60); and
4. that the methods of use are very rapid, the total number of pebbles needed very small, and the line abacus very portable.
Pebble Puzzle (first posted 6/14/2010)  Try to answer the questions on the first page before looking at the answers on subsequent pages.

News Flash! My latest thinking, as of August 14, 2009, is in these videos (each < 10 min.)
(I apologize for the amateurish video and audio production quality. Please concentrate on content.)

 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.

One major change between the methods in the videos and the rest of this site is in the position of the unit lines. The better method in the videos is to use the lines next to the semicircles as the unit lines. Then there is no need for complicated rules for radix point positioning.

That also makes the meaning of the semicircles clearer: i.e., a number is composed of two parts, a fraction and an exponent that represents the "whole" unit that the fraction is a part of, each with positive and negative parts. This interpretation resolves the "ambiguity" of Babylonian cuneiform numbers: all of them had the radix point at the left because they were all fractions of a whole; a whole that was explained by the context!

I've changed the text on this page, but not the figures, to reflect this. No changes have been made to the rest of the site (so much to do, so little time).

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:
• teach place value and the basis for the standard algorithms used to perform all four arithmetic operations;
• teach an appreciation of ancient numerology (Roman, Greek, Egyptian, and Babylonian) by associating ancient number symbols with the lines on The Stephenson Abacus™;
• foster an interest in math history; and
• provide scholars with a robust working model of what could be done with counting-board type abaci so that they can compare it to their historic artifacts.
The development of The Stephenson Abacus™ was inspired by:
• pictures of "the oldest surviving counting-board ... The Salamis Tablet, ... discovered in 1846 on the island of Salamis";
• subtractive number notations in Roman Numerals and ancient Babylonian numbers;
• the striking similarity between the ancient Roman Hand Abacus and the modern Japanese Soroban;
• Roman Numerals V, L, and D, and their counterparts in ancient Greek numerals,
• "5" beads on the Roman Hand Abacus and the Japanese Soroban;
• use of five and tens complements when manipulating the Soroban, a relatively modern and practical use of subtractive notation;
• doubling and halving arithmetic of the Ancient Egyptians, et al;
• lack of a "decimal" point in cuneiform sexagesimal numbers; and
• the very large exponential number system described in Archimedes' The Sand Reckoner.

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.

For a base-10, or decimal, counting board, a token on the units line represents a "one", a token on the line above it a "ten", and a token on the line below it "one tenth", etc. A token in the space between horizontal lines represents a value one half the line above it; e.g., a token in the space above the units line represents 5 (one half of 10). The vertical bar, the median, separates positive from negative. Tokens placed to the right of the median are positive, to the left negative.

The number 123,456,789 is represented in Figure 3. To differentiate them, digits are alternately registered away from or next to the median. The digits 3, 4, 8, and 9 are represented with both positive and negative parts, which we will call subtractive form: 3 = 5 – 2, 4 = 5 – 1, 8 = 10 – 2, and 9 = 10 – 1. Use of subtractive form dramatically reduces the number of tokens needed to represent many-digit numbers, and is essential for practical arithmetic on many-digit numbers (see analysis on pages 3-4 and 8-9 in an earlier paper). 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.    © 2006-2010 Stephen Kent Stephenson. Some Rights Reserved. 