Introduction
My first Abacus publication was The Roman Hand Abacus (2003). But then my studies continued.As an engineer and math teacher, my approach was to use historical clues to figure out the structure of an ancient counting board abacus and the way it was used. The clues were: The Salamis Tablet, the Roman Hand Abacus, and the Japanese Abacus (Soroban). The sources listed below this Introduction contain my analyses and conclusions.
Some surprising results:
 All four arithmetic operations are accomplished with surprising ease.
 Multiple number bases can be used (e.g., decimal, sexagesimal, or duodecimal).
 Numbers are held in exponential form; i.e., each number has two parts, a fraction and a radix shift (exponent),
 so no decimal point is needed, and no trailing zeros.
 Each number part can contain both positive and negative values.
 There's some builtin error checking in the number entry process.
 There is extreme pebble/token efficiency:
 Multiplication or division of two base10 numbers with 10 digits in their fraction part and 4 digits in their exponent part can be expressed with an average of about 1.9 x 14 x 4 ~= 100 pebbles or tokens. Just a small bag of pebbles for a Roman and two rolls of pennies for a modern user.
 The Ancient Babylonians using base60 numbers with 5 digit fractions and 2 digit exponents would only need on average about 2.9 x 7 x 4 ~= 80 pebbles or tokens.
 In the videos I use Heron's method to calculate the square root of 2 in sexagesimal to the precision on Yale tablet YBC 7289 in only 25 minutes; blindingly fast for that time (c. 18001600 BCE). Especially since the Babylonians wrote their numerals on clay tablets with reed styluses that they cut with bronze knives sharpened on stones  no paper or pencil.
Ancient Computers, Part I  Rediscovery, Edition 2:
Free online prepublish paper. (+ older version at http://arxiv.org/abs/1206.4349.) 

Paperback Book, ISBN 1490964371 Kindle eBook, ASIN: B00DVPPQ78 

Back and Front Covers 
Ancient Computers, Part II  Video Users' Manual:
DVD1. How To Use A Counting Board Abacus (First 12 of these free YouTube videos; but DVD has no ads.) 

DVD2. How Romans Used A Counting Board Abacus (Remainder of these free YouTube videos; but DVD has no ads.) 
Ancient Computers, Part III  You Do It:
Three Tablet Abacus on Paper, 36 in. x 24 in. You may have the file printed for your personal use, or I'll have it printed and mailed to you for $25. 

Three Tablet Abacus on Wood (later)  
Three Tablet Abacus on Fired Clay (later)  
Three Tablet Abacus on Marble (later)  
Three Tablet Abacus in GeoGebra 
Ancient Computers, Part IV  You Do It, Advanced:
In video 9.2, position 2:20, I issue a challenge to historians. I ask them to calculate the square root of 2 in sexagesimal their way using clay and reeds and to record their entire process in a publicly viewable internet video; then to compare their time to my time of 25 minutes using a counting board abacus.I point out that I'm a rank beginner abacist and that speeds approaching a Soroban master may be possible.
I challenge you to use a counting board abacus to beat my time and publish your video on YouTube. Please leave a comment on video 9.2 with a URL link to your video so people can verify the correctness of your solution and your time.