The Stephenson Abacus(TM)
See how to use this line-abacus, and its 4000+ year history, at http://bit.ly/sks23cuMyACPubs. Drag pebbles from the cache to the right onto appropriate lines and spaces. Note that 3 = 5 - 2, 4 = 5 - 1, 8 = 10 - 2, and 9 = 10 - 1. The X's mark the unit lines. 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 built-in error checking in the number entry process. There is extreme pebble/token efficiency: Multiplication or division of two base-10 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 base-60 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. Using Heron's method to calculate the square root of 2 in sexagesimal to the precision on Yale tablet YBC 7289 takes only 25 minutes; blindingly fast for that time (c. 1800-1600 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.