4.1 POSITIONAL NUMBER SYSTEMS 183 1685), 18-22, 301. (Web server info)
4.1 POSITIONAL NUMBER SYSTEMS 183 1685), 18-22, 301. The fact that any integer greater than 1 could serve as radix was apparently first stated in print by Blaise Pascal in De numeris multiplicibus, which was written about 1658 [see Pascal s QXuvres Compl&tes (Paris: hditions de Seuil, 1963), 84-891. Pascal wrote, Denaria enim ex institute hominum, non ex necessitate nature ut vulgus arbitratur, et sane satis inepte, posita est ; i.e., The decimal system has been established, somewhat foolishly to be sure, according to man s custom, not from a natural necessity as most people would think. He stated that the duodecimal (radix twelve) system would be a welcome change, and he gave a rule for testing a duodecimal number for divisibility by nine. Erhard Weigel tried to drum up enthusiasm for the quaternary (radix four) system in a series of publications beginning in 1673. A detailed discussion of radix-twelve arithmetic was given by Joshua Jordaine, Duodecimal Arithmetick (London, 1687). Although decimal notation was almost exclusively used for arithmetic during that era, other systems of weights and measures were rarely if ever based on multiples of 10, and many business transactions required a good deal of skill in adding quantities such as pounds, shillings, and pence. For centuries merchants had therefore learned to compute sums and differences of quantities expressed in peculiar units of currency, weights, and measures; and this was actually arithmetic in a nondecimal number system. The common units of liquid measure in England, dating from the 13th century or earlier, are particularly noteworthy: 2 gills = 1 chopin 2 demibushels = 1 bushel or firkin 2 chopins = 1 pint 2 firkins = 1 kilderkin 2 pints = 1 quart 2 kilderkins = 1 barrel 2 quarts = 1 pottle 2 pottles = 1 gallon 2 barrels = 1 hogshead 2 gallons = 1 peck 2 hogsheads = 1 pipe 2 pecks = 1 demibushel 2 pipes = 1 tun Quantities of liquid expressed in gallons, pottles, quarts, pints, etc. were essen- tially written in binary notation. Perhaps the true inventors of binary arithmetic were English wine merchants! The first known appearance of pure binary notation was about 1605 in some unpublished manuscripts of Thomas Harriot (1560-1621). Harriot was a creative man, who first became famous by coming to America as a representative of Sir Walter Raleigh. He invented (among other things) a notation like that now used for less than and greater than relations; but for some reason he chose not to publish many of his discoveries. Excerpts from his notes on binary arithmetic have been reproduced by John W. Shirley, Amer. J. Physics 19 (1951), 452-454. The first published discussion of the binary system was given in a comparatively little-known work by a Spanish bishop, Juan Caramuel Lobkowitz, Mathesis biceps 1 (Campan&, 1670), 45-48; Caramuel discussed the representation of numbers in radices 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, and 60 at some length, but gave no examples of arithmetic operations in nondecimal systems (except for the trivial operation of adding unity).