Location arithmetic

Location arithmetic (Latin arithmeticae localis) is the additive (non-positional) binary numeral systems, which John Napier explored as a computation technique in his treatise Rabdology (1617), both symbolically and on a chessboard-like grid.

Napier's terminology, derived from using the positions of counters on the board to represent numbers, is potentially misleading because the numbering system is, in facts, non-positional in current vocabulary.

During Napier's time, most of the computations were made on boards with tally-marks or jetons. So, unlike how it may be seen by the modern reader, his goal was not to use moves of counters on a board to multiply, divide and find square roots, but rather to find a way to compute symbolically with pen and paper.

However, when reproduced on the board, this new technique did not require mental trial-and-error computations nor complex carry memorization (unlike base 10 computations). He was so pleased by his discovery that he said in his preface:

it might be well described as more of a lark than a labor, for it carries out addition, subtraction, multiplication, division and the extraction of square roots purely by moving counters from place to place.