Жон Напиердің арифметикалық есептеу құралы

One of three devices to aid arithmetic calculation described by John Napier in a treatise

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:

Napier proceeded to the rest of arithmetic, that is multiplication, division and square root, on an abacus, as it was common in his times. However, since the development of micro processor computer, a lot of applicable algorithms have been developed or revived based on doubling and halving. Doubling is done by adding a numeral to itself, which mean doubling each of its digit. This gives an extended form, which has to be abbreviated if needed. This operation can be done in one step by changing each digit of a numeral to the next larger digit. For example, the double of a is b, the double of b is c, the double of ab is bc, the double of acfg is bdgh, etc. Similarly, multiplying by a power of two, is just translating its digits. To multiply by c = 4, forexample, is transforming the digits a → c, b → d, c → e,

Halving is the reverse of doubling: change each digit to the next smaller digit. For example, the half of bdgh is acfg. One sees immediately that it is only feasible when the numeral to be halved does not contain an a (or, if the numeral is extended, an odd number of as). In other words, an abbreviated numeral is odd if it contains an a and even if it does not. With these basic operations (doubling and halving), we can adapt all the binary algorithms starting by, but not limited to, the Bisection method and Dichotomic search.

Division can be carried out by successive subtractions: the quotient is the number of time the divisor can be subtracted from the dividend, and the remainder is what is left rest after all the possible subtractions. This process, which can be very long, may be made efficient if instead of the divisor we subtract multiple of the divisor, and computations are easier if we restrict to multiple by a power of 2. In fact, this is what we do in the long division method.

This process requires one to add counters to the abacus (board) to make square figures. The top of page 149 shows diagrams that explain this process. Begin by placing a single counter on the board (it will actually go on one of the dotted squares). Adding three other counters adjacent (or with blank rows and columns between them and the first one placed) will result in another square figure on the abacus. Similarly adding another five counters to this (with or without the blank rows and columns shown) will result in an even bigger square. Take the number to be considered and put counters along one margin that represent its value. From the position of the largest counter in that value, follow the diagonal lines (bishop’s moves) across the board until you come to a square with a dot. Place a counter on that square. Subtract the value represented by this single counter from the original number in the margin. Add three (five, seven, for subsequent steps) to create a square on the board and subtract the value of the
added counters from the number in the margin until the number is either too large to be subtracted or there is no space left on the board. You should be left with a large square of counters (perhaps with blank rows and columns between them) on the board. Move one of the counters in each row of the square to the margin and the positions of these marginal counters will yield the square root of the number. Napier provides an example of determining the square root of 1238. The largest counter is in the 1024 position so the first counter is placed on the dot found by moving down the 1024 diagonal (at the 32,32 position). Subtracting this value (1024) from the original number leaves counters at 128,
64, 16, 4 and 2 (= 214). Placing three counters on the board to form a square with the first counter but whose value can still be subtracted from 214, results in counters at positions 32,2; 2,2; and 2,32 (whose values are 64, 4 and 64, which when subtracted
from the remainder of 214 = 82. The next square that can be constructed from five counters, yet the values of those five counters still being capable of being subtracted from 82 results in counters in positions 32,1; 2,1; 1,1; 1.2; and 1,32. The values of these five counters total 69 which when subtracted from 82 leave 13 as a remainder. As there is no more room on the board we have to stop. Move one counter from each row to the margin (rows 32, 2 and 1) and this value (35) is the square root required, or at least the integer part of it (the actual value is 35.1852 ). Napier provides a second example for calculating the square root of 2209 (= 47).