page_title: Location arithmetic
section_title: Napier's Method
hierarchical_section_title: Location arithmetic / Square roots / Napier's Method
caption_attribution_description: English: Example of finding the square root of 2209 using Napier's method for finding square roots provided in Rabdology on page 153
context_page_description: Location arithmetic is the additive binary numeral systems, which John Napier explored as a computation technique in his treatise Rabdology, 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 in current vocabulary because the numbering system is non-positional.
  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.
  However, when reproduced on the board, this new technique did not require mental trial-and-error computations nor complex carry memorization. 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.
context_section_description: 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).
caption: Example of finding the square root of 2209 using Napier's method for finding square roots provided in Rabdology on page 153
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