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John Napier's Binary Chessboard Calculator - Simplified

Sidney J. Kolpas and Erwin Tomash

Napier's Chessboard Calculator is described in the final section of his Rabdologiae. Titled "Location Arithmetic as Performed on a Chessboard" ("Arithmeticae localis, quae in Scacchiae abaco exercetur"), this 42-page section consists of 11 short chapters with diagrams. The remainder of this article amplifies and simplifies Napier's rather terse exposition of his invention, beginning with a brief discussion of his pioneering use of binary arithmetic and a physical description of the Chessboard Calculator. It then presents the axioms for its use and describes how to use it to add, subtract, multiply, divide, and take square roots. The article concludes with some potential educational applications.

From a modern perspective, the required background knowledge for understanding the operation of the Binary Chessboard Calculator is minimal, the sole prerequisite being a rudimentary understanding of the binary number system. From Napier's early 17th century perspective, the binary number system was unknown (Glaser, 1971) and neither exponential notation nor the concept of base had as yet crystallized. However, the arithmetic processes known as mediation (halving) and duplation (doubling) were well known and familiar. These are of ancient origin and their use persisted throughout the Middle Ages and well past Napier's time. (Early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his La Géométrie of 1637.)

The Chessboard Calculator is based on mediation and duplation handled through an intermediate set of alphabetic "location" numbers. Napier created an alphabetic notation for the successive powers of 2 which he described as "ordinary numbers expressed in a new form." The use of alphabetic number systems was, of course, commonplace in Napier's time, the most obvious example being Roman numerals, which were used in England well into the 17th century. This article will discuss a simplified version of the Chessboard Calculator without the alphabetic "location" numbers.

Figure 11. Napier's Chessboard Calculator from his 1617 Rabdologiae. From the Tomash Library on the History of Computing, where the authors of the commentary on the digitized copy of the book argued that Napier himself described the orientation of the chessboard calculator as it is shown here (Tomash & Williams, 2009) 

In its utilization of simple, everyday components, the Chessboard Calculator is yet another instance of Napier's faculty of uniting basic elements with great theoretical insight. Both chessboards and counters were commonplace in Napier's milieu. Boards for the games of chess and checkers, the latter known in England as "draughts", were to be found everywhere. Casting counters were likewise ubiquitous because they were used in daily life as aids to commercial calculation. We suggest using bingo chips, checkers, candy, or the like as counters. 

Napier's two-dimensional calculating board is a chessboard (or checkerboard) with the addition of a lower horizontal margin and right vertical margin, the margins being used to represent the values in a given arithmetic problem. The value of the lower right square is 1, with powers of two progressing both horizontally and vertically.  There is no limit on the size of the board, which may be increased for use with larger numbers.

Figure 12. Napier's Chessboard Calculator, updated for modern use. (Download a copy of this diagram here.)

To better understand the fundamentals of Napier's Chessboard Calculator, the board itself (see Figure 12), as well as its bottom and right edges, has been labeled using powers of 2. Equivalence lines are the Southwest-Northeast (SW-NE) diagonal lines on the board, which pass through squares of equal values. The Northwest-Southeast main diagonal passes through powers of two with even exponents; that is, powers of 2 that are perfect squares: 1, 4, 16, 64, 256, 1024, 4096, 16384.

Axioms of the Chessboard Calculator

The following axioms relate the Chessboard Calculator to the game of chess. However, the most important of the axioms below is the last, which points out that squares on the chessboard are labeled so that the square at the intersection of a horizontal margin number and a vertical margin number holds the product of those two numbers.

In the game of chess, a Bishop moves diagonally in any direction from the square it currently occupies, and a Rook moves in any direction horizontally or vertically from the square it currently occupies. The following axioms may be verified by inspection of the Binary Chessboard.

Diagonal (Bishop) Moves:

Axiom 1:  Moving diagonally up and to the left multiplies by four with each square moved.

Axiom 2:  Moving diagonally down and to the right divides by four with each square moved.

Axiom 3:  Moving diagonally along an equivalence line (SW-NE) makes no change in value with each square moved.

Horizontal/Vertical (Rook) Moves: Duplation and Mediation

Axiom 4:  Moving horizontally to the left multiplies by two with each square moved.

Axiom 5:  Moving horizontally to the right divides by two with each square moved.

Axiom 6:  Moving vertically up multiplies by two with each square moved.

Axiom 7:  Moving vertically down divides by two with each square moved.

Multiplication Axiom:

Axiom 8:  The entry in the square at the intersection of a horizontal margin number and a vertical margin number is the product of those two numbers.

In succeeding pages of this article, we will learn how to use Napier's binary chessboard calculator to add, subtract, multiply, divide, and extract square roots.

Sidney J. Kolpas and Erwin Tomash, "John Napier's Binary Chessboard Calculator - Simplified," Convergence (December 2018)