Napier's Binary Chessboard Calculator - Napier's 'Rabdologiae'

Sidney J. Kolpas and Erwin Tomash

Napier's 'Rabdologiae'

Logarithms were by no means the only computational aid invented by Napier. In 1617, the year of his death, Napier published Rabdologiae, a slender volume describing a trilogy of mechanical calculating aids. Remarkably, while none of the three are based upon logarithms, one is based on the binary number system, predating by almost a century Gottfried Wilhelm Leibniz's better known treatise on the binary number system, "Explication de l'Arithmétique Binaire" (1703). (To view this treatise in Convergence, see the article "Russian Multiplication, Microprocessors, and Leibniz".)

Figure 8. John Napier's Rabdologiae (1617). From the Erwin Tomash Library on the History of Computing. An entire copy of the Rabdologiae, with commentary by Tomash and Michael R. Williams, can be viewed here, courtesy of the Tomash Library. Another copy is available via Google Books.

Napier's 'bones'

The best known of the trilogy of topics in Rabdologiae, the so-called 'Napier's bones', was a set of rods used for multiplying, dividing, and extracting roots. As Napier noted in the introduction to Rabdologiae, his rods were in quite common use by the time the book itself was published. Napier's rods are simply portions of the multiplication table inscribed in gelosia- or lattice-format on sticks made of wood or bone. These 'bones' surely represent the first, albeit primitive, mechanical pocket calculator.

For an example of multiplication using Napier's rods, see the Convergence article, "John Napier: His Life, His Logs, and His Bones". 

Gelosia multiplication is also known as latticed multiplication. The term 'gelosia' is due to Luca Pacioli (ca. 1445-ca. 1514), who used it in his Suma de Arithmetica, Geometria, Proportioni et Proportionalita (Venice, 1494). The pattern suggests the gratings placed in Venetian windows to protect residents from the public gaze. Known as gelosia, the Italian word for jealousy, they were designed to permit women to peep out and yet remain unseen from the street. The English word 'jalousie' for a type of Venetian blind has come down to us.

Figure 9. Napier's bones (or rods)

The widespread acceptance of Napier's rods in a few short years is perhaps better understood through J. W. L. Glaisher's comment (1874):

There is abundant evidence that, till comparatively recent times (say the beginning of the eighteenth century), multiplication was regarded as a most laborious operation; this is testified not only indirectly by the very simple examples given in old arithmetics, but explicitly by Decker in his 'Erste Deel van de Nieuwe Telkonst'. The great popularity of Napier's bones, and the eagerness with which they were received all over Europe, show how great an assistance the simplest contrivance for reducing the labour of multiplications was considered to be.

Napier's 'promptuary'

In the first appendix to the Rabdologia, Napier introduced a second, more complex rod machine capable of much higher-speed multiplication. This was a two-dimensional configuration of rods which he named the Promptuary. The sole known early example of a Promptuary resides in a museum in Madrid, Spain, where the second author happily discovered it over three decades ago (Tomash, 1988).

Figure 10. Napier's Promptuary

Napier's 'chessboard calculator'

The third and final section of the Rabdologia is a 37-page addendum that describes what Napier called "Location Arithmetic as Performed on a Chessboard", and this arithmetic is the focus of the present article. Napier's enthusiasm for his Chessboard Calculator is apparent from his remarks in the preface to the addendum, which is worth quoting in its entirety (Napier, 1990):

While working in my spare time on these short methods and seeking ways in which the tedium of calculation might be removed, I developed (not only my logarithms, my Rabdology, my Promptuary for multiplication and other things, but also) a method of arithmetic on a flat surface. As this performs all the more difficult operations of common arithmetic on a chessboard, it might well be described as more of a lark than a labor, for it carries out addition, subtraction, multiplication, division, and [even] the extraction of roots purely by moving counters from place to place. There is one small difficulty in working with it, and that is that the numbers it uses differ from ordinary numbers, so that one must begin by expressing ordinary numbers in the new form and end by reducing them to common form. Either process is simple enough even during the course of the calculation, and overall my method is easier and more accurate than almost any other short method of Arithmetic. I therefore resolved not to bury it in silence nor (as it is so short) to publish it on its own but to subjoin it to my Rabdology after the aforementioned promptuary for the benefit of the studious and the scrutiny of the learned.

Despite Napier's enthusiasm, the Chessboard Calculator never seems to have found favor. "Location Arithmetic as Performed on a Chessboard" was largely ignored and in subsequent years was reported on as a mere curiosity.  It is somewhat ironic that within Napier's unheralded Chessboard Calculator lies buried one of the earliest applications of the binary number system—described as "a pioneering explanation of binary arithmetic as a computational aid" by Robin Rider in the Introduction to a translation of Napier's Rabdologiae (Napier 1990, p. xxiii). We find Napier's invention to be worthy of explication for at least two reasons: first, his early use of binary arithmetic, which he identified as the "one small difficulty in working with" his Chessboard Calculator, and, second, its potential as a source of enriching educational experiences for today's students.