Napier's Binary Chessboard Calculator - Napier and Logarithms

John Napier (1550-1617)

John Napier first announced his new 'artificial numbers', for which he had coined the name logarithms (from Greek meaning 'ratio numbers'), in his Descriptio published in 1614. This work was limited in scope to the practical application of logarithms and included the first published logarithmic table or what Napier referred to as a 'Canon' (a general law, rule, or principle). This was a table not of the logarithms of numbers but of the logarithms of sines of angles for every minute of the quadrant (the first 90 degrees). It was not until 1619, in his Constructio, that Napier presented the concept of logarithms and the method he used to construct rules for their use in calculation. The preparation of the table and the two books must have taken Napier many years. That he had made significant progress at least ten years before the date of publication is evidenced by Johannes Kepler's 1594 report that Tycho Brahe had recently learned of Napier's canon and hoped it would soon be published. W. W. Rouse Ball stated that Napier “had privately communicated a summary of his results to Tycho Brahe as early as 1594” (1912/2001, p. 195).

Figure 1. Mirifici Logarithmorum Canonis Descriptio (1614). From the Erwin Tomash Library on the History of Computing.

Figure 2. Mirifici Logarithmorum Canonis Constructio (1620). From the collection of Sidney J. Kolpas.

The groundbreaking nature of logarithms was widely recognized by Napier's peers. Logarithms replaced clumsy methods by reducing multiplication to addition and division to subtraction. Napier's canon represented a major improvement that in principle provided practitioners of mathematics, such as astronomers, surveyors, and navigators, with a new, powerful tool with which to carry out large scale computation much more easily. In practice, however, Naperian logarithms were inconvenient because their base was not decimal. Napier quickly became aware of this difficulty for in the second issue of the first edition of the Descriptio, also published in 1614, he included a brief closing 'Admonitio' in which he mentioned that another system of logarithms more suitable for practical calculation in the decimal system would soon be available. This was a reference to the decimal logarithms that we know today, sometimes referred to as Briggsian logarithms or common logarithms.

Figure 3. Henry Briggs' proposal for base 10 logarithms from Napier's Canonis Constructio (1620). From the collection of Sidney J. Kolpas

Soon after the initial publication of the Descriptio, Henry Briggs (1561-1630), the first Gresham Professor of Geometry in London and later the first Savilian Professor of Geometry at Oxford, undertook, with the active cooperation of Napier, the calculation of a set of decimal-based logarithms. Briggs' work was taken up and completed by the Dutch mathematician, Adriaan Vlacq, so that by 1628, only 14 years after Napier's initial announcement, convenient, multi-place decimal logarithmic tables were available to simplify computation.

Figure 4. Vlacq's Tabulae Sinuum, Tangentium, et Secantium, et Logarithmorum (1670). From the Erwin Tomash Library on the History of Computing.

The calculation and application of logarithms occupied practitioners of mathematics well into the next century. Even two centuries later, one of the major influences motivating Charles Babbage to undertake the design of his Difference Engine was his desire to automatically generate error-free, easy-to-read logarithmic tables.

Figure 5. Modern execution of Babbage's plans for his Difference Engine

Figure 6. Babbage's Table of Logarithms (1827). From the Erwin Tomash Library on the History of Computing.

The invention of logarithms led in short order to the invention of mechanical calculating devices based upon them. In 1620, Edmund Gunter described his 'line of numbers', the physical expression of decimal logarithms as a sequence of lengths on a straight edge. He then added more 'lines' for the logarithms of the trigonometric functions, creating a popular device commonly referred to as Gunter's scale.  In 1630, Edmund Wingate described the sliding of two such scales against each other, thus inventing the linear slide rule. Only two years later, William Oughtred marked a Gunter's line on the edge of a circle and rotated two such circles against each other, thus inventing the circular slide rule.

Figure 7. Circular slide rule

However, Napier's interest in developing computational aids was not restricted to logarithms. In his 1617 Rabdologiae, to be discussed in the next section of this article, he presented three other inventions for this purpose.