John Napier

Scottish mathematician, inventor, and theologian

• Born: 1550
• Birthplace: Merchiston Castle, near Edinburgh, Scotland
• Died: April 4, 1617
• Place of death: Merchiston Castle, near Edinburgh, Scotland

Working alone, without the benefit of earlier work and the encouragement of mentors, Napier invented logarithms, which revolutionized arithmetic calculation and was the greatest boon to experimental science produced during the Renaissance.

Early Life

John Napier (NAY-pyuhr), eighth lord of Merchiston, was born at Merchiston Castle, the son of Sir Archibald Napier by his first wife, Janet Bothwell. He was born into a family notable for several famous soldiers at a time when religious controversy was rife in Scotland.

Little is known of his childhood, but when Napier was thirteen, his mother died. He was subsequently sent to St. Salvator’s College, St. Andrews University, a school not noted for its quiet academic environment. Although Napier remained at St. Andrews for only one year, he developed two intense interests that were to continue for the remainder of his life: theology and arithmetic. Because of the nonacademic environment, the bishop of Orkney advised that young John could better pursue an academic career at schools on the Continent. Although no direct evidence remains to confirm this, it is highly probable that he followed this course.

As young Napier traveled through a Europe divided into warring factions by the Protestant Reformation, he became a strong adherent of the Calvinist movement then sweeping Scotland. He was to remain a fervent and uncompromising believer, active in Protestant politics throughout his life, much of which was spent embroiled in bitter religious dissension aggravated by the embarrassing political activities of his papist father-in-law, Sir James Chisholm.

By 1571, he had returned to Scotland. The following year, he married Elizabeth Stirling and occupied a castle at Gartnes. In 1579, his wife died, leaving two children. Subsequently, Napier married Agnes Chisholm and fathered ten additional children. On the death of his father in 1608, Napier moved into Merchiston Castle, where he remained for the rest of his life.

As a member of the Scottish landed aristocracy, he had the time and resources to pursue his many interests. These included theology, agricultural improvements, and military science. In the latter field, he anticipated inventions three centuries before they were actually fabricated, and he invented an artillery so powerfully destructive that he refused, in horror, to develop or even to publicize it. Napier also experimented with fertilizers for crops and invented a mechanical device to pump water out of coal pits.

Life’s Work

Napier’s first literary work, A Plaine Discovery of the Whole Revelation of St. John , published in 1593 after five years of toil, was the first important work of biblical interpretation written in Scotland. In the book’s introduction, the Scottish king James VI (the future James I of England) is entreated to safeguard the Scottish Protestant church and to purge and punish the Roman Catholic nobility. The body of this bitterly anti-Catholic exposition, among other things, identifies the pope as the anti-Christ described in the biblical Book of Revelations. Although from the perspective of history this enterprise may appear to be little more than fruitless theological supposition, it established Napier’s reputation as both scholar and theologian. His theological interpretations followed the Greek form of mathematical argument, a form of theological reasoning that would not become popular for several centuries.

Although Napier’s public life during these tumultuous times has been amply documented, the development of his mathematical work, conducted alone and almost in secret, is more difficult to trace. It seems as though mathematics was for Napier a solitary pursuit of leisure, while his highly visible public life focused on anti-Catholic proclamations meant to keep Catholicism out of Scotland.

An early treatise concerned with arithmetic and algebra was apparently assembled during his first marriage but remained unpublished until 1839. About 1590, he set out to make arithmetic easier; the task required twenty years of labor, but he succeeded by inventing logarithms, a system that simplified the computation of products, quotients, and roots. His fanatical dedication to Calvinist Protestantism shows the same obsessive persistence that enabled him to finish the grueling task of producing a usable set of logarithmic tables.

Logarithms, or “logs,” are the exponents of a stated number, the “base,” and are used to represent powers (exponents) of the base. Consider, for example, the powers of 2: 2¹, 2², 2³, 2⁴, 2⁵, and so on. These correspond to 2, 4, 8, 16, 32, and so on. The exponents 1, 2, 3, 4, and 5 are the logs of these numbers to the base 2. To multiply any two numbers in the series, it is necessary only to add the exponents (or logs) of the numbers and then find to the antilog of the result, which corresponds to the sum desired. Thus, to multiply 32 by 4, take the log of 32, which is 5, and add it to the log of 4, which is 2, to get 7. The antilog of 7 (that is, the number with a log of 7) is 128, the desired result. Division is performed by subtracting logs.

By extension, numbers not found in the above series can be used if a noninteger number can be found such that when 2 is raised to this power the desired number is produced. For example, since 2 = 2¹ and 4 = 2², it follows that 3 = 2^x, where x must be a number greater than 1 but less than 2. In fact, x is approximately equal to 1.585. Since any number can be expressed to a good approximation as a power of 2, any arithmetic operations can be performed provided a table of powers of 2 is provided.

Although Napier did not use 2 as the base of his logarithms, the principle is the same. Whereas logs make arithmetic computation considerably easier, Napier set himself the grueling task of computing, by various mathematical means, a complete set of log tables, that is, sufficient powers of the base to generate a complete set of numbers, including decimal fractions. The calculation of the tables occupied Napier for almost twenty years. While not entirely error-free, the calculations were basically accurate, forming the foundation for all subsequent log tables.

In 1614, Napier published the description of his logarithms together with a set of log tables, several uses for them, and rules for the solution of both plane and spherical triangles using the tables. This work, titled Mirifici Logarithmorum Canonis Descriptio (Description of a Marvelous Canon of Logarithms , 1857), omitted any explanation of his methods of calculation. Although the common folk who were Napier’s neighbors had always suspected him of being a warlock who delved into the black arts behind his thick castle walls, his miraculous technique of logarithms, presented unexpectedly without explanation or rationale, seemed like black magic even to the relatively sophisticated people who had the occasion to use them.

A later work, published posthumously in 1619, Mirifici Logarithmorum Canonis Constructio (Construction of a Marvelous Canon of Logarithms , 1889), provides the explanation of his calculations, an outline of the steps leading to his invention, and the properties of his logarithmic function.

Napier sent a copy of his 1614 work to Henry Briggs, a professor at Gresham College. Briggs had the idea of making the base of the log tables 10, an innovation of which Napier approved because it simplified calculations. In 1624, Briggs published his tables of common logs (base 10 logarithms), but he gave full credit to Napier for the original idea.

Napier also invested considerable time in deriving complicated equations and exponential forms of trigonometric functions, since these played such important roles in astronomical computations. By mathematical manipulation, he was able to reduce the requisite number of spherical trigonometry equations from ten to just two general statements.

Napier’s tables of logarithms were greeted with great enthusiasm by astronomers, since they simplified computations and removed some of the drudgery from analyzing data. Johannes Kepler (1571-1630), who inherited several decades of extremely accurate data on planetary motions from the great Danish astronomerTycho Brahe, used Napier’s logarithms to simplify the analysis. The results of his work led to Kepler’s three laws of planetary motion, the first correct and accurate statement of planetary motion. Later, Isaac Newton (1642-1727) used Kepler’s laws in formulating his theory of gravity.

In 1617, Napier published the results of his work on a mechanical system to simplify arithmetic computation, Rabdologiae, seu numerationis per virgulas libri duo (Study of Divining Rods: Or, Two Books of Numbering by Means of Rods , 1667). This involved manipulating a set of small counting rods (later termed “Napier’s Bones”) to multiply and divide numbers. This device could be considered the precursor of the slide rule (a set of sliding logarithmic scales that enabled rapid multiplication and division), a device widely used by scientists and students until the latter half of the twentieth century. Last but not least, Napier standardized and popularized the system now universally used for decimal notation, in which a decimal point is used to separate the integer from the fractional part of a number.

Significance

The 1614 publication of Napier’s canon of logarithms is one of those extraordinary and exceptional events in the history of science whereby a new invention of great importance appears, seemingly out of thin air, with no obvious precursors foreshadowing its creation. Napier’s invention removed much of the drudgery from reducing scientific data, particularly for astronomers attempting to use accurate measurements to predict planetary motions. When Johann Kepler used Tycho Brahe’s accurate data to deduce his laws of planetary motion, Napier’s logarithms helped make the arduous task possible.

In the centuries following their invention, log tables grew more detailed and more accurate, culminating in 1964 with the publication of a table of logarithms accurate to 110 decimal places. Until the 1970’s, when inexpensive hand-held calculators and personal computers rendered them obsolete, log tables formed an essential component of college-preparatory secondary education, and no reputable engineer would be without his slide rule, a portable version of the log tables.

As a titled landowner, Napier, lord of Merchiston, devoted considerable energy to agricultural products to improve his crops and cattle. He tinkered with inventions and was granted a patent for a hydraulic screw to pump water from coal pits, and he outlined plans for (but never constructed) four new weapons of war, including an artillery piece that was designed to kill anything within a one-mile radius. Napier’s first literary work, an interpretation of the Book of Revelation, secured his reputation as a scholar and as a theologian, although outside Scotland this work no longer commands high regard.

Bibliography

Burton, David M. The History of Mathematics: An Introduction. 5th ed. Boston: McGraw-Hill, 2003. Survey of important developments in math and the people behind those developments. This edition adds broader coverage of important mathematicians, including women in math. Includes illustrations, bibliographic references, and index.

Gladstone-Millar, Lynne. John Napier: Logarithm John. Edinburgh: National Museums of Scotland, 2003. Biography of Napier emphasizes his importance, not just to mathematics, but to astronomy as well. Includes illustrations and bibliographic references.

Hobson, E. W. John Napier and the Invention of Logarithms. Cambridge, England: Cambridge University Press, 1914. This lecture is the most useful of the various reconstructions of Napier’s invention of logarithms. Highly recommended.

Knott, C. G., ed. Napier Tercentenary Memorial Volume. London: Dawson’s of Pall Mall, 1966. A reprint of a 1915 original. Contains a set of articles detailing different aspects of Napier’s accomplishments by experts in various fields of mathematics, as well as some considerable detail on the historical background to his work. Also included is a complete bibliography of books exhibited at the July, 1914, Napier Tercentenary Celebration.

McLeish, John. Number. New York: Fawcett Columbine, 1991. Chapter 12, “John Napier: The Rationalization of Arithmetic,” details his work on logarithms and Napier’s Bones. Included are examples detailing the construction and use of both these inventions.

Napier, John. Napier’s Mathematical Works. Translated by William F. Hawkins. 3 vols. Auckland, New Zealand: University of Auckland Press, 1982. A translation from Latin of all of Napier’s writings on mathematics, including those published posthumously. Volumes 2 and 3 include a commentary on Napier’s work and how it fits into the history of mathematics.

Napier, Mark. Memoirs of John Napier of Merchiston: His Lineage, Life, and Times. Edinburgh: W. Blackwood, 1834. Written by a direct descendant of John Napier with access to the family’s private papers, this carefully researched work provides the original source material from which most later books were derived.

Neal, Katherine. From Discrete to Continuous: The Broadening of Number Concepts in Early Modern England. Boston: Kluwer Academic, 2002. Places Napier’s invention of logarithms in the context of other changes in number theory in early modern England. Discusses Napier alongside such contemporaries and successors as Isaac Barrow and John Wallis. Includes illustrations, bibliographic references, and index.