John Napier in relevance to log tables?

Answer 1

The method of logarithms was first publicly propounded in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier, Baron of Merchiston in Scotland (Joost Bürgi independently discovered logarithms; however, he did not publish his discovery until four years after Napier). This method contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible. Prior to the advent of calculators and computers, it was used constantly in surveying, navigation, and other branches of practical mathematics. It supplanted the more involved method of prosthaphaeresis, which relied on trigonometric identities as a quick method of computing products. Besides their usefulness in computation, logarithms also fill an important place in higher theoretical mathematics.

At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers". Later, Napier formed the word logarithm to mean a number that indicates a ratio: λόγος (logos) meaning proportion, and ἀριθμός (arithmos) meaning number. Napier chose that because the difference of two logarithms determines the ratio of the numbers for which they stand, so that an arithmetic series of logarithms corresponds to a geometric series of numbers. The term antilogarithm was introduced in the late 17th century and, while never used extensively in mathematics, persisted in collections of tables until they fell into disuse.

Napier did not use a base as we now understand it, but his logarithms were, up to a scaling factor, effectively to base 1/e. For interpolation purposes and ease of calculation, it is useful to make the ratio r in the geometric series close to 1. Napier chose r = 1 - 10−7 = 0.999999 (Bürgi chose r = 1 + 10−4 = 1.0001). Napier's original logarithms did not have log 1 = 0 but rather log 107 = 0. Thus if N is a number and L is its logarithm as calculated by Napier, N = 107(1 − 10−7)L. Since (1 − 10−7)107 is approximately 1/e, this makes L/107 approximately equal to log1/e N/107.

Answer 2

John Napier is the first person to have made an extensive tabulation of logarithms.
From Wikipedia:
"Napier is relatively little-known outside mathematical and engineering circles, where he made what is undoubtedly a key advance in the use of mathematics. Logarithms made calculations by hand much easier and quicker, and thereby opened the way to many later scientific advances. His work, Mirifici Logarithmorum Canonis Descriptio, contained thirty-seven pages of explanatory matter and ninety pages of tables, which facilitated the furtherment of astronomy, dynamics, physics, and astrology. He also invented Napier's bones, a multiplication tool using a set of numbered rods."

Answer 3

As already stated, John Napier acquired international fame for his contribution to mathematics, primarily by the invention of logarithms in 1614 and to a lesser extent by the development of Napier’s bones or rods and a mnemonic for formulas used in solving spherical triangles. Napier also found exponential expressions for trigonometric functions and was the first who used and then popularised the decimal point to separate the whole number part from the fractional part of a number.

English mathematician Henry Briggs went to Edinburgh in 1616 and later to discuss the logarithmic tables with Napier. Together they worked out improvements, such as the idea of using the base ten. Napier’s discussion of logarithms appears in Mirifici logarithmorum canonis descriptio / Description of the Marvellous Canon of Logarithms (1614), the first important work on mathematics produced in Great Britain. Napier’s hope was that his logarithms would greatly facilitate the task of astronomers by saving them time and enabling them to avoid errors in calculations. Two hundred years later, Laplace verified this hope, stating that logarithms, by reducing significantly the labours, doubled the astronomer’s life. Other mathematical works by John Napier include: De arte logistica (1573 but not published until 1839), Mirifici logarithmorum canonis constructio, published two years after his death and Rabdologiae (‘numeration by little rods’, 1617). In the latter, he explained his calculating system constructed of 10 rods on which was engraved the multiplication table. This simple system made the process of multiplying and dividing numbers (even very large ones) faster and easier.

Inventions For War And Peace
Mathematics was not the only scientific preoccupation of John Napier. A document from Napier’s own hand, illustrates the restless ingenuity of his mind. It is a list of war engines which “by the grace of god and worke of expert craftsmen” he hoped to produce “for defense of this Island”. These terrific engines were as follows: a burning mirror which would consume an enemy’s ship “at whatever appointed distance”; another mirror constructed on a different principle which would produce like effects; a piece of artillery which would sweep a whole field clear of the enemy; a chariot which would be like the moving mouth of a mettle and scatter destruction on all sides; and finally “devices for sailing under water, with divers other devises and stratagems for harming the enemyes.” [3]

Napier’s ingenuity was also turned to more peaceful applications. He made advances in scientific farming, especially by the use of salt as a fertiliser. In 1597, he patented a hydraulic screw by means of which water could be removed from flooded coal pits.

Answer 4

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Answer 5

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