# John Napier

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John Napier of Merchistoun (1550 – 4 April 1617), the son of Sir Archibald Napier of Merchiston and Janet Bothwell, was a mathematician, physicist, astrologer and 8th Laird of Merchistoun. His surname appears in many different spellings - Napeir, Nepair, Nepeir, Neper, Napare, Naper, Naipper - but John Napier would most commonly have been written Jhone Neper in his time. Napier is most remembered as the inventor of logarithms and "Napier's bones", (a multiplication tool using a set of numbered ivory rods)[1], and for popularizing the use of the decimal point. He also contributed a mnemonic for formulas used in solving spherical triangles[2], and two formulas known as Napier's analogies[3]. Napier's birth place, Merchiston Tower, Edinburgh, Scotland, is now part of Edinburgh's Napier University [4] Neper crater, on the Moon, is named after him [5] as is an asteroid, 7096 Napier. After dying of gout, Napier was buried in St Cuthbert's Church, Edinburgh.

© Image: The History for Computing Project
Portrait of John Napier.[6]

## Life

Little is known about John Napier's early years, but a letter survives written by the Bishop of Orkney, who was John's uncle, to Archibald Napier when John was eleven years old:-

I pray you, schir, to send your son Jhone to the schuyllis; oyer to France or Flandaris; for he can leyr na guid at hame, nor get na proffeitt in this maist perullous worlde ... [7]

Napier entered St Andrew's University a the age of 13, where he became passionately interested in theology. There is no record of his having graduated, and it appears that he probably learned his mathematics elsewhere, possibly during his travels in Europe. In 1571, Napier returned to Scotland, and the following year married Elizabeth Stirling, daughter of Scottish mathematician James Stirling (1692-1770), and bought a castle at Gartnes. The couple had two children before Elizabeth died in 1579. Napier later married Agnes Chisholm, with whom he had ten children. When his father died in 1608, Napier and his family moved into Merchiston Castle, where he lived the rest of his life. There, he became known as "Marvellous Merchiston" for the mechanisms he built to improve his crops and cattle.

## Logarithms

In 1614, Napier described how logarithms could be used to simplify the multiplication of large numbers, and published tables of logarithms to facilitate this[8]. A number, n, is the logarithm of x to "base" b if ${\displaystyle \scriptstyle x\,=\,b^{n}}$. For common logarithms, the base, b is 10, and for "natural" or "hyperbolic" logarithms the base is the number e (2.7182818...)[9]. Napier did not think of logarithms in this algebraic way - indeed algebra was not well advanced at the time - but he calculated them in a way approximately equivalent to calculating them to base 1/e[10], although, confusingly natural logarithms are sometimes called Naperian (or Napierian) logarithms.

Napier's invention of logarithms was an extremely important advance, and the ease of calculation that it introduced allowed Kepler to make the breakthrough that underpinned Isaac Newton's theory of gravitation. In the preface to the Mirifici logarithmorum canonis descriptio, Napier says

Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of (perhaps) hereafter. But amongst all, none more profitable than this which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away from the work itself even the very numbers themselves that are to be multiplied, divided and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and subtraction, division by two or division by three.[11]

The book contained just thirty-seven pages of explanation but importantly, ninety pages of tables. Laplace, 200 year later, was to say that logarithms: "...by shortening the labours, doubled the life of the astronomer."[12].

Ten years after Napier's work, the English mathematician Henry Briggs (1561-1631) in 1624 published a set of tables that gave the common logarithm (i.e. to base 10) of integers up to 20,000 and from 90,000 to 101,000 with values given to fourteen digits [13]; the gap between 20,000 and 90,000 was filled by the Dutchman Adrian Vlacq (1600-1667) in 1628[14].

## Napier's rods

In 1617 Napier published Rabdologia, where he explained how to use 'Napier's rods' which could be used to multiply numbers together. He explains in the introduction that he: "... was induced to publish a description of the construction and use of the numbersing rods because many of my friends, to whom I have already shown them, were so pleased with them that the rods are already almost common and are even being carried to foreign countries."

Napier's "rods" or "bones" were essentially multiplication tables inscribed on the four sides of ten oblong sticks of wood or bone, with square ends; by placing the rods side by side, large numbers can be multiplied with ease. As well as for multiplication, the rods could also be used to find square roots and cube roots.[15]

## "Secret Inventions"

Napier produced several designs for machines intended to be used in warfare, and in particular to defend the country from invasion by King Philip of Spain: “devices for sailing under water, with divers other devises and stratagems for harming the enemyes.” Amongst his 'Secret Inventions' was a round chariot whereby its occupants could move while firing through holes in its sides and a 'burning mirror' which would consume enemy ships 'at whatever appointed distance'. In 1796, Etienne-Gaspard Robert (1763 - 1837) made a similar proposal to the French government to burn the invading ships of the English navy.[16] Neither suggestion was adopted. The notion of using a parabolic mirror in this way is based on the myth that Archimedes had used such mirrors to set fire to a fleet of Roman ships as they sought to capture Syracuse in 213BCE.[17])

## Anti-Catholicism

Napier was a fervent Protestant (and rabidly anti-Catholic); and in A Plaine Discovery of the Whole Revelation of St. John, which he regarded as his most important work, Napier used the Book of Revelation to predict the Apocalypse, which he predicted would occur in 1688 or 1700, and he declared that his studies pointed to Pope Clement VIII as being the "Antichrist. Accordingly, he wrote to King James VI: "Let it be your Majesty's continuall study to reforme the universall enormities of your country, and first to begin at your Majesty's owne house, familie and court, and purge the same of all suspicion of Papists and Atheists and Newtrals, whereof this Revelation forthtelleth that the number shall greatly increase in these latter daies."

## Napier and the Powers of Darkness

"It is said that Napier adopted the policy of Mahomet

to control his own domestics, and impressed them with a belief that he and chanticleer together could detect them in their most secret doings. Having missed some property, and suspecting his servants, he ordered them one by one into a dark room, where his favourite was confined, and declared that the cock would crow when the guilty one stroked his back, as each was required to do. The cock remained silent during all the ceremony ; but the hands of one of the servants were found to be entirely free from the soot with which

the feathers of the mysterious bird had been anointed."

Rumours circulated that Napier was "in league with the powers of darkness", and these are taken seriously in a biography written by Mark Napier (a descendant), who claimed (see quote above)that John Napier deliberately played to the superstitions of his servants by going round with a cock which he had covered with soot, pretending it to be his familiar spirit. [18] [19][20]

Mark Napier also recounts a tale of his ancestor "bewitching the pigeons" of Merchiston. Being annoyed by the flocks that ate up his grain, he threatened to impound them. " Do so, if you can catch them," challenged his neighbour, the Lord of Roslin; the next morning the fields about Merchiston were full of reeling pigeons, who were easily captured, having been intoxicated by pease saturated with alcohol.

## References

1. published in his Rabdologiae in 1617
2. Herschel AS (1871) Proof of Napier's rules. Nature5:24
3. Napier's Analogis
4. Napier University Homepage
5. Napier is just one of 28 Scots whose nanes are given to craters on the moon; the others are Alexander Graham Bell (1847-1922); David Brewster (1781-1868); Sir Thomas Makdougall Brisbane (1773-1860); Williamina Paton Fleming (1857-1911); Alexander Gerard (1792-1839); James Gregory (1638-1675); Thomas Henderson (1798-1844); David Hume (1711-1776); James Hutton (1726-1797); Robert Thornton Ayton Innes (1861-1933); John Jackson (1887-1958); Johann von Lamont (1805-1879); Sir Charles Lyell (1797-1875); Colin Maclaurin (1698-1746); Thomas Logie McDonald 1901-1973); Arthur Butler Phillips Mee (1860-1926); Sir Roderick Impey Murchison (1792-1871); James Nasmyth (1808-1890); Robert Methven Petrie (1906-1966); Charles Piazzi Smyth (1819-1900);John Playfair (1748-1819); William John Macquorn Rankine (1820-1872); James Short (1710-1768); Hugh Sempill (1596-1654); Mary Fairfax Somerville (1780-1872); James Watt (1736-1819) and Alexander Wilson (1714-1786)
6. John Napier The History of Computing Project
7. John Napier biography]
8. The Swiss clockmaker Jost Bürgi (1552-1632) independently invented logarithms but his work remained unpublished until 1620
9. The natural log of e is 1 and the natural log of 1 is 0
10. Bossetti A (1904) The base of Napier's logarithms. Nature69:582
11. English translation in 1616 of original text in Latin (1614) of the preface to Mirifici logarithmorum canonis descriptio Mirifici logarithmorum canonis descriptio
12. Quoted in In Mathematical Circles by H. Eves; Boston, 1969
13. H. Briggs, Arithmetica logarithmica sive logarithmorum Chiliades Triginta, London, (1624)
14. A. Vlacq, Arithmetica logarithmica, Gouda, (1628)
15. Napier's Rods
16. E.G.Robertson and the Phantasmagoria The Glass Armonica - Benjamin Franklin's Magical Musical Invention
17. Doubt cast on Archimedes' killer mirrors The Guardian, 24th October 2005
18. Memoirs of John Napier of Merchiston: His Lineage, Life, and Times by Mark Napier (1834), original from Harvard University, digitized by Google books
19. John Napier and the Devil