John Napier of Merchiston ( NAY-pee-ər; Latinized as Ioannes Neper; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist and astronomer. He was the 8th Laird of Merchiston. Napier is best known as the discoverer of logarithms. He also invented the Napier's bones calculating device and popularised the use of the decimal point in arithmetic.
Napier's birthplace, Merchiston Tower in Edinburgh, is now part of the facilities of Edinburgh Napier University. There is a memorial to him at St Cuthbert's Parish Church at the west end of Princes Street Gardens in Edinburgh.
Napier's father was Sir Archibald Napier of Merchiston Castle and his mother was Janet Bothwell, daughter of the politician and judge Francis Bothwell, and a sister of Adam Bothwell who became the Bishop of Orkney. Archibald Napier was 16 years old when John Napier was born.
There are no records of Napier's early learning, but many believe that he was privately tutored during early childhood. At age 13 he was enrolled in St Salvator's College, St Andrews. Near the time of his matriculation the quality of the education provided by the university was poor, owing in part to the Reformation's causing strife between those of the old faith and the growing numbers of Protestants. There are no records showing that John Napier completed his education at St Andrews. It is believed he left Scotland to further his education in mainland Europe, following the advice given by his uncle Adam Bothwell in a letter written to John Napier's father on 5 December 1560, saying, "I pray you, sir, to send John to the schools either to France or Flanders, for he can learn no good at home." It is not known which university Napier attended in Europe, but when he returned to Scotland in 1571 he was fluent in Greek, a language not commonly taught in European universities at the time. There are no records showing his enrollment in the premier universities in Paris or Geneva during this time either.
In 1571 Napier, aged 21, returned to Scotland and bought a castle at Gartness in 1574. On the death of his father in 1608 Napier and his family moved into Merchiston Castle in Edinburgh, where he lived for the rest of his life. He had a property within Edinburgh city as well, on Borthwick's Close off the Royal Mile.
On 7 June 1596 Napier wrote a paper Secret inventions, profitable and necessary in these days for defence of this island. He describes two kinds of burning mirror for use against ships at a distance, a special kind of artillery shot and a musket-proof metal chariot.
Napier died from the effects of gout at home at Merchiston Castle at the age of 67. He was buried in the kirkyard of St Giles in Edinburgh. Following the loss of the kirkyard of St Giles to build Parliament House, his remains were transferred to an underground vault on the north side of St Cuthbert's Parish Church on the west side of Edinburgh. There is also a wall monument to Napier at St Cuthbert's.
Like many mathematicians at the time Napier worked on methods to reduce the labour required for calculations, and he became famous for the devices that he invented to assist with these issues of computation, for example the numbering rods more quaintly known as Napier's bones.
In addition Napier recognised the potential of the recent developments in mathematics, particularly those of prosthaphaeresis, decimal fractions and symbolic index arithmetic, to tackle the issue of reducing computation. He appreciated that for the most part practitioners who had laborious computations generally did them in the context of trigonometry. Therefore, as well as developing the logarithmic relation, Napier set it in a trigonometric context so it would be even more relevant.
His work Mirifici Logarithmorum Canonis Descriptio (1614) contained fifty-seven pages of explanatory matter and ninety pages of tables listing the natural logarithms of trigonometric functions. The book also has a discussion of theorems in spherical trigonometry, usually known as Napier's Rules of Circular Parts.
Modern English translations of both Napier's books on logarithms and their description can be found on the web, as well as a discussion of Napier's bones and Promptuary (another early calculating device).
His invention of logarithms was quickly taken up at Gresham College, and prominent English mathematician Henry Briggs visited Napier in 1615. Among the matters they discussed was a re-scaling of Napier's logarithms, in which the presence of the mathematical constant now known as e (more accurately, e times a large power of 10 rounded to an integer) was a practical difficulty. Neither Napier nor Briggs actually discovered the constant e; that discovery was made decades later by Jacob Bernoulli.
Napier delegated to Briggs the computation of a revised table. The computational advance available via logarithms, the inverse of powered numbers or exponential notation, was such that it made calculations by hand much quicker. The way was opened to later scientific advances, in astronomy, dynamics, and other areas of physics.
Napier made further contributions. He improved Simon Stevin's decimal notation, introducing the full stop (.) as the delimiter for the fractional part. Lattice multiplication, used by Fibonacci, was made more convenient by his introduction of Napier's bones, a multiplication tool using a set of numbered rods.
Napier may have worked largely in isolation, but he had contact with Tycho Brahe who corresponded with his friend John Craig. Craig certainly announced the discovery of logarithms to Brahe in the 1590s (the name itself came later); there is a story from Anthony à Wood, perhaps not well substantiated, that Napier had a hint from Craig that Longomontanus, a follower of Brahe, was working in a similar direction. Craig had notes on a method of Paul Wittich that used trigonometric identities to reduce a multiplication formula for the sine function to additions.
When one of the angles, say C, of a spherical triangle is equal to π/2 the various identities given above are considerably simplified. There are ten identities relating three elements chosen from the set a, b, c, A, B.
Napier provided an elegant mnemonic aid for the ten independent equations: the mnemonic is called Napier's circle or Napier's pentagon (when the circle in the above figure, right, is replaced by a pentagon).
First, write the six parts of the triangle (three vertex angles, three arc angles for the sides) in the order they occur around any circuit of the triangle: for the triangle shown above left, going clockwise starting with a gives aCbAcB. Next replace the parts that are not adjacent to C (that is A, c, B) by their complements and then delete the angle C from the list. The remaining parts can then be drawn as five ordered, equal slices of a pentagram, or circle, as shown in the above figure (right). For any choice of three contiguous parts, one (the middle part) will be adjacent to two parts and opposite the other two parts. The ten Napier's Rules are given by