He did not stay in college very long. It is believed that he dropped out of school in Scotland and perhaps travelled in mainland Europe to better continue his studies. Little is known about those years, where, when, or with whom he might have studied, although his uncle Adam Bothwell wrote a letter to John'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", and it is believed that this advice was followed.
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 resided the remainder of his life. He died at the age of 67.
In 1572, Napier married 16-year-old Elizabeth, daughter of James Stirling, the 4th Laird of Keir and of Cadder. They had two children. Elizabeth died in 1579, and Napier then married Agnes Chisholm, with whom he had ten more children. His father-in-law James Chisholm of Cromlix was later mixed up in the Spanish blanks plot, over which Napier with others petitioned the king.
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.
Napier had an interest in the Book of Revelation, from his student days at St Salvator's College, St Andrews. Under the influence of the sermons of Christopher Goodman, he developed a strongly anti-papal reading. He further used the Book of Revelation for chronography, to predict the Apocalypse, in A Plaine Discovery of the Whole Revelation of St. John (1593), which he regarded as his most important work; he also applied the Sibylline Oracles, to calculate the date of the end of the world. Napier believed that would occur in 1688 or 1700. He dated the seventh trumpet to 1541.
In his dedication of the Plaine Discovery to James VI, dated 29 Jan 1594, Napier urged the king to see "that justice be done against the enemies of God's church," and counselled the King "to reform the universal enormities of his country, and first to begin at his own house, family, and court." The volume includes nine pages of Napier's English verse. It met with success at home and abroad. In 1600 Michiel Panneel produced a Dutch translation, and this reached a second edition in 1607. In 1602 the work appeared at La Rochelle in a French version, by Georges Thomson, revised by Napier, and that also went through several editions (1603, 1605, and 1607). A new edition of the English original was called for in 1611, when it was revised and corrected by the author, and enlarged by the addition of With a resolution of certain doubts, moved by some well affected brethren.; this appeared simultaneously at Edinburgh and London. The author stated that he still intended to publish a Latin edition, but it never appeared. A German translation, by Leo de Dromna, of the first part of Napier's work appeared at Gera in 1611, and of the whole by Wolfgang Meyer at Frankfurt-am-Main, in 1615. Among Napier's followers was Matthew Cotterius (Matthieu Cottière).
His work, Mirifici Logarithmorum Canonis Descriptio (1614) contained fifty-seven pages of explanatory matter and ninety pages of tables of numbers related to natural logarithms (see Napierian logarithm). The book also has an excellent discussion of theorems in spherical trigonometry, usually known as Napier's Rules of Circular Parts.
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 were 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.
Among Napier's early followers were the instrument makers Edmund Gunter and John Speidell. The development of logarithms is given credit as the largest single factor in the general adoption of decimal arithmetic. The Trissotetras (1645) of Thomas Urquhart builds on Napier's work, in trigonometry.
Many mathematicians at the time were acutely aware of the issues of computation and were dedicated to relieving practitioners of the calculation burden. In particular, the Scottish Mathematician John Napier was famous for his devices to assist with computation. He invented a well-known mathematical artifact, the ingenious numbering rods more quaintly known as “Napier's bones,” that offered mechanical means for facilitating computation. (For additional information on “Napier's bones,” see the article, “John Napier: His Life, His Logs, and His Bones” (2006).)