John Napier

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John Napier of Merchiston (1550–1617) was a Scottish laird, mathematician, and Protestant theologian, remembered now for inventing logarithms but known in his own lifetime above all as the man who calculated, from the Book of Revelation, when the world would end. The two reputations belong to one mind, and to one conviction: that number was the key to hidden order, whether in the heavens’ arithmetic or in the sealed text of prophecy.

John Napier (1550–1617), engraving by Samuel Freeman after a 1616 original portrait — via Wikimedia Commons (public domain).

He was born at Merchiston Tower, on the southern edge of Edinburgh, the heir of a family long settled there — by inheritance the eighth laird of Merchiston, a landed Scot of the Reformation generation rather than a professional scholar. He matriculated at St Andrews in 1563, at thirteen, but took no degree; on the counsel of his uncle Adam Bothwell, who judged that the boy would learn nothing good at home, he went abroad to continue his studies on the Continent before returning to Scotland and to the management of his estates. The intellectual formation that mattered most happened away from the universities, in a long private labor of calculation and exegesis carried on by a man who was, first and last, a working landowner: an improver of soils and a tester of manures, a designer of hydraulic engines for draining the coal pits on his ground, and — in a memorandum of 1596 listing “secret inventions” for the defense of the realm — a proposer of war machines against the feared Spanish: a burning mirror, an artillery piece, an armored chariot, a vessel that could move under water. The same temperament that drafted those devices set itself, across two decades, to the reduction of arithmetic and the deciphering of the Apocalypse.

Merchiston Tower, Edinburgh, the fifteenth-century seat of the Napiers and John Napier’s birthplace — M J Richardson, via Wikimedia Commons (CC BY-SA 2.0).

The reduction of number

Napier’s standing as a scientist rests on real and lasting work. The Mirifici Logarithmorum Canonis Descriptio — the “description of the marvelous canon of logarithms” — appeared at Edinburgh in 1614 after some twenty years of private calculation. It is a slim book: a few dozen pages of explanation followed by ninety pages of tables, giving the logarithms of the sines through every minute of the quadrant. Its premise is one of the most consequential simplifications in the history of computation. Multiplication and division of large numbers — the ruinous, error-breeding labor at the heart of astronomy and navigation — could be replaced by addition and subtraction, if one first translated each number into its logarithm, did the easy operation, and translated back. The work of days became the work of an afternoon.

Napier did not arrive at this by the modern route of exponents. He built it from motion. Imagine two points set going at the same instant and the same speed. One travels along a line of finite length — Napier took it as ten million units, the radius of his circle of sines — and slows continuously, its velocity at each moment proportional to the distance still ahead of it, so that the lengths it has yet to cross fall away in geometric proportion. The other travels along an endless line at unvarying speed, adding equal lengths in equal times. At any instant, the distance remaining on the first line is a sine; the distance covered on the second is its logarithm. As one quantity shrinks by constant ratio, the other grows by constant difference — and this is precisely the correspondence that turns a product into a sum. The full account of how the tables were constructed Napier did not live to publish; it appeared posthumously in 1619 as the Mirifici Logarithmorum Canonis Constructio, prepared with his son Robert, the kinematic definition laid out for those who wished to verify the canon themselves. The geometry of two racing points was, for him, no metaphor but the literal generation of the numbers.

Title page of the posthumous Mirifici Logarithmorum Canonis Constructio (Edinburgh, 1619), which set out the kinematic construction of the tables — via Wikimedia Commons (public domain).

The reception was swift and decisive, and it came first from a mathematician hundreds of miles south. Henry Briggs, professor of geometry at Gresham College in London, read the Descriptio and, by his own report, could think of little else. In the summers of 1615 and 1616 he made the long northward journey to Merchiston — four hard days on the road — to sit with its author. Out of those two visits came the form of the logarithm the world has used ever since. Briggs proposed, and Napier agreed, that the table should be rebuilt on base ten, with the logarithm of one set at zero and the logarithm of ten at one; these are the common, or Briggsian, logarithms, and Briggs spent the rest of his life computing them, publishing the great Arithmetica Logarithmica of fourteen-place tables in 1624. Napier had also done much to put the decimal point into general use as the mark dividing a whole number from its fractional tail — a notation so small and so universal that its invention is easy to forget.

He left, besides, a small museum of calculating instruments. The Rabdologiae of 1617, the last work to appear in his lifetime, described the set of numbered rods soon known everywhere as “Napier’s bones”: strips carrying the multiples of each digit in a diagonal lattice, so that the products of large numbers could be read off and added by anyone, learned or not. The same book set out a more elaborate strip device he called the promptuary, and a method of reckoning by counters on a gridded board — a binary “location arithmetic” — that converted multiplication into the shifting of markers. Where the logarithms served the astronomer, the bones served the merchant and the surveyor; together they expressed a single ambition, to lift the burden of computation off the human mind and lodge it in a procedure.

A set of “Napier’s bones,” the numbered multiplication rods described in the Rabdologiae of 1617 — Kim Traynor, via Wikimedia Commons (CC BY-SA 3.0).

None of this was disinterested play. The burden of celestial reckoning was, for Napier, partly a religious burden, the universe being God’s work and therefore intelligible by measure. The man who shortened the astronomer’s labor and the man who counted the years to Judgment were not two figures awkwardly housed in one body; they were a single calculator turning the same conviction on two texts — the book of the heavens and the book of the seer of Patmos.

The arithmetic of the end

The book Napier himself regarded as his most important was not mathematical at all. A Plaine Discovery of the Whole Revelation of Saint John, published at Edinburgh in 1593, was a verse-by-verse exegesis of the Apocalypse, and he wrote it not in the Latin of the learned but in English, so that — in his own phrase — the simple of the island might be instructed. He dedicated it to the young James VI of Scotland, the dedication dated January 1594, with an exhortation to the king to see justice done against the enemies of God’s church. It was a book of its moment: post-Reformation Scotland, the Kirk militant, the realm braced against Spain and Rome and against the recusant nobility within. The anti-papal force of its reading had a local source in the preaching of Christopher Goodman, the radical Marian exile whose sermons shaped Napier’s conviction. Its central and most notorious thesis was stated without qualification: the Papacy is the Antichrist of prophecy, the Roman church the Babylon of the Revelation. This identification was not Napier’s invention — it was common Reformation polemic, the shared property of Martin Luther and the wider Protestant exegetical tradition — but Napier pressed it through the whole text with a thoroughness that made his book the standard British statement of the case.

The Plaine Discovery is built in two treatises. The first is a sequence of thirty-six propositions, each stated and then proved in the manner of a geometer — the same Euclidean architecture Napier would later bring to the logarithms, here laid over scripture. The second applies those propositions, paraphrastically and historically, to the Apocalypse verse by verse, matching the seals, the trumpets, and the vials to the events of church history as Napier read them. What turns the work from polemic into computation is its numerology. Napier took the standard apocalyptic convention, shared with readings of the Book of Daniel, that a prophetic “day” stands for a year, so that the famous twelve hundred and sixty days of the Beast’s reign become twelve hundred and sixty years. From such equivalences he built a chronology of salvation history with dated stations: the seventh trumpet, by his reckoning, had sounded in 1541, and the great periods running out from the early church converged on a narrow window at the close of the seventeenth century. The Day of God’s judgment, he concluded, would fall between the years 1688 and 1700. To this arithmetic he appended a set of Sibylline oracles which he read as confirming, from a second and pagan source, what the Revelation had told him.

The logic was the logic of the tables. A prophecy, like a sine, was a magnitude encoded; the prophetic numbers were data to be reduced by a known rule to a result. The same faith that the world is counted ran through both books — that behind the surface of things, whether the motions of the planets or the visions of the seer, lay a quantity that the patient reckoner could recover. Napier set his apocalyptic mathematics in what later scholarship has called the “middle future”: not the imminent end of the fanatic nor the indefinite deferral of the cautious, but a datable horizon a century off, near enough to compel and far enough to survive the reckoner. It belonged to the lineage that reads number as the cipher of eschatology — the conviction, older than the Reformation and traceable through Pythagorean arithmology and Christian prophecy alike, that the second coming is written in measurable time and can be read out by whoever holds the key.

It was the Plaine Discovery, not the logarithms, that made Napier’s name in his century. The book was a sustained popular success: a Dutch translation by Michiel Panneel appeared in 1600 and went to a second edition; a French version by Georges Thomson came out at La Rochelle in 1602, revised by Napier himself, and ran through several printings; a German rendering followed; and a corrected and enlarged English edition, with answers to objections raised by sympathetic readers, appeared at Edinburgh and London in 1611. While the Descriptio waited for the small company of mathematicians who could use it, the Plaine Discovery was read across Protestant Europe by anyone who feared Rome and wished to know the hour. To Napier the precedence was not strange. The apocalyptic arithmetic and the logarithmic tables were two applications of one faith, and of the two he judged the deciphering of prophecy the graver work.

The sorcerer of Merchiston

Around this serious figure gathered a second, folkloric one. Napier’s neighbours took him for a sorcerer — a man in a long black cloak, master of hidden arts, keeping company with the uncanny. Stories survive of a black spider carried about in a small box, of pigeons stupefied so that they could be gathered by hand, and, best known, of a black cockerel used to catch a thief: Napier, the tale runs, shut the soot-covered bird in a dark room and had each servant go in and stroke it, telling them the cock would crow at the guilty touch — and found his thief in the one man who came out with clean hands, having been afraid to lay a finger on the bird at all. The pigeons, by another account, had been eating his grain, and he stilled them by sowing the field with peas soaked in spirits, then walked among the drunken birds collecting them at leisure.

These tales, repeated long after his death, say more about how a reclusive man of unusual learning struck a rural seventeenth-century community than about anything Napier did; historians treat them as legend, not biography. What the stories record is a recognizable transformation — the conversion of a quiet experimenter into a wonder-worker by neighbors who could not tell calculation from conjuring. The brandied peas are an improver’s trick, the soot-and-cockerel a shrewd reading of a guilty conscience; folk memory rendered both as magic. Yet the legend did its work. It fixed Napier, in popular imagination, among the magicians, and it is partly that reputation — alongside the prophetic computation — that draws him into the history of Western esotericism rather than the history of science alone, beside the other learned men of his age, John Dee the nearest among them, in whom the new mathematics and the old hidden wisdom were not yet rivals.

The mind at the joint

What makes Napier worth attention is the joint he stands at. In him the new quantitative science and the old conviction that scripture conceals a decipherable plan were not yet rivals but allies, prosecuted by the same hand with the same tools. He is not alone in this. The century around him is full of calculators who read more than one book of the world: Johannes Kepler, who seized on Napier’s logarithms within two years and built them into the Rudolphine Tables of 1627 — the most accurate planetary tables yet made — while pursuing his own search for the harmonic, geometric order of the cosmos; Girolamo Cardano, the Milanese physician who solved the cubic and cast horoscopes with the same conviction that nature was a single legible science; Cecco d’Ascoli, the Bologna astrologer who paid for his determinism at the stake. Napier’s distinct polarity is that he turned number not toward the stars but against Rome — his joint is calculation joined to decipherable scripture, the Euclidean proof laid over the Apocalypse, the apologetics of the militant Kirk armored in arithmetic.

The end-times he reckoned did not come. The logarithms remained.

Texts and scholarship

The four mathematical works and the one theological work form a compact corpus. The Mirifici Logarithmorum Canonis Descriptio (Edinburgh, 1614) and the posthumous Constructio (1619) carry the logarithms; the Rabdologiae (1617) carries the rods, the promptuary, and location arithmetic; the De Arte Logistica, a treatise on arithmetic and algebra, survived only in manuscript, transcribed by Napier’s son Robert for the use of Henry Briggs, and was not printed until 1839, when his descendant Mark Napier issued it. The Plaine Discovery of the Whole Revelation of Saint John (1593, revised 1611) is the prophetic work; the original text of the first edition is freely available in facsimile at the Internet Archive.

The definitive modern resource is the collected volume edited by Brian Rice, Enrique González-Velasco, and Alexander Corrigan, The Life and Works of John Napier (Springer, 2017), which gathers all five works in English translation for the first time, with introductory studies of the man, the mathematics, and the religious writing, the biographical chapter prepared by a descendant from archival research. For the mathematics in particular, Denis Roegel’s careful reconstruction, Napier’s ideal construction of the logarithms (LOCOMAT, 2012), recovers the kinematic argument of the Constructio and rebuilds the tables. The University of St Andrews maintains a detailed biography in its MacTutor History of Mathematics archive, including the early life, the Briggs visits, and the catalog of works.

The prophetic side has its own scholarship, attentive to setting the Plaine Discovery within the dense apocalyptic culture of late-sixteenth-century Britain rather than treating it as an eccentric appendage to a mathematician’s career. Recent studies read Napier’s chronology as a deliberate intervention in that tradition — a “middle future” apocalypticism analyzed in the Scottish Journal of Theology article “John Napier and the mathematics of the ‘middle future’ apocalypse”, and traced as a challenge to its sixteenth-century predecessors in the Edinburgh doctoral study John Napier of Merchiston’s Plaine Discovery: a challenge to the sixteenth-century apocalyptic tradition. Together this work has dissolved the old image of two unrelated reputations — the mathematician and the prophet — and restored the single calculator who held them both.

→ Related: Daniel · Apologetics · Numerology · Eschatology · Apocalypse · Second Coming · Prophecy · Martin Luther · Pythagoreanism · Girolamo Cardan · Cecco D Ascoli · John Dee · Johannes Kepler