Napier, John, of Merchiston

(1550-1617), mathematician

by George Molland

© Oxford University Press 2004 All rights reserved

Napier, John, of Merchiston (1550-1617), mathematician, was born at Merchiston Castle, the first child of Sir Archibald Napier of Merchiston (d. 1608), who was only sixteen or less, and his first wife, Janet Bothwell. On his father's side he came from a line of minor Scottish nobles, who for over a hundred years had been lairds of Merchiston (on the outskirts of Edinburgh). His mother came from a notable Edinburgh family, many of whom held burghal positions. In 1559 her brother Adam became bishop of Orkney, favouring in general whichever party he perceived to be in the ascendant. In the year following his appointment he recommended a foreign education for his nephew John, either in France or Flanders, 'for he can leyr na guid at hame, nor get na proffeit in this meist perullus worlde' (M. Napier, Memoirs, 67). In 1563 John enrolled at St Andrews University, but probably spent only a short time there, and then proceeded abroad in accord with his uncle's advice. The place (or places) of his foreign education is again conjectural: some have suggested Paris, but K. R. Firth has also made an interesting case for the Collège de Guyenne in Bordeaux (Firth, 135-8).

Family matters, technology, and magic
Back in Scotland, in 1572 Napier married Elizabeth Stirling (d. 1579), the daughter of a prominent lawyer of noble family, and the couple took up residence at Gartness in Stirlingshire, not far from the banks of Loch Lomond. There he built a much admired castle (now a ruin), which remained his principal place of abode until his father's death in 1608, when he moved to Merchiston Castle. His first wife died in 1579 after bearing two children, Archibald Napier and Jane, and a few years later he married Agnes, daughter of Sir James Chisholm of Cromlix in Perthshire, with whom he had five sons and five daughters. Embarrassingly for Napier, his second father-in-law was of strongly Catholic inclinations, and in 1592 he was involved in the so-called Spanish blanks affair, in which purportedly he was in collusion with the earls of Huntly, Angus, and Erroll in a plot to encourage a Spanish invasion of Scotland and England. The king was disposed to be lenient, but, although the evidence is a little ambiguous between himself and his father, it seems that John Napier, who had become a commissioner to the general assembly of the Church of Scotland in 1588, was one of a group of delegates appointed to urge stronger measures against the supposed conspirators.

Involvement in public affairs was almost inevitable at the time for someone of Napier's status, but, perhaps because of his multifarious intellectual interests, he did not allow it to occupy too much of his energy. One of his main concerns during his time at Gartness was the management of his own and his father's numerous estates. This sometimes involved him in legal disputes about land rights, but more importantly he experimented with ways of improving the fertility of the fields, and in 1598 a patent was taken out in the name of his oldest son Archibald (who by this time was attached to the court of James VI) for a method that involved manuring land with common salt. Napier's practical philosophy, which puts him in a tradition represented earlier by Archimedes, Roger Bacon, and Leonard and Thomas Digges, and later by Cornelius Drebbel, Athanasius Kircher, and John Wilkins, also characteristically projected machines that might be useful in warfare. In a document of 1596 (transcript with facsimile, M. Napier, Memoirs, 247-8), when the Spanish threat could still seem real, he mentioned four such. The first two were burning mirrors, which might be of service for burning enemy ships, but the others were less traditional. One was:

a piece of artillery, which, shott, passeth not linallie through the enemie, destroying onlie those that stand on the randon thereof, and fra them forth flying idly, as utheris do; but passeth superficially, ranging abrode within the whole appointed place, and not departing furth of the place till it hath executed his whole strength, by destroying those that be within the boundes of the said place.
Of this, the eccentric Sir Thomas Urquhart said:
he gave proof upon a large plaine in Scotland to the destruction of a great many herds of cattel and flocks of sheep, whereof some were distant from other half a mile on all sides and some a whole mile. (Urquhart, 100)
The final invention was:
a round chariot of mettle made of the proofe of dooble muskett, which motion shall be by those that be within the same, more easie, more light, and more spedie by much then so manie armed men would be otherwayes. The use hereof as well, in moving, serveth to breake the array of the enemies battle and to make passage, as also in staying and abiding within the enemies battle, it serveth to destroy the environed enemy by continuall charge and shott of harquebush through small hoalles.
On a more mundane (actually sub-mundane) level he had patented at about the same time a device involving a revolving screw and axle for the draining of mines. This had at least a family resemblance--the word cochlea is even used in its description--to the cochlias sometimes ascribed to Archimedes, and said by Diodorus Siculus to have been used in antiquity for removing water from Spanish mines.

In his own time the projection and production of such devices resonantly evoked images of magical practices and, even if ill-founded legends are discounted, it cannot be asserted that Napier was a stranger to the occult, in at least some senses of that term. There is extant a manuscript record of a consultation between him and the German adept Daniel Müller about esoteric alchemy, and his son Robert (who became his literary executor) left behind him a work on the subject, in which he strongly emphasized the need for secrecy with regard to such a sacred science. These familial interests may have been encouraged by the fact that in 1582 John Napier's father became 'General of his Majesty's Cunzie House' (or what south of the border would be called master of the mint), with implications of an interest in gold-making, and had the 'spirit of divination' attributed to him. Also, John's uncle, the bishop of Orkney, was described as a 'sorcerer and execrable magician', and his cousin Richard Napier gained in England a very strong magical reputation. Another incident involving both valuables and probably divinatory practices was the strange contract that John Napier entered into in 1594 with a very dubious character called Robert Logan of Restalrig, to find hidden treasure at the latter's seaside property of Fastcastle, near Berwick. The venture seems to have come to a bad end, for in a somewhat later document Napier explicitly discriminated against people with the name of Logan.

Napier's most widely diffused work was his Plaine Discovery of the Whole Revelation of Saint Iohn. This was first published in 1593, and, besides numerous editions in English, was also published several times in Dutch, French, and German. Its stance is vehemently anti-papal and in the dedication to James VI the king is urged to purge his house 'of all suspicion of Papists, and Atheists or Newtrals, wherof this Revelation foretelleth, that the number shall greatly increase in these latter daies' (J. Napier, Plaine Discovery of the Whole Revelation of Saint Iohn, 1593, sig. A3v). The text of the first part is presented in quasi-mathematical form as a series of propositions, starting with the manner of equating Biblical days with mundane time. Proposition 26 maintains that 'The Pope is that only Antichrist, prophecied of, in particular' (ibid., 41), and Proposition 32 that 'Gog is the Pope, and Magog is the Turkes and Mahometanes' (ibid., 59). Proposition 34 moves to more historical considerations, for the beginning of Satan's bondage was about the year AD 300, but in 1300, being freed:

He passeth foorth to stirre up and seduce, these Papistical and Mahometike armies of Gog and Magog, to strife and warrefare. But in that 1300, yeare, began (by Sathans instigation) that proude strife betwixt them for supremacie, both of them chalenging to themselves the Empire of the whole earth: for Pope Boniface the eight, instituted the first Iubelee, that yeare, and clothed himselfe the one day in the Pontificals of a Bishop, and the other day in the robe-royall, of an Emperour, and having borne before him two swords, proclaimed these words, Ecce duo gladii hic: meaning thereby he was ful Monarch and more than Monarch: For, thereby he usurped to himselfe universallie both spirituall and temporall power. (ibid., 62-3)
The troubles continued until about 1560 in the time of the Reformation, when the Antichrist's 1260-year universal reign became abrogated; the way was even now being prepared for the final and utter destruction of Rome, and the Last Judgement, according to Napier, was due to occur between 1688 and 1700.

At first glance, and especially to modern eyes, Napier's account is bigotedly anti-Catholic, but not only the normally virulent theological rhetoric of the age but also special circumstances may mitigate this judgement. In an oft-quoted passage from one of the prefaces to the work, Napier wrote:

In my tender yeares, and barneage in Sanct-Androis at the Schooles, having on the one parte contracted a loving familiaritie with a certaine Gentleman, &c. a Papist: And on the other part, being attentive to the Sermons of that worthie man of God, Maister Christopher Goodman [a close associate of John Knox], teaching upon the Apocalyps, I was so mooved in admiration, against the blindnes of Papists, that could not most evidently see their seven hilled citie Rome, painted out there so lively by Saint Iohn, as the mother of all spirituall whoredome, that not onely bursted I out in continual reasoning against my said familiar, but also from thenceforth, I determined with my selfe (by the assistance of Gods spirit) to employ my studie and diligence to search out the remanent mysteries of that holy book: as to this houre (praised be the Lorde) I have bin doing at al such times, as conveniently I might have occasion.
This points to the origin of the work in intelligent undergraduate, or (given the age of the participants) schoolboy, debating: as such, untempered gusto should come as no surprise, and this was, at least later, combined with the rigorously systematic approach of a mathematician. The passage also shows how Napier could have very close friendships with papists, and indeed he dedicated a work published in the year of his death (the Rabdologia) to the crypto-Catholic Alexander Seton, earl of Dunfermline. Again, the Plaine Discovery was published just after his father-in-law's implication in the Spanish blanks affair, and as such it served to demonstrate a clear theological-cum-political divide between the men. In short, nuance, if not confusion, abounded.

Besides exciting great interest among his contemporaries Napier's apocalypticism has also occupied the attention of several modern scholars. But, although by no means as 'popular', it is his mathematics that displayed the greatest originality and provoked the greatest admiration, both with the cognoscenti of his time and with subsequent specialist historians. What is extant of this all centres on the subject of complicated numerical calculations. In the sixteenth century such calculations were largely at the service of trigonometry, which itself served astronomy, and it is therefore almost certain that a principal motivation for Napier's mathematical work was astronomical; although there is little evidence of his own astronomical doings, the whole family was famous for its penchant for astrology, which depended heavily upon mathematical astronomy.

Napier's extant mathematical writings may be divided into three classes: relatively traditional arithmetical (logistical) and algebraic treatises; concrete aids to calculations; and logarithms. Although there would have been overlap, the order of development was probably the arithmetic, followed by logarithms and then concrete aids to calculation. The papers relating to the arithmetical and algebraic treatises were not published until the nineteenth century, and have not been much studied. Further work may indicate whether they contain significant new approaches, and in particular whether they may be seen as inviting the idea of logarithms. It may also be significant for such a study that, at least in his earlier days, Napier had evinced considerable interest in Indian mathematics.

Rods and other artefacts
Napier's concrete aids to calculation lie in the tradition of finger reckoning, the abacus or counting board, and so on. His treatise Rabdologiae ... libri duo was first published in 1617 (the year of his death), and included appendices on the promptuarium and on arithmetica localis. In the preface he refers to logarithms, but says that he has worked out these three methods for those who prefer instead to work with 'natural numbers' as they present themselves. The main body of the text, the rabdology, deals with what became popularly known as 'Napier's rods' or 'Napier's bones'. This must be seen against the background of the ancient gelosia or lattice method for multiplication of large numbers (which probably derived initially from India). In this a large square was divided into a grid of smaller squares and then these squares divided by parallel diagonal lines, so that, for instance, in a four-by-four square there were thirty-two triangular spaces. It was then possible, by entering single digits into these spaces, to reduce the multiplication of two four-digit numbers to a series of simple multiplications (up to 9 × 9) and additions, without, it was sometimes said, too much thought. In this method the triangles were filled in as the process of multiplication proceeded, but Napier's bones present them as already supplied with digits. The basic set consists of ten quadrilateral columns with square bases: on each lateral face of each there are ten squares, each bisected diagonally, with digits being inscribed in the triangles according to prescribed rules. By placing rods side by side large numbers could be represented (up to ten digits in some cases), and their multiples up to 9 × easily read off. To multiply two numbers together, one of them is set up on the rods, its multiples by the digits of the other written down on paper with appropriate placement, and the results added together to achieve the result. Arguably this is rather more complicated than the gelosia method, but the rods had the great advantage that long divisions could be performed on them, and (with a small addition of equipment) extraction of roots. They were also adaptable to geometrical and other mensurational purposes.

The promptuary was a more complicated device also related to the gelosia method of multiplication. For it Napier prescribed 200 strips of ivory or similar material, each patterned with diagonally divided squares, which were stored in order in a box. To multiply large numbers some of the strips (most of whose triangles contained digits) were placed on top of the box and regarded as pointing towards the operator; other strips were placed on top of these and perpendicular to them. These upper strips were perforated by triangular holes, through which (with the aid of some simple addition) the multiple could be read off.

Local arithmetic was something rather different. In it numbers were translated into a binary system: that is, they were represented as sums of powers of two, rather than as sums of powers of ten, as in decimal numeration. In the full development of the method a sort of chessboard was formed (Napier describes one of 24 × 24 squares), and each square assigned a simple binary value according to rule. Multiplications, divisions, and extractions of square roots were then performed by moving counters around on the board.

Napier's greatest achievement is universally and rightly regarded as the invention of logarithms. This was first made public in his Mirifici logarithmorum canonis descriptio (1614), but before this was written he had composed another work, published posthumously by his son Robert as Mirifici logarithmorum canonis constructio (1619). (These easily become confused in library catalogues.) The Descriptio swiftly appeared in English in a translation by Edward Wright (1616), and other editions, translations, and adaptations of both works followed.

The basic idea is quite simple, and depends on the difference between arithmetical and geometrical progression, or simple and proportional increase and decrease. Manifestations of it could earlier be found in musical theory, in medieval ratio theory, in Archimedes' representation of large numbers in his Sand-Reckoner, and perhaps most significantly in the very principles of place-value numeration, upon which Napier clearly reflected deeply, a meditation which probably facilitated his introduction of the decimal point in a manner that later became standard. Napier imagined two straight lines, one indefinitely extended from point A, and the other of finite length az, on each of which a point is conceived to move, a process which again has Archimedean echoes. On the second line point b starts from a and moves in such a way that the length bz loses equal ratios in equal times, while on the first line point B starts from point A at the same time and moves uniformly at the same speed as that which b had initially. Then at any time the number representing the length AB is the logarithm (number of the ratio) of that representing bz. The crucial advantage of this conception is that it allows complex multiplications and divisions to be performed by far simpler additions and subtractions of the corresponding logarithms, a process which until very recently led to tables of logarithms being the constant companion of every school pupil with anything more than a very modest claim to mathematical competence.

But thereby comes the rub. Napier's conception may have been essentially simple, although directed to an end with the boldness of genius, but constructing tables demanded a huge amount of more mundane labour, as well as virtuosity in the framing of techniques. The Descriptio, which gave the principles and the actual tables, did not say how they were made. This was the central burden of the earlier Constructio, in the text of which what were later referred to as logarithms are called numeri artificiales ('artful numbers') . Because of the astronomical connection, Napier uses trigonometrical terminology in his exposition. The finite line (called az above) is the whole sine (sinus totus) or radius of a circle--sines were then seen as lines rather than as ratios--and Napier assigns the length 10,000,000 to it, which is the sine of 90°; Napier's table proceeds by intervals of 1', but, as it includes in each line the complementary angle with respect to 90°, it does not have to go on beyond forty-five parts. Napier's method was to form three auxiliary tables of numbers decreasing by small proportional intervals from 10,000,000. By means of these, and a number of rules for assigning upper and lower bounds to logarithms, he was able to form his final table. As has often been noted, it does include a small replicated error, but not one that radically mars the outcome.

Publication of the Descriptio was received rapturously in certain quarters, and by two men in particular. The first reaction of the great German astronomer Johannes Kepler was somewhat muted: early in 1618 he spoke of the work of a certain Scottish baron whose name escaped him, and which he clearly did not think would be of much use to him, but quite soon, spurred on by the publication in that year of his sometime associate Benjamin Ursinus's Cursus mathematici practici which, as the title explained, contained a version of Napier's trigonometria logarithmica, his reaction became ecstatic. In his Ephemeris for 1620 Kepler included a eulogistic dedicatory letter to Napier dated July 1619, in which he clearly did not realize that the addressee had long been dead. Kepler also composed his own treatise on logarithms, published in 1624 complete with tables.

In England the most noteworthy respondent to the Descriptio was Henry Briggs, then professor of geometry at Gresham College, London, who very soon travelled to Scotland to visit Napier. In a charming if somewhat implausible account of their first meeting the astrologer William Lilly wrote that 'almost one quarter of an hour was spent, each beholding other with admiration, before one word was spoken' (M. Napier, Memoirs, 409). It must be noted that Napier's logarithms were not even what are now often called Naperian logarithms, but (translated into modern notation) the logarithm of x was 107loge107 - 107logex. However, it seems that even before the publication of the Descriptio Napier had been thinking of an improved system, in which the logarithm of unity was zero, and in which the number ten occupied a special position, so that the logarithm either of ten itself, or of one tenth, or of the whole sine was 10,000,000,000. Napier was by then in ill health; he died on 4 April 1617, and was probably buried at St Cuthbert's Church, Edinburgh. It was left to Briggs to develop decimal logarithms as the direct ancestor of most later logarithmic tables.

Napier's work in retrospect
At roughly the same time as Napier's work a form of logarithmic tables was being developed by the Swiss instrument maker Jost Bürgi, but there is no suggestion of influence in either direction. Bürgi did not publish on the subject until 1620; he was apparently very secretive, and even his friend Kepler did not know his work in this direction until later on.

An intriguing and oft-quoted story was propagated by the none too reliable annalist Anthony Wood, who wrote:

It must be now known, that one Dr. Craig a Scotchman ... coming out of Denmark into his own country, called upon Joh. Neper baron of Marcheston near Edinburgh, and told him among other discourses of a new invention in Denmark (by Longomontanus as 'tis said) to save the tedious multiplication and division in astronomical calculations. Neper being solicitous to know farther of him concerning this matter, he could give no other account of it, than that it was by proportionable numbers. Which hint Neper taking, he desired him at his return to call on him again. Craig, after some weeks had passed, did so, and Neper then shew'd him a rude draught of what he called Canon mirabilis Logarithmorum. (Wood, Ath. Oxon., 2.491-2)
The informant referred to is clearly John Craig, brother to the famous jurist Thomas Craig. John had taught medicine in Germany before returning to Scotland about 1582, and O. Gingerich and R. S. Westman have recently shown that while abroad Craig copied out specimens of the so-called method of prosthaphaeresis as developed by Paul Wittich (not Longomontanus), and, despite Wood's probably proleptic reference to 'proportionable numbers', it has usually been assumed that prosthaphaeresis was the subject of his remarks. This made use of equalities such as (in modern notation) sin A sin B = 12 cos (A - B) - 12 cos (A + B), and was used for easing certain trigonometrical calculations. It clearly has little resemblance to Napier's more universal method, except for the general idea of replacing multiplications and divisions by additions and subtractions, but it could be that discussions with Craig spurred Napier to intensive work along his own lines. In 1592 Craig informed Tycho Brahe that a canon mirificus was being constructed by a certain noble compatriot of his (a generoso quodam consanguineo nostro; Opera omnia, 7.335) , and so, even if the conception did not occur much before this date, the pre-publication process of maturation was clearly long.

Early in the nineteenth century numerous papers relating to John Napier and the Napier family were lost by fire and by shipwreck. Historiographically this was grievous, for with them a far richer and better rounded picture of this early modern Scottish Archimedes might well have been achieved.


M. Napier, Memoirs of John Napier of Merchiston, his lineage, life, and times, with a history of the invention of logarithms (1834)
M. E. Baron, 'Napier, John', DSB
C. G. Knott, ed., Napier tercentenary memorial volume (1915)
J. Napier, The construction of the wonderful canon of logarithms, trans. W. R. Maconald (1889) [incl. bibliography of Napier's works]
J. Small, 'Sketches of later Scottish alchemists: John Napier of Merchiston, Robert Napier, Sir David Lindsay, first earl of Balcarres, Patrick Ruthven, Alexander Seton, and Patrick Scot', Proceedings of the Society of Antiquaries of Scotland, 11 (1874-6), 410-38
K. R. Firth, The apocalyptic tradition in Reformation Britain, 1530-1645 (1979)
A. Keller, 'The physical nature of man: science, medicine, mathematics', Humanism in Renaissance Scotland, ed. J. MacQueen (1990), 97-122
W. F. Hawkins, 'The mathematical work of John Napier, 1550-1617', Bulletin of the Australian Mathematical Society, 26 (1982), 455-68
J. Napier, Rabdology, trans. W. F. Richardson with introduction by R. E. Rider (1990)
W. F. Hawkins, 'The first calculating machine (John Napier, 1617)', New Zealand Mathematical Society, supplement to newsletter number 16 (Dec 1979), 1-23
K. W. Menninger, Number words and number symbols: a cultural history of numbers, trans. P. Broneer (1969)
C. Naux, Histoire des logarithmes de Neper à Euler, 2 vols. (1966)
W. Kaunzer, 'Logarithms', Companion encyclopaedia of the history and philosophy of the mathematical sciences, ed. I. Grattan-Guinness (1994), 1.210-28
F. Maseres, Scriptores logarithmici (1791-1807)
J. M. Thomson and others, eds., Registrum magni sigilli regum Scotorum / The register of the great seal of Scotland, 11 vols. (1882-1914), vol. 6
T. Urquhart, The jewel, ed. R. D. S. Jack and R. J. Lyall (1983)
Wood, Ath. Oxon.
O. Gingerich and R. S. Westman, The Wittich connection: conflict and priority in late sixteenth-century cosmology (1988)
Tycho Brahe: opera omnia, ed. J. L. E. Dreyer, 15 vols. in 7 (1913-29); repr. (Amsterdam, 1972)
Johannes Kepler: Gesammelte Werke, ed. W. von Dyck and M. Caspar (München, 1938-)
H. S. Carslaw, 'The discovery of logarithms by Napier of Merchistoun', Journal of Proceedings of the Royal Society of New South Wales, 48 (1914), 42-72

LPL, papers

portrait, 1616, U. Edin. [see illus.]

Wealth at death  
considerable: will, repr. in Napier, Memoirs of John Napier, 427-31

Oxford University Press 2004 All rights reserved


GO TO THE OUP ARTICLE (Sign-in required)