John Napier

How to use this course

John Napier published his invention of logarithms in 1614, when he was 64. He defined them without exponents, without the number \(e\), and without algebraic notation, using two points moving along lines. Within ten years Henry Briggs had recast them in base 10, Johannes Kepler had used them to compute the Rudolphine Tables, and calculators everywhere had stopped multiplying long numbers by hand. Napier himself regarded a commentary on the Book of Revelation as his most important work.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Napier worked with a “whole sine” (radius) of \(R=10^7\), so that sines were integers; we follow him where it helps and translate into modern notation throughout.

Timeline

Merchiston and St Andrews, 1550–1571

John Napier was born in 1550 at Merchiston Castle, a tower house then just outside Edinburgh which his family had held since the fifteenth century. His father, Archibald Napier, was about sixteen at the time. Archibald was knighted in 1565 and became Master of the Mint in 1582. His mother, Janet Bothwell, was the sister of Adam Bothwell, who became Bishop of Orkney and was one of the Protestant reformers of the Scottish church.

Scotland in the 1550s and 1560s was in religious and political upheaval: the Reformation Parliament of 1560, the return and fall of Mary, Queen of Scots, and civil war between her party and that of the infant James VI. When John was about eleven, Adam Bothwell wrote to Archibald urging him to send the boy to school in France or Flanders, because he would learn little good at home.

In 1563, aged thirteen, John matriculated at St Salvator’s College in the University of St Andrews. His mother died shortly afterwards. He left without a degree, and nothing certain is known of the next seven or so years. The usual inference, from his uncle’s advice, from his later command of Greek and from the common practice of Scottish gentry, is that he studied on the Continent, perhaps in Paris. He was back in Scotland by 1571.

Laird of Gartness, 1571–1608

In 1572 most of the family estates were made over to John, and he married Elizabeth Stirling, daughter of James Stirling of Keir. They had two children before her death in 1579. He then married Agnes Chisholm, with whom he had ten more. From 1574 the family lived at a new castle at Gartness in Stirlingshire, on the Endrick Water.

Napier ran his estates actively. He experimented with manures, including common salt as a fertilizer. Local stories made the laird a sorcerer: neighbours told of a black cockerel that detected thieves and of a spider he kept in a box. Such tales say more about how a man who calculated and experimented looked to his neighbours than about Napier himself.

The period was dangerous for a committed Protestant landowner. Catholic earls conspired with Spain in the 1580s and 1590s, and the fear of a Spanish landing after the Armada of 1588 was real. Napier also signed a curious contract with Robert Logan of Restalrig to search for treasure believed hidden at Fast Castle; nothing was found. After his father’s death in 1608 he moved the household to Merchiston, where he lived for the rest of his life.

The Plaine Discovery and the inventions, 1593–1596

Napier’s first and, in his own eyes, most important book was A Plaine Discovery of the Whole Revelation of Saint John, published in Edinburgh in 1593 and dedicated to James VI. Written in English for a wide readership, it argued that the pope was the Antichrist and used a systematic, almost mathematical analysis of the prophetic numbers to date the events of Revelation. Napier concluded that the world would end between 1688 and 1700. The book was a European success, with translations into Dutch, French and German and many editions.

In 1596 he set down a short paper of “secret inventions” for the defence of the island against invaders. It described burning mirrors to set ships alight, a kind of artillery, and a protected war chariot. There is no evidence that any was built.

Mathematics as a pastime

Throughout these decades Napier pursued arithmetic and trigonometry as a private study. He is traditionally said to have worked on logarithms for some twenty years before publishing. The Constructio, which explains how the tables were computed, was written before the Descriptio that announced them. A manuscript on arithmetic and algebra, later published as De arte logistica (1839), shows a working command of the algebra of his day. His approach to the central problem of practical mathematics in his age, reducing the labour of multiplying large numbers, was shaped by the needs of astronomy and navigation, where long products of sines were routine.

Calculation before logarithms

For the general reader. Before logarithms, an astronomer who needed the product of two seven-digit sines had to multiply them out by hand, and a long computation involved hundreds of such products. Two ideas promised relief: a trigonometric identity that turns products into sums, and the observation that adding exponents multiplies powers. Napier’s logarithms grew from the second.

Prosthaphaeresis

In the 1580s astronomers around Tycho Brahe used the identity

to multiply. To form \(xy\) with \(0<x,y<1\), look up \(\alpha=\arccos x,\ \beta=\arccos y\), add and subtract the angles, look up two cosines, and halve their sum. Two table look-ups and an addition replace a long multiplication. The method was called prosthaphaeresis, from the Greek for addition and subtraction.

Two progressions

Michael Stifel, in his Arithmetica integra (1544), set the powers of 2 against their exponents:

Multiplication in the lower row corresponds to addition in the upper, division to subtraction, and roots to division. The idea goes back to Archimedes. The difficulty is that the lower row has huge gaps: most numbers do not appear in it. A usable table needs a geometric progression whose ratio is so close to 1 that its terms fill the line densely, and an arithmetic progression matched to it.

Two moving points

For the general reader. Napier did not define logarithms as exponents. He imagined two points setting off at the same speed. One slows down as it approaches its goal, always moving at a speed equal to the distance still to go; the other keeps its speed. The distance the steady point has covered is the logarithm of the distance the slowing point has left. This motion-based definition makes his logarithm a continuous function, not just a table.

The definition

Let \(R=10^7\), the whole sine. A point P starts at distance \(R\) from a fixed point Q and moves toward Q with speed equal to its current distance from Q. A point L starts at the same moment on another line and moves at the constant speed \(R\). If \(y\) is the distance PQ at some moment, Napier defines the logarithm of \(y\) as the distance \(x\) that L has travelled.

In modern terms \(dy/dt=-y\), \(y(0)=R\), so \(y=Re^{-t}\), while \(x=Rt\). Eliminating \(t\):

The factor \(1/R\) in the product rule is the price of not having \(\operatorname{NapLog}(1)=0\). For Napier it was harmless, since his numbers were sines scaled by \(R\), and a product of sines must be divided by \(R\) to be a sine again. Napier stated the property as a rule of proportion: if \(a:b=c:d\), then \(\operatorname{NapLog}a-\operatorname{NapLog}b=\operatorname{NapLog}c-\operatorname{NapLog}d\). It is sometimes said that Napier’s logarithms have base \(1/e\); more exactly, \(\operatorname{NapLog}(y)/R=\log_{1/e}(y/R)\). Napier had no concept of a base.

Bounds

Proof (Napier’s). Let \(T\) be the time P takes to go from distance \(R\) to distance \(y\). During that time P’s speed lies strictly between \(y\) and \(R\), and it covers the distance \(R-y\). Hence \(yT<R-y<RT\). Since L moves at speed \(R\), \(\operatorname{NapLog}(y)=RT\), and the two inequalities give the lemma.

Building the canon

For the general reader. Napier’s table gave the logarithms of the sines of every minute of arc from \(0^\circ\) to \(90^\circ\), 5400 values to about seven figures. He computed them by hand, using only subtraction, halving and clever bookkeeping. The Constructio, published two years after his death, explains how.

Three tables

• First table. 101 terms \(10^7(1-10^{-7})^n\), \(n=0,\dots,100\). Each term is obtained from the previous by subtracting it shifted seven decimal places, so no multiplication is needed. The last term is \(9\,999\,900.0004950\).
• Second table. 51 terms \(10^7(1-10^{-5})^n\), \(n=0,\dots,50\). The ratio \(1-10^{-5}\) is very nearly the ratio between the first and last terms of the first table. Napier’s last entry, \(9\,995\,001.222927\), contains a small arithmetic slip; the correct value is \(9\,995\,001.224804\).
• Third, or radical, table. 69 columns and 21 rows. Down each column the ratio is \(1-\frac{1}{2000}=0.9995\), close to the span of the second table; across the rows it is \(0.99\). The 1449 entries run from \(10^7\) down to about \(4\,998\,609\), roughly half the radius, the sine of \(30^\circ\).

The logarithm of each entry is known from the logarithms of the ratios, which Napier fixed by the bounds lemma: \(\operatorname{NapLog}(9\,999\,999)\approx1.00000005\), whence the logarithms of the steps of the second and third tables follow by addition, again corrected by the lemma. Any sine between \(R/2\) and \(R\) then lies between two entries of the radical table, and its logarithm is found by the bounds applied to the small ratio between it and the nearest entry. Sines below \(R/2\) are brought into range by multiplying by known factors such as 2 or 10, whose logarithms are tabulated.

Proof. \(\operatorname{NapLog}(y)-\operatorname{NapLog}(y')=\operatorname{NapLog}(Ry/y')\) by the proportion rule. Apply the lemma to \(Ry/y'\): \(R-Ry/y'=R(y'-y)/y'\), and \(R(R-Ry/y')/(Ry/y')=R(y'-y)/y\).

The printed table

The 1614 table is laid out on facing pages, one degree per pair, with seven columns: minutes; the sine; its logarithm; a column of differentiae, the difference between the logarithms of sine and cosine, which is the logarithm of the tangent; the logarithm of the cosine; the cosine; and the complementary minutes. One opening therefore serves an angle and its complement.

Briggs and common logarithms

For the general reader. Napier’s logarithms worked, but they had awkward features: the logarithm of 1 was not zero, and powers of ten did not have simple logarithms. Henry Briggs, a London professor, travelled to Scotland to meet Napier, and together they agreed on a better scale. The result, logarithms to base 10, was the form used by everyone who calculated until electronic calculators replaced the tables.

Henry Briggs, the first Gresham Professor of Geometry in London, read the Descriptio soon after it appeared. In a letter of 10 March 1615 he wrote that he had “never saw a book which pleased me better or made me more wonder.” He spent a month at Merchiston in the summer of 1615 and returned in 1616. In their discussions both men proposed changes of scale; the version adopted was that the logarithm of 1 should be 0 and that of 10 should be a power of ten. Napier, ageing and unwell, left the computation to Briggs.

Briggs’s method

Briggs extracted 54 successive square roots of 10, to more than thirty decimal places. If \(10^{1/2^{k}}=1+\delta_k\), then for large \(k\), \(\log_{10}(1+\delta_k)=2^{-k}\) and \(\delta_k\) is very small, so the logarithm of a number near 1 is proportional to its excess over 1:

To find \(\log_{10}x\), take enough square roots of \(x\) to bring it near 1, and use the proportionality. His Logarithmorum chilias prima (1617) gave the logarithms of 1 to 1000 to fourteen places; Arithmetica logarithmica (1624) extended them to 1–20,000 and 90,000–100,000. Adriaan Vlacq filled the gap in 1628.

From Napier to \(\ln\) and \(e\)

For the general reader. Napier’s definition already contains the natural logarithm, but the number \(e\) and the idea of a logarithm as an area appeared only later in the seventeenth century. This lesson follows the path from his two moving points to the function \(\ln x\) of calculus.

Bürgi

The Swiss instrument maker Jost Bürgi, working in Prague, computed tables pairing the terms of the progression \(10^8(1.0001)^n\) with \(10n\). They were printed in 1620 as Arithmetische und Geometrische Progress Tabulen. Bürgi’s idea may have been as early as Napier’s, but he published later and without explanation, and his tables had little influence.

Logarithms as areas

Proof. The map \((x,y)\mapsto(kx,y/k)\) preserves area and sends the hyperbola to itself. Then \(L(xy)=A(1,x)+A(x,xy)=L(x)+A(1,y)\). Saint-Vincent proved the scaling property; his student de Sarasa recognized the logarithm. In modern notation \(L=\ln\), and \(\operatorname{NapLog}(y)=R\big(L(R)-L(y)\big)\).

In 1668 Nicholas Mercator published the series obtained by integrating \(1/(1+t)=1-t+t^2-\cdots\):

The number \(e\) appeared later: Jacob Bernoulli met \(\lim(1+1/n)^n\) in a problem on compound interest in 1683, and Leonhard Euler, who used the letter \(e\) for it, defined the exponential and logarithm as inverse functions in his Introductio in analysin infinitorum (1748).

Spherical trigonometry

For the general reader. Astronomers and navigators solve triangles drawn on a sphere, whose sides are arcs of great circles. Napier found a memory device that compresses the ten formulas for right-angled spherical triangles into two rules. With logarithms, each rule needs only additions.

The rules of circular parts

Let a spherical triangle on the unit sphere have a right angle at \(C\), legs \(a,b\), hypotenuse \(c\), and angles \(A,B\). Arrange five “circular parts” around a circle in the order \(a,\ b,\ \bar A,\ \bar c,\ \bar B\), where a bar means the complement: \(\bar A=90^\circ-A\). Any one part is the middle part; the two beside it are adjacent, the other two opposite.

Ten formulas result (five middle parts, two rules each), for example

Derivation of two. The spherical law of cosines \(\cos c=\cos a\cos b+\sin a\sin b\cos C\) with \(C=90^\circ\) gives the spherical Pythagorean theorem \(\cos c=\cos a\cos b\). The spherical law of sines \(\sin a/\sin A=\sin c/\sin C\) gives \(\sin a=\sin c\sin A\). The rest follow by combining these with the law of cosines for angles.

Napier’s analogies

For general spherical triangles, the formulas known as Napier’s analogies (“analogy” meaning proportion) are attributed to him and were published in the posthumous volume of 1619:

Given two sides and the included angle, the first pair yields the other two angles with logarithms alone.

Rods, strips and a chessboard

For the general reader. In the year of his death Napier published a small book describing three calculating aids. One of them, a set of numbered rods, became popular across Europe as “Napier’s bones”. Another, played on a chessboard, worked in binary arithmetic more than eighty years before Leibniz wrote about the binary system.

Rabdologiae, seu numerationis per virgulas libri duo (“Two books of rabdology, or counting with rods”) appeared in Edinburgh in 1617. Napier published it partly to secure credit for the rods, which were already in use.

The bones

Each rod carries the multiples \(1d,2d,\dots,9d\) of one digit \(d\), each written in a square split by a diagonal, tens above-left and units below-right. To multiply a number by a digit \(k\), place the rods for its digits side by side, read row \(k\), and add along the diagonals. This is the old lattice method of multiplication, with the table pre-written. Further rods and procedures handle square and cube roots.

The promptuary

The multiplicationis promptuarium is a box of thin strips carrying digit products, with masks that expose only the needed products for multi-digit multipliers. It multiplies without writing intermediate products, but it was harder to make than the bones and saw little use.

Location arithmetic

To multiply, place one factor’s binary digits along an edge of the board and the other’s along another edge, put a counter at each crossing of a 1 with a 1, slide the counters diagonally to the edge (diagonal cells all have the same value), and simplify pairs. It remained a curiosity, mainly because converting numbers into and out of binary cost more effort than it saved.

1614–1617: reception and death

Mirifici logarithmorum canonis descriptio (“Description of the wonderful canon of logarithms”) appeared in Edinburgh in 1614: 57 pages of explanation and 90 pages of tables. Napier explained the definition and the uses, especially in trigonometry, and promised to publish the construction if the invention was well received. It was. Edward Wright, the leading English authority on navigation, translated it into English; Wright died in 1615, and the translation appeared in 1616.

Briggs’s two visits in 1615 and 1616 began the transformation into common logarithms (Lesson 07). He planned a third visit in 1617. Napier, long suffering from gout, died at Merchiston on 4 April 1617, aged 67. He was buried in the churchyard of St Giles in Edinburgh; a memorial to him stands in St Cuthbert’s Church. The Rabdologiae appeared the same year. In 1619 his son Robert published the Constructio, with additional material on spherical trigonometry.

Kepler

Johannes Kepler soon became its most powerful advocate. He dedicated his Ephemerides for 1620 to Napier and wrote his own treatise on logarithms, Chilias logarithmorum (1624). The Rudolphine Tables of 1627, computed with logarithms, gave planetary positions far more accurate than any before and secured the acceptance of Kepler’s elliptical orbits.

Legacy

Logarithms spread with unusual speed. Edmund Gunter published tables of logarithms of sines and tangents in 1620 and engraved a logarithmic scale on a rule, so that products could be found with a pair of dividers. William Oughtred placed two such scales side by side, inventing the slide rule. From then until the 1970s, logarithm tables and slide rules were the standard tools of scientific and engineering computation. Pierre-Simon Laplace later remarked that logarithms, by shortening the labours, had doubled the life of the astronomer.

• Computation: logarithm tables, the slide rule and the reduction of multiplication to addition, central to astronomy, navigation, surveying and engineering for three and a half centuries.
• Analysis: the logarithm as a continuous function defined by motion, a step toward the calculus; through Saint-Vincent, Mercator, Newton and Euler, it became \(\ln x\) and \(e^x\).
• Trigonometry: the rules of circular parts and Napier’s analogies, used in navigation and astronomy into the twentieth century.
• Notation: wide adoption of the decimal point.
• Names: Edinburgh Napier University, whose Merchiston campus incorporates his birthplace tower; the neper, a logarithmic unit of ratio; the lunar crater Neper. In French and Spanish the natural logarithm is still called the Neperian logarithm.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, John Napier, MacTutor History of Mathematics.
2. John Napier, Wikipedia.
3. M. Napier, Memoirs of John Napier of Merchiston, Edinburgh, 1834.
4. C. G. Knott (ed.), Napier Tercentenary Memorial Volume, Royal Society of Edinburgh, 1915.
5. J. Havil, John Napier: Life, Logarithms, and Legacy, Princeton University Press, 2014.

Mathematics

1. J. Napier, Mirifici logarithmorum canonis descriptio, Edinburgh, 1614. Overview and table structure.
2. J. Napier, Mirifici logarithmorum canonis constructio, Edinburgh, 1619; English translation by W. R. Macdonald, 1889. Reconstruction of the tables (LOCOMAT).
3. J. Napier, Rabdologiae, Edinburgh, 1617. Rabdology, Wikipedia; English translation by W. F. Richardson, MIT Press, 1990.
4. I. Bruce, Napier’s logarithms: translation and notes, 17centurymaths.com.
5. H. Briggs, Arithmetica logarithmica, London, 1624.
6. E. W. Hobson, John Napier and the Invention of Logarithms, 1614, Cambridge University Press, 1914.
7. E. Maor, e: The Story of a Number, Princeton University Press, 1994, chapters 1–2.