Logarithms, Napier's Bones & the Decimal Point: John Napier's 1614 Invention

The Problem Napier Solved

To appreciate what Napier achieved, it helps to understand the sheer drudgery of calculation in the late sixteenth century. Astronomy, navigation and gunnery all demanded multiplying and dividing enormous multi-digit numbers — work that could consume hours or days by hand and was riddled with opportunities for what Napier himself called “slippery errors”. Astronomers such as Tycho Brahe were producing vast tables of observations that needed processing; navigators relied on trigonometric calculation to fix a ship's position, and a single arithmetical slip could cost a crew its life.

Napier put the problem with real feeling in the preface to his great work: “There is nothing…so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors.” His goal was simply stated and revolutionary in effect: reduce laborious multiplication and division to easy addition and subtraction.

The 1614 Descriptio

Napier worked on the problem for roughly twenty years before publishing. A 1594 letter to Tycho Brahe shows he had already grasped the abstract principle, but converting that insight into a usable table took two more decades of solitary calculation, much of it at Gartness, where local tradition holds that the constant roar of the mill's water wheel did not disturb his concentration, but the occasional clack of the mechanism did — so he would ask the miller to stop the mill while he thought.

The result, Mirifici Logarithmorum Canonis Descriptio (“A Description of the Wonderful Canon of Logarithms”), was printed in Edinburgh by Andrew Hart in 1614: 57 pages of explanation followed by 90 pages of tables. The core idea is elegantly simple. A logarithm turns multiplication into addition: instead of multiplying two large numbers directly, a user looks up the logarithm of each, adds them, and converts back — in modern notation, log(a × b) = log(a) + log(b). Napier coined the word “logarithm” himself, from the Greek logos (ratio) and arithmos (number).

It is worth being precise about what Napier actually built, because it differs from the logarithm taught in schools today. He did not think in terms of a base raised to a power; algebra of that period was not developed enough to express the idea that way. Instead he defined his logarithm kinematically, imagining two points moving along lines — one travelling at constant speed, the other slowing in proportion to the distance it still had to cover. His system was anchored to 10,000,000, the largest entry then found in the best trigonometric tables, and in his original scheme the logarithm of 1 was not zero, which made his tables considerably less convenient than the ones that followed.

“There is nothing…so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers.”

Was Napier First? The Bürgi Question

A properly calibrated account of Napier cannot treat him as the sole, uncontested inventor of logarithms. That would flatter national pride at the expense of accuracy — and the honest history is, in any case, the more interesting one.

The case for Jost Bürgi. The Swiss clockmaker, instrument-maker and astronomer Jost Bürgi (1552–1632), who worked for the Landgrave of Hesse-Kassel and later at the imperial court in Prague alongside Johannes Kepler, is understood by most historians of mathematics to have devised his own logarithm-like system of progressions independently of Napier — and by some accounts as early as the late 1580s or 1590s, which would put his conception before Napier's earliest documented statement of the idea in 1594. Bürgi's method grew out of practical work on geometric and arithmetic progressions used in astronomical calculation, a different route to a mathematically related destination. Kepler himself, who used Bürgi's tables, praised his skill and regretted that Bürgi kept the method to himself for so long.

The case for Napier. Whatever Bürgi worked out privately, he did not publish it. His Arithmetische und geometrische Progress-Tabulen did not appear in print until 1620, in Prague — six years after Napier's Descriptio had already reached Edinburgh's presses, circulated to mathematicians across Europe, and prompted Henry Briggs's celebrated 1615 journey north to meet its author. In the normal conventions of the history of science, priority is generally awarded on the basis of publication and public disclosure, not private, undated invention that cannot be independently dated or verified beyond an author's later account of his own earlier work. By the time Bürgi's tables appeared, Napier's system was already established and spreading, and it was Napier's terminology — not Bürgi's — that the mathematical world adopted.

There is no firm evidence either man had heard of the other's project before 1614. Historians generally treat the two discoveries as genuinely, independently arrived at. The fairest summary is this: Napier earned the credit that publication confers, and which the historical record has attached to his name ever since; Bürgi has a serious, well-documented claim to having reasoned his way to an equivalent idea on his own, and quite possibly first, but the six years he let pass before publishing cost him the priority that history recognises. Both things are true, and neither cancels the other.

Briggs and Common Logarithms

The awkwardness of Napier's original system — a logarithm of 1 that was not zero, a base pinned to 10,000,000 — was resolved through one of the more celebrated meetings in the history of mathematics. Henry Briggs, professor of geometry at Gresham College in London, read the Descriptio and was electrified, writing to a friend that he “never saw a book which pleased me better or made me more wonder.” In the summer of 1615 he made the roughly four-day journey north to meet Napier in person at Merchiston. By one well-known contemporary account, the two men stood in silent mutual admiration for almost a quarter of an hour before either spoke.

Together they agreed that the new tables should be rebuilt on base 10, with the logarithm of 1 set to zero — the “common” logarithm still taught in schools today. Briggs then spent years computing these tables, publishing Arithmetica Logarithmica in 1624, seven years after Napier's death. This part of the story deserves to be told as a shared achievement rather than folded silently into Napier's biography: Napier supplied the founding insight and the name; Briggs supplied the practical, mathematically superior form in which logarithms actually entered everyday use.

Napier's Bones

Logarithms were not Napier's only calculating invention, and the two should not be confused — they were separate ideas published three years apart. In 1617, the year of his death, Napier published Rabdologiae, seu Numerationis per Virgulas (“the art of numbering by means of rods”), printed in Edinburgh and dedicated to Alexander Seton, Earl of Dunfermline. It described a set of numbered rods, quickly nicknamed “Napier's Bones” or “Napier's Rods”, that simplified multiplication, division and even the extraction of square and cube roots.

The principle drew on the old “lattice” or gelosia method of multiplication. Each rod, typically made of wood, bone or ivory, carried the multiples of a single digit. By laying out the rods for the number to be multiplied and reading off the correct row, adding figures along the diagonals, a user could complete a long multiplication almost mechanically. The bones were an immediate success: Napier himself wrote that even before publication his friends found them so pleasing that the rods were “already almost common and are even being carried to foreign countries.”

Napier's Bones are rightly seen as an ancestor of the slide rule and, more distantly, of the mechanical calculator — a portable aid to arithmetic that spread across Europe within Napier's own lifetime.

The Decimal Point

Napier's third contribution to practical mathematics is more modest, and needs to be stated carefully. He did not invent decimal notation from nothing — forms of decimal fractions were already in use on the Continent, most notably through the work of the Flemish mathematician Simon Stevin in his 1585 treatise De Thiende (“The Tenth”). What Napier did was become one of the earliest and most influential advocates of the modern decimal point in print, particularly in his posthumously published Mirifici Logarithmorum Canonis Constructio (1619), where the notation appears in a form very close to what is used today. That advocacy, arriving alongside his logarithms, helped popularise the decimal point in English and Scottish mathematical writing at a formative moment.

Kepler, the Slide Rule and Apollo

The reaction to Napier's logarithms was immediate and, among those who needed them most, close to rapturous. The most striking example is Johannes Kepler, who called the invention a “happy calamity” — recognising at once that it would transform the punishing computations of astronomy. Kepler put logarithms to work on his monumental Rudolphine Tables, finally published in 1627, and dedicated a 1620 astronomical work to Napier. It is a nice historical irony that Kepler, who used both Napier's published logarithms and Bürgi's private ones, sat at the exact crossing point of the priority dispute.

From logarithms grew the slide rule. Building on Napier's logarithms and Edmund Gunter's logarithmic scale of 1620, the English clergyman William Oughtred placed two such scales side by side and slid them against each other to multiply and divide directly — inventing the slide rule around 1622. That humble “slip stick” remained the essential calculating instrument of engineers and scientists for roughly 350 years, until electronic calculators became widely available in the early 1970s.

The slide rule's finest hour came with the Apollo programme. NASA engineers used slide rules to help design the rockets and plan the Moon missions, and according to the Smithsonian's National Air and Space Museum, Apollo crews carried a slide rule — the compact Pickett N600-ES — for “more routine calculations” as backup to onboard computers. Buzz Aldrin's own Apollo 11 example later sold at auction for tens of thousands of dollars, a direct four-century thread back to Merchiston.

Logarithms also underpin huge swathes of modern mathematics, science and technology beyond calculation itself. Logarithmic scales describe earthquake magnitude on the Richter scale, the loudness of sound in decibels, the acidity of solutions on the pH scale, and the intervals of musical tuning; they sit at the foundations of compound-interest mathematics and of Claude Shannon's measure of information entropy. Pierre-Simon Laplace summed up their value in a phrase still quoted today: logarithms, he said, “by shortening the labours, doubled the life of the astronomer.”

Theology and the Wizard of Merchiston

It is easy to forget, looking back from an age that reveres him as a mathematician, that Napier considered his theological writing his life's real work. A Plaine Discovery of the Whole Revelation of Saint John (1593) was an urgent, fiercely argued Protestant reading of the Book of Revelation, aimed squarely at the papacy, and it made Napier a name across Reformed Europe long before his logarithms did. He also drew up, in 1596, designs for national defence against a feared Spanish invasion — burning mirrors, novel artillery, and a musket-proof armoured war chariot — preserved today among the papers at Lambeth Palace.

To his own household and tenants at Merchiston, however, Napier's intense, secretive study habits earned him a very different reputation: that of a magician, in league with dark powers. Locally he was said to walk the grounds at night in a long gown, to keep a black spider shut in a box, and to keep a black cockerel as a “familiar” spirit. The story of the soot-blackened cockerel, used to expose a servant stealing from the household, has been told and retold for four centuries — a vivid, human counterpoint to the abstract brilliance of the mathematics for which he is now remembered.

Legacy

John Napier died on 4 April 1617 at Merchiston Castle, aged about 67, from the effects of gout. He was buried in the kirkyard of St Giles; when that ground was later lost to the building of Parliament House, his remains were moved, and he is now memorialised at St Cuthbert's Parish Church at the west end of Princes Street Gardens, Edinburgh.

His name is everywhere in Scottish mathematical memory. Edinburgh Napier University, built around the restored shell of Merchiston Tower where he was born, takes its name from him, and a statue of Napier stands within the tower today. An oil portrait of 1616, held by the University of Edinburgh, survives in several copies, including one in the National Galleries of Scotland captioned “Discoverer of Logarithms.” In 1914 the Royal Society of Edinburgh marked the tercentenary of the Descriptio with a major celebration of one of the great events in the history of science. His name even reaches the heavens, in the lunar crater Neper, and into electrical engineering, in the unit called the neper; several languages still call the natural logarithm “Napierian.”

Historians have struggled to find words grand enough for the achievement. J. W. L. Glaisher judged that, “with the exception of the Principia of Newton, there is no mathematical work published in this country which has produced such important consequences” as Napier's Descriptio. E. W. Hobson, in his 1914 tercentenary lecture, called the invention of logarithms “one of the very greatest scientific discoveries that the world has seen.” None of that need come at Jost Bürgi's expense — it is simply the historical verdict on the man who got the idea into print first, and into the hands of everyone who needed it.

Did You Know?

• Napier spent roughly twenty years computing his logarithm tables, much of it, tradition says, at a mill in Gartness — where he had the miller stop the wheel whenever its clacking, not its roar, broke his concentration.
• Jost Bürgi, who may have worked out an equivalent system before Napier, waited so long to publish that Kepler himself lamented the delay.
• Henry Briggs travelled roughly four days from London to meet Napier in 1615 — and the two men reportedly stood in silent admiration of one another for nearly fifteen minutes before speaking.
• Napier's Bones and logarithms are often confused, but they are two separate inventions, published three years apart, by the same man.
• Buzz Aldrin carried a slide rule — a direct descendant of Napier's logarithms — on Apollo 11; his own example later sold at auction for tens of thousands of dollars.
• Napier's own tenants believed he was a wizard, and used a soot-blackened black cockerel, treated as his “familiar”, to unmask a thief in his household.

Timeline

1. 1550

   John Napier born at Merchiston Castle, Edinburgh

   Eldest son of Sir Archibald Napier, future Master of the Mint

2. 1563

   Enters the University of St Andrews aged 13

   Leaves without a degree; likely continues study on the Continent, probably Paris

3. c. 1588

   Jost Bürgi, working in Kassel and later Prague, is understood to have devised his own logarithm-like system

   Not published for another three decades — the heart of the priority dispute

4. 1593

   Napier publishes A Plaine Discovery of the Whole Revelation of Saint John

   The work he himself considered his most important; a Protestant sensation across Europe

5. 1594

   A letter to Tycho Brahe shows Napier already grasps the abstract principle of logarithms

   Two more decades of solitary calculation lie ahead before publication

6. 1596

   Drafts 'Secret Inventions' for national defence

   Designs for burning mirrors, artillery and a musket-proof chariot, now at Lambeth Palace

7. 1614

   Mirifici Logarithmorum Canonis Descriptio printed in Edinburgh by Andrew Hart

   The first published table of logarithms — the work that fixes Napier's priority in the historical record

8. 1615

   Henry Briggs travels from London to meet Napier at Merchiston

   The two agree on base-10 'common' logarithms with log(1) = 0

9. 1617

   Napier publishes Rabdologiae, describing Napier's Bones

   Dies at Merchiston on 4 April, aged about 67

10. 1619

    Mirifici Logarithmorum Canonis Constructio published posthumously

    Contains some of the earliest systematic use of the decimal point in print

11. 1620

    Jost Bürgi finally publishes his Arithmetische und geometrische Progress-Tabulen in Prague

    Six years after Napier — too late to contest the priority Napier's earlier publication had already secured

12. 1624

    Briggs's Arithmetica Logarithmica establishes the modern common logarithm

    The standard form used by scientists and engineers until electronic calculators

13. c. 1622

    William Oughtred combines two logarithmic scales to build the slide rule

    In use by engineers and scientists for roughly 350 years

14. 1969

    Apollo-era engineers still carry slide rules as backup calculating tools

    Buzz Aldrin's own example later sold at auction

Notes on the Evidence

The Bürgi dating is not precise. Bürgi's own account, and later scholarship reconstructing his working papers, place his conception of a logarithm-like system somewhere between the late 1580s and the mid-1590s. No dated manuscript from that period survives to fix the date exactly; the estimate rests on later testimony and internal evidence in his 1620 publication. This is treated here as a credible, well-supported claim rather than a certainty.

Napier's own priority is dated by publication, not conception. A 1594 letter to Tycho Brahe shows Napier had grasped the underlying principle by that date, but this is evidence of an idea in progress, not a finished, published system. His claim to priority rests specifically on the 1614 Descriptio being in print before Bürgi's 1620 tables.

Napier's logarithms and Briggs's logarithms are not the same object. Popular accounts sometimes elide the two. Napier's 1614 system used a different base and did not set log(1) = 0; the base-10 “common” logarithm familiar today is a joint Napier–Briggs development, finalised and published by Briggs alone in 1624, after Napier's death.

The decimal point claim is bounded deliberately. This article does not claim Napier invented decimal notation; Simon Stevin's 1585 work predates Napier's use of it. Napier is credited here only with being an early and influential populariser of the modern decimal-point form, chiefly through the 1619 Constructio.

Anecdotal material is flagged as such. The mill story, the black-cockerel story and the Napier–Briggs fifteen-minutes-of-silence anecdote are long-standing traditions recorded in early biographical accounts of Napier; they are presented here as tradition, not as documented fact with primary-source verification.