Logarithms

Introduction — a tower house on the edge of Edinburgh

When a Scotsman working alone in a draughty tower house on the edge of Edinburgh sat down to make arithmetic easier, he could not have known that he was about to hand the modern world one of its most powerful tools. Yet that is exactly what John Napier of Merchiston did. His invention of logarithms in 1614 transformed the labour of calculation, accelerated the scientific revolution, and remained the backbone of practical mathematics for more than three and a half centuries — right up to the slide rules that helped put men on the Moon.

He was a Protestant aristocrat, a scientific farmer, a designer of secret weapons against a feared Spanish invasion, and — to his neighbours — a man suspected of dabbling in the black arts. But above all he was a mathematician of singular patience and originality, working in isolation on a problem that had defeated everyone else: how to tame the monstrous arithmetic of the age.

Early Life at Merchiston

John Napier was born in 1550 at Merchiston Castle, then a fortified tower house on the outskirts of Edinburgh and now the heart of the university campus that bears his name. He was the eldest son of Sir Archibald Napier, who became Laird of Merchiston and Master of the Mint in 1582. His mother, Janet Bothwell, was sister to Adam Bothwell, Bishop of Orkney. The Napiers were one of the most powerful families in late-sixteenth-century Scotland; John would become the 8th Laird of Merchiston.

He entered the University of St Andrews in 1563 aged just thirteen, living at St Salvator's College under the personal care of the Principal, John Rutherford. He left without a degree — entirely normal for the sons of the Scottish gentry — and historians believe he acquired his deep knowledge of mathematics and classical literature on the Continent, most likely at Paris. By 1571 he was back in Scotland, married to Elizabeth Stirling, and living at a castle built at Gartness in Stirlingshire.

There Napier lived the life of a scientific landowner. He experimented with 'manuring' field land using common salts to boost yields. He was a fiercely committed Protestant: his 1593 anti-Catholic interpretation of the Book of Revelation, A Plaine Discovery of the Whole Revelation of Saint John, was a sensation, translated into Dutch, French and German, and made his name across Protestant Europe — far more famous in his own lifetime than his mathematics.

To his tenants, though, Napier was something stranger: a man widely suspected of being a magician. He was said to walk the grounds in a long gown, to keep a black spider in a box, and to treat his black cockerel as a 'familiar' spirit. The famous tale of the soot-covered cockerel — used to flush out a thieving servant — captures the contradiction at the heart of the man: eccentric, brilliant, deeply devout, and utterly fascinated by numbers.

The Problem Logarithms Solved

To appreciate Napier's achievement, you have to understand the sheer drudgery of calculation in the late sixteenth century. Astronomy, navigation and warfare all demanded the multiplication and division of enormous multi-digit numbers — work that could take hours or even days by hand, and which was riddled with opportunities for 'slippery errors'. Astronomers such as Tycho Brahe were producing vast tables of observations that needed processing. Navigators relied on trigonometric calculations to fix their positions at sea, and a single arithmetical slip could send a ship and its crew to the bottom of the ocean.

Napier described the problem with feeling in the preface to his great work, lamenting that '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 simple to state and revolutionary in effect: to find a method that would reduce these laborious multiplications and divisions to the far easier operations of addition and subtraction.

The Invention of Logarithms

Napier worked on the concept for around twenty years before publishing. A letter to Tycho Brahe shows that he had grasped the abstract principle by 1594, but turning that insight into usable tables took roughly two decades of solitary calculation, much of it at Gartness, where local tradition held that the constant roar of the mill's water wheel did not disturb him but the occasional clack of the mill did, so he would ask the miller to stop the mill while he thought.

The result was Mirifici Logarithmorum Canonis Descriptio ('A Description of the Wonderful Canon of Logarithms'), printed in Edinburgh by Andrew Hart in 1614. It contained 57 pages of explanation followed by 90 pages of tables. The core idea is beautiful in its simplicity. A logarithm turns multiplication into addition: instead of multiplying two large numbers directly, you look up the logarithm of each, add them together, and then convert 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 did, because it differs from the logarithms we use today. He did not think in terms of a base raised to a power — algebra was not yet developed enough for that. Instead he defined his logarithm kinematically, by 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 built around 10⁷, and in his original scheme the logarithm of 1 was not zero — which made his logarithms less convenient than modern ones.

That difficulty was resolved through one of the most celebrated meetings in the history of science. The English mathematician Henry Briggs, professor of geometry at Gresham College in London, read the Descriptio and was electrified, writing to a friend: 'I never saw a book which pleased me better or made me more wonder.' That summer he made the four-day journey north to meet Napier in person. By one contemporary account, the two men stood in silent admiration for almost a quarter of an hour before either spoke. Together they agreed that the new tables should use base 10 and set the logarithm of 1 to zero. Briggs went on to compute these 'common' logarithms, publishing his Arithmetica Logarithmica in 1624.

Napier's Bones

Logarithms were not Napier's only calculating invention. In 1617 he 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. In it he 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 was based on the old 'lattice' or gelosia method of multiplication. Each rod, typically made of wood, bone or ivory, was engraved with the multiples of a digit. By laying out the rods corresponding to the number you wished to multiply and reading off the appropriate row — adding the figures along the diagonals — you could perform a long multiplication almost mechanically. The bones were an instant success. Napier wrote that even before publication his friends were so pleased with them that the rods were 'already almost common and are even being carried to foreign countries'.

Napier's Bones are rightly regarded as an ancestor of the slide rule and the mechanical calculator — the first portable, pocket-sized aid to arithmetic in history, and a direct ancestor of every calculating machine that followed.

The Impact of Logarithms

The reaction to Napier's logarithms was immediate and rapturous. Astronomers and navigators seized on the new tool with relief. The most striking example is Johannes Kepler, who called it a 'happy calamity' — recognising that it would transform the back-breaking computations of astronomy. Kepler put logarithms to work on the monumental Rudolphine Tables, finally published in 1627, and dedicated a 1620 astronomical work to Napier.

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 arrived in the early 1970s.

The slide rule's finest hour came with the Apollo programme: NASA engineers used slide rules to design the rockets and plan the Moon missions, and according to the Smithsonian's National Air and Space Museum, the crews 'carried a slide rule for more routine calculations' — the five-inch metal Pickett N600-ES. Buzz Aldrin's own Apollo 11 example later sold at auction for tens of thousands of dollars.

Logarithms went on to underpin huge swathes of modern mathematics, science and technology. 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 appear in the mathematics of compound interest and at the foundations of information theory, including Claude Shannon's measure of entropy. The French mathematician Pierre-Simon Laplace summed up their value in a phrase that has echoed ever since: logarithms, he said, 'by shortening the labours, doubled the life of the astronomer.'

Legacy

John Napier died on 4 April 1617 at Merchiston Castle, aged 67, from the effects of gout. He was buried in the kirkyard of St Giles; after that ground was 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 in Edinburgh.

His legacy is everywhere. Edinburgh Napier University, built around the restored shell of Merchiston Tower where he was born, traces its name to him; a statue of Napier stands within the tower. His portrait, an oil painting of 1616 held by the University of Edinburgh, survives in copies — including one in the National Galleries of Scotland, which describes him as the 'Discoverer of Logarithms.' In 1914 the Royal Society of Edinburgh marked the tercentenary of the Descriptio with a 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, while 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 the 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.' Working alone in his tower, the Marvellous Merchiston had given the world a tool that made modern science possible.

Frequently Asked Questions

Who invented logarithms? Logarithms were invented by the Scottish mathematician John Napier of Merchiston, who published his Mirifici Logarithmorum Canonis Descriptio in Edinburgh in 1614 after roughly twenty years of solitary work. The base-10 'common' logarithm used by scientists and engineers ever since was developed shortly afterwards by the English mathematician Henry Briggs, working closely with Napier.

What are logarithms? A logarithm is the power to which a base must be raised to produce a given number. Their critical practical property is that log(a × b) = log(a) + log(b) — they turn multiplication into addition. This makes them an immensely powerful labour-saving device in any calculation involving large numbers, and the basis of scales like the Richter scale, the decibel and the pH scale.

What are Napier's Bones? Napier's Bones are a set of numbered rods, published by John Napier in 1617, that allow the user to perform multiplication, division and root extraction quickly and reliably. They are considered the world's first pocket calculator and the direct ancestor of the slide rule and the mechanical calculating machines that followed.

How did logarithms help science? By collapsing days of arithmetic into minutes, logarithms made it possible to process vast astronomical and navigational tables that had previously been impractical. Kepler used them to complete his Rudolphine Tables; navigators used them to fix their positions at sea; engineers used the slide rule — built on Napier's logarithms — to design every major piece of infrastructure from the Industrial Revolution until the 1970s, including the rockets and spacecraft of the Apollo Moon missions.