John Napier

Chapter 1 · The Laird of Merchiston

An estate, a reputation for sorcery, and a black cockerel

1550 – 1590 · Merchiston Castle · Edinburgh · St Andrews

John Napier was born in 1550 at Merchiston Castle, just outside Edinburgh, the son of Sir Archibald Napier. He was sent to St Andrews at thirteen, seems not to have completed a degree, and studied abroad before returning to manage the family estate, which he did for the rest of his life.

He was never a professional scholar. There was no post, no salary, no students. He was a landowner with an income, and mathematics happened in the hours the estate left him.

He took the estate seriously and improved it: he experimented with fertilisers, particularly with common salt, and took out a patent on a hydraulic screw for clearing water out of coal pits. When a Spanish invasion was feared in the 1590s he drew up designs for engines of war, including a burning mirror and a metal-plated chariot, none of which was built.

Locally he was thought to be a sorcerer, and he appears to have found this useful rather than alarming. The best-known story has him detecting a thief among his servants by sending each of them alone into a dark room to stroke a black cockerel, which he said would crow when the guilty one touched it. He had covered the bird in soot. The innocent servants stroked it and came out with black hands; the thief, who had not dared to touch it, came out clean.

The story may be improved in the telling. It is entirely characteristic of the man: practical, indirect, and quietly pleased with itself.

Why this matters

The mathematics that abolished a century of drudgery was done by an amateur, in his own time, on his own land.

A laird, an estate, and a reputation for magic. What would you ask him?

Ask Napier

• “Did you really catch a thief with a sooty cockerel?”
• “What was it like to do mathematics with no university post?”
• “Why did people think you were a sorcerer?”
• “What were the engines of war you designed?”
• “How much of your time did the estate actually take?”

Chapter 2 · The Book He Cared About

A Plaine Discovery, 1593

1593 · Edinburgh

In 1593 Napier published A Plaine Discovery of the Whole Revelation of Saint John. It is a work of Protestant polemic, arguing from the text of Revelation that the Pope is Antichrist, and it includes a calculation of when the world would end — he suggested some point between 1688 and 1700.

He considered this the important thing he had done. It went through many editions and was translated into Dutch, French and German, and in his own time it made him better known than any table ever did.

This is worth taking seriously rather than laughing at. Napier lived through the Scottish Reformation and its aftermath, in a country where the religious settlement was contested and could be reversed, where people were executed over these questions, and where a Spanish invasion in support of Catholicism was a live fear. The book is dedicated to King James VI with a warning to purge his household of papists. It was a political act as much as a theological one.

It also shows the mind that produced the tables. The method of the Plaine Discovery is to take a text everybody found bewildering and impose an exact system on it — numbered propositions, dated periods, a chronology worked out arithmetically. He is doing to Revelation what he would later do to multiplication: taking something that resisted method and forcing a method onto it.

He was wrong about the end of the world. The habit of mind was the same one, and in the other application it worked.

Why this matters

People are unreliable judges of their own work — and the same cast of mind produced the theology and the tables.

The book he thought would last. What would you ask him?

Ask Napier

• “Why did you think the Revelation book was the important one?”
• “What was at stake religiously in Scotland in the 1590s?”
• “You dated the end of the world. How did you calculate it?”
• “Is the method of that book the same as the method of your tables?”
• “Does being wrong about the date undermine the rest?”

Chapter 3 · Twenty Years of Arithmetic

Abolishing a labour by enduring it

c. 1594 – 1614 · Merchiston Castle

In Napier's day an astronomer or a navigator spent a large fraction of a working life multiplying and dividing six- and seven-figure numbers by hand. Every reduction of an observation, every position fixed at sea, meant hours of that labour, with an error anywhere in it fatal to the result.

Napier's idea was to replace that labour with a different one. If you can arrange numbers in a table so that multiplying two of them corresponds to ADDING two entries, then every multiplication becomes an addition, which is quick and where a mistake is easy to spot. Division becomes a subtraction the same way.

The arrangement is elegant. Building the table was not. It required an immense quantity of hand computation — the very labour he was trying to abolish, done once, in bulk, by him, so that nobody else would ever have to do it again. He worked at it for about twenty years, and the Mirifici Logarithmorum Canonis Descriptio was published in 1614.

The reception was immediate and international. Kepler, who was grinding through exactly this kind of computation on Tycho Brahe's observations, seized on it. Navigators adopted it. Within a few years the tables were the standard equipment of every serious computation in Europe, and they stayed so for three hundred years, until electronic calculation.

There is something worth noticing in the shape of it. He did not find a shortcut. He accepted the whole burden himself, in advance, on behalf of everybody who came after — which is what building an instrument means.

Why this matters

Every table, every slide rule and every calculator descends from the decision to suffer a labour once so that nobody need suffer it again.

Twenty years of hand computation, done deliberately. What would you ask him?

Ask Napier

• “Why spend twenty years computing a table by hand?”
• “How does adding two entries do the work of multiplying?”
• “Who was suffering most from the arithmetic you set out to abolish?”
• “How did you check the entries as you built them?”
• “Did you ever think of giving up partway through?”

Chapter 4 · Briggs Comes North

A quarter of an hour of silence

1615 – 1616 · Merchiston Castle · London

Henry Briggs was professor of geometry at Gresham College in London, and when the Descriptio reached him he could think of little else. In 1615 he made the journey to Edinburgh — four days on horseback each way — simply to meet its author.

The meeting is described by William Lilly, who had it at second hand, and the account is famous. Briggs arrived, and the two men looked at each other for almost a quarter of an hour without speaking, before Briggs said that he had undertaken this long journey purposely to see the person who had first found out so excellent a help in astronomy.

What they then agreed mattered as much as the original tables. Napier's first arrangement was awkward to use in practice, and both men had independently seen how to improve it: the table should be arranged so that the entry for one is nothing and the entry for ten is one. That makes the whole system fit the decimal way of writing numbers, so that the same table serves numbers of any size.

Briggs went home and computed the revised tables. Napier died in 1617, and Briggs completed the work alone, publishing tables to fourteen places for thirty thousand numbers. Napier said openly that Briggs had proposed the change, and Briggs said equally openly that the idea was Napier's; each credited the other, which is not the usual pattern.

In 1617, the year of his death, Napier also published the Rabdologiae, describing a set of numbered rods for multiplying — Napier's bones — which put a simplified version of the same labour-saving into the hands of anybody with a wooden box.

Why this matters

A famous collaboration in which each man insisted the good idea was the other's — and the improvement is what made the tables usable.

Four days on horseback, and a quarter of an hour of silence. What would you ask him?

Ask Napier

• “What did Henry Briggs persuade you to change?”
• “Is the story of the silent quarter of an hour true?”
• “Why did each of you insist the idea was the other's?”
• “What are the rods, and who were they for?”
• “Did you know you would not live to see the new tables finished?”

Chapter 5 · The Point After the Units

The smallest thing he did, and the one everybody uses

1617 and after · Merchiston Castle · Europe

Inside the Rabdologiae, almost in passing, Napier does something that every person reading this does several times a day.

Fractions of a unit had been a nuisance for centuries. Astronomers used sixtieths, inherited from Babylon. Merchants used halves, thirds and quarters. Simon Stevin, in a short pamphlet of 1585 called De Thiende, had argued powerfully that fractions should be written in tenths, hundredths and thousandths like the whole numbers — the same idea we use — but his notation was clumsy, marking each place with a circled number above it, and it did not catch on.

Napier writes the units, then a point, then the tenths, hundredths and thousandths straight on. He recommends the practice explicitly as the plainest of all. That is the whole innovation, and it is now so ordinary that nobody realises it had to be proposed.

What is easy to miss is why he cared. Every number in his canon is written to a stated number of places and no further, because it was computed to that accuracy and not beyond it. The point is what makes that statement possible: it separates the part of a number that has been established from the part that has not, and it lets a computer say precisely how far a value is to be trusted. A notation for fractions is also a notation for honesty about precision.

He died at Merchiston on 4 April 1617. He believed he had written one important book and one useful one, and he had them the wrong way round.

Why this matters

The decimal point was proposed, argued for and adopted — and it exists partly to say how far a number can be trusted.

A dot, recommended in passing, now used by everybody. What would you ask him?

Ask Napier

• “Why is a point after the units better than the older notations?”
• “Did you invent decimal fractions?”
• “To how many places is a number in your canon true?”
• “Is a decimal that never ends a quantity you can write down?”
• “Which of your two books would you defend now?”

What Napier changed

Napier's canon of 1614 turned multiplication into addition, and for three hundred years — until electronic calculation — almost every serious computation in astronomy, navigation, surveying and engineering went through tables descended from it; the slide rule is the same idea in wood. The Rabdologiae of 1617 fixed the decimal point after the units figure as the standard notation for fractions, which is now so universal that nobody remembers it was a proposal. His collaboration with Henry Briggs is the model case of two people each crediting the other.

A debate that continues

He did not invent decimal fractions; Simon Stevin argued for them in 1585, and Napier's contribution was the point as separator. The change of arrangement that made the tables usable was agreed jointly with Briggs and completed after Napier's death, so the credit is genuinely shared. His confident dating of the end of the world between 1688 and 1700, in the book he valued most, was wrong.

Keep exploring — ask Napier

• “Can a person be trusted to judge which of their own works matters?”
• “Is a notation an invention in the same way a method is?”
• “What is worth twenty years of tedious work, if it spares thousands of people the same?”

Related lives

• Simon Stevin — Whose Wreath Of Balls Would Not Move
• Jacob Bernoulli — Against My Father's Will
• Carl Friedrich Gauss — Few, But Ripe

Related themes

Decimal notation · Computation before calculators · Precision and accuracy