Jacob Bernoulli

Chapter 1 · Against My Father's Will

A theology degree, taken under protest

1654 – 1687 · Basel · Geneva · France · England

Jacob Bernoulli was born at Basel in 1654 into a family of merchants and civic officials — Protestant refugees from Antwerp two generations earlier. His father intended him for the ministry, which was a respectable career and paid.

Jacob took the theology degree, and simultaneously studied mathematics and astronomy, which he had been told not to do. When he later adopted a personal device it showed the sun's rays and the motto Invito patre sidera verso: against my father's will, I study the stars. It is a remarkably unforgiving thing to put on a family crest.

After graduating he travelled, which was how a young scholar without a post acquired one. He tutored in Geneva, went to France to study the Cartesian philosophy, and then to the Low Countries and England, where he met Robert Boyle and Robert Hooke. He came back convinced that the future lay in the new mathematics rather than in the old.

He began lecturing on experimental physics at Basel in 1683 and was appointed to the chair of mathematics in 1687, which he held until his death. His younger brother Johann, thirteen years his junior, was at that point studying medicine at his father's insistence, and Jacob taught him mathematics privately. That decision would cost them both a great deal.

Why this matters

A career in mathematics had to be taken rather than offered, and taking it meant defying a family with other plans.

A theology degree taken under protest, and a motto about it. What would you ask him?

Ask Bernoulli

• “What did your father want for you, and why did you refuse?”
• “Why put a motto about defying your father on your own device?”
• “What did travelling get a young scholar in the 1670s?”
• “What did you make of Boyle and Hooke in England?”
• “Why did you teach mathematics to your brother?”

Chapter 2 · Reading Leibniz

A method almost nobody could use

1684 – 1700 · Basel

Leibniz published his differential calculus in 1684, in a paper of six pages in the Acta Eruditorum. It is extremely short, the notation was new, several of the arguments are compressed to the point of unintelligibility, and it made almost no impression. Leibniz himself later admitted it was an enigma rather than an explanation.

Jacob and Johann Bernoulli read it, worked out what it meant, and within a few years were using the method more fluently than anybody except its inventor — and arguably more fluently than him, because they applied it to problems he had not attempted.

Jacob solved the catenary: the shape taken by a chain hanging under its own weight, which Galileo had wrongly guessed was a parabola. He worked on the isochrone, the curve down which a body falls so as to descend equal heights in equal times. He introduced the word 'integral' into the subject. He worked on the curve of a stretched elastic band, the sail of a ship under wind, and the problem of finding the curve of quickest descent, which was set as a challenge by Johann and answered by Newton, Leibniz, l'Hôpital and both brothers. Out of those problems came the calculus of variations.

He was captivated by the logarithmic spiral, which reproduces itself under a whole family of operations — scale it, and you get the same spiral turned. He asked for it to be engraved on his tombstone with the motto eadem mutata resurgo, 'though changed, I rise the same'. The stonemason, not being a mathematician, cut an Archimedean spiral instead, and it is still there.

Why this matters

A great method can sit unread for years; somebody has to do the work of making it usable, and that work is itself a contribution.

Six unreadable pages, turned into a working method. What would you ask him?

Ask Bernoulli

• “What was it like reading Leibniz's first paper on the calculus?”
• “What shape does a hanging chain actually take?”
• “Why the logarithmic spiral on your tombstone?”
• “Does it bother you that the mason cut the wrong spiral?”
• “What is a curve of quickest descent, and who solved it?”

Chapter 3 · Interest Upon Interest

1690, and a total that refuses to run away

1690 · Basel

In 1690 Jacob published a short piece in the Acta Eruditorum about a question from commercial arithmetic, and it turned out to be about something much larger.

Suppose a sum is lent at interest, and the interest, as soon as it is earned, is itself lent out on the same terms. If the reckoning is made once at the end of the year, the answer is obvious. But suppose it is made twice a year, each time at half the rate. The result is a little more, because the first half-year's interest earns interest in the second. Reckon it four times, and it is more again. Reckon it monthly, daily, hourly.

The natural guess is that the total climbs without limit — that if the interest is compounded often enough, any sum becomes any other sum. Jacob showed that it does not. However often the reckoning is made, the total for one year at a rate of one hundred per cent is trapped between two and three times the original.

That bounded quantity is the constant now written as e, which governs continuous growth wherever it occurs: in populations, in cooling bodies, in radioactive decay, in charge leaking from a capacitor. Jacob did not name it and did not compute it beyond the bounds; Euler did both, decades later, and the letter is his.

It is a good example of a small, practical, almost dull question opening onto something enormous. Jacob was asking about a loan. What he had found was the number that describes how anything grows when its growth feeds on itself.

Why this matters

The constant behind every process of continuous growth was first met in a question about lending money and reckoning the interest more often.

A question about a loan that opened onto a constant of nature. What would you ask him?

Ask Bernoulli

• “What happens if you reckon the interest more and more often?”
• “Why doesn't the total simply run away without limit?”
• “How can you bound a quantity you cannot name?”
• “Did you know you had found something bigger than a banking question?”
• “How do you feel about Euler getting the letter?”

Chapter 4 · The Brothers

A decade of public warfare

1690s – 1705 · Basel · Groningen

Jacob taught Johann mathematics. Johann was thirteen years younger, at least as talented, and did not stay a pupil for long. Within a few years the two were working on the same problems, and by the 1690s they were at war.

The pattern was public. One would set a problem as a challenge in a journal, framed so that the other's methods would struggle with it. Solutions were published with remarks about the incompetence of rivals who were not named but were unmistakable. Priority was disputed over the catenary, over the isochrone, over the calculus of variations. Johann accused Jacob of stealing his results; Jacob accused Johann of misunderstanding results he had been given as a student.

Both accusations had something in them. Both men behaved badly. When Johann was appointed at Groningen in 1695 the distance did not help; the letters continued and the tone got worse.

The standard reading is that Jacob could not accept that the brother he had taught had overtaken him in some things, and Johann could not accept being treated as a pupil for the rest of his life. What is certain is that it consumed enormous energy for a decade, that it made the Basel mathematical world unpleasant, and that Johann's own son Daniel would later have a comparable quarrel with Johann — including one occasion when Johann threw him out of the house for winning a prize they had both entered.

Jacob died in 1705. Johann succeeded him in the Basel chair.

Why this matters

Two of the best mathematicians in Europe spent a decade attacking each other, which is a useful corrective to the idea that scientific work is naturally cooperative.

A brother taught, and then fought, for ten years. What would you ask him?

Ask Bernoulli

• “What was the quarrel with Johann actually about?”
• “Why set problems as public challenges to each other?”
• “Was it hard to be overtaken by somebody you taught?”
• “What did the quarrel cost the mathematics?”
• “Would you do it differently?”

Chapter 5 · The Golden Theorem

Ars Conjectandi, unfinished

1685 – 1713 · Basel

For the last twenty years of his life Jacob worked on a book about chance, and he did not finish it.

The Ars Conjectandi has four parts. The first reprints Huygens's treatise on games of chance with a commentary. The second treats permutations and combinations, and contains the numbers now named after him. The third works through problems about games. The fourth is why the book matters, and it is the part he left incomplete.

He wanted to apply the mathematics of dice to what he called civil, moral and economic questions — to the things where the underlying chances are not known in advance and can only be estimated from experience. To do that he needed to establish something nobody had: that estimating from experience actually works.

What he proved is that as the number of trials increases, the proportion observed comes as close to the true proportion as you please, with as much certainty as you please, provided you take enough trials. He called it his golden theorem. He did not merely assert it; he computed how many trials would be needed to reach a given closeness with a given confidence, and the numbers came out large enough that he seems to have been discouraged by them.

He died in 1705 with the fourth part unfinished. The family quarrel followed him: Johann was asked to complete the book and declined, or was thought unsuitable, and it was eventually published in 1713 by Nicolaus Bernoulli, Jacob's nephew, largely as it stood.

Every poll, every clinical trial, every quality-control sample rests on the theorem in that unfinished fourth part.

Why this matters

The reason a sample tells you anything about a population was proved in a book its author never finished and never saw printed.

Twenty years' work, unfinished at his death. What would you ask him?

Ask Bernoulli

• “What is your golden theorem, and what does it promise?”
• “How many trials does it actually take?”
• “Why apply the mathematics of dice to civil and moral questions?”
• “Why did you never finish it?”
• “What would you have written in the part you never wrote?”

What Bernoulli changed

Jacob Bernoulli's golden theorem — that an observed proportion approaches the true one as trials increase, with computable certainty — is the reason any sample tells you anything, and underlies every poll, trial and quality check ever run. His 1690 question about interest reckoned ever more often was the first encounter with the constant that governs continuous growth throughout science. With his brother he made Leibniz's calculus into a usable method and opened the calculus of variations, and the family he began produced mathematicians for four generations.

A debate that continues

The decade-long quarrel with his brother Johann was ugly on both sides, and who took what from whom cannot now be settled; both accusations of appropriation had something in them. He bounded the constant in his interest question but neither named nor computed it, which is why it carries Euler's letter and not his. The Ars Conjectandi was left unfinished and published eight years after his death, so its final form is not entirely his.

Keep exploring — ask Bernoulli

• “How many observations are enough to be confident of a proportion?”
• “Can a quantity be understood before it can be named or computed?”
• “What does rivalry cost a subject, and does it ever repay it?”

Related lives

• Leonhard Euler — Who Gave Us The Notation
• Blaise Pascal — Who Carried A Barometer Up A Mountain
• Christiaan Huygens — Who Watched From A Moving Boat

Related themes

Compound growth · Probability and sampling · The early calculus