Jacob Bernoulli

Against My Father's Will · Mathematician · 1654–1705

The first of the Bernoulli dynasty: he took interest charged on interest to its limit and trapped the answer between two and three, and he proved that in enough trials an observed proportion approaches the true one.

Who was Bernoulli?

Jacob Bernoulli was born at Basel in 1654 into a merchant family. His father intended him for the ministry and he took a degree in theology to comply, studying mathematics and astronomy against instructions; the motto he adopted — against my father's will, I study the stars — is the whole of his temperament in six words. He travelled, met Boyle and Hooke in England and Huygens's circle in the Low Countries, and returned to a chair of mathematics at Basel which he held until his death. When Leibniz published his calculus in 1684 the papers were short and nearly unreadable, and hardly anybody made anything of them. Jacob and his younger brother Johann did, and within a few years were extending the method further than its inventor had taken it in print. Jacob solved the catenary — the shape a hanging chain takes — the isochrone, and problems that became the calculus of variations. He asked for the logarithmic spiral to be cut on his tombstone with the motto eadem mutata resurgo, 'though changed, I rise the same'; the mason cut an Archimedean spiral instead. In 1690, in a paper in the Acta Eruditorum, he considered a sum lent at interest where the interest is itself lent out the moment it is earned. What happens, he asked, if the reckoning is made not yearly but half-yearly, then quarterly, then continually? The obvious guess is that the total runs away without limit. It does not: he showed it is trapped between two and three. The constant it approaches was named and computed by Euler decades later. His major work is the Ars Conjectandi, on the doctrine of chance, and he never finished it. Its fourth part contains what he called his golden theorem: as the number of trials increases, the proportion observed comes as close to the true proportion as anybody could require, and he computed how many trials that would take. His nephew Nicolaus published the book in 1713, eight years after his death. He and his brother Johann quarrelled publicly and viciously for the last decade of his life, over priority, over problems set as challenges to each other, and over who had taught whom.

Major achievements

• Took up Leibniz's calculus almost alone and extended it within a few years
• Solved the catenary, the isochrone, and the problems that began the calculus of variations
• Showed in 1690 that interest reckoned ever more often is bounded rather than unlimited
• Proved the theorem that observed proportions approach true ones as trials increase
• Computed how many trials would be needed to reach a given closeness

Life in brief

• 1654 — Born at Basel: Into a merchant family; intended for the ministry.
• 1676–1682 — Travels: France, the Low Countries and England; meets Boyle and Hooke.
• 1687 — Chair at Basel: Professor of mathematics, a post held until his death.
• 1690 — The interest question: Reckoning ever more often, and the answer bounded between two and three.
• 1690s — The quarrel: A long and public dispute with his brother Johann.
• 1705 — Died at Basel: Leaving the Ars Conjectandi unfinished.
• 1713 — Ars Conjectandi: Published by his nephew Nicolaus, with the golden theorem in Part IV.

Explore the life of Bernoulli — five chapters

1. Against My Father's Will — A theology degree, taken under protest
2. Reading Leibniz — A method almost nobody could use
3. Interest Upon Interest — 1690, and a total that refuses to run away
4. The Brothers — A decade of public warfare
5. The Golden Theorem — Ars Conjectandi, unfinished

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