Born on 6th January **1655**, son of
Niklaus Bernoulli and Margarethe Schönauer

He studied philosophy and theology at Basel University, receiving his master's degree in philosophy in **1671** and a licentiate in theology in **1676**, but he had also started to lean towards mathematics which he had been teaching himself.

Between **1676** and **1679**, he worked variously as tutor in Geneva, initiating a friendship with Niclaus Fatio. He traveled several times to France (in 1678 he was house tutor for a short time at the Chateau Nède (in Limousin), the home of the Marquis de Lestanges) and in **1681/82** he made a tour thru Holland, Britain (visiting for a few weeks in **1682**) and Deutschland, during which time he met the likes of Johann Hudde, Robert Boyle, Robert Hooke and Flamsteed. In Holland, he met Pierre Bayle, who emigrated from France after the revocation of the Edicy of Nantes.

It was these latter visits that introduced Jakob to the higher mathematics of the time and in 1682/3 he produced two works - (Newly Discovered Methods.... etc.) on comets (more particularly the Great Comet of 1680 discovered by Kirch) and on 'the ether'. He predicted the Comet would return in 1719, a prediction which was ridiculed later by Voltaire when it proved to be untrue).

He turned down a clerical job in Strasbourg.

In **1683** he became a lecturer in Maths and Natural Philosophy at the University of Basel, and in
**1687** he was appointed Professor, successor to Peter Megerlin.

In **1684** he had married Judith Stupanus with whom he had two children, neither of whom followed the upcoming family tradition into Mathematics.

Although *e* is closely connected with logarithms, it was first "discovered" by Jakob when studying compound interest. In **1683**, in examining continuous compound interest, he tried to find the limit of (1 + ^{1}/*n*)^{n} as *n* tends to infinity. He used the binomial theorem to show that the limit had to lie between 2 and 3 and this is generally considered to be the first approximation found to *e*. Also if we accept this as a definition of *e*, it is the first time that a number was defined by a limiting process. However, Jakob certainly did not recognise any connection between his work and logarithms.

Beginnings of the Calculus

Leibniz's ideas on the calculus were published in **1684** and **1686** under the name 'calculus summatorius'.

On being appointed Professor in **1687**, Jakob wrote to Leibniz but only received a reply three years later (because of Leibniz's commitments/travels elsewhere). This left Jakob to learn the methods himself.

In the meantime he discovered the isochrone (or curve of constant descent), publishing in May **1690** - finding it to be a semi-cubic parabola. Although Huygens had discovered the solution in 1687 using synthetic methods, Jakob used the calculus for his solution and was thus able to say he was on top of the subject (coincidentally it was Leibniz himself who had posed the problem in the first place).

The isochrone is the curve along which a pendulum takes the same time to make a complete oscillation whether it swings thru a wide or small arc, or the curve along which a particle will descend under gravity from any point to the bottom in exactly the same time, no matter what the starting point. Jakob showed it was equivalent to solving a first-order nonlinear differential equation. After finding the differential equation, Bernoulli then solved it by what we now call separation of variables.

The paper on the isochrone contained the first use in print of the word 'integral', a word which had actually been invented by Johann and which was then taken on by Leibniz himself.

Jakob joined with his younger brother, Johann, in recognizing the importance of Leibniz's highly abbreviated analysis of infinitesimals. In the late 1600s this trio produced almost all of what we now call elementary calculus as well as the beginnings of ordinary differential equations.

**1685** issued a joint publication with Johann (not an official 'paper' as such - a feature of their history is that they did not issue joint papers)

In **1687** he presented a geometric construction whereby any triangle can be divided into four parts of equal area by virtue of two perpendicular lines.

**1689** infinite series

the harmonic series Σ 1/n diverges

he showed Σ 1/n^{2}< 2. It was Euler in 1737 who showed it summed to π^{2}/6

1690, Differential geometry

His **1691** study of the catenary, or the curve formed by a chain suspended between its two extremities, was soon applied in the building of suspension bridges. Johann had first come up with the solution but Jakob's later independent effort gave a more complete solution.

around **1692** geometric studies included evolutes, caustic curves, associated curves of parabola, logarithmic spiral (spira mirabilis), epicycloids, elastica. Klein W-curve

parabolic spiral

r = a-b √φ

**1694** golden theorem, formula for radius of curvature

In **1695** he also applied calculus to the design of bridges. sails.

Jacob Bernoulli also discovered a general method to determine evolutes of a curve as the envelope of its circles of curvature. He also investigated caustic curves and in particular he studied these associated curves of the parabola, the logarithmic spiral and epicycloids around 1692.

The lemniscate of Bernoulli was first announced by Jacob Bernoulli in 1694. He described it as "being shaped like a figure 8, or a knot, or bow of a ribbon" and the name lemniscate derives from a latin term for a type of ribbon.

His investigations on the length of the arc of the lemniscate laid the foundations for later work on elliptic functions. In **1694**, he examined the shape the an elastic rod will take if compressed at the ends. He showed that the curve satisfied

dy/dx= x^{2}/√(a^{2}-x^{2})

ds/dt= 1/√(1 -t^{4})

then introduced the lemniscate curve

(

x^{2}+y^{2})^{2}= (x^{2}-y^{2})

whose arc length is given by the integral from 0 to *x* of

dt/√(1 -t^{4})

He was unaware that the lemniscate was a special case of the Ovals of Cassini, which have the cartesian equation

(

x^{2}+y^{2})^{2}- 2a^{2}(x^{2}-y^{2}) +a^{4}-c^{4}= 0

The Cassinian ovals are the locus of a point P that moves so that the product of its distances from two fixed points S ( a, 0) and T (-a,0) is a constant c^{2}. The shape of the curve depends on c/a. If c > a then the curve consists of two loops. If c < a then the curve consists of a single loop. The case where c = a produces the lemniscate of Bernoulli.

The curve was first investigated by Giovanni Cassini in 1680 when he was studying the relative motions of the Earth and the Sun. Cassini believed that the Sun travelled round the Earth on one of these ovals, with the Earth at one focus of the oval.

Cassinian Ovals are anallagmatic curves. They are defined by the bipolar equation rr' = k2.

In **1695** he investigated the drawbridge problem which seeks the curve required so that a weight sliding along the cable always keeps the drawbridge balanced.

In Acta **1695** (or **1696**), problem of solving

(Bernoulli's Equation)

by Leibniz, although Acta 1696, Jakob solved by separation of variables

**1696** Transcendental curves

By **1697**, he had broken with his brother Johann.

**1700, 1701** Isoperimetry

research in compressibility of air, vibration

He was an early user of polar coordinates.

Jakob Hermann

Investigations into loaded beam

Bernoulli trials

law of large numbers

Bernoulli Distribution

bernoulli theorem heat???

### Books

**Ars Conjectandi **
*(1713)*, published 8 years after his death, 4 sections, it includes treatises on Probability, Bernoulli Numbers which resulted from a series expansion (B_{1}=1/6, B_{2}=-1/30, B_{3}=1/42, B_{4}=-1/30) infinite series, among others. Law of large numbers uncompleted. The book was produced under the agency of Nicolaus Bernoulli, who wrote a foreword to it. Assisted by Jacob Hermann. Included reprints of five disputations, the third with Jakob Hermann. The final one is with Nicolaus himself. The book is dedicated to Jakob's father, another Nicolaus who outlived Jakob.

**Opera ** 2 vols
*(1744)* edited by Cramer in Geneva, with the assistance of Nicolaus