# Johann Bernoulli

Date of Birth: 07/27/1677

Age: 70

Place of birth: Basel

Citizenship: Switzerland

**Leader of European mathematicians**

Johann became a master (arts) at age 18, went on to study medicine, but at the same time became interested in mathematics (although medicine has not thrown). Together with his brother Jakob examines Leibniz first article on the methods of differential and integral calculus, starting your own in-depth research.

1691: while in France, promoting the new calculus, creating the first analysis of the Paris school. On his return to Switzerland he corresponds with his student Marquis de l`Hopital, who left the substantive outline of the new doctrine of two parts: the infinitesimal calculus and integral calculus.

As a conceptual framework for action with infinitesimal Johann formulated at the beginning of lectures three postulates (the first attempt to study the analysis):

1. The amount reduced or increased by an infinitesimal amount is not reduced and not increased.

2. Every curve consists of an infinite number of lines, which themselves are infinitesimal.

3. The figure included between two ordinates, the difference between the horizontal and the infinitely small piece of any curve is regarded as a parallelogram.

Later at L`Hopital edition of his textbook dropped 3rd postulate as superfluous, follows from the first.

In the same 1691 came the first printed work by Johann in Acta Eruditorum: he found the equation "catenary" (due to the absence at the time of construction of the exponential function performed by a logarithmic function). At the same time a detailed study of Leibniz and Huygens given curve.

1692: received the classical expression for the radius of curvature of the curve.

1693: joined the correspondence with his brother Leibnitz.

1694: married; doctorate in medicine. In response to a letter from L`Hopital tells him indeterminate form method, now known as "L`Hospital`s rule."

Prints in Acta Eruditorum article "The general method of construction of the first-order differential equations." There appeared the phrase "order of the equation" and "separation of variables" - the latter term Johann enjoyed even in his Paris lectures. Expressing doubts as reducibility to the form of any equation with multiple variables, Johann offers for the first order equations a general method for constructing all the integral curves using isoclines defined in the equation box directions.

1695: On the recommendation of the Huygens became a professor of mathematics at Groningen.

1696: L`Hopital released in Paris under his own name a textbook on mathematical analysis in the history of the first: "Analysis of the infinitely small to the study of curves" (in French), in which was based on the first part of the outline of Bernoulli.

The value of this book is to spread the new doctrine is difficult to overestimate - not only because it was the first, but also due to the clear presentation, excellent syllable, the abundance of examples. As abstract Bernoulli, l`Hopital textbook contained a variety of applications; in fact, they occupied the lion`s share of the book - 95%.

Almost all the material contained L`Hopital was drawn from the work of Leibniz and Johann Bernoulli (the authorship of which in its general form has been recognized in the preface). Something, however, added L`Hospital and from its own findings in the field of solutions of differential equations.

The explanation for this unusual situation - in financial difficulties after the marriage of Johann [1].

Two years earlier, in a letter dated March 17, 1694 L`Hopital Johann proposed an annual pension of 300 livres, promising to then increase it, provided that Johann will take over the development of their questions and will tell him, and only him, his new opening, and no one will send a copy of his works left in his time at L`Hopital.

This unusual agreement punctiliously observed 2 years before the publication of the book L`Hospital. Later, Johann Bernoulli - first in letters to his friends, and after the death of L`Hospital (1704) and in the press - began to defend their copyrights.

Bernoulli, l`Hopital The book was a resounding success with the public at large, withstood four editions (the last - in 1781), it has acquired the comments were even (in 1730) translated into English, with the replacement of terminology in Newton (differentials on fluxions, etc...) . In England, the first general textbook on analysis released only in 1706 (Ditton).

1696: the brachistochrone Iogannpublikuet task: to find the shape of the curve, at which point the material will likely slide from one given point to another. More Galileo thought about it, but mistakenly believed that brachistochrone - arc.

It was the first in the history of the variational problem, and mathematics brilliantly coped with it. Johann formulated the problem in a letter to Leibniz, who once decided and advised her to put up for competition. Then Johann published it in Acta Eruditorum. The contest came three decisions, all the faithful from L`Hopital, and Jacob Bernoulli (published anonymously in London without proof) by Newton. The curve was a cycloid. His own decision Johann also published.

1699: together with Jacob was elected a foreign member of the Paris Academy of Sciences.

1702: Together with Leibniz discovered technique of decomposition of rational fractions in the amount of protozoa.

1705: returned to the University of Basel, Professor of Greek.

1708 After the death of his brother Jacob (1705) was invited to his chair at Basel and holds it up to his death (1748).

Other scientific achievements: Johann Bernoulli put the classical problem of geodesic lines and found a characteristic geometric property of these lines, and later brought them to the differential equation. It should also be noted that he has trained many disciples, among them - Euler and Daniel Bernoulli.

In honor of Jacob and Johann Bernoulli named a crater on the Moon.