Johann Bernoulli Mathematician Date of Birth: 27.07.1677 Country: Switzerland

1. Johann Bernoulli: The Swiss Mathematician and Mechanic
2. Early Life and Education
3. Contributions to Mathematics
4. Academic Career and Legacy

Johann Bernoulli: The Swiss Mathematician and Mechanic

Johann Bernoulli, a Swiss mathematician and mechanic, was one of the greatest mathematicians of his time. He was the most famous member of the Bernoulli family and the younger brother of Jacob Bernoulli, as well as the father of Daniel Bernoulli.

Early Life and Education

Johann became a Master of Arts at the age of 18 and then pursued the study of medicine while also developing a passion for mathematics. Despite his interest in mathematics, he did not abandon his medical studies. Together with his brother Jacob, he studied the first articles by Leibniz on the methods of differential and integral calculus, which sparked their own deep research.

Contributions to Mathematics

In 1691, while in France, Johann advocated for the new calculus and established the first Parisian school of analysis. Upon his return to Switzerland, he corresponded with his student, Marquis de l'Hôpital, and provided him with a comprehensive summary of the new calculus in two parts: differential calculus and integral calculus. Johann formulated three postulates at the beginning of his lectures as the conceptual basis for dealing with infinitesimals, making it the first attempt to justify analysis. However, L'Hôpital later omitted the third postulate as redundant in his textbook.

In 1691, Johann's first printed work appeared in Acta Eruditorum, where he found the equation for the "catenary" curve (as there was no exponential function at that time, the construction was performed through the logarithmic function). Detailed studies of the curve were also conducted by Leibniz and Huygens. In 1692, the classical expression for the radius of curvature of a curve was obtained.

Johann joined his brother's correspondence with Leibniz in 1693, and in 1694, he got married and defended his doctoral dissertation in medicine. In response to a letter from L'Hôpital, he informed him of a method for resolving uncertainties, now known as "L'Hôpital's Rule." Johann published an article in Acta Eruditorum titled "A General Method for Constructing All Differential Equations of the First Order." It was in this work that the terms "order of an equation" and "separation of variables" first appeared. Expressing doubt about the reducibility of any equation to a form with separable variables, Johann proposed a general method for constructing all integral curves using isoclines within the equation-defined field of directions.

Academic Career and Legacy

In 1695, Johann became a professor of mathematics in Groningen, recommended by Huygens. In 1696, L'Hôpital published the first textbook on mathematical analysis under his name, titled "An Analysis of Infinitely Small Quantities for the Study of Curved Lines," which was based on the first part of Johann's summary. The significance of this book for the dissemination of the new discipline cannot be overstated, not only because it was the first of its kind but also due to its clear exposition, excellent style, and abundance of examples. Like Johann's summary, L'Hôpital's textbook included numerous applications, which constituted the majority of the book—approximately 95%.

Almost all the material presented by L'Hôpital was drawn from the works of Leibniz and Johann Bernoulli (whose authorship was acknowledged in the preface). However, L'Hôpital also added some of his own findings in the field of solving differential equations. The explanation for this unusual situation lies in Johann's financial difficulties after his marriage.

Two years prior, in a letter dated March 17, 1694, L'Hôpital offered Johann an annual pension of 300 livres, with the promise to increase it later, on the condition that Johann would undertake the development of questions that interested him and would inform only L'Hôpital about his new discoveries, refraining from sending copies of his works to anyone else. This unusual agreement was punctually observed for two years until the publication of L'Hôpital's book. Later, Johann Bernoulli, initially in letters to friends and later in print, began to defend his authorship rights.

The Bernoulli-L'Hôpital book had resounding success among a wide range of readers, undergoing four editions (the last in 1781) and being enriched with commentaries. It was even translated into English in 1730, with Newtonian terminology replacing certain terms (e.g., "differentials" became "fluxions"). In England, the first comprehensive textbook on analysis was only published in 1706 by Dittone.

In 1696, Johann published the brachistochrone problem, which involved finding the shape of a curve along which a material point would descend from one given point to another in the shortest time possible. Galileo had also contemplated this problem, but mistakenly believed that the brachistochrone curve was a circular arc. This was the first variational problem in history, and mathematicians solved it brilliantly. Johann formulated the problem in a letter to Leibniz, who promptly solved it and advised him to announce it as a competition. Johann then published it in Acta Eruditorum, and three correct solutions were submitted: from L'Hôpital, Jacob Bernoulli, and an anonymous submission published in London without a proof by Newton. The curve was found to be a cycloid. Johann also published his own solution.

In 1699, Johann and Jacob were elected as foreign members of the Paris Academy of Sciences. In 1702, Johann, along with Leibniz, discovered the method of partial fraction decomposition for rational functions. In 1705, he returned to the University of Basel as a professor of Greek language. In 1708, after the death of his brother Jacob in 1705, Johann was invited to take over his chair in Basel, which he held until his death in 1748.

Other Scientific Achievements
Johann Bernoulli posed the classical problem of geodesic lines and found their characteristic geometric property, later deriving their differential equation. It is also worth noting that he mentored numerous students, including Euler and Daniel Bernoulli.

In honor of Jacob and Johann Bernoulli, a crater on the Moon is named after them.