# Leonhard Euler: Mathematical Genius in the Enlightenment

Leonhard Euler (1707-1783) is not only the prominent mathematician of the 18th century, he was also physicist, astronomer, engineer, and administrator. He contributed to mathematics in analysis, graph theory, number theory, geometry and many others, but he also was quite active in mechanics, fluid dynamics, optics, music theory, and astronomical observations. Many papers and books were devoted to his life and work already. Some were written, revived or translated on the occasion of the tercentenary year 2007 which brought us an English translation of E. Fellmann's German biography from 1995 and even a graphical novel by Heyne and Heyne (see e.g. here and here for a review). His biography in French by Ph. Henry, put emphasizing on Euler as a geometer, and there are of course the MAA tercentenary volumes and there are more. But besides the books about Euler, there is the enormous output of Euler himself, being collected in more than eighty volumes of the *Opera omnia*. Many of his papers are available in electronic form at the euler archive of the MAA and some were discussed in E. Sandifer's monthly column How Euler did it. Of all the biographical publications, the present book is the most complete from an historical perspective. Both Euler's life and his work is discussed, but the the technicalities of the mathematics as such, like for example the discussions of Sandifer's columns, are not included. What we do get is the background of when Euler did what, how, and why.

Much about Euler's life and scientific contributions can be looked up on the Web, which I will not repeat in detail here. The way Calinger has organized the book is by subdividing the life of Euler into chapters that discuss time slices of only a few successive years. Since the focus is on his scientifically active life, the first chapter collects the first twenty years that Euler spent in Basel. Then there are four chapters about Euler's first stay (1727-1741) at the Imperial Russian Academy of Sciences in Saint Petersburg. He accepted an invitation from this Academy because he was not appointed professor in Basel as he had hoped and because that Academy attracted the best scientists from Europe, among which two Bernoulli brothers, sons of Euler's teacher Johann I Bernoulli. But when the climate became unfavorable at the Academy around 1740, Euler accepted an invitation from Frederick the Great who wanted to start a prestigious Prussian Academy in Berlin. Seven chapters deal with the apogee years of Euler in Berlin (1741-1766). Then, because the relation with the king was not optimal, he was not appointed president, even though he had done a lot for the Academy and had been a director of the Mathematical Section for many years. Therefore he accepted an invitation of Catherine the Great to return to Saint Petersburg where he stayed till the end of his life. Calinger covers the latter period with three chapters.

Euler was plagued with health problems. From the well known portraits of Euler, it is obvious that he had problems with eyesight. From 1738 he lost sight in his right eye and after a cataract operation on his other eye he seized an infection so that from 1766 on he was almost completely blind. He married Katharina Gsell in 1734. During their long marriage, they had 13 children of which only five reached adulthood. When his wife died in 1773, he remarried the younger half sister of his first wife. All these facts are pretty well known.

Calinger's main sources are Euler's scientific publications but even more so the surviving notebooks that Euler kept during his lifetime (some 4000 pages) and the extensive correspondence that Euler had with many of his peers. Quite often, some ideas were first formulated, or problems and solutions were discussed in letters. Only later the same ideas appeared in print in journals or were presented to the Academies or, if wrapped up in a book, that took even longer, often years, before the book was printed. As a consequence, several topics return and evolve in the successive time slices in the book. For example Euler as a sincere religious person was fighting the monad theory that has its roots with Pythagoras, but that was revived by Leibniz. So till the middle of the century Euler spent quite some energy in disputes with the Wolffians, on religious grounds. These shared their interpretation of Leibniz's *Monadology* with their main spokesman Christian Wolff. Euler argued that the monadic belief that reason is the basis of all knowledge would lead to atheism, which was unacceptable for him. This theme reappears several times in separate chapters. Euler didn't shy away from controversy and he chose sides in several involving Voltaire, d'Alembert, Clairaut, Maupertuis, König and others, and again these often span several chapters. While in Berlin, he still kept corresponding with the group in Saint Petersburg and pulled strings there as well. It happened on occasion that a lively correspondence with a person suddenly drops to zero for a longer period. Calinger then of course explains what caused this.

Most chapters start with a survey of what happened in the period considered in that chapter, what publications were written or published, what problems Euler had to deal with, the general political background, etc. We also find in most chapters a section on Euler's life, with an account of his health problems, or of what happened with his family. With the detailed sources available, we learn for instance the date and even the hour of the day when Euler left Basel for his travel to Saint Petersburg. Chess and music where his only recreations when he was not working. His ability to concentrate, even in a hectic family environment is legendary. Fortunately his wife took care of all practical problems of the household, so that he could fully concentrate on his job. From his letters, we also learn that Euler has repeatedly negotiated his salary. His financial agreement with Catherine II for his second period in Saint Petersburg was outrageous with extras besides his regular salary and a survival pension for his wife. We also learn that Euler was not always the most tactical and polite opponent. Networking in the company of noblemen was necessary, but himself not being a noble (his father was a pastor), made the communication not always easy. It is explained how he organized his work when he became totally blind: he arranged for the young Nicolas Fuss from Basel to be hired in Saint Petersburg as his personal assistant, and we read that he enjoyed using his old Basel accent. We are informed about his maneuvering to acquire, much against the likings of his sons, a new wife after his spouse had died in 1773. It is made clear how strongly Frederick II personally supervised and fostered the Berlin Academy, but how he was regularly involved in wars so that his attention was diverted from the Academy. Calinger tells us how Euler came to his results. Early in his career Euler is just making a lot of computations, until a pattern starts to show. More computations are done to verify an assumption, and from this finally the abstraction and the general result could be concluded. It is also explained how Euler operated as a teacher, how he managed business at the Academies (he thought that members should be expelled when they were not productive), what prizes he won at the Paris Academy of Sciences. We learn about the wars that went on and on (where the camaraderie between officers of both sides was more important than dedication to the soldiers of their own army), even the number of inhabitants of the cities that Euler lived in are included.

Of course also Euler's mathematical and other scientific achievements are mentioned. For example where and when he introduced modern notation that is still used today like the capital Σ for the summation sign, the introduction of the Euler constant e as basis of the natural logarithm, the Euler-Mascheroni constant γ, the notation *f(x)* for a function, notation for trigonometric functions such as sin, cos, sec,..., i and j were originally used for infinity large numbers, but later he used i for the square root of -1 and ∞ for infinity. The origin and discussion of famous problems connected to Euler such as the Basel problem (a 100 year old problem to sum the reciprocals of the squares of the natural numbers, solved by Euler in 1935, giving him immediate recognition), the Saint Petersburg paradox (a paradox related to probability in a lottery problem), the Bridges of Königsberg (a problem that he solved and that is considered to be the origin of graph theory), etc.

The book is so rich in information that it makes it the best reference work on Euler that is currently available. There are quotations, but not too many so that the text is not a collection of quotes which sometimes happens with biographies, but it is a fluent story of a remarkable man, placed on the detailed political and scientific background of an exciting period that is populated by all these masterminds that formed modern science. The readability is obtained because short quotes are often just placed in a sentence, and not as separate quotes. The quotes and the titles of the papers and books are in the original language (with an English translation following between parenthesis). The Academy in Berlin had adopted French as the official language of science. There are many grey-scale illustrations (these are often showing the title page of a book, or the portrait of a relevant person). Reference is made to notes collected at the end of the book, where we find also the Enström index of Euler's papers, the numbering used in his *Omnia Opera*, a list of facsimile reproductions and publications of Euler's publications that became recently available, and many other sources about Euler and his work. The price one has to pay for a readable text is that it might be somewhat more difficult to find something in a dense text. To this end, the name index and general index that is provided with detailed lists of the pages where the name or the term is used is very useful. As mentioned before, for the detailed mathematical analysis like in Sandifer's *How Euler did it* columns are not to be found here (that would be a completely different encyclopedia). Nevertheless, for all other information about Euler, this book will be a standard for many years to come.

**Submitted by Adhemar Bultheel |

**22 / Dec / 2015