There were many good mathematicians during the 18th century. The discoveries of Newton and Leibniz, Fermat and Descartes, Pascal and Galileo, among others, had opened up a new mathe- matical landscape, and they had provided many of the conceptual tools required to explore it. Many individuals took advantage of these opportunities and made creative and useful discoveries in one or more branches of mathematics. Some of their stories are recounted in this series, but in the 18th century one individual stood out from all others. He was the Swiss mathematician and scientist Leonhard Euler (1707–83). Some histories of mathemat- ics even call the time when Euler was active the Age of Euler.
Almost every branch of mathematics that existed in the 18th cen- tury includes a set of theorems attributed to Euler. He was unique. Many mathematicians make contributions to their chosen field when they are young and the subject and its challenges are new to them but later lose interest or enthusiasm for their chosen field. By contrast, Euler lived a long life, and his output, as measured by the number of publications that he wrote, continued to increase right to the end. In
the years from 1733 to 1743, for example, he published 49 papers. During the last decade of his life, beginning in 1773, he published 355. It is worth noting that he was blind the last 17 years of his life and even that had no apparent effect on his ever-increasing output.
Euler’s father, Paul Euler, was a minister with a mathe- matical background. His mother, Margaret Brucker, was a member of a family of scholars. Paul studied mathe- matics at the University of Basel. He anticipated that his son, Leonhard, would also become a minister, but this idea did not prevent the father from tutoring his
young son in mathematics. These classes were enough to get the son started on a lifetime of mathematical exploration. By the time he was 13 years of age Leonhard Euler was a student at the University of Basel, and by the time he was 16 he had earned a master’s degree. He studied languages, theology, philosophy, and mathematics at Basel. Later, he briefly studied medicine as well. Euler spent most of his adult life in two cities. He lived in Saint Petersburg, Russia, where he was a member of the Academy of Sciences from 1726 until 1741. For a period of time both Euler and his friends, Daniel and Nicolaus Bernoulli, worked at the academy together. In 1741 Euler left Saint Petersburg for Berlin to work at the Berlin Academy of Sciences under the patronage of Frederick the Great. Euler was not happy in Berlin, and in 1766 he returned to Russia, where he lived the rest of his life.
Euler’s contributions to probability involved the study of games of chance and research on certain specialized functions that would
Leonhard Euler, one of the most productive mathematicians of all time (Library of Congress, Prints and Photographs Division)
later play an important role in the mathematical expression of probability. With respect to games of chance, for example, Euler considered the following problem: There are two players. Each player has a deck of cards. The cards are turned up in pairs, one from each deck. If the cards are always different, then one player wins the game and the bet. If it should happen that one pair of identical cards are turned face up simultaneously, then the second player wins. Euler computed the odds of winning for each player. This kind of problem was similar in spirit to those already consid- ered by de Moivre. Euler’s best-known work on probability involved the analysis of various state lottery schemes.
While in Berlin, Euler wrote several articles on lotteries appar- ently at the behest of Frederick the Great. It was a common prac- tice then as now for governments to raise money by sponsoring lotteries. One state under Frederick’s control, for example, spon- sored a lottery to raise money to pay off its war debts. The goal of all these lotteries was, of course, to turn a profit for the lottery’s sponsor rather than the players. Euler investigated the odds of winning various types of lotteries as well as the risk that the state incurred in offering large prizes. He wrote at least two reports to Frederick on the risks associated with various schemes.
Part of the difficulty in this type of work is that these kinds of problems can be computationally intensive. That was certainly the case for a few of the problems that Euler undertook to solve. To make his work easier Euler invented the symbol
[
qp]
to represent the expression p( p–1)(p–2) . . . (p–q+1), an expression thatq(q–1)(q–2) . . . 1
commonly arises in problems involving probability. It represents the number of ways that distinct subsets with q elements can be chosen from a set of p objects. Although the expression is now usually written as
(
qp)
the basic notation originates with Euler.Euler also was one of the first to make progress in the study of the so-called beta function and in hypergeometrical series. These functions play an important role in the theory of probabil- ity. The mathematical properties of these functions are not easy to
identify. They are generally expressed in terms of fairly compli- cated formulas, and that, in part, is what makes them difficult to use. All contemporary mathematicians interested in probability acquire some skill in manipulating these functions, but Euler was one of the first to make headway in understanding their basic mathematical properties. He did not study these functions because of their value to the theory of probability, but his discoveries have found a lasting place in this branch of mathematics.
Euler’s work in the theory of probability extended our under- standing about games of chance, but he did not branch out into new applications of the theory. Eighteenth-century probability theory was marked by many divergent lines of thought. There was still a lot of work done on games of chance, but the new ideas were being extended to other areas of science as well. Bernoulli’s work on smallpox is the most prominent example. Mathematicians and scientists were inspired by the tremendous advances in the physical sciences, and many of them tried to apply quantitative methods, and especially probabilistic methods, to problems in the social sciences. Even theology was not exempt from attempts to prove various ideas through the use of clever probabilistic techniques. The field of probability had fragmented. Many new ideas were developed during this time, but there was no unifying concept. There was no broad treatment of probability that joined all these ideas in a single conceptual framework. D’Alembert, in particular, generated a lot of heat criticizing the work of others, but criticizing others was easy because there was a general lack of insight into the underpinnings of the subject. It would be many years before the first axiomatic treatment of probability was completed.