One of the greatest mathematicians of all time with over 800 publications to his name, Leonhard Euler (1707–1783) also played a decisive role in the development of geometry, calculus, mechanics, and number theory.
Professor of Mathematics and Computer Science at Adelphi University, Robert Bradley is a co-author of Leonhard Euler: Life, Work and Legacy (2007). His research focuses on the historical development of mathematical analysis from the mid-17th to the mid-19th century.
Simply Charly: In your personal statement on the Adelphi website, you mention that your priority is the “preparation of excellent future math teachers.” What makes Euler an important subject for potential math teachers to learn?
Robert Bradley: The history of mathematics has recently come to be considered an essential element in the preparation of future math teachers. The National Council of Teachers of Mathematics (NCTM) recommends a course in the history of mathematics as part of a high school math teacher’s preparation. I frequently teach such a course at Adelphi University, aimed at upper-division undergraduates or masters students.
Like many people teaching such a course, I appreciate the value of using sources to enrich the teaching of historical mathematics. One challenge of doing this is that a lot of early mathematics is very difficult for modern readers to understand. The notation (when there is notation) is totally unfamiliar and the arguments, frequently done entirely in words, seem convoluted or incomplete by modern standards. By the time we get to the 19th century, mathematics begins to resemble its modern form. As much as anyone, Euler was the pivotal person in this evolution. His mathematics is largely comprehensible to modern readers, but still distant enough to illustrate the fact that mathematics is an organic and evolving field of knowledge.
Euler did not write about mathematics education, but he was deeply interested in this subject. He wrote a number of textbooks at all levels, from elementary arithmetic and algebra books to cutting-edge research monographs. Euler was the consummate pedagogue: his textbooks, as well as many of his research papers, introduced new concepts gently, working through simple examples thoroughly to afford insight and build gradually to rich and deep results in a logically self-contained way. Modern research papers, by contrast, are streamlined and rarely include logically complete arguments, relying instead on references back to previously published results. To read Euler’s mathematics is to learn good pedagogical practice form one of the masters.
SC: As one of Euler’s modern-day biographers, what would you say his most important contributions to mathematics were?
RB: He established himself as a great mathematician early on in his career by solving an open problem known as the Basel Problem. The goal was to find the sum of the infinite series
\(\frac{1}{1}+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}\) ⋯
That is, start with the sequence of the squares of the whole numbers—1, 4, 9, 16, 25, and so on—take the reciprocal of each term and add them all up. Even though there are infinitely many terms, they sum up to a finite number. A student in a modern-day calculus class can easily show that this infinite sum has a limit, which is a little bit larger than 1.64, but there’s nothing in the usual calculus curriculum that gives you the tools to discover what the sum of the series is exactly.
That was the state of knowledge in the early 1700s when Euler applied himself to the problem. In 1735, he discovered that the sum of the series is precisely \(\frac{\pi^{2}}{6}\). This was a tour de force, and would have assured him a place in the history of mathematics, even if he had achieved nothing else in his career.
Not content with just one discovery, Euler considered the more general problem of finding the sum of the infinite series
\(\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\frac{1}{5^s}\)⋯
Note that the case s=2 is the Basel Problem. Euler found that he could sum up the series for any even-numbered exponent s=2, 4, 6, 8, … Although he could not calculate the precise sum for odd numbers, he actually considered the problem in its full generality, where s could represent any real number or even a complex number.
This infinite series, with the exponent in the denominators as considered to be the variable, is now called the Riemann Zeta Function, named for Bernhard Riemann (1826-1866), who worked on the series a century after Euler. Riemann formulated a conjecture about this function, which is called the Riemann Hypothesis. Settling the Riemann Hypothesis—either providing a proof in the case that it is true or giving a counterexample in the case that it is false—is considered to be one of the most important open problems in mathematics today. In the year 2000, the Clay Institute offered a prize of $1,000,000 to anyone who could settle the Riemann Hypothesis. The prize money is still waiting to be claimed!
One reason that the Riemann Zeta Function, and the corresponding hypothesis, is considered to be so important is a theorem that Euler discovered in 1737, relating this infinite sum to an infinite product involving the prime numbers. Anything involving prime numbers and their distribution is usually considered to be of great interest to mathematicians. Although this interest is usually purely intellectual, it’s important to note that prime numbers are used in cryptography and make secure communication and commerce possible on the Internet.
On a completely different level, Euler popularized the idea of a function in mathematics and gave us the notation we use today for functions. The idea of a function goes back to Leibniz (1646-1716), who was one of the inventors of the calculus. In the days before Euler, calculus was considered to be a set of techniques or algorithms that were applied to equations. It was Euler who saw that the true objects of calculus are actually functions and not equations. At about the same time, he introduced the functional notation we use in calculus and related courses, e.g. \(f(x)=\sqrt{x}\) or \(f(x,y)= x^2+y^2\), which allow us to distinguish between independent and dependent variables, to transform variables from one system to another and to combine functions by composition.
In fact, Euler had a keen sense for good notation and gave us a lot of the notation we use today, including e for the base of natural logarithms, i for the imaginary root of –1, and π for the ratio of the circumference of a circle to its diameter (in the case of π, Euler wasn’t the first person to use this symbol in this way, but he was the one to popularize its use).
SC: You’re also the current sitting President of the Euler Society. What kind of work does the Society do? What do your duties as President entail?
RB: The Euler Society promotes the teaching, learning, and wider understanding of Leonhard Euler’s role in mathematics and the history of science. The Society uses the life and works of Euler as a foundation for its efforts to examine connections among mathematics, mechanics, astronomy, and technology, from the 18th century onward. The Euler Society was founded in 2002, and its initial focus was to promote Euler scholarship in the years leading up to Euler’s tercentenary in 2007.
The Society holds annual meetings in which members present research papers on Euler’s mathematics, physics, and philosophy. The central focus of each meeting is the Euler Lecture, a keynote address given by a noted Euler scholar. Speakers in the past have included Craig Fraser, Henk Bos, and William Dunham. Other activities include group readings of Euler’s works in the original languages (mostly Latin and French) to promote the translation of his book and papers into English.
Because of Euler’s central role in the development of mathematics, there was a great deal of interest in Euler scholarship within the mathematical community in the year 2007. The Mathematical Association of America marked the event with a series of five volumes of Euler scholarship called The Euler Collection. Although the Euler Society was not directly involved in this venture, many of our members contributed to the collection as both authors and editors, most notably our founding secretary, Ed Sandifer. In the same year, the Euler Society held its annual meeting as a joint meeting with the Mathematical Association of America.
Another collaboration is with the Euler Archive (eulerarchive.org), an online repository of Leonhard Euler’s original works, along with modern Euler scholarship. The archive is directed by three of our executive members, Dominic Klyve, Lee Stemkowski, and Erik Tou. Lee and Dominic attended the first meeting of the Euler Society in 2002 when they were still graduate students at Dartmouth College, where they were inspired to create this website. Along with their fellow student Erik, they were soon providing access to Euler’s original publications via this dynamic library and database. By 2007, virtually all of Euler’s publications were available. The website continues to grow with the addition of new scholarship and translations. The Euler Society promotes the translation of Euler’s writings into English, and the Euler Archive provides an accessible forum for the publication of these translations.
The Euler Society should not be confused with the Euler Commission. The Euler Commission of the Swiss Academy of Sciences was founded in 1907 (Euler’s bicentennial) with the goal of publishing all of Euler’s books, articles, and correspondence. The task is largely complete, but the Commission continues the work of publishing further volumes of Euler’s extensive correspondence.
SC: In addition to Euler, who would you cite as being vital to the development of mathematics between the 17th and 19th centuries?
RB: The 17th century is sometimes called the Heroic Period in the history of mathematics. This view sees the century as dominated by a few eminent men who made essential discoveries in the development of mathematics: the invention of coordinate geometry by Descartes (1596-1650), or the independent discovery of calculus by Newton (1643-1727) and Leibniz.
To a great extent, the 18th century was the century of Euler. His mathematical career began in 1725 and continued, with posthumous publications, well into the 1800s. In the early 1700s, Euler’s teacher Johann Bernoulli (1667-1748) was the pre-eminent figure in mathematics on the European continent. Johann’s son, Daniel Bernoulli (1700-1782), was in the only one of Euler’s contemporaries to be in the same league as him. Colin Maclaurin (1698-1746), was Euler’s most influential contemporary in Great Britain. The Swiss mathematician Gabriel Cramer (1704-1752) was another mathematician of significant influence. Like Maclaurin, Cramer was interested in many of the same problems that occupied Euler’s attention, but neither man enjoyed the long life that Euler did.
The most important figure in the development of mathematics in the generation following Euler was Joseph-Louis Lagrange (1736-1813). Lagrange became the President of the Berlin Academy after Euler left Berlin in 1666 and was in many ways Euler’s intellectual descendant. He worked in most of the same branches of mathematics as Euler had done, and was in many ways the dominant figure in mathematics in the late 18th century. In some ways, the mantle then passed to Augustin-Louis Cauchy (1789-1857), who was one of the most prolific mathematicians of all time. Cauchy and Carl Friedrich Gauss (1777-1855) both worked in areas of mathematics pioneered by Euler.
SC: One of Euler’s claims to fame was solving the “Seven Bridges of Königsberg,” or rather proving the problem unsolvable. Why was Euler the first to discover the problem had no solution?
RB: The Prussian city of Königsberg (now in Russia and called Kaliningrad) is situated on the river Pregel (now called Pregolya). The river winds through the city and around an island, essentially dividing the town into four regions, connected by seven bridges. According to tradition, on a Sunday afternoon, the townspeople would try to take a walk through the city, crossing each bridge exactly once, and ending up back where they had started. As far as we know, Euler was the first person to prove that it is, in fact, impossible to cross all the bridges and return to one’s starting point without crossing some bridge twice. (In fact, at least two of the bridges have to be crossed more than once.)
The problem seems to have first been brought to Euler’s attention in a letter he received in 1735 from Karl Ehler, the mayor of Gdansk—which was at that time also part of Prussia and called Danzig, although it’s possible Euler had heard of the problem before then. Ehler communicated a request from a mathematician named Kühn that Euler consider the problem and send him a solution, along with a proof.
In a letter he wrote to another mathematician at about the same time, Euler said he had been told that the problem had hitherto been unsolved and that it was “worthy of attention, in that neither geometry, nor algebra, nor even the art of counting was sufficient to solve it.” Euler considered this to be a problem in Geometriam Situs (the geometry of position), which was essentially a new branch of mathematics, foreseen by Leibniz. This branch of mathematics is now called topology.
In any case, modern mathematicians see Euler’s solution of the Bridges of Königsberg Problem as the first-ever result in Graph Theory. Although Euler himself did not express the problem in this way, his proof can be adapted to an argument about finite graphs, or networks, in a way that foreshadows this important branch of modern mathematics.
SC: Euler’s mentor was none other than Johann Bernoulli, himself a prominent Swiss mathematician. What was the relationship between Euler and the Bernoulli family? Can Bernoulli’s influence be seen in Euler’s works?
RB: Johann Bernoulli was the most famous and prolific mathematician in the celebrated Bernoulli family. Together with his older brother Jakob (1654-1705), Johann mastered Leibniz’s infinitesimal calculus in the 1680s and shared it with the world. Euler’s father Paul went to the University of Basel in 1685, where he roomed with Johann and studied under Jakob, who was professor of mathematics at the time. So there was already a connection between the Bernoullis and the Eulers well before Leonhard was born.
By the time Leonhard attended the University of Basel in the 1720s, Johann Bernoulli had succeeded his late brother as professor of mathematics. Euler did not attend Bernoulli’s elementary mathematics lectures. Instead, because of the ties between their families, Johann Bernoulli offered the young Euler something much more valuable—weekly tutorials at his house on Saturday afternoons. Bernoulli told Euler what mathematics books he should read, then made himself available to help Euler with any portions that he might have difficulty understanding, or any exercises that he might not be able to solve. In this way, Euler not only learned his mathematics from the masterworks but also developed his talents for self-directed study.
Euler’s father was a minister in the Basler Evangelical-Reformed Church and expected Leonhard to follow in his footsteps. However, in about 1725 Johann Bernoulli convinced the elder Euler that his son had an exceptional talent for mathematics and should be allowed to pursue a career as an academic. Two years later, Leonhard secured a position at the St. Petersburg Academy. This was almost certainly due to the influence of Bernoulli, whose son Daniel was assistant professor of mathematics at the Academy.
Euler never returned to Basel after his departure for Russia in 1727 and never saw his mentor again, but the two kept up a lively correspondence until Bernoulli’s death in 1748. Bernoulli had been a pioneer in applying calculus to problems in physics, and much of Euler’s time early in his career was spent working in this area. Bernoulli had taught Euler the importance of publishing his work in Latin, which was the universal language of learning at that time, although the trend towards writing in vernacular languages had begun. Most of Euler’s books and articles were written in Latin, although he was required to write in French, the language of the Prussian court while he was at the Berlin Academy.
During his first few years at the St. Petersburg Academy, Euler worked closely with Daniel Bernoulli, who became professor of mathematics in 1730. Euler also roomed at Bernoulli’s house, until Daniel returned to Basel in 1733, at which time Euler succeeded him in the professor’s chair. The two had developed a deep friendship, but also became rivals for the biennial Prize of the Paris Academy, which each man won on a number of occasions. They carried a fruitful and mostly friendly correspondence for decades.
SC: Was Euler’s high output unusual for a mathematician of his time? Has he been matched by anyone since?
RB: Euler published hundreds of articles during his lifetime, as well as 20 books, many of which were multi-volume. In the early 20th century, the historian of mathematics Gustaf Eneström cataloged 866 publications, although this admittedly includes a number of Euler’s letters that had been published in the 19th century. Euler’s Opera Omnia, the modern edition of his collected works, consists of 72 large quarto volumes, filling nearly 30,000 pages. Prior to the 20th century, no mathematician ever came close to matching such a prolific level of output.
The Hungarian mathematician Paul Erdős (1913–1996) published more than 1,500 mathematical articles during his lifetime, more than any other person in history. Erdös’ publication record is very different from Euler’s. For one thing, he published no books, only research articles. Another important difference is that virtually all of Erdös’ articles had two or more authors. Erdös was the consummate collaborator. He roamed the world, living out of a suitcase and working with a large network of friends and colleagues. Mathematicians these days count their “Erdös Number,” or collaboration distance from Erdös. More than 500 mathematicians co-authored papers with him and, therefore, have an Erdös number of one. Anyone who has co-authored a paper with one of them has an Erdős number of two—over 9000 people to date. The shortest distance from you to Erdös, through collaborative authorship, defines your Erdös number. (Mine is four, by the way.)
By contrast, Euler mostly worked alone and had a very stable family life. He rarely traveled and only moved from one city to another on three occasions.
SC: Could you summarize some of Euler’s mathematical inventions? For example, what are Euler’s formula and Euler’s identity?
RB: There is no perfect agreement about what the names “Euler’s Formula” and “Euler’s Identity” refer to. Most of the time the name “Euler’s Formula” is used to mean the Polyhedral Formula: V – E + F = 2. A polyhedron is a three-dimensional figure with flat faces in the shape of polygons. A cube is a familiar example of a polyhedron, as is any rectangular box. On the other hand, spheres and cylinders are three-dimensional objects that are not polyhedra.
Euler’s Formula relates the number of vertices (V), edges (E) and faces (F) of a polyhedron. Take the cube, for example: it has six faces (numbered with one to six dots on dice), 12 edges and eight vertices (the corners where the various edges meet). So in this case, V – E + F = 8 – 12 + 6 = 2. The formula holds for any reasonable polyhedron. The pyramid with a square base has V = 5, E = 8 and F = 5, while the tetrahedron – the pyramid with a triangular base – has V = 4, E = 6 and F = 4.
Of course, there are infinitely many distinct polyhedra, some of which are very complicated indeed. So after discovering his formula, Euler needed to come up with a general proof that could apply to any possible polyhedron. His original proof of 1750 was based on a complicated geometric argument. The challenge of proving the Polyhedral Formula captured the imagination of a variety of other mathematicians over the course of the next century. In the process of developing ever-clearer arguments to prove the validity of the formula, the mathematical community gradually came to see the Polyhedral Formula as a combinatorial statement, and not a geometric proposition. Because of this, Professor David Richeson, in his book Euler’s Gem (Princeton 2008), argues convincingly that Euler’s Formula was the first step on the road to the modern branch of mathematics known as topology.
On the other hand, “Euler’s Identity” usually refers to the equation \(e^{i\pi}+1=0\). Mathematicians often wax lyrical about the simplicity and elegance of this surprising equation, which relates the fundamental numbers zero and one—the identity elements of addition and multiplication—to the irrational constants—π from geometry, the base e of natural logarithms from analysis—and i, the imaginary square root -1. In fact, some people even attach a sort of mystical significance to this unlikely equation.
But there’s a dirty little secret about Euler’s Identity. In all the tens of thousands of pages of mathematics that Euler wrote during his long career, he never once wrote \(e^{i\pi}+1=0\)! What he did discover, and published in 1748 in his textbook Introductio in analysin infinitorum, was the identity \(e^{i\theta} = cos(\theta) + i sin(\theta)\). By this time, many mathematicians had come to appreciate the value of complex numbers, particularly their role in the solution of equations of degree three and higher. Mathematicians knew how to add, subtract, multiply, and divide complex numbers. The identity \(e^{i\theta} = cos(\theta) + i sin(\theta)\) makes exponentiation of complex numbers possible and gives us the polar form for complex numbers. If you make the substitution \(\theta= \pi\) in this identity, you get \(e^{i\pi}=1+i \cdot 0=1\). In the early 19th century, this special case came to be called Euler’s Identity and was rewritten as \(e^{i\pi}+1=0\).
SC: What is the difference between Venn diagrams and Euler diagrams? Is there any reason Venn’s design is better known?
RB: The short answer is that Venn diagrams are descended from Euler diagrams. In his 1880 article, Venn says: “Schemes of diagrammatic representation have been so familiarly introduced into logical treatises during the last century or so… Of these schemes one only, viz. that commonly called Eulerian circles, has met with any general acceptance.”
In the 1760s, Euler composed a series of lessons in science and philosophy for Friederike Charlotte (1745-1808), who was the niece of King Frederick the Great. Because the Seven Years’ War was raging, the princess could not be present at court, so Euler wrote his lessons up as letters, which he sent twice a week or so for about two years. Later on, he collected these lessons and published them as Letters to a German Princess. This was Euler’s great work of popular science writing. The letters were written in French, the language of Frederick’s court, but this book was translated into all of the major European languages. Henry Hunter produced the first English translation in 1795, and this went through a number of further editions in the 19th century.
Today, Venn diagrams are mostly used in set theory. However, Venn, like Euler before him, used them for logic. Venn diagrams can be used to illustrate all of the standard forms of the syllogism, as well as to reason with even more complicated arguments. Both Euler and Venn did this, but Venn did more with his diagrams than Euler had done with his circles in the Letters to a German Princess. Among Venn’s innovations was the use of a single diagram with three overlapping circles to illustrate all possible arguments involving three propositions. He also showed how to use a diagram with four overlapping ellipses, which divide the plane into 16 regions, for arguments involving four propositions.
When these diagrams became popular in the 20th century to deal with set theory and related mathematical topics, it was natural to associate Venn’s name with them. Although he did not invent the diagrams, Venn popularized them and increased the scope of their applicability. And besides, Euler already had his name attached to many things mathematical.
SC: Of Euler’s works, which would you say contained his most important theories?
RB: Euler was almost single-handedly responsible for raising two fields of mathematics out of their infancy and delivering them into the mainstream: differential equations and number theory.
Euler was born shortly after the invention of calculus. Newton and Leibniz are credited with the independent discovery of the principles of calculus in the 1660s and 1670s, respectively. One reason this was such an important breakthrough is that Newton and Leibniz, along with the brothers Jakob and Johann Bernoulli and a few others, were able to use the new calculus to solve difficult and important problems in physics, involving the dynamics of both earthly objects and of the celestial bodies. However, these applications of calculus to physical problems generally relied on particular geometric arguments which, when completed, rarely gave any insight into how to solve other physical problems.
In the early years of his career, Euler worked on a more general approach to solving problems from physics, which involved first extracting an equation from a physical situation (which always involved the differential calculus), and then solving this equation using a standard set of techniques. This process involved classifying these differential equations and developing methods of solution specific to each type. Developing this field of mathematics was a lifelong pursuit for Euler, but he worked out much of the basic theory during his early years. Many years later he gathered his results together and organized them in the three-volume Foundations of Integral Calculus (1768-1770).
Another one of his lifelong interests that flourished during his early years was number theory. Pierre de Fermat (1601-1665) had made many discoveries about the theory of whole numbers, including his celebrated conjecture, which we now call Fermat’s Last Theorem. By and large, Fermat did not publish his results but communicated them without proof in his correspondence with other mathematicians. Sometimes Fermat would claim he had a proof and challenge his correspondent to come up with one of his own, while other times he made no such claim. Many of Fermat’s letters were collected and published in the 17th century.
Before Euler’s day, the sort of problems Fermat worked on would have been considered recreational and not belonging to a serious branch of mathematics. However, by the 19th century, the great mathematician Gauss would claim that “mathematics is the queen of the sciences, and number theory is the queen of mathematics.” It was almost entirely due to Euler’s efforts that number theory came to have such a high status. During his early years, Euler tackled many of Fermat’s conjectures, supplying proofs to many of them and occasionally a counterexample to refute one. As he continued to work in this branch of mathematics, Euler used the new calculus to reach far beyond what Fermat could have done. Euler is credited with establishing the field of analytic number theory and raising the profile of the entire subject.
