A hundred years ago, if you’d asked people why Leonhard Euler was famous, those who had an answer would very likely have mentioned his discovery of the Euler line, the remarkable property that the orthocenter, the center of gravity and the circum center of a triangle are collinear. But times change, and so do fashions and the standards by which we interpret history.

At the end of the 19th century, triangle geometry was regarded as one of the crowning achievements of mathematics, and the Euler line was one of its finest jewels. Mathematicians who neglected triangle geometry to study exotic...

The early French mathematician Pierre de Fermat (1601–1665) is well known for his misnamed “Last Theorem,” the conjecture that ifn> 2, then the equation

xⁿ+ⁿ= zⁿ

has no nonzero integer solutions. Many of us also know “Fermat’s Little Theorem,” that ifpis a prime number that does not divide a numbera, thenpdivides a^p–1–1.

It is less well known that Fermat left dozens, perhaps even hundreds of problems and conjectures for his successors. Typically, Fermat would state a problem in a letter to a friend. Sometimes he would claim that he had a had a...

When an interesting illustration catches our eye, we sometimes stop to figure out what it is. But when I first saw this illustration I was in a hurry. I resolved to come back to it “later.” Now that later has finally arrived, I’m glad I remembered to go back.

The picture that caught my eye was the squarish-looking spiral below. It was part of theSummariumof [E275 ], “Notes on a certain passage of Descartes for looking at the quadrature of the circle.” TheSummariumis a summary of an article, usually written by the editor of the journal,...

In the two centuries between Descartes (1596–1650) and Dirichlet (1805–1859), the mathematics of curves gradually shifted from the study of the means by which the curves were constructed to a study of the functions that define those curves. Indeed, Descartes’ great insight, achieved around 1637, was that curves, at least the curves he knew about, had associated equations, and some properties of the curves could be revealed by studying those equations. Almost exactly 200 years later, in 1837, Dirichlet gave his famous example of a function defined on the closed interval [0, 1] that is discontinuous at every...

The 200th anniversary of Euler’s death took place in September 1983. The milestone was marked in a reasonable number of publications, although fewer than the number that celebrated the tercentenary of his birth two years ago. The MAA devoted an entire issue of theMathematics Magazineto Euler’s life and work [Vol. 56, no. 4, Nov. 1983] and there were at least two edited volumes of essays published to mark the event.

Among the many essays included in [Burckhardt 1983 ] is a piece by Pierre Speziali on Euler and Gabriel Cramer (1704–1752), the same Cramer whose name is...

Two names stand large in the history of number theory, Pierre de Fermat (1601–1665) and Leonhard Euler (1707–1783). Fermat, sometimes called The Great Amateur, was a part-time mathematician, a contemporary and rival of Descartes. His “real job” was as a judge in Toulouse, France. At the time, judges were expected to avoid the company of people on whom they might be required to pass judgment, so Fermat lived in comparative isolation, away from the people of Toulouse, with plenty of time to work on his mathematics. He kept in touch with current developments through his correspondence with Marin...

Triangles are one of the most basic objects in mathematics. We have been studying them for thousands of years, and the study of triangles, trigonometry, is, to some extent, a part of every mathematical curriculum. Our oldest named theorem, the Pythagorean theorem, is about triangles, though the theorem was known long before Pythagoras. It is probably our most famous and most often proved theorem as well. Hundreds of different proofs are known, [Loomis 1940] and good writers still find interesting things to say about the theorem. [Maor 2007]

The particular branch of trigonometry where we ask that certain parts of...

Late in his life, Euler devoted a lot of time and effort to number theory. Indeed, almost 40% of his papers in number theory were published after he had died. Many of these late papers were on Diophantine equations, usually involving square numbers in one way or another.

Usually, Euler does not tell us what motivates the particular Diophantine equations that he chooses to study. Sometimes he does tell us, as in [E754], Problème de geometrie resolu par l’analyse de Diophante, “A problem in geometry solved by Diophantine analysis”. In that paper, Euler seeks three integer numbers,x, yand...

When I was first learning probability as an undergraduate, I learned about something called “the St. Petersburg Paradox.” One version of this paradox is as follows: [J]

A man is to throw a coin until he throws head. If he throws head at thenth throw,and not before,he is to receive £2ⁿ.

What is the value of his expectation?

We learned about this just after we learned about geometric distributions, so we knew that the probability of throwing n heads in a row was 1²n. Our instructor, David Griffeath,¹ asked us to do two things with this problem,...

The history of mortality tables and life insurance is sprinkled with the names of people more famous for other things. American composer Charles Ives, who, like your columnist, worked in Danbury, Connecticut, also invented the insurance agency, so that insurance customers themselves no longer had to negotiate directly with the insurance companies. Edmund Halley, of comet fame, devoted a good deal of energy to calculating one of the earlier mortality tables. [Halley 1693] Henry Briggs, better known for his pioneering work with logarithms, calculated interest tables. Swiss religious reformer John Calvin preached that life insurance was not necessarily immoral usury,...

Last month we began this two part series on Euler’s work in actuarial science with an account of his study of mortality, the “death” part of “Life and Death.” This month we turn to the other half of the equation and ask the mathematical question, “Where do those babies come from?”

To seek his answers, Euler begins with a number of assumptions. Some of them are just to simplify the beginnings of his analysis and will be replaced with more sophisticated assumptions later. Others are simple because he sees no way to gather the data to support more complex ones....

One of the most famous formulas in mathematics, indeed in all of science is commonly written in two different ways:

e^πi= -1 or e^πi+ 1 = 0.

Moreover, it is variously known as the Euler identity (the name we will use in this column), the Euler formula or the Euler equation. Whatever its name or form, it consistently appears at or near the top of lists of people’s “favorite” results. It finished first in a 1988 survey by David Wells forMathematical Intelligencerof “most beautiful theorems.” It finished second in a 2004 survey by the editors of...

Two of Euler’s best known and most influential discoveries involve what we now call the Riemann zeta function. The first of these discoveries made him famous when he solved the Basel problem. He showed [E41] that the sum of the reciprocals of the square numbers was

$1 - \frac{1}{4} + \frac{1}{9} + \frac{1}{{16}} + ... + \frac{1}{{{n^2}}} + etc. = \frac{{{\pi ^2}}}{6}$.

Euler’s second great result [E72] on this topic was what we now call the Euler product formula, and we write it as

$\sum\limits_{k = 1}^\infty {\frac{1}{{{k^n}}} = \prod\limits_{\rho prime} {\frac{1}{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {{p^n}}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{p^n}}$}}}}}}$.

For the readers unfamiliar with the zeta function, we’ll give a brief introduction.

It has long been known that the harmonic series, 1 + ½ + ⅓ + ¼ +...

Many people have been interested in summing powers of integers. Most readers of this column know, for example, that

$\sum\limits_{i - 1}^k {i = \frac{{k(k + 1)}}{2}}$,

and many of us even have our favorite proofs. Those of us who have studied or taught mathematical induction lately are likely to recall that

$\sum\limits_{i = 1}^k {{i^2} = \frac{{k(k + 1)(2k + 1)}}{6}}$,

and it is obvious that

$\sum\limits_{i = 1}^k {{i^0} = k}$.

Indeed, some days it seems like the main reason mathematical induction exists is so that we can make students prove identities like this. Euler knew all of these identities as well. In fact, he knew them at least up to the eighth exponent:

Jakob Bernoulli [Bernoulli 1713] and...

Early in our algebra careers we learn the basic relationship between the coefficients of a monic quadratic polynomial and the roots of that polynomial. If the roots are α and β and if the polynomial is${x^2} - Ax + B$, then$A = \alpha + \beta $and$B = \alpha \beta $. Not too long afterwards, we learn that this fact generalizes to higher degree polynomials. As Euler said it, if a polynomial

${x^n} - A{x^{n - 1}} + B{x^{n - 2}} - C{x^{n - 3}} + D{x^{n - 4}} - E{x^{n - 5}} + ... \bot N - 0$

has roots α,β,γ, δ, . . . , v, then

A= sum of all the roots = α + β + γ + δ + . . + v,

B= sum of products taken two...

On Friday, June 7, 1779, Leonhard Euler sent a paper [E705] to the regular twice-weekly meeting of the St. Petersburg Academy. Euler, blind and disillusioned with the corruption of Domaschneff, the President of the Academy, seldom attended the meetings himself, so he sent one of his assistants, Nicolas Fuss, to read the paper to the ten members of the Academy who attended the meeting.

The paper bore the cumbersome title “Investigatio quarundam serierum quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae” (Investigation of certain series which are designed to approximate the true ratio of the...

To look at familiar things in new ways is one of the most fruitful techniques in the creative process. When we look at the Pythagorean theorem, for example, and we ask, “what if it’snota right triangle?” it leads us to the Law of Cosines. When we ask, “what if −1didhave a square root?” we discover complex variables. To find variations on a familiar theme isn’t the only tool in our creative repertoire, but it is one of the most reliable.

But it doesn’t always work. This month we’ll look at a variation on the Taylor series...

The story of Euler and complex numbers is a complicated one. Earlier in his career, Euler was a champion of equal rights for complex numbers, treating them just like real numbers whenever he could. For example, he showed how to integrate$\int {{x^{{2^1}}}} |{1^{dx}}$without using inverse trigonometric functions. He factored${x^2} + 1 - (x + \sqrt 1),$then used partial fractions to rewrite

$\frac{1}{{{x^2} - 1}} = \frac{{\frac{1}{2}\sqrt { - 1} }}{{x + \sqrt { - 1} }} - \frac{{\frac{1}{2}\sqrt { - 1} }}{{x - \sqrt { - 1} }}$

then integrated this difference to get

$\int {\frac{1}{{{x^2} + 1}}dx = } \frac{1}{2}\sqrt { - 1} \ln \frac{{x + \sqrt { - 1} }}{{x - \sqrt { - 1} }}$

Euler typically omitted constants of integeration until he needed them, and also seldom used i in place of 1. His role in that particular notational is exaggerated.

Euler struck a second, and better-known blow for justice...

Euler gave us two mathematical objects now known as “gamma.” One is a function and the other is a constant. The function, Г(x), generalizes the sequence of factorial numbers, and is the subject of this month’s column. A nice history of the gamma function is found in a 1959 article by Philip Davis, [D] and a shorter one is online at [Anon.]. The second gamma, denotedγ, is a constant, approximately equal to 0.577, and, if things go as planned, it will be the subject of next month’s column. In 2003, Julian Havel wrote a book about gamma the constant....

Sam Kutler, now retired from St. John’s College in Annapolis, once pointed out that there are three great constants in mathematics, π, e and γ, and that Euler had a role in all three of them. Euler did not discover e or π, but he gave both of them their names. In contrast, Euler discovered, but did not name γ, the third and least known of these constants.

This γ is usually known as the Euler-Mascheroni constant, acknowledging both the work Euler did in discovering the constant in about 1734, (more on this later) and the work of Lorenzo Mascheroni...

Sometimes Euler has a nice sense of showmanship and a flair for a “big finish.” At the end of a long, sometimes difficult work, he’ll put a beautiful or particularly interesting result to reward his reader for making it clear to the end. He doesn’t always do this, but when he does, it seems like a real treat.

For example, at the end of one of his papers on number theory [E228] in which Euler is studying numbers that are sums of two squares, he shows that 1,000,009, a number that had appeared on several lists of primes among the...

Imagine my surprise when I was looking at Euler’sCalculi differentialis. [E212] There, deep into part 2 (the part that John Blanton hasn’t translated yet), I saw the odd title of chapter 16,De differentiatione functionum inexplicabilium, “On the differentiation of inexplicable functions.” That made me curious. What was an “inexplicable function?”

In the nine chapters of part 1, Euler had taught us how to take derivatives of polynomials, of algebraic functions, of transcendental functions, to take higher derivatives, and to solve certain kinds of differential equations. Part 2 is about twice as long as part 1, both in...

Solving a good research question should open more doors than it closes. One of Euler’s lesser papers,Methodus generalis summandi progressiones(“General methods of summing progressions”) [E25] is more noteworthy for the things it started than the things it finished. The principal role of the paper is as one of a sequence of papers that led to Euler’s development of the Euler-Maclaurin summation formula. That sequence began in 1729 with a letter to Goldbach containing results that Euler later published in 1738 in [E20], and continued through [E25], [E43], [E46], [E47] up to [E55],Methodus universalis series summandi ulterius promota,...

On Monday, March 20, 1777 the Imperial Academy of Sciences of St. Petersburg had one of its regular meetings. Except for holidays and occasional special meetings, they met twice a week on Mondays and Fridays, a total of 70 or 80 meetings per year.

This particular meeting wasn’t much different from the other meetings they had that year, though it was a little shorter than most. The minutes from that meeting are shown in the photograph below. [SPA] It opened with a report from the Academy’s translator, a Mr. Jaehrig, including his account of some letters between the Dalai Lama...

Euler’s output was split fairly evenly between pure and applied mathematics, the latter including many topics that we would today classify as physics. Most of his papers fall decisively into one category or the other, but it wasn’t at all rare for one of his works of applied mathematics to include new techniques or results in analysis. This frequently happened in the Paris Prize competition, for example, where the questions were generally of a practical nature. This month, we’ll look at an astronomical paper [E401] that proposes numerical techniques for approximating a body’s velocity and acceleration. Remarkably, one of the...

Leonhard Euler did an immense amount of work in optics, but that work is not very well known among mathematicians. Seven volumes in Series III of theOpera omniaare devoted to Euler’s optics, two volumes to his 1769 book theDioptricaeand five volumes containing the 56 papers he wrote on the subject. All but six of those papers were published during Euler’s lifetime, evidence of how important his work was considered at the time. Several of Euler’sLetters to a German Princesswere devoted to optics as well.

The two volumes of theDioptricaeand the 56 papers...

In our last column we began a study of one of Euler’s papers on how the Earth’s atmosphere refracts light,Sur l’effet de la réfraction dans les observations terrestres,“On the effect of refraction on terrestrial observations,” [E502] written about 1777 and published in 1780. We learned ”that the rays of light do not always go in straight lines to our eyes, as we ordinarily suppose, but they are found to be a little bit curved, and their concavity is turned downward” and that this phenomenon is due in part to refraction as the rays pass between the rarified air...

Euler seemed to be interested in everything, and when he was interested in something, he sought to understand it with mathematics. Somehow, he got interested in saws, and in 1756, while working at the Berlin Academy, he wrote a 25-page paper,Sur l’action des scies,“On the action of saws” [E235].

I can only speculate on why Euler decided to write this paper. In their Editors’ Introduction to the volume of theOpera omniawhere this paper is reprinted, Charles Blanc and Pierre de Haller suggest that “its main purpose was to show one more time the possibility of putting...

For his whole life Euler was interested in fluids and fluid mechanics, especially their applications to shipbuilding and navigation. He first wrote on fluid mechanics in his Paris Prize essay of 1727, [E4],Meditationes super problemate nautico, de implantatione malorum. . . , (Thoughts about a navigational problem on the placement of masts), an essay that earned the young Euler an acessit, roughly an honorable mention, from the Paris Academy. Euler’s last book,Théorie complete de la construction et de la manoeuvre des vaisseaux,(Complete theory of the construction and maneuvering of ships) [E426], published in 1773, also dealt...

The popular myth of the discovery of gravity goes something like this: one day, an apple fell on the head of a young Isaac Newton. After pondering this event, Newton wrote down an equation describing an invisible force, which he called gravity. This equation united ideas about the paths of cannon balls and apples (terrestrial motion) with the paths of moons and planets (celestial motion). Once it was written down, it elegantly and easily explained the motion of all the planets and moons, and remained unquestioned, revered, and perfect for centuries (at least until Einstein).

Readers who are familiar with...

Is the earth hollow? Is there a sun 600 miles in diameter at the center of the hollow earth? Is the inside of the shell of the hollow earth covered with mountains larger than the ones we see on the outside? Is there a hole in the shell of the hollow earth through which flying saucers from Venus and space ships from other galaxies fly to get to their bases inside the hollow earth? Are there secret passages from the bases of the Great Pyramid and other locations around the earth that connect the outside to the inside of the...

By now, regular readers of this columnmight have come to believe that, except for occasional computational errors beyond the 15th decimal place, and except for a regular and flagrant disregard of the issues of convergence when dealing with series, Euler was always right about everything. Now that 2007, the so-called “Euler year” is over and the celebrations of the 300th anniversary of his birth are winding down, perhaps we will be forgiven if we admit an uncomfortable fact: Euler was sometimes wrong. We are devoting this month’s column to a few of the things Euler was wrong about.

Euler thought...

We sometimes celebrate the first of April with a column on the lighter side of Euler scholarship. We continue that occasional tradition with some stories intended to help perpetuate the idea that no matter where we look, we can find a connection with Euler.

Once is amusing. Twice is a coincidence. Three times is worth remarking about.

We’ve recently come across a third, and maybe a fourth, person with connections both to Euler and to piracy or privateering. For our collective amusement, we thought we’d share them with you.

The first, of course, was Maupertuis (1698–1759), President of the...

What would it have been like to be a student of Leonhard Euler? Until we invent time machines, it will be impossible to answer this question. Still, it is almost impossible not to ask it anyway.

By all accounts, Euler was regarded as an excellent teacher. His eulogists all mention that he was a kind and pious man, a great genius and wonderful teacher. There are stories that students from abroad, particularly from France and Russia, studying in Berlin would rent rooms in Euler’s house and that they would talk about science and mathematics at meal times. Accounts of the...

In Part 1 of this column [Sandifer Jan 2010] we looked at what is known about Euler the Teacher during his first St. Petersburg period (1727–1741) and his time in Berlin (1741–1766). Condorcet [Condorcet 1786] gives us some accounts of Euler’s teaching in his second St. Petersburg period: His sons and students copied his calculations and wrote by dictation the remainingMemoires.

It can be seen that he much preferred the education of his students than the small satisfaction derived from astonishment; he never believed that he had truly done enough for Science if he did not feel...