Euler and his wife Katharina had thirteen children together, but only five survived to their teens. While it is true that infant mortality rates were high during the eighteenth century when Euler lived, the death of eight young children would have stung him particularly. Euler was an involved and affectionate father, writing that some of his most important mathematical ideas came when he was holding a baby in his lap. Some of these children, it is likely, were in one stage or another of some fatal disease.
Another significant challenge was the loss of sight in one of Euler’s eyes. He carried on in spite of this, but it was a real blow when he lost the use of his other eye as well, two years before his wife died. Because he had a phenomenal memory, a spirit to match, and supportive friends and family who could be scribes for him, he continued to make contributions to mathematics right through his last day on earth.
People who have learned to maintain a positive outlook toward life in spite of deep pain often learn compassion as well, and are unwilling to knowingly inflict suffering, not even in the form of mild retaliation. Even if Euler had been inclined to cruelty, though, he was aware of far too many impressive mathematical curiosities to have resorted to the nonsense equation mentioned in the Euler-Diderot Incident.
An unusual mathematical imagination allowed him to visualize, between two apparently unrelated concepts, relationships that are generally non-intuitive. He had insight into the significance, in terms of mathematical modeling of observable phenomena, of what came to be known as the natural logarithm, and saw connections between it and infinite series. He calculated a numerical approximation for the base, which he named e, of its inverse function. He also related various infinite series to the natural constant pi (the ratio of the circumference of a circle to its diameter) which yielded many pleasingly symmetrical mathematical formulae. Another beautiful curiosity Euler discovered was the link between harmonic series and prime numbers, which he used to give a new proof for Euclid’s assertion that there are an infinite number of primes.
Euler assigned the notation i to the square root of -1. Euler was largely responsible for making the formerly skeptical mathematical world comfortable with this “imaginary” number by describing the essential role it played in the field of analytical algebra. Euler’s facility with algebraic analysis carried into his explorations in the field of geometry. There was a general feeling among geometers in the eighteenth century that results derived from calculations involving algebraic symbols were less elegant than proofs made up primarily of logical constructions. Leave it to Euler, though, to intuit with his mind’s eye the geometric relationship within the elements of triangles that no other geometer, not even Euclid, had seen, and to use analytic geometry to prove this observation that might have defied proof without Cartesian analysis. This result is the discovery that every triangle has what is called the “Euler line” containing the circumcenter, the orthocenter, and the centroid, and the distance from the centroid to the orthocenter is exactly twice the distance from the orthocenter to the circumcenter. Here again is another beautiful and surprising result that delighted the mathematical aesthete while winning him over to the power of the newer computational innovations, in this case, algebraic analysis within the framework of the Cartesian coordinate system.
Article comments
1 - duane
An article about Euler! Thank you. Fascinating. e^(i*pi)+1=0.
2 - Dr Dreadful
Good piece, Irene. I'd never have dreamed, slouched over my desk in the maths classroom on a dreary Monday afternoon[mumble]ty years ago, that mathematics could be interesting!
3 - Irene Athena
Thanks Duane and Dr. Dreadful.
Post-prandial math classes are never a good idea. Pie yes, pi, no.
4 - John Wilson
Good article! Interesting and lively.
5 - wet blanket
...the same serene guy who translated Benjamin Robins' work into German (with annotations) and helped advance the technology of warfare.
What a saint.
No, really. See May 24th on the Lutheran Calender.
(fun article, Irene)
6 - Irene Athena
Thanks John. Wet Blanket, Scientia non habit inimicum prater Ignoratem. See the footnote at the bottom of page 2; you'll dig the irony. I will say this about that, though. He recommended Robins' book to King Frederick even after Robins had written a scathing review of his own.
There. I've put him back on his pedestal as...the Patron Saint of Paintball Trajectory Planning.
7 - Richard Bunbury
It's "Maupertuis", with the "u" and "i" in this order. I may be fussy but I must say that after seeing the repeated misspelling I lost confidence in the rest of the article. Maupertuis is a well-known and important historical figures, and inability to spell his name doesn't bode well for the author's competence. I may be unfair, but please correct the error! Thanks.
8 - Irene Athena
Maupertuis is a well-known and important historical figures
Figure"s." You have no more chance of correcting that error in your already published comment than I have of changing, in an already published article, Maupertius to Maupertuis. Ptooey.
9 - Irene Athena
And while we're on the subject of "youse" being where you aren't supposed to be (which is where I am right now) BUnbury is a rather unconventional spelling.
Ride a cock-horse to Banbury Cross,
To see a fine lady upon a white horse;
Rings on her fingers and bells on her toes,
And she shall have music wherever she goes.
10 - Irene Athena
Richard Bunbury, I have sent the editors a request to change Maupertius to Maupertuis. Someone with a particular sensitivity to that sort of thing might have been driven up the wall, I suppose, to see that error occur five times on a single page. I'm sorry.
11 - zingzing
great article, irene. math is a foreign language to me a great deal of the time, but i always seem to enjoy it when it's slathered in drama.
12 - Irene Athena
I just took something out of the library that you might enjoy*, ZingZing.
* Correction. BORROWED. Happy Banned Books Week. :)
13 - John Lake
My heavens. Can any of us imagine living at a time when crippling and often fatal diseases were inescapable? A ruptured appendix or a swelling dental abscess often brought about an early death. Fighting influenza was a fight for life. And Euler, whom I know nothing more of than what your insight brings, worked on in spite of blindness. He was investigating you say the orbit of Uranus to the day of his death.
Utterly amazing. I could easily come to cherish the Euler-Diderot rumor; life must have been a tedious ache, a hard to ignore pain in those days, in 19th century Russia. I could say it may have been “bittersweet”, but these things seldom are.
14 - irene athena
John Lake, amazing it was. Euler's reaction to losing the sight in his right eye is reported to have been: "Now I will have less distraction."
Source: H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
That's some kind of attitude! As far as other comforts go, he did have it a lot better than the Russian peasants of his day, because Catherine the Great had given him a post at St. Petersburg Academy.