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A Nasty Mathematical Myth

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One who investigates the life of Leonhard Euler does not proceed far without encountering details of the “Euler-Diderot Incident.”  So the story goes, Czarina Catherine the Great of Russia was concerned about the deleterious effect the philosopher Diderot was having on the religious faith of the nobility who were listening to him hold forth on atheism in her court. She encouraged famous mathematician Leonard Euler to confront him, and he did, with the following challenge: “Sir, (a + bn)/z = x, hence God exists—reply!” Diderot, who, according to the story, was completely mystified by all things mathematical, fled the court and Russia in deep humiliation.

Diderot and Euler actually were in Russia at the same time, both at the invitation of the Czarina, but this is a joke at Diderot’s expense that neither Euler the man nor Euler the mathematician would have made. Even if it had been, Diderot—who was actually a fairly capable mathematician himself—would not have been stumped. Who might have started this rumor, and why? Bear in mind, accounts of it are found in literature predating the advent of the Internet message board.

Years before, Euler had been slated to follow in his father’s footsteps as a Protestant minister. While pursuing a master’s degree in philosophy at the University of Basel, Leonhard met Johann Bernoulli who recognized and fostered the young man’s mathematical talent. He convinced Euler’s father that the church was not the best showplace for his son’s extraordinary gift. Euler’s connections with the Bernoulli family led him to the St. Petersburg Academy in Russia, where he became the chairman of the mathematics department.

Euler’s correspondence with Jean le Rond d’Alembert gives evidence of the falseness of the Euler-Diderot Incident tale.  D’Alembert was Diderot’s close associate, and contributed articles on mathematics, philosophy and religion to the Encyclopedia on which he and Diderot were collaborating. Euler, too, was a prolific writer.  His very popular Letters to a German Princess—a compendium of what he considered essential knowledge for a young member of the Prussian nobility who had been entrusted to his tutelage—contained, just as the French Encyclopedia did, treatments of topics mathematical as well as metaphysical. While Euler remained, throughout his life, a devout and traditional Christian, d’Alembert rejected traditional religion and tended toward atheism. Though Euler and d’Alembert held to and defended in writing quite divergent views on Deity, their mutual interest in mathematics, and significantly, the mutual respect each man had for the mathematical contributions of the other, laid the foundation for a friendship unmarred by dismissive disrespect over religious matters. Their friendship did hit a snag, but it concerned a mathematical disagreement over “the vibrating string problem,” nothing to do with religion.

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.

All this fulfilling mathematical research proceeded through times of more personal trials. He had to leave his post at the St. Petersburg Academy when the political climate in Russia made it an unsafe place for foreigners. At this time, Euler accepted an invitation by Frederick the Great of Prussia to fill a post at the Berlin Academy of Science. At first, Euler’s mathematical reputation had impressed the monarch, but soon after Euler’s arrival at Frederick’s court, the difference between the quiet, traditionally religious scholar and the witty free-thinking French philosophes with whom Frederick surrounded himself became apparent, and Euler was devalued in the Prussian King’s eyes.

Voltaire, who was a favorite at Frederick’s court, was particularly contemptuous of Euler. Matters worsened when a local mathematician claimed that Euler’s friend, Maupertius—who was the president of the Berlin Academy—had stolen his work on the “Least Effort Principle.” Euler, who had personally helped Maupertius develop it, knew this to be false, and worked hard to vindicate his friend against the slander. Voltaire had a personal grievance against Maupertius to settle, and he took advantage of the dust-up to get revenge. Voltaire used his influence as a popular novelist to discredit Maupertius, whom he mocked viciously as the main character in his Diatribes. Maupertius had been made a fool of in front of all of Europe, and he left the Berlin Academy, irreparably disgraced. William Dunham, in the biography Euler: Master of Us All, quotes Voltaire, who, in a sequel to his Diatribes, gives his opinion, as arrogant as it is ignorant, of Euler:

           “…never learnt philosophy…fame [consisted only] of being the mathematician who in a given time has filled more sheets of paper with calculations than any other.”

Euler bore up under it patiently, acting as de facto President of the Berlin Academy after his friend left, filling an administrative gap in spite of Frederick the Great’s continued refusal to grant him the permanent presidency position. The King had developed a nickname for Euler, alluding cruelly to the deforming nature of Euler’s eye troubles: “our Cyclops.” In time, the political situation in Russia reversed again, and Catherine the Great extended to Euler another invitation to the St. Petersburg Academy. Grateful for an escape from Frederick the Great’s abuse, Euler accepted. D’Alembert tried to convince Frederick the Great to grant Euler the presidency, so that this renowned mathematician would stay, but to no avail.

Having come full circle in the study of Euler’s life, one finds himself back in the Czarina’s court. Here is the site of Euler’s alleged confrontation with the good friend, Diderot, of Euler’s good friend d’Alembert. With the additional knowledge that Voltaire’s good friend Frederick the Great had developed an antipathy to Diderot (jealous over the latter’s devotion to the court of Catherine the Great, perhaps, rather than his own), one begins to form a hypothesis about the genesis of the Euler-Diderot rumor—which puts both gentlemen in such a bad light—and how it came to be spread throughout Europe, believed as fact, and passed down through the generations as such.

Euler rose above it all and spent his very productive final years in Russia, where, on the last day of his life, September 18, 1873, a brain hemorrhage interrupted investigations the now virtually blind Euler was making into the orbit of Uranus. He was gone within hours, but he lives on in the work of the many mathematical masters who built upon his work.

Who knows? Voltaire, much to his very great surprise, may be living on, too, in a small library somewhere in the cosmos, with all 25,000 pages of Euler’s quite impressive Opera Omnia as his only companions.

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About Irene Athena

Wonderfully ordinary.
  • 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.

  • 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.

  • I just took something out of the library that you might enjoy*, ZingZing.

    * Correction. BORROWED. Happy Banned Books Week. 🙂

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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)

  • John Wilson

    Good article! Interesting and lively.

  • Irene Athena

    Thanks Duane and Dr. Dreadful.
    Post-prandial math classes are never a good idea. Pie yes, pi, no.

  • 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!

  • duane

    An article about Euler! Thank you. Fascinating. e^(i*pi)+1=0.