Angle of Incidence.

This is actually based on the Kaverin novel I posted about yesterday, but that post was long enough already, so I figured I’d let this stand on its own. At one point Liza is having coffee in the Rotonde with her husband Georgii and Kostya, the love of her life, who’s visiting from Russia; she and Kostya have a few precious moments alone when Georgii wanders off too far to hear, and they have the following exchange (Kostya speaks first):

— Он не умеет читать по губам?
— Нет. Кроме того, для нас с тобой угол падения не равен углу отражения.

“He doesn’t know how to read lips?”
“No. Besides, for us the angle of incidence isn’t equal to the angle of reflection.”

The line about the angles is repeated twice more in the course of the novel, so it’s not merely a passing remark (and it occurs to me that it’s probably not irrelevant to the titular mirror). It may seem a bit random to the reader unfamiliar with its antecedents, but it is part of a tradition — a meme, as we say nowadays — in Russian literature. The immediate source is Marina Tsvetaeva’s long essay on the painter Natalia Goncharova (both she and especially Tsvetaeva are characters in the novel); she writes that the world is reflected косвенно (‘indirectly, obliquely’) in Goncharova’s paintings, and continues:

То, что я как-то сказала о поэте, можно сказать о каждом творчестве: угол падения не равен углу отражения.

What I once said about a poet could be said about all creative work: the angle of incidence is not equal to the angle of reflection.

This, of course, is a poetic contradiction of the Law of Reflection (“the angle of incidence is equal to the angle of reflection”; for detailed explanations, see this Stack Exchange thread), but why does she bring it up? Because it was in one of the most important Russian works about painting, Merezhkovsky’s Воскресшие боги. Леонардо да Винчи (Resurrected Gods: Leonardo da Vinci; see this LH post) — describing Leonardo’s ability to discuss frightful subjects calmly and dispassionately, Merezhkovsky writes:

Говоря о мертвых телах, которые сталкиваются в водоворотах, прибавил: «изображая эти удары и столкновения, не забывай закона механики, по которому угол падения равен углу отражения».

Speaking about dead bodies colliding in whirlpools, he added: “in depicting these blows and collisions, do not forget the mechanical law according to which the angle of incidence is equal to the angle of reflection.”

Again, this is repeated several times; for example, when Leonardo realizes that waves in water, sound waves, and light all obey the same laws, he exclaims:

«Единая воля и справедливость Твоя, Первый Двигатель: угол падения равен углу отражения!»

“Thy will and justice are one and indivisible, First Mover: the angle of incidence is equal to the angle of reflection!”

(You can see the other occurrences at this National Corpus page.) But two decades earlier, Turgenev had used it in a very different way in one of his poems in prose, Истина и Правда [Scientific truth and real truth], where one person says scientific truth is vitally important and the other mocks him, asking if he can imagine someone running into a gathering and saying excitedly:

«Друзья мои, послушайте, что я узнал, какую истину! Угол падения равен углу отражения! Или вот еще: между двумя точками самый краткий путь — прямая линия!»

“Friends, listen to what I’ve just found out, what truth! The angle of incidence is equal to the angle of reflection! Or this: the shortest path between two points is a straight line!”

Here it represents the height of banality, on the level of 2 × 2 = 4 (which of course has its own Russian literary history). Why did this law of physics become a meme in Russian and not in English? I speculate because the Russian words are normal, everyday words: падение, translated in this context as “incidence,” is the normal word for ‘fall(ing)’ (both physical and in the Fall of Man), so the law sounds like a popular saying.


  1. Of course one can’t skip Grebenschikov’s angle on this:

  2. Nice!

    Мы стояли на плоскости
    С переменным углом отраженья…

    We stood on a plane
    With variable angle of reflection…

  3. Surely two plus two is more banal than two times two, except when it makes five?

  4. Yes, but it’s “дважды два ― четыре” that’s a meme in Russian. A few examples of many:

    Sologub, Мелкий бес (The Petty Demon, 1902):
    ― Ну вот, дважды два ― четыре, что тебе следует жениться на моей сестре.
    [Well, twice two is four, so you should marry my sister.]

    Gorky, Фома Гордеев (Foma Gordeev, 1899):
    Значит, оно-то и есть самое настоящее ― самое полезное и обязательное… ― Как дважды два! ― Просто!
    [So it’s the most real, the most useful and obligatory… ― Like twice two is four! ― Simple!]

    Chekhov, Ариадна (Ariadna, 1895):
    […] его покойный дед Иларион положительно, как дважды два, подтверждал, что Ариадна будет его женой.
    […his late grandfather Ilarion positively, like twice two, confirmed that Ariadna would be his wife.]

    Chekhov, Огни (Fires, 1888):
    Словами можно доказать и опровергнуть всё, что угодно, и скоро люди усовершенствуют технику языка до такой степени, что будут доказывать математически верно, что дважды два ― семь.
    [By means of words you can prove and disprove anything you like, and soon people will perfect the technique of language to such an extent that they’ll prove mathematically that twice two is seven.]

    Dostoevsky, Подросток (The Adolescent, 1875):
    И, главное, все было так ясно, как дважды два, а я ― я все еще упорно верил.
    [And, the main thing, everything was as clear as twice two, and I — I still stubbornly believed.]

    Dostoevsky, Идиот (The Idiot, 1869):
    Но что всего вернее, как дважды два, это то, что Лебедев обожает и своего племянника!
    [But what’s even truer, like twice two, is that Lebedev worships his nephew too!]

    Dostoevsky, Преступление и наказание (Crime and Punishment, 1866):
    […] хотелось бы следствие, так сказать, математически ясно представить, хотелось бы такую уличку достать, чтоб на дважды два ― четыре походило!
    […you’d want the investigation to present, so to say, mathematically clearly, you’d want to get such a nice little piece of evidence that it would be like twice two is four!]

    And of course it’s an important element of the Underground Man’s argument in Dostoevsky’s Записки из подполья (Notes from Underground, 1864), e.g.
    Дважды два и без моей воли четыре будет.
    [Twice two will be four whether I want it or not.]

  5. [But what’s even truer, like twice two, is that Lebedev worships his niece too!]
    Nitpick: nephew

  6. Woops, fixed, thanks! That’s what comes of posting hastily.

  7. The Russian stuff about two times two immediately made me think of Asimov:

    Nine times seven, thought Shuman with deep satisfaction, is sixty-three, and I don’t need a computer to tell me so. The computer is in my own head.

    And it was amazing the feeling of power that gave him.

    However, since Asimov was never a fluent speaker—much less reader—of Russian (his cradle tongue was Yiddish, and his family left Russia when he was three), I suppose this is probably unrelated.

  8. Stu Clayton says

    If you had a song for a fraction with a repeating decimal representation, it would thus have a refrain to sing over and over. Something to keep the kids busy until dinner is ready.

  9. January First-of-May says

    I suspected that the link was for the song. I used to really like it as a kid; IIRC it was the first song I ever fully memorized.

    Meanwhile, the “scientific knowledge” scene in one of the favorite cartoons of my childhood, Island of Errors, features both 2×2=4 (at the very start) and the angle of incidence (at the very end).

  10. Aw, what a great cartoon! I wish I could understand more than a few words.

  11. Delightful! The angle of incidence is used to ward off a 2 (bad grade) at 22:18.

  12. Perhaps unrelated, but Stegner’s Angle of Repose came out in 1971, also a novel by a male writer drawing heavily from the actual correspondence of a real woman who was the basis for the central character.

    Men understanding their relation to the perspective of their heroines in the terms of geometry, or something.

  13. David Marjanović says

    Heh. “3 x 3 is forever 9, nothing to be done about it.”

  14. Athel Cornish-Bowden says

    It doesn’t work for billiard balls (because of energy lost in the impact), as I learned when I tried to play billiards many years ago.

  15. Stu Clayton says

    to ward off a 2 (bad grade)

    So grades run in the opposite direction as compared with Germany ? I think this came up here years ago.

  16. Yes, the Russian system goes form “1” (worst, and almost never applied in practice) up to “5” (best grade). The lowest passing grade is “3”.

  17. In the cartoons the terrifying “1”s are guarding the entrance gate, but it’s the “2”s that actually hinder the protagonist.

  18. About some non-numerical satisfactions of banality, there’s the Flaubert who gave us the Dictionary of Received Ideas and M. Binet with his lathe. And Sherwood Anderson’s “A Man of Ideas,” one of the episodes of Winesburg,Ohio, has a comic ending but for most of its length is a variation on Gray’s Elegy. Its title character is a man who keeps noticing and poignantly falling in love with the universe’s banal miracles, this way.

    When George Willard went to work for the Winesburg Eagle he was besieged by Joe Welling. Joe envied the boy. It seemed to him that he was meant by Nature to be a reporter on a newspaper. “It is what I should be doing, there is no doubt of that,” he declared, stopping George Willard on the sidewalk before Daugherty’s Feed Store. His eyes began to glisten and his forefinger to tremble. “Of course I make more money with the Standard Oil Company and I’m only telling you,” he added. “I’ve got nothing against you but I should have your place. I could do the work at odd moments. Here and there I would run finding out things you’ll never see.”

    Becoming more excited Joe Welling crowded the young reporter against the front of the feed store. He appeared to be lost in thought, rolling his eyes about and running a thin nervous hand through his hair. A smile spread over his face and his gold teeth glittered. “You get out your note book,” he commanded. “You carry a little pad of paper in your pocket, don’t you? I knew you did. Well, you set this down. I thought of it the other day. Let’s take decay. Now what is decay? It’s fire. It burns up wood and other things. You never thought of that? Of course not. This sidewalk here and this feed store, the trees down the street there—they’re all on fire. They’re burning up. Decay you see is always going on. It doesn’t stop. Water and paint can’t stop it. If a thing is iron, then what? It rusts, you see. That’s fire, too. The world is on fire. Start your pieces in the paper that way. Just say in big letters ‘The World Is On Fire.’ That will make ’em look up. They’ll say you’re a smart one. I don’t care. I don’t envy you. I just snatched that idea out of the air. I would make a newspaper hum. You got to admit that.”

    Turning quickly, Joe Welling walked rapidly away. When he had taken several steps he stopped and looked back. “I’m going to stick to you,” he said. “I’m going to make you a regular hummer. I should start a newspaper myself, that’s what I should do. I’d be a marvel. Everybody knows that.”

  19. “Why did this law of physics become a meme in Russian and not in English?”
    I suppose the meme that serves a similar function in English writing is Newton’s 3rd law: For every action there is an equal and opposite reaction.

  20. Lars Mathiesen says

    I think the Archimedean principle has that status in Denmark.

    “When a body is lowered into water, wholly or partially, the phone rings.”

  21. @Zongora: I tell my students I don’t ever want to hear about “action and reaction.” Newton’s Third Law is about forces, not “action”; the term “reaction force” is acceptable, but only in the proper context.

  22. Action-réaction, which goes as well as can be expected.

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