A harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression (so an example is 1/1, 1/2, 1/3, 1/4 …). When tutoring mathematics at Oxford, Charles Dodgson had a favorite example to illustrate this:
According to him, it is (or was) the rule at Christ Church that, if an undergraduate is absent for a night during term-time without leave, he is for the first offence sent down for a term; if he commits the offence a second time, he is sent down for two terms; if a third time, Christ Church knows him no more. This last calamity Dodgson designated as ‘infinite.’ Here, then, the three degrees of punishment may be reckoned as 1, 2, infinity. These three figures represent three terms in an ascending series of Harmonic Progression, being the counterparts of 1, 1/2, 0, which are three terms in a descending Arithmetical Progression.
— Lionel A. Tollemache, “Reminiscences of ‘Lewis Carroll,'” Literature, Feb. 5, 1898
In 1965, as they were writing the first draft of 2001: A Space Odyssey, Stanley Kubrick showed Arthur C. Clarke a set of 12 plastic tiles. Each tile consisted of five squares joined along their edges. These are known as pentominoes, and a set of 12 includes every possible such configuration, if rotations and reflections aren’t considered distinct. The challenge, Kubrick explained, is to fit the 12 tiles together into a tidy rectangle. Because 12 five-square tiles cover 60 squares altogether, there are four possible rectangular solutions: 6 × 10, 5 × 12, 4 × 15, and 3 × 20. (A 2 × 30 rectangle would be too narrow to accommodate all the shapes.)
Clarke, who rarely played intellectual games, found that this challenge “can rather rapidly escalate — if you have that sort of mind — into a way of life.” He stole a set of tiles from his niece, spent hundreds of hours playing with it, and even worked the shapes into the design of a rug for his office. “That a jigsaw puzzle consisting of only 12 pieces cannot be quickly solved seems incredible, and no one will believe it until he has tried,” he wrote in the Sunday Telegraph Magazine. It took him a full month to arrange the 12 shapes into a 6 × 10 rectangle — a task that he was later abashed to learn can be done in 2339 different ways. There are 1010 solutions to the 5 × 12 rectangle and 368 solutions to the 4 × 15.
But “The most interesting case, however, is that of the long, thin rectangle only 3 units wide and 20 long.” Clarke became fascinated with this challenge when Martin Gardner revealed that only two solutions exist. He offered 10 rupees to anyone who could find the solutions, and was delighted when a friend produced them, as he’d calculated that solving the problem by blind permutation would take more than 20 billion years.
Clarke even worked the 3 × 20 problem into his 1975 novel Imperial Earth. Challenged by his grandmother, the character Duncan struggles with the task and declares it impossible. “I’m glad you made the effort,” she says. “Generalizing — exploring every possibility — is what mathematics is all about. But you’re wrong. It can be done. There are just two solutions; and if you find one, you’ll also have the other.”
On Nov. 21, 1897, Mark Twain addressed the Vienna Press Club on “The Horrors of the German Language.” He spoke in German; here’s his literal translation:
It has me deeply touched, my gentlemen, here so hospitably received to be. From colleagues out of my own profession, in this from my own home so far distant land. My heart is full of gratitude, but my poverty of German words forces me to great economy of expression. Excuse you, my gentlemen, that I read off, what I you say will.
The German language speak I not good, but have numerous connoisseurs me assured that I her write like an angel. Maybe — I know not. Have till now no acquaintance with the angels had. That comes later — when it the dear God please — it has no hurry.
Since long, my gentlemen, have I the passionate longing nursed a speech on German to hold, but one has me not permitted. Men, who no feeling for the art had, laid me ever hindrance in the way and made naught my desire — sometimes by excuses, often by force. Always said these men to me: ‘Keep you still, your Highness! Silence! For God’s sake seek another way and means yourself obnoxious to make.’
In the present case, as usual it is me difficult become, for me the permission to obtain. The committee sorrowed deeply, but could me the permission not grant on account of a law which from the Concordia demands she shall the German language protect. Du liebe Zeit! How so had one to me this say could — might — dared — should? I am indeed the truest friend of the German language — and not only now, but from long since — yes, before twenty years already. And never have I the desire had the noble language to hurt; to the contrary, only wished she to improve — I would her only reform. It is the dream of my life been. I have already visits by the various German governments paid and for contracts prayed. I am now to Austria in the same task come. I would only some changes effect. I would only the language method — the luxurious, elaborate construction — compress, the eternal parenthesis suppress, do away with, annihilate; the introduction of more than thirteen subjects in one sentence forbid; the verb so far to the front pull that one it without a telescope discover can. With one word, my gentlemen, I would your beloved language simplify so that, my gentlemen, when you her for prayer need, One her yonder-up understands.
I beseech you, from me yourself counsel to let, execute these mentioned reforms. Then will you an elegant language possess, and afterward, when you some thing say will, will you at least yourself understand what you said had. But often nowadays, when you a mile-long sentence from you given and you yourself somewhat have rested, then must you a touching inquisitiveness have yourself to determine what you actually spoken have. Before several days has the correspondent of a local paper a sentence constructed which hundred and twelve words contained, and therein were seven parentheses smuggled in, and the subject seven times changed. Think you only, my gentlemen, in the course of the voyage of a single sentence must the poor, persecuted, fatigued subject seven times change position!
Now, when we the mentioned reforms execute, will it no longer so bad be. Doch noch eins. I might gladly the separable verb also a little bit reform. I might none do let what Schiller did: he has the whole history of the Thirty Years’ War between the two members of a separable verb in-pushed. That has even Germany itself aroused, and one has Schiller the permission refused the History of the Hundred Years’ War to compose — God be it thanked! After all these reforms established be will, will the German language the noblest and the prettiest on the world be.
Since to you now, my gentlemen, the character of my mission known is, beseech I you so friendly to be and to me your valuable help grant. Mr. Potzl has the public believed make would that I to Vienna come am in order the bridges to clog up and the traffic to hinder, while I observations gather and note. Allow you yourselves but not from him deceived. My frequent presence on the bridges has an entirely innocent ground. Yonder gives it the necessary space, yonder can one a noble long German sentence elaborate, the bridge-railing along, and his whole contents with one glance overlook. On the one end of the railing pasted I the first member of a separable verb and the final member cleave I to the other end — then spread the body of the sentence between it out! Usually are for my purposes the bridges of the city long enough; when I but Potzl’s writings study will I ride out and use the glorious endless imperial bridge. But this is a calumny; Potzl writes the prettiest German. Perhaps not so pliable as the mine, but in many details much better. Excuse you these flatteries. These are well deserved.
Now I my speech execute — no, I would say I bring her to the close. I am a foreigner — but here, under you, have I it entirely forgotten. And so again and yet again proffer I you my heartiest thanks.
Reportedly his spoken German was actually excellent (PDF), and he delivered the address without reading the text.
He’d been sparring with German for some time — his essay “The Awful German Language” had appeared as an appendix to A Tramp Abroad in 1880.
Here are two transcriptions of a speech by Home Secretary Sir William Harcourt, reprinted in the London Times on Jan. 23, 1882. At left is the column as it originally appeared; at right is the same speech in a hastily issued replacement edition. What’s the difference between them?
In the column on the left, about midway down, a disgruntled compositor has inserted the line “The speaker then said he felt inclined for a bit of fucking.”
The paper issued an apology and suppressed the offending edition as well as it could, but that only increased public interest, driving the price of a copy up from threepence to £5 in some areas (it would reach £100 by the 1990s). The Times’ quarterly index recorded the offense:
Harcourt (Sir W.) at Burton on Trent, 23 j 7 c
———Gross Line Maliciously Interpolated in a
Few Copies only of the Issue, 23 j 7 d — 27 j 9 f
The paper tried to rise above all this, but it made a new rule: If you sack a compositor, get him off the premises immediately.
When the Scottish writer William Sharp died in 1905, his wife revealed a surprising secret: For 10 years he had kept up a second career as a reclusive novelist named Fiona Macleod, carrying on correspondences and writing works in two distinctly different styles. In this week’s episode of the Futility Closet podcast we’ll explore Sharp’s curious relationship with his feminine alter ego, whose sporadic appearances perplexed even him.
We’ll also hunt tigers in Singapore and puzzle over a surprisingly unsuccessful bank robber.
In 1904 Mrs. Membury, of Hyde Corner, Bridport, Dorset, set out to make a snake of stamps.
Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and we’ve set up some rewards to help thank you for your support. You can also make a one-time donation on the Support Us page of the Futility Closet website.
Many thanks to Doug Ross for the music in this episode.
Cheering news from India. My press clipping agency has sent me a letter from the correspondence column of an Indian paper about a cow that came into the bungalow of a Mr. Verrier Elwyn, who lives at Patengarth, Mandla District, and ate his copy of Carry On, Jeeves, ‘selecting it from a shelf which contained, among other works, books by Shakespeare, Thomas Hardy and Henry Fielding.’ A pretty striking tribute I look on that as.
— P.G. Wodehouse to William Townend, Sept. 3, 1929
After his daughter Jean’s death in 1909, Mark Twain began to write:
Would I bring her back to life if I could do it? I would not. If a word would do it, I would beg for strength to withhold the word. And I would have the strength; I am sure of it. In her loss I am almost bankrupt, and my life is a bitterness, but I am content: for she has been enriched with the most precious of all gifts — that gift which makes all other gifts mean and poor — death. I have never wanted any released friend of mine restored to life since I reached manhood. I felt in this way when Susy passed away; and later my wife, and later Mr. Rogers. When Clara met me at the station in New York and told me Mr. Rogers had died suddenly that morning, my thought was, Oh, favorite of fortune — fortunate all his long and lovely life — fortunate to his latest moment! The reporters said there were tears of sorrow in my eyes. True — but they were for ME, not for him. He had suffered no loss. All the fortunes he had ever made before were poverty compared with this one.
“I am setting it down,” he told his friend Albert Bigelow Paine, “everything. It is a relief to me to write it. It furnishes me an excuse for thinking.”
He wrote for three days, handed the manuscript to Paine, and told him to make it the final chapter of his autobiography. Four months later he was dead.
At the end of his 1986 book Paradoxes in Probability Theory and Mathematical Statistics, statistician Gábor J. Székely offers a final paradox from his late professor Alfréd Rényi:
Since I started to deal with information theory I have often meditated upon the conciseness of poems; how can a single line of verse contain far more ‘information’ than a highly concise telegram of the same length. The surprising richness of meaning of literary works seems to be in contradiction with the laws of information theory. The key to this paradox is, I think, the notion of ‘resonance.’ The writer does not merely give us information, but also plays on the strings of the language with such virtuosity, that our mind, and even the subconscious self resonate. A poet can recall chains of ideas, emotions and memories with a well-turned word. In this sense, writing is magic.
Edgar Allan Poe’s “The Cask of Amontillado” may be a classic horror story, but it’s full of “weird wine howlers,” according to Clifton Fadiman.
Fortunato, who is immured in the story, “prided himself on his connoisseurship in wine,” and Montresor, who does the immuring, adds, “I was skilful in the Italian vintages myself, and bought largely whenever I could.”
But Fortunato tells him, “Luchesi is quite incapable of telling Amontillado from Sherry,” and, later, “Amontillado! You have been imposed upon; and as for Luchesi, he cannot distinguish Sherry from Amontillado.”
But Amontillado is a sherry! H. Warner Allen points out that André Simon’s wine encyclopedia defines Amontillado as “one of the most popular types of Sherry, neither too dry nor too sweet.”
Compounding this error, Montresor tells Fortunato that he wants Luchesi’s opinion of a pipe of Amontillado that he has received. But a pipe is a cask of port; a cask of sherry is a butt.
Also, Poe seems to have thought that Amontillado is an Italian wine, perhaps judging by the look of the word. Fadiman writes, “What he thought ‘a flagon of De Grâve’ could be is almost beyond conjecture.”
(Clifton Fadiman, Dionysus: A Case of Vintage Tales About Wine, 1962.)