If a sphere is inscribed in a tetrahedron, and segments are drawn on each face connecting its vertices to the point of tangency, then the same three angles will appear on each face.
And the 12 triangles produced will form congruent pairs across each edge of the tetrahedron.
Into an empty vase drop balls numbered 1 to 10. Remove ball 1. Add balls numbered 11 to 20. Remove ball 2. Continue in this way, spending half an hour on the first transaction, 15 minutes on the next, and so on. After one hour all the transactions will be finished.
Obviously, in the end the vase will contain infinitely many balls, since with each step more balls have been added than removed.
But, equally obviously, after an hour the vase will be empty — since the time of each ball’s removal is known.
He, then, who says that something true exists either only asserts that something true exists or proves it. And if he merely asserts it, he will be told the opposite of his mere assertion, namely, that nothing is true. But if he proves that something is true, he proves it either by a true proof or by one that is not true. But he will not say that it is by one not true, for such a proof is not to be trusted. And if it is by a true proof, whence comes it that the proof which proves that something is true is itself true? If it is true of itself, it will be possible also to state as true of itself that truth does not exist; while if it is derived from proof, the question will again be asked ‘How is it that this proof is true?’ and so on ad infinitum. Since, then, in order to learn that there is something true, an infinite series must first be grasped, and it is not possible for an infinite series to be grasped, it is not possible to know for a surety that something true exists.
— Sextus Empiricus, Against the Logicians
From Bismarck’s Reflections and Reminiscences, 1898:
At the time of my first stay at St. Petersburg, in 1859, I had an example of another Russian peculiarity. During the first spring days it was then the custom for every one connected with the Court to promenade in the Summer Garden between Paul’s Palace and the Neva. There the Emperor had noticed a sentry standing in the middle of a grass plot; in reply to the question why he was standing there, the soldier could only answer, ‘Those are my orders.’ The Emperor therefore sent one of his adjutants to the guard-room to make inquiries; but no explanation was forthcoming except that a sentry had to stand there winter and summer. The source of the original order could no longer be discovered. The matter was talked of at Court, and reached the ears of the servants. One of these, an old pensioner, came forward and stated that his father had once said to him as they passed the sentry in the Summer Garden: ‘There he is, still standing to guard the flower; on that spot the Empress Catherine once noticed a snowdrop in bloom unusually early, and gave orders that it was not to be plucked.’ This command had been carried out by placing a sentry on the spot, and ever since then one had stood there all the year round.
“Stories of this sort excite our amusement and criticism, but they are an expression of the elementary force and persistence on which the strength of the Russian nature depends in its attitude towards the rest of Europe.”
The audience at … a play are spectators of a world they are not in. They see what they may well describe as, say, Othello in front of a certain palace in Venice … [b]ut they are not themselves at any specifiable distance from that palace. …
I see Othello strangle Desdemona; but that will not entail that I, as part of my biography, have ever seen anyone strangle anyone. Nor need the actress who plays Emilia ever see a dead body; but Emilia does, for she sees the dead body of Desdemona.
“[W]e can in fact even visualise the unseen, because the fact that in visualisation I am as it were seeing is not itself necessarily an element of what is visualised.”
— Bernard Williams, Problems of the Self, 1973
A puzzle from the 18th century — punctuate this sentence so that it makes sense:
King Charles the First walked and talked half an hour after his head was cut off.
Play with work blend, keep warmish feet,
Away drive trouble, slowly eat;
Air pure breathe, and early rise;
Beware excess, take exercise.
Exercise take, excess beware;
Rise early and breathe pure air;
Eat slowly; trouble drive away;
Feet warmish keep, blend work with play.
— “W.E.R.,” in Truth, Jan. 13, 1881
There are several simple little drawing tricks which the nurse may use to arouse the interest of her patient as she uses puzzles and catches. The oldest of these is by Hogarth and represents a soldier and his dog going through a doorway. As is seen by the diagram, it consists of three straight lines and one curved one.
— William Rush Dunton, Occupation Therapy, 1915
In the 1950s, humorist Roger Price invented “Droodles,” simple enigmatic drawings explained by their captions. Frank Zappa used one on the cover of a 1982 album:
It’s called Ship Arriving Too Late to Save a Drowning Witch.
In March 1939, students at McGill University dictated this sentence to a dozen faculty members:
Outside a cemetery sat a harassed cobbler and an embarrassed oculist, picknicking on a desiccated apple, and gazing at the symmetry of a lady’s ankle with unparalleled ecstasy.
The participants included three English professors, the head of the journalism department, and a proofreading instructor.
Only Esther Korstad, instructor of typewriting and shorthand, spelled everything correctly. The average participant misspelled 4.25 words.
In the classic Indian rope trick, a rope rises into the sky, its end lost to view. A boy disappears up the rope, and when he fails to return the angry magician climbs up after him. Body parts fall to the ground, the magician descends and places the parts in a basket, and the boy reappears uninjured.
This is all thought to be a legend, but in 1979 mathematician J.L.G. Pinhey of The Perse Boys’ School worked out that levitating a rope is possible, at least in principle. If the top of the fakir’s rope is 1.5 × 108 meters above Earth’s surface, it will simply stand erect, its position sustained by the motion of the planet.
“Since the rope between its ends is in tension the configuration is stable, and the faqir and his boy-victim can climb it in safety. However, in order to drop the bits to earth, the pair must not climb even a quarter of the way to the top.”
(J.L.G. Pinhey, “63.12 The Indian Rope Trick,” Mathematical Gazette 63:424 [June 1979], 110-111.)
