Descartes’ Theorem

If three circles “kiss,” like the black circles above, then a fourth circle can be drawn that’s tangent to all three. In 1643 René Descartes showed that if the curvature or “bend” of a circle is defined as k = 1/r, then the radius of the fourth circle can be found by

descartes' theorem

The ± sign reflects the fact that two solutions are generally possible — the plus sign corresponds to the smaller red circle, the minus sign to the larger (circumscribing) one.

Frederick Soddy summed this up in a poem in Nature (June 20, 1936):

The Kiss Precise

For pairs of lips to kiss maybe
Involves no trigonometry.
‘Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.

Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There’s now no need for rule of thumb.
Since zero bend’s a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.

To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.

(Thanks, Sean.)

In a Word

micropsychy
n. faint-heartedness

abulia
n. an inability to act decisively

quakebuttock
n. a coward

Unquote

http://commons.wikimedia.org/wiki/File:Burning-of-a-heretic--_Sassetta--Melburn_museum.jpg

“Heresy is the side that loses.” — J.V. Fleming

Charge!

https://docs.google.com/viewer?url=patentimages.storage.googleapis.com/pdfs/US695903.pdf

Horses are difficult to manage onstage, so in 1901 music hall performer Alexander Braatz invented this ingenious alternative. The actor’s own legs masquerade as those of his horse, and a system of levers and cords moves the hind legs as well.

False human legs are attached at each side to give the appearance that the performer is mounted; his real feet “are advantageously covered by large clumsy hoof-like coverings.” I suppose you could even race these things between rehearsals.

https://docs.google.com/viewer?url=patentimages.storage.googleapis.com/pdfs/US695903.pdf

A Point of Law

A gentleman, who had been described as a ‘Pettifogger,’ accused another gentleman, whom he had styled a ‘Fish-fag,’ with an assault. It being a very intricate point, it was of course referred to the Lord Mayor. It stood as follows: — ‘Whether puffing a cloud of tobacco-smoke in a man’s face constituted an assault?’ After some grave consultation with that encyclopaedia of wisdom, Mr. Hobler, the decision ran thus — The Lord Mayor: ‘There has been no assault; nothing but words, words.’ — Complainant: ‘I beg pardon, my Lord.’ — The Lord Mayor: ‘Well, then, all smoke, if you please, or words and puffs. There have been no blows.’ — Now we beg his Lordship’s pardon. Pray what is a puff but a blow?

The Age, Aug. 8, 1830

Pointing Fingers

Only one of these statements is true. Which is it?

A. All of the below
B. None of the below
C. One of the above
D. All of the above
E. None of the above
F. None of the above

Click for Answer

Advice

http://commons.wikimedia.org/wiki/File:Franklin-Benjamin-LOC-head.jpeg

More wisdom from Poor Richard’s Almanack:

  • Thirst after desert — not reward.
  • Gifts much expected, are paid not given.
  • Children and princes will quarrel for trifles.
  • Praise little, dispraise less.
  • The Child thinks 20 Shillings and 20 Years can scarce ever be spent.
  • Industry need not wish.
  • ‘Tis great Confidence in a Friend to tell him your Faults, greater to tell him his.
  • Neglect kills Injuries, Revenge increases them.
  • Observe all men; thyself most.
  • Do not do that which you would not have known.
  • He that resolves to mend hereafter, resolves not to mend now.
  • The honest Man takes Pains, and then enjoys Pleasures; the knave takes Pleasure, and then suffers Pains.
  • All would live long, but none would be old.
  • Nothing more like a fool, than a drunken man.
  • What e’er’s begun in anger, ends in shame.
  • As often as we do good, we sacrifice.
  • There is much difference between imitating a good man, and counterfeiting him.

And “Cut the Wings of your Hens and Hopes, lest they lead you a weary Dance after them.”

Podcast Episode 14: The Unsinkable Violet Jessop

http://commons.wikimedia.org/wiki/File:Violet_jessop_titanic.jpg

Stewardess Violet Jessop was both cursed and blessed — during the 1910s she met disaster on all three of the White Star Line’s Olympic class of gigantic ocean liners, but she managed to escape each time.

In this episode of the Futility Closet podcast we’ll accompany Violet on her three ill-fated voyages, including the famous sinkings of the Titanic and the Britannic, and learn the importance of toothbrushes in ocean disasters.

We’ll also play with the International Date Line and puzzle over the identity of Salvador Dalí’s brother.

University of Chicago economist Steven Levitt discusses his coin-flipping experiment about halfway through this BBC podcast. The associated website is here.

We first wrote about Violet Jessop on March 11, 2009. Maritime historian John Maxtone-Graham interviewed her in 1970 for The Only Way to Cross, his 1978 book about the era of ocean liners. When Violet died in 1971 she left a manuscript to her daughters, which, edited by Maxtone-Graham, came to light in 1997 as Titanic Survivor: The Newly Discovered Memoirs of Violet Jessop, Who Survived Both the Titanic and Britannic Disasters. A poetic note from Maxtone-Graham in that book:

“One particular service commemorates the 1500 lost on the Titanic: Every 14th of April, a United States Coast Guard cutter comes to pay the homage of the Ice Patrol, which owes its inception to the disaster. With engines stilled and church pennant at the masthead, officers and men line the deck in full dress, while the commander reads the burial service. Three volleys of rifle fire can be heard, then the cutter passes on, leaving a lone wreath on the waves above the broken hull.”

Lewis Carroll underscored the need for an international date line with this conundrum, which he presented among the mathematical puzzle stories he wrote for the Monthly Packet in the 1880s:

The day changes only at midnight. Suppose it’s midnight in Chelsea; Wednesday has concluded and Thursday is about to begin. It’s still Wednesday in Ireland and America, and it’s already Thursday in Germany and Russia.

That’s fine. But continue in both directions. If it’s Wednesday in America, is it Wednesday in Hawaii? If it’s Thursday in Russia, is it Thursday in Japan? Mustn’t the two days “meet” on the farther side of the globe?

“It isn’t midnight anywhere else; so it can’t be changing from one day to another anywhere else. And yet, if Ireland and America and so on call it Wednesday, and Germany and Russia and so on call it Thursday, there must be some place, not Chelsea, that has different days on the two sides of it. And the worst of it is, the people there get their days in the wrong order: they’ve got Wednesday east of them, and Thursday west — just as if their day had changed from Thursday to Wednesday!”

Carroll normally presented the solution to each problem in the following month’s number. In this case he postponed the solution, “partly because I am myself so entirely puzzled by it,” and then discontinued the column without resolving the problem.

Further curiosities regarding the International Date Line:

Paul Sloane and Des MacHale have written a whole series of books of lateral thinking puzzles. This week’s puzzle on Salvador Dalí’s brother comes from their Ingenious Lateral Thinking Puzzles (1998).

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at http://feedpress.me/futilitycloset. The show notes are on the blog. Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

Art Appreciation

thurber drawing

When James Thurber tried to improve his drawings, E.B. White told him, “Don’t do that. If you ever got good you’d be mediocre.”

Milestones

https://commons.wikimedia.org/wiki/File:Julius_LeBlanc_Stewart_-_Les_Dames_Goldsmith_au_blois_de_Boulogne_en_1897_sur_une_voiturette.jpg

A conundrum by French puzzle maven Pierre Berloquin:

Caroline leaves town driving at a constant speed. After some time she passes a milestone displaying a two-digit number. An hour later she passes a milestone displaying the same two digits but in reverse order. In another hour she passes a third milestone with the same two digits (in some order) separated by a zero.

What is the speed of Caroline’s car?

Click for Answer
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