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Proof Without Words

schaer proof

In the January 2001 issue of the College Mathematics Journal, University of Calgary mathematician Jonathan Schaer offers this simple proof that arctan 1 + arctan 2 + arctan 3 = π. The sum of the angles of this large triangle is 180°. And the diagram shows that its lower left angle is arctan 3, its lower right angle is arctan 2, and its top angle is part of an isosceles right triangle. So arctan 1 + arctan 2 + arctan 3 = 180°, or π radians. “No words, even no symbols!”

(“Miscellanea,” College Mathematics Journal 32:1, 68-71.)

| Science & Math

Last Words

On Sept. 8, 1880, an explosion tore through the Seaham Colliery, a coal mine in County Durham in the North of England, filling both shafts with debris and trapping scores of men in the burning seams. Most of the 164 dead were found where they had been working, suggesting that death had come quickly, but some of those farther from the shafts appear to have survived for hours or even longer in pockets of air — the oil in some of their lamps was exhausted. Some of these men had had time to leave messages to those who might find them:

September 8 1880
E.Hall and J.Lonsdale died at half-past 3 in the morning.W.Murray and W.Morris and James Clarke visited the rest on half-past nine in the morn and all living in the incline,
Yours truly,
W.Murray, Master-Shifter

Five o’ clock, we have been praying to God

The Lord has been with us, we are all ready for heaven – Ric Cole, half past 2 o’ clock Thursday

Bless the Lord we have had a jolly prayer meeting, every man ready for glory. Praise the Lord. Sign.R.Cole

Dear Margaret,
There was 40 of us altogether at 7am. Some was singing hymns but my thoughts was on my little Michael that him and I would meet in heaven at the same time. Oh Dear wife, God save you and the children, and pray for me … Dear wife, Farewell. My last thoughts are about you and the children. Be sure and learn the children to pray for me. Oh what an awful position we are in ! Michael Smith, 54 Henry Street

The last had been scratched on a tin water bottle with a rusty nail by Michael Smith, who had left his dying infant son to go to work in the mine that morning. The son died on the same day.

(From Durham Records Online.)

| Death

A Word for Everything

In the 17th century natural philosopher John Wilkins set out to create a universal language for scholars, to replace Latin, whose sometimes arbitrary features made it difficult to learn. His solution, laid out in the 1668 Essay Towards a Real Character, is a system of symbols that could classify every thing and idea in the world, somewhat like the Linnaean system in biology. There are 40 main genera, each of which supplies the first two letters of a word. Each genus is divided into “differences,” which supply the next letter. And a “species” gives the fourth letter. So, for example, Zi indicates the genus “beasts,” or mammals; Zit specifies the difference “rapacious beasts of the dog kind”; and Zita gives the species “dogs.” The symbols weren’t necessarily meant to be spoken, though Wilkins later assigned phonetic values to the various characters — rather they were meant to provide an unambiguous way of classifying the world so that scholars (and, perhaps, diplomats, travelers, and merchants) could communicate clearly.

Wilkins presented the system to the Royal Society, who were impressed but concluded that it was impractical — its descriptions of similar (and confusable) items might differ by a single letter, and it would be difficult to remember all the distinctions, which seemed to invite trouble. Borges later lampooned it with his (fictional) Celestial Emporium of Benevolent Knowledge, which divides animals into 14 categories:

  1. Those that belong to the emperor
  2. Embalmed ones
  3. Those that are trained
  4. Suckling pigs
  5. Mermaids (or Sirens)
  6. Fabulous ones
  7. Stray dogs
  8. Those that are included in this classification
  9. Those that tremble as if they were mad
  10. Innumerable ones
  11. Those drawn with a very fine camel hair brush
  12. Et cetera
  13. Those that have just broken the flower vase
  14. Those that, at a distance, resemble flies

(Thanks, Alex.)

| Language

Last Word

https://commons.wikimedia.org/wiki/File:Avro_Avro_691_Lancastrian_3_G-AGWH_cn_1280_%27Stardust%27_BSAA_(British_South_American_Airways)_(15215624954).jpg

In August 1947 a British South American Airways airliner en route from Buenos Aires to Santiago crashed into Mount Tupungato in the Argentine Andes. The wreckage wasn’t located for 50 years, but it’s believed today that, hindered by the jet stream, the pilots had started their descent before they’d cleared the mountaintops and crashed into Tupungato.

The last Morse code message received by the Santiago airport was “ETA SANTIAGO 17.45 HRS STENDEC.” The operator didn’t recognize the last word and asked that it be sent again. The flight transmitted STENDEC twice more and then was lost. The meaning of that last transmission is unknown — despite much speculation, it’s never been definitively explained.

| Oddities

“Forgotten Words Are Mighty Hard to Rhyme”

Quoth I to me, “A chant royal I’ll dite,
With much ado of words long laid away,
And make windsuckers of the bards who cite
The sloomy phrases of the present day.
My song, though it encompass but a page,
Will man illume from April bud till snow —
A song all merry-sorry, con and pro.”
(I would have pulled it off, too, given time,
Except for one small catch that didn’t show:
Forgotten words are mighty hard to rhyme.)

Ah, hadavist, in younghede, when from night
There dawned abluscent some fair morn in May
(The word for dawning, ‘sparrowfart,’ won’t quite
Work in here) — hadavist, I say,
That I would ever by stoopgallant age
Be shabbed, adushed, pitchkettled, suggiled so,
I’d not have been so redmod! Could I know? —
One scantling piece of outwit’s all that I’m
Still sure of, after all this catch-and-throw:
Forgotten words are mighty hard to rhyme.

In younghede ne’er a thrip gave I for blight
Of cark or ribble; I was ycore, gay;
I matched boonfellows hum for hum, each wight
By eelpots aimcried, till we’d swerve and sway,
Turngiddy. Blashy ale could not assuage
My thirst, nor kill-priest, even. No Lothario
Could overpass me on Poplolly Row.
A fairhead who eyebit me in my prime
Soon shared my donge. (The meaning’s clear, although
Forgotten words are mighty hard to rhyme.)

Fair draggle-tails once spurred my appetite;
Then walking morts and drossels shared my play.
Bedswerver, smellsmock, housebreak was I hight —
Poop-noddy at poop-noddy. Now I pray
That other fonkins reach safe anchorage —
Find bellibone, straight-fingered, to bestow
True love, till truehead in their own hearts grow.
Still, umbecasting friends who scrowward climb,
I’m swerked by mubblefubbles. Wit grows slow;
Forgotten words are mighty hard to rhyme.

Dim on the wong at cockshut falls the light;
Birds’ sleepy croodles cease. Not long to stay …
Once nesh as open-tide, I now affright;
I’m lennow, spittle-ready — samdead clay,
One clutched bell-penny left of all my wage.
Acclumsied now, I dare no more the scrow,
But look downsteepy to the Pit below.
Ah, hadavist! … Yet silly is the chime;
Such squiddle is no longer apropos.
Forgotten words are mighty hard to rhyme.

— Willard R. Espy

| Language · Poems

“Fine Words Butter No Parsnips”

English proverbs:

Experience keeps a dear school. (1743)
Everybody stretches his legs according to the length of his coverlet. (1550)
He that would the daughter win, must with the mother first begin. (1578)
A still tongue makes a wise head. (1562)
Speak not of my debts unless you mean to pay them. (1640)
One of these days is none of these days. (1658)
One hand for yourself and one for the ship. (1799)
It’s never too late to mend. (1590)
The highest branch is not the safest roost. (1563)
He who is absent is always in the wrong. (1640)
The golden age was never the present age. (1732)
Example is better than precept. (1400)
Sweep your own doorstep clean. (1624)
Idle people have the least leisure. (1678)
He that would have eggs must endure the cackling of hens. (1659)

| Quotations

Words and Motion

https://commons.wikimedia.org/wiki/File:RO(1875)_P195_HERON.jpg

In a 1778 letter, English naturalist Gilbert White captured the characteristic movement of almost 50 birds:

Owls move in a buoyant manner, as if lighter than air … herons seem incumbered with too much sail for their light bodies … the green-finch … exhibits such languishing and faltering gestures as to appear like a wounded and dying bird … fernowls, or goat-suckers, glance in the dusk over the tops of trees like a meteor; starlings as it were swim along.

Biographer Richard Mabey writes, “What is striking is the way Gilbert often arranges his sentence structure to echo the physical style of a bird’s flight. So, ‘The white-throat uses odd jerks and gesticulations over the tops of hedges and bushes’; and ‘woodpeckers fly volatu undosu [in an undulating flight], opening and closing their wings at every stroke, and so are always rising or falling in curves.”

(From Mabey’s Gilbert White: A Biography of the Author of The Natural History of Selborne, 2007.)

| Language · Literature

Last Words

In 1899, six years before her death at age 70, Aboriginal Tasmanian Fanny Cochrane Smith made five wax cylinder recordings of traditional Aboriginal songs and language.

They are the only recorded example of Tasmanian Aboriginal songs and the only recorded example of any Tasmanian Aboriginal language.

Emily Keene, who was present at the recording, said that when the cylinder was played back for her, Smith cried, “My poor race. What have I done.”

“We could not pacify her for a long time,” Keene said. “She thought the voice she had heard was that of her mother.”

| History · Language

The River Witham Sword

https://britishlibrary.typepad.co.uk/digitisedmanuscripts/2015/08/help-us-decipher-this-inscription.html

This 13th-century double-edged sword, possibly of German manufacture, was found in the River Witham, Lincolnshire, in 1825. Inlaid in gold wire along one of its edges is a curious inscription:

+NDXOXCHWDRGHDXORVI+

It’s been speculated that this is a religious invocation, but its full meaning is not clear. In 2015 the British Library invited readers to offer their thoughts, but no conclusive solution was reached. Medieval historian Marc van Hasselt of Utrecht University says it may be the product of a sophisticated workshop that made swords for the elite, as similar blades have been found throughout Europe. “These similarities go so far as to suggest the same hand in making the inscriptions. However, their contents are still a mystery, regardless of their origins.”

| History · Technology

Spreading Word

When a Western scrub jay discovers the body of a dead jay, it summons other birds to screech over the body for up to half an hour. It’s not clear why they do this — the birds are territorial and not normally social. Possibly it’s a way to share news of danger, concentrate attention to find a predator, or teach young about dangers in the environment.

The gatherings are sometimes called funerals, though we don’t know enough to understand the reasons behind them. But UC Davis student Teresa Iglesias said, “I think there’s a huge possibility that there is much more to learn about the social and emotional lives of birds.”

| Science & Math