Face Value

Suppose that Schweitzer and Gandhi are equally saintly and that Green and White are equally unsavory characters with long criminal records. Suppose that on separate occasions Green gratuitously slaps Schweitzer in the face, Schweitzer gratuitously slaps White in the face, and Gandhi gratuitously slaps Schweitzer in the face. If guilt were proportional, not just to the offence, but to the moral uprightness of the offended party, then Green would incur more guilt and liability to punishment than would Schweitzer. For since Schweitzer is worthier than White, Green’s failure to show respect for Schweitzer was more grievous than Schweitzer’s failure to show respect for White. Similarly, Gandhi’s action would be more culpable than Schweitzer’s. In fact, I think we are more apt to consider guilt as directly proportional to the nature of the offender than to the nature of the offended party. Schweitzer’s action in slapping White is, if anything, more culpable than Green’s action in slapping Schweitzer. In view of Schweitzer’s long-standing habits of self-control and moral behaviour, we should expect more from him than from Green who has never developed these habits. Similarly, we should expect more from Gandhi. Nor would we say that Gandhi’s act was more culpable than Schweitzer’s. We might even have some inclination to be less outraged at Gandhi, since he was at least ‘picking on someone’ of his own moral stature.

— Marilyn McCord Adams, “Hell and the God of Justice,” Religious Studies 11:4 [December 1975], 433-447


The second act of Alban Berg’s 1937 opera Lulu includes a three-minute sequence that’s a musical palindrome — after the first 90 seconds, there’s an ascending piano arpeggio, a pause … and then the music unfolds in reverse, a perfect mirror image back to the start.

This music accompanies a silent film that is itself a palindrome in some ways — for example, three people arrest Lulu and put her in prison, and then three liberate her. And Lulu’s husbands in Act I are played by the same singers as her clients in Act III.

“Hiawatha’s Photographing”


Lewis Carroll was an early enthusiast of photography, though he seems to have found the social aspects trying — he published this poem in 1857:

From his shoulder Hiawatha
Took the camera of rosewood,
Made of sliding, folding rosewood;
Neatly put it all together.
In its case it lay compactly,
Folded into nearly nothing;
But he opened out the hinges,
Pushed and pulled the joints and hinges,
Till it looked all squares and oblongs,
Like a complicated figure
In the Second Book of Euclid.
This he perched upon a tripod —
Crouched beneath its dusky cover —
Stretched his hand, enforcing silence —
Said, “Be motionless, I beg you!”
Mystic, awful was the process.
All the family in order
Sat before him for their pictures:
Each in turn, as he was taken,
Volunteered his own suggestions,
His ingenious suggestions.
First the Governor, the Father:
He suggested velvet curtains
Looped about a massy pillar;
And the corner of a table,
Of a rosewood dining-table.
He would hold a scroll of something,
Hold it firmly in his left-hand;
He would keep his right-hand buried
(Like Napoleon) in his waistcoat;
He would contemplate the distance
With a look of pensive meaning,
As of ducks that die ill tempests.
Grand, heroic was the notion:
Yet the picture failed entirely:
Failed, because he moved a little,
Moved, because he couldn’t help it.
Next, his better half took courage;
She would have her picture taken.
She came dressed beyond description,
Dressed in jewels and in satin
Far too gorgeous for an empress.
Gracefully she sat down sideways,
With a simper scarcely human,
Holding in her hand a bouquet
Rather larger than a cabbage.
All the while that she was sitting,
Still the lady chattered, chattered,
Like a monkey in the forest.
“Am I sitting still?” she asked him.
“Is my face enough in profile?
Shall I hold the bouquet higher?
Will it came into the picture?”
And the picture failed completely.
Next the Son, the Stunning-Cantab:
He suggested curves of beauty,
Curves pervading all his figure,
Which the eye might follow onward,
Till they centered in the breast-pin,
Centered in the golden breast-pin.
He had learnt it all from Ruskin
(Author of ‘The Stones of Venice,’
‘Seven Lamps of Architecture,’
‘Modern Painters,’ and some others);
And perhaps he had not fully
Understood his author’s meaning;
But, whatever was the reason,
All was fruitless, as the picture
Ended in an utter failure.
Next to him the eldest daughter:
She suggested very little,
Only asked if he would take her
With her look of ‘passive beauty.’
Her idea of passive beauty
Was a squinting of the left-eye,
Was a drooping of the right-eye,
Was a smile that went up sideways
To the corner of the nostrils.
Hiawatha, when she asked him,
Took no notice of the question,
Looked as if he hadn’t heard it;
But, when pointedly appealed to,
Smiled in his peculiar manner,
Coughed and said it ‘didn’t matter,’
Bit his lip and changed the subject.
Nor in this was he mistaken,
As the picture failed completely.
So in turn the other sisters.
Last, the youngest son was taken:
Very rough and thick his hair was,
Very round and red his face was,
Very dusty was his jacket,
Very fidgety his manner.
And his overbearing sisters
Called him names he disapproved of:
Called him Johnny, ‘Daddy’s Darling,’
Called him Jacky, ‘Scrubby School-boy.’
And, so awful was the picture,
In comparison the others
Seemed, to one’s bewildered fancy,
To have partially succeeded.
Finally my Hiawatha
Tumbled all the tribe together,
(‘Grouped’ is not the right expression),
And, as happy chance would have it
Did at last obtain a picture
Where the faces all succeeded:
Each came out a perfect likeness.
Then they joined and all abused it,
Unrestrainedly abused it,
As the worst and ugliest picture
They could possibly have dreamed of.
‘Giving one such strange expressions —
Sullen, stupid, pert expressions.
Really any one would take us
(Any one that did not know us)
For the most unpleasant people!’
(Hiawatha seemed to think so,
Seemed to think it not unlikely.)
All together rang their voices,
Angry, loud, discordant voices,
As of dogs that howl in concert,
As of cats that wail in chorus.
But my Hiawatha’s patience,
His politeness and his patience,
Unaccountably had vanished,
And he left that happy party.
Neither did he leave them slowly,
With the calm deliberation,
The intense deliberation
Of a photographic artist:
But he left them in a hurry,
Left them in a mighty hurry,
Stating that he would not stand it,
Stating in emphatic language
What he’d be before he’d stand it.
Hurriedly he packed his boxes:
Hurriedly the porter trundled
On a barrow all his boxes:
Hurriedly he took his ticket:
Hurriedly the train received him:
Thus departed Hiawatha.

He introduced it by writing, “In an age of imitation, I can claim no special merit for this slight attempt at doing what is known to be so easy. Any fairly practised writer, with the slightest ear for rhythm, could compose, for hours together, in the easy running metre of The Song of Hiawatha. Having then distinctly stated that I challenge no attention in the following little poem to its merely verbal jingle, I must beg the candid reader to confine his criticism to its treatment of the subject.”

Double Magic

Image: Wikimedia Commons

This style of compound magic square was first devised by Kenneth Kelsey of Great Britain. The numbers 70-94 appear in the blue boxes, making a pandiagonal magic square. The numbers 95-110 appear in the yellow circles, making a pandiagonal magic square of their own. And embedding one in the other produces a compound square — the numbers in the circles can be added to the numbers in the squares in either of the adjoining sections. So, for example, the second row, 87 + 79 + 91 + 83 + 70 = 410, can include the row of circles above it (87 + 108 + 79 + 105 + 91 + 99 + 83 + 98 + 70 = 820) or below it (87 + 95 + 79 + 102 + 91 + 104 + 83 + 109 + 70 = 820).

In the finished figure, every number from 70 to 110 appears once, and the blue square and the yellow square have the same magic constant — 410!

Early Days

In her 1914 book Una Mary: The Inner Life of a Child, Una Hunt, the daughter of geologist Frank Wigglesworth Clarke, set out to describe the subjective world of her young girlhood. Here’s an example — she had created an imaginary land she called My Country in which her alternate self, Una Mary, lived, and then established it in the Persian rug in the parlor, where her chessmen could play out their adventures:

A very yellow palm-leaf in one corner of the pattern was the Holy Land. I thought it was holey, full of holes. I had simply heard some one speak of having been there the winter before, and the name sounded sunny and yellow, a cheerful sort of place, full of caves in the soft rock. I thought the whole country must look rather like Swiss cheese to deserve its name. The Holy Land was, of course, simply infested by robbers. The Forty Thieves lived there, each with a cave to himself, all in a row, and for some reason it was always there that we hid from pirates.

The outside border of the rug was the sea. I felt sure, of course, that the world was bounded by the sea and if you sailed to the edge the ship would fall off, so the chessmen were always careful not to go beyond the second stripe of the border outside. …

The stem of one flower was the Charles River, where I had found the turtle eggs, and another was The Amazon. Always that name has fascinated me, The Amazon, and I feel sure the river itself is a tawny orange zigzag with huge, many-colored leaves and flowers growing out of it at unexpected angles. It was like that on the rug, and I chose that particular stem to be The Amazon because its color was like the sound of the word. There was another reason besides the fascination of the name itself which later made me include it in the geography of My Country, and that was because Brazil was my only association with Royalty.

Psychologist G. Stanley Hall said, “I would rather have written it myself than to have made any study of childhood that has ever appeared.” The whole thing is here.

The Bicycle Puzzle

Stand a bicycle so that one pedal is in its lowest position and one in its highest. Now if we pull backward on the low pedal, will the bicycle move forward or backward?

Intuition suggests it will move forward: We’re turning the pedals in the same direction that a rider would, and normally this motion drives the rear wheel to propel the bike forward.

Surprisingly, though, in the experiment the bike moves backward. That’s because (in most bikes, in most gears) each pedal is constantly moving forward with respect to the ground. So pulling backward on the pedal doesn’t produce the intuitive result — it moves the pedal backward relative to the ground, and so produces the opposite result to the one we expect.

An exception: If the bike is in a sufficiently low gear, pulling backward on the pedal will drive the bike forward. But the sprocket ratio must be so low that really we’re betraying intuition rather than the reverse (see the video).

Southern Exposure


I forgot to mention that I donned a kilt for the Highland Ball at Glenferness. It was anxious work at first, as it is a garment with no notion of privacy, and delights in giving all present tantalising glimpses of things unseen. However, with careful manipulation and a pair of drawers, I got through the evening tolerably. It is quite comfortable to dance in, but should be a godsend to mosquitoes.

— Sir Alan Lascelles, diary, September 15, 1907

Mixed Emotions

A brainteaser by S. Ageyev, from the November-December 1991 issue of Quantum:

ageyev problem

Suppose that we change the signs of 50 of these numbers such that exactly half the numbers in each row and each column get a minus sign. Prove that the sum of all the numbers in the resulting table is zero.

Click for Answer


“The Mathematician in Love,” by Scottish mechanical engineer William Rankine (1820-1872):

A mathematician fell madly in love
With a lady, young, handsome, and charming:
By angles and ratios harmonic he strove
Her curves and proportions all faultless to prove
As he scrawled hieroglyphics alarming.

He measured with care, from the ends of a base,
The arcs which her features subtended:
Then he framed transcendental equations, to trace
The flowing outlines of her figure and face,
And thought the result very splendid.

He studied (since music has charms for the fair)
The theory of fiddles and whistles,–
Then composed, by acoustic equations, an air,
Which, when ’twas performed, made the lady’s long hair
Stand on end, like a porcupine’s bristles.

The lady loved dancing:–he therefore applied,
To the polka and waltz, an equation;
But when to rotate on his axis he tried,
His centre of gravity swayed to one side,
And he fell, by the earth’s gravitation.

No doubts of the fate of his suit made him pause,
For he proved, to his own satisfaction,
That the fair one returned his affection;–“because,
“As every one knows, by mechanical laws,
“Re-action is equal to action.”

“Let x denote beauty,–y, manners well-bred,–
z, Fortune,–(this last is essential),–
“Let L stand for love”–our philosopher said,–
“Then L is a function of x, y, and z,
“Of the kind which is known as potential.”

“Now integrate L with respect to d t,
“(t standing for time and persuasion);
“Then, between proper limits, ’tis easy to see,
“The definite integral Marriage must be:–
“(A very concise demonstration).”

Said he–“If the wandering course of the moon
“By Algebra can be predicted,
“The female affections must yield to it soon”–
–But the lady ran off with a dashing dragoon,
And left him amazed and afflicted.

The Mindset List

In 1998, Tom McBride and Ron Nief of Wisconsin’s Beloit College began compiling lists of what had “always” or “never” been true in the lives of each incoming class of students, to remind faculty to be mindful of the references they made in class.

For example, that first class, born in 1980, had been 11 years old when the Soviet Union broke up and did not remember the Cold War. They had never had a polio shot and never owned a record player. Their popcorn had always been cooked in a microwave, and they’d always had cable television. Here are some details of the worldview of the class of 2022:

  • Outer space has never been without human habitation.
  • They will never fly TWA, Swissair, or Sabena airlines.
  • The Prius has always been on the road in the U.S.
  • They never used a spit bowl in a dentist’s office.
  • “You’ve got mail” would sound as ancient to them as “number, please” would have sounded to their parents.
  • Mass market books have always been available exclusively as Ebooks.
  • There have always been more than a billion people in India.
  • Films have always been distributed on the Internet.
  • The detachable computer mouse is almost extinct.
  • The Mir space station has always been at the bottom of the South Pacific.

Other recent lists are here.