# Stature

Venice’s Museo Correr exhibits a pair of wooden implements whose use isn’t immediately clear — they’re chopines, a type of platform shoe popular in the 15th, 16th, and 17th centuries. Worn under a woman’s skirt they could add up to 20 inches to her height, giving her an impressive eminence but an uncertain gait. Shakespeare mocked the trend in Hamlet’s greeting to a visiting player:

“By’r lady, your ladyship is nearer to heaven than when I saw you last, by the altitude of a chopine.”

08/15/2024 UPDATE: Reader Peter Kidd notes this even more impressive pair, now at the Museo Civico Medievale in Bologna:

(Thanks, Peter.)

# Cube Route

A centered hexagonal number is a number that can be represented by a hexagonal lattice with a dot in the center, like so:

Starting at the center, successive hexagons contain 1, 7, 19, and 37 dots. The sequence goes on forever.

The sum of the first n centered hexagonal numbers is n3, and there’s a pretty “proof without words” to show that this is so:

Instead of regarding each figure as a hexagon, think of it as a perspective view of a cube, looking down along a space diagonal. The first cube here contains a single dot. How many dots must we add to produce the next larger cube? Seven, and from our bird’s-eye perspective this pattern of 7 added dots matches the 7-dot hexagon shown above. The same thing happens when we advance to a 3×3×3 cube: This requires surrounding the 2×2×2 cube with 19 additional dots, and from our imagined vantage point these again take the form of a hexagonal lattice. In the last image our 33 cube must accrete another 37 dots to become a 43 cube … and the pattern continues.

# Reflections

Epigrams of poet Ralph Hodgson:

• Oaths in anguish rank with prayers.
• The wink was not our best invention.
• When crises pall, humdrum is sensational.
• But Woman — in whose image made?
• A sparrow in a snowstorm with a feather in his bill: that is Faith.
• Forget the slush, but keep the snow / Of Christmasses of long ago.
• Anniversary: Familiarity breeds content.
• Some things have to be believed to be seen.
• Who shall paraphrase a tear!
• There’s one thing to be said for sin — it does give conscience exercise.
• Why not Foremothers?
• The Golden Rule was called new-fangled, once upon a time.
• Blessed are the children of a nobody.
• The handwriting on the wall may be a forgery.

And “The ‘last word’ is only the latest.”

# Liberty

Reader Derek Christie sent in this surprising curiosity after Wednesday’s post about Borromean rings:

Both the ring and the karabiner clip are attached to the cord and can’t be removed.

Now complicate matters by clipping the karabiner onto the ring:

Quite unexpectedly, the cord can now be just pulled away:

(Thanks, Derek.) (A related perplexity: The Prisoners’ Release Puzzle.)

# Step by Step

“Rules for the direction of the mind,” from an unfinished treatise by René Descartes:

1. The aim of our studies must be the direction of our mind so that it may form solid and true judgments on whatever matters arise.
2. We must occupy ourselves only with those objects that our intellectual powers appear competent to know certainly and indubitably.
3. As regards any subject we propose to investigate, we must inquire not what other people have thought, or what we ourselves conjecture, but what we can clearly and manifestly perceive by intuition or deduce with certainty. For there is no other way of acquiring knowledge.
4. There is need of a method for finding out the truth.
5. Method consists entirely in the order and disposition of the objects towards which our mental vision must be directed if we would find out any truth. We shall comply with it exactly if we reduce involved and obscure propositions step by step to those that are simpler, and then starting with the intuitive apprehension of all those that are absolutely simple, attempt to ascend to the knowledge of all others by precisely similar steps.
6. In order to separate out what is quite simple from what is complex, and to arrange these matters methodically, we ought, in the case of every series in which we have deduced certain facts the one from the other, to notice which fact is simple, and to mark the interval, greater, less, or equal, which separates all the others from this.
7. If we wish our science to be complete, those matters which promote the end we have in view must one and all be scrutinized by a movement of thought which is continuous and nowhere interrupted; they must also be included in an enumeration which is both adequate and methodical.
8. If in the matters to be examined we come to a step in the series of which our understanding is not sufficiently well able to have an intuitive cognition, we must stop short there. We must make no attempt to examine what follows; thus we shall spare ourselves superfluous labour.
9. We ought to give the whole of our attention to the most insignificant and most easily mastered facts, and remain a long time in contemplation of them until we are accustomed to behold the truth clearly and distinctly.
10. In order that it may acquire sagacity the mind should be exercised in pursuing just those inquiries of which the solution has already been found by others; and it ought to traverse in a systematic way even the most trifling of men’s inventions though those ought to be preferred in which order is explained or implied.
11. If, after we have recognized intuitively a number of simple truths, we wish to draw any inference from them, it is useful to run them over in a continuous and uninterrupted act of thought, to reflect upon their relations to one another, and to grasp together distinctly a number of these propositions so far as is possible at the same time. For this is a way of making our knowledge much more certain, and of greatly increasing the power of the mind.
12. Finally we ought to employ all the help of understanding, imagination, sense and memory, first for the purpose of having a distinct intuition of simple propositions; partly also in order to compare the propositions.
13. If we perfectly understand a problem we must abstract it from every superfluous conception, reduce it to its simplest terms and, by means of an enumeration, divide it up into the smallest possible parts.
14. The problem should be re-expressed in terms of the real extension of bodies and should be pictured in our imagination entirely by means of bare figures. Thus it will be perceived much more distinctly by our intellect.
15. It is generally helpful if we draw these figures and display them before our external senses. In this way it will be easier for us to keep our mind alert.
16. As for things which do not require the immediate attention of the mind, however necessary they may be for the conclusion, it is better to represent them by very concise symbols rather than by complete figures. It will thus be impossible for our memory to go wrong, and our mind will not be distracted by having to retain these while it is taken up with deducing other matters.

He’d planned a further 15 but did not finish the work. These 21 were published posthumously in 1701.

# Temper

It is time to bury the nonsense of the ‘incomplete animal.’ As Julian Huxley, the eminent British biologist, once observed concerning human toughness, man is the only creature that can walk twenty miles, run two miles, swim a river, and then climb a tree. Physiologically, he has one of the toughest bodies known; no other species could survive weeks of exposure on the open sea, or in deserts, or the Arctic. Man’s superior exploits are not evidence of cultural inventions: clothing on a giraffe will not allow it to survive in Antarctica, and neither shade nor shoes will help a salamander in the Sahara. I am not speaking of living in those places permanently, but simply as a measure of the durability of men under stress.

— Paul Shepard, The Tender Carnivore and the Sacred Game, 1973

# Misc

• Vatican City has 2.27 popes per square kilometer.
• Skylab was fined for littering.
• Five-syllable rhyming words in English: vocabulary, constabulary
• 8767122 + 3287682 = 876712328768
• “We die only once, and for such a long time!” — Molière

Above is the only known film footage of Mark Twain, shot at Twain’s Connecticut home in 1909. The women are thought to be his daughters Clara and Jean.

# All Together Now

German scientist Gaspar Schott’s 1657 Magia universalis naturæ et artis includes a description of “the music of donkeys”: “the trick, according to Schott, lay in using male donkeys of particular natural pitches and stimulating them to bray with the urine of a female donkey, which will induce the males to make ‘most contented’ noises that the generous might construe as a kind of music.” Schott had argued that “the excessively discordant singing” of men and animals becomes sweeter when encountered rarely.

From Mark A. Waddell, Jesuit Science and the End of Nature’s Secrets, 2015.

Related: the cat organ and the piganino.

# In a Word

rarissima
n. extremely rare books, manuscripts, or prints

In The Book Hunter (1863), John Hill Burton identifies five types of “persons who meddle with books”:

• “A bibliognoste, from the Greek, is one knowing in title-pages and colophons, and in editions; the place and year when printed; the presses whence issued; and all the minutiae of a book.”
• “A bibliographe is a describer of books and other literary arrangements.”
• “A bibliomane is an indiscriminate accumulator, who blunders faster than he buys, cock-brained and purse-heavy.”
• “A bibliophile, the lover of books, is the only one in the class who appears to read them for his own pleasure.”
• “A bibliotaphe buries his books, by keeping them under lock, or framing them in glass cases.”

These groups seem to have been proposed by French librarian Jean Joseph Rive. Bibliographer Gabriel Peignot added four more:

• bibliolyte, a destroyer of books
• bibliologue, one who discourses about books
• bibliotacte, a classifier of books
• bibliopée, “‘l’art d’écrire ou de composer des livres,’ or, as the unlearned would say, the function of an author.”

# The Champ

Proposed cabinet of Dizzy Gillespie, who ran for president in 1964:

• Secretary of State: Duke Ellington
• Director of the CIA: Miles Davis
• Secretary of Defense: Max Roach
• Secretary of Peace: Charles Mingus
• Librarian of Congress: Ray Charles
• Secretary of Agriculture: Louis Armstrong
• Ambassador to the Vatican: Mary Lou Williams