Either know, or listen to someone who does. You can’t live without understanding, whether your own or someone else’s. There are many, however, who don’t know that they don’t know, and others who think they know, but don’t. Stupidity’s faults are incurable, for since the ignorant don’t know what they are, they don’t search for what they lack. Some individuals would be wise if they didn’t believe that they already were. Given all this, although oracles of good sense are rare, they sit idle, because nobody consults them. Seeking advice will neither diminish your greatness nor refute your ability. In fact, it will enhance your reputation. Engage with reason so misfortune doesn’t contend with you.

— Baltasar Gracián, The Pocket Oracle and Art of Prudence, 1647


When Sarah Davis of Hiawatha, Kansas, died in 1930, her husband John spent most of his wealth on her grave, including 11 life-size statues of Italian marble consorting under a canopy that weighs 50 tons.

The extravagance won John some criticism during the Depression, but today the monument brings a steady flow of tourists to the town.

Image: Flickr

Leonardo noted that studies of shadow appeal to the same geometric laws as those of perspective — “in matters perspectival, a light source is no different from the eye [… since] the visual ray is similar to the shadowy ray in terms of speed and converging lines.” Just as perspective considers a scene as regarded from a particular point of view, a shadow is a record of what “the lamp cannot see.”

Philosopher Roberto Casati writes, “This is a strong analogy. There is shadow where the light cannot see. Indeed, shadow is exactly what the light source cannot see. A lamp sees only the things that it illuminates; objects in shadow are behind lighted objects that screen the shadow from the lamp’s perspective. For this same reason, when we examine a shadow, we discover the profile of things from the point of view of the lamp. … Shadow allows us to see a point of view other than our own without even getting up from our seats.”

(From Casati’s Shadows, 2000, via Marko Uršič, Shadows of Being: Four Philosophical Essays, 2019.)

Art and Commerce

Before the 19th century, containers did not come in standard sizes, and students in the 1400s were taught to “gauge” their capacity as part of their standard mathematical education:

There is a barrel, each of its ends being 2 bracci in diameter; the diameter at its bung is 2 1/4 bracci and halfway between bung and end it is 2 2/9 bracci. The barrel is 2 bracci long. What is its cubic measure?

This is like a pair of truncated cones. Square the diameter at the ends: 2 × 2 = 4. Then square the median diameter 2 2/9 × 2 2/9 = 4 76/81. Add them together: 8 76/81. Multiply 2 × 2 2/9 = 4 4/9. Add this to 8 76/81 = 13 31/81. Divide by 3 = 4 112/243 … Now square 2 1/4 = 2 1/4 × 2 1/4 = 5 1/16. Add it to the square of the median diameter: 5 5/16 + 4 76/81 = 10 1/129. Multiply 2 2/9 × 2 1/4 = 5. Add this to the previous sum: 15 1/129. Divide by 3: 5 1/3888. Add it to the first result: 4 112/243 + 5 1/3888 = 9 1792/3888. Multiply this by 11 and then divide by 14 [i.e. multiply by π/4]: the final result is 7 23600/54432. This is the cubic measure of the barrel.

Interestingly, this practice informed the art of the time — this exercise is from a mathematical handbook for merchants by Piero della Francesca, the Renaissance painter. Because many artists had attended the same lay schools as business people, they could invoke the same mathematical training in their work, and visual references that recalled these skills became a way to appeal to an educated audience. “The literate public had these same geometrical skills to look at pictures with,” writes art historian Michael Baxandall. “It was a medium in which they were equipped to make discriminations, and the painters knew this.”

(Michael Baxandall, Painting and Experience in Fifteenth Century Italy, 1988.)

04/10/2021 UPDATE: A reader suggests that there’s a typo in the original reference here. If 9 1792/3888 is changed to 9 1793/3888, the final result is 7 23611/54432, which is exactly the result obtained by integration using the approximation π = 22/7. (Thanks, Mariano.)

Out of the Way

Thomas Heatherwick designed this “rolling bridge” in 2002 as part of a redevelopment of London’s Paddington Basin. When not in use it curls into an octagon half the width of the waterway, and when extended the eight sections form a 12-meter footbridge.

It won the 2005 British Structural Steel Design Award in 2005.

“American Ships Head to Gulf”

In a forum on Testy Copy Editors in 2009, editor Mike O’Connell posted a headline from the newspaper Japan Today: “Violinist Linked to JAL Crash Blossoms.” He asked, “what do you call these kinds of strangely phrased hedlines? is there a word for them?”

The answer suggested itself — a crash blossom is headline that’s painfully ambiguous, usually due to unwise ellipsis, double meaning, or tortured syntax. Linguist Ben Zimmer gave some examples in the New York Times the following year:

Giant Waves Down Queen Mary’s Funnel
MacArthur Flies Back to Front
Eighth Army Push Bottles Up Germans
McDonald’s Fries the Holy Grail for Potato Farmers
British Left Waffles on Falklands
Gator Attacks Puzzle Experts

And the Language Log blog lists examples from time to time:

Infant Pulled From Wrecked Car Involved in Short Police Pursuit
Letter Bombs Accused in Court
Mexico Mine Missing Declared Dead
Queen Mary Having Bottom Scraped
Two Soviet Ships Collide — One Dies
Soviet Virgin Lands Short of Goal
Smoking Riskier Than Thought
Headless Corpse Accused in Court

Here’s an archive.

Double Duty

On Nov. 5, 1996, Election Day in the United States, the New York Times crossword puzzle carried a surprising clue:

39. Lead story in tomorrow’s newspaper (!), with 43A

43 across turned out to be ELECTED, but 39 across might be either CLINTON or BOBDOLE — both possibilities had seven letters. Was the Times venturing to guess the outcome of the day’s election?

No. Composer Jeremiah Farrell had contrived each of the seven down clues to admit of two possible answers, so that no matter which candidate won, the newspaper might claim a “correct” result.

Crossword editor Will Shortz called Farrell’s ambiguous effort his favorite puzzle of all time.

(Thanks, Andrew.)

A Reindeer Cyclone

Russian photographer Lev Fedoseyev captured this drone footage on March 24 on the Kola Peninsula in the Arctic Circle.

When threatened, a herd of reindeer runs in a circle, making it hard for a predator to target any individual. The fawns are at the center.