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

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.)

Easter Fare

Wolfram Alpha offers some surprising seasonal equations — a bunny:

max(min(-51/25 abs(-(21 x)/(22 a) – (5 y)/(17 a) + 2/11)^(29/16) – 37/17 abs((5 x)/(17 a) – (21 y)/(22 a) + 15/17)^(35/23) + 1, -75/22 abs(-(12 x)/(17 a) – (12 y)/(17 a) + 19/24)^(34/15) – 105/13 abs(-(12 x)/(17 a) + (12 y)/(17 a) + 1/34)^(123/62) + 1, x/a), min(-51/25 abs((21 x)/(22 a) – (5 y)/(17 a) + 2/11)^(29/16) – 37/17 abs(-(5 x)/(17 a) – (21 y)/(22 a) + 15/17)^(35/23) + 1, -75/22 abs((12 x)/(17 a) – (12 y)/(17 a) + 19/24)^(34/15) – 105/13 abs((12 x)/(17 a) + (12 y)/(17 a) + 1/34)^(123/62) + 1, -x/a), min(max(-(177 x^2)/(13 a^2) – 46/15 (y/a + 1/24)^2 + 1, (690 x^2)/(29 a^2) + 63/4 (y/a + 8/17)^2 – 1), 1/10 – ((79 x^2)/(16 a^2) + 16 (y/a + 1/2)^2 – 1) ((16 x^2)/a^2 + (79 y^2)/(16 a^2) – 1), 6287/17 (x/a – 1/9)^2 + 100 (y/a + 1/16)^2 – 1, 6287/17 (x/a + 1/9)^2 + 100 (y/a + 1/16)^2 – 1), -31550/23 (x/a – 2/19)^2 – 62500/49 (y/a + 1/11)^2 + 1, -31550/23 (x/a + 2/19)^2 – 62500/49 (y/a + 1/11)^2 + 1, -18407811/17 (x/a – 1/25)^4 – 250127/15 (y/a + 13/22)^4 + 1, -18407811/17 (x/a + 1/25)^4 – 250127/15 (y/a + 13/22)^4 + 1, -(x/a – 1/2)^2 – (y/a + 5/4)^2 + 1/30, 11/20 – ((y^4/(63 a^4) – y^3/(11 a^3) – y^2/(7 a^2) + (13 y)/(15 a) + x^2/a^2 + 29/43) ((142 x^2)/(15 a^2) + ((-(304 y)/(23 a) – 878/31) x)/a + (1019 y)/(25 a) + (184 y^2)/(13 a^2) + 349/11) ((142 x^2)/(15 a^2) + (((304 y)/(23 a) + 878/31) x)/a + (1019 y)/(25 a) + (184 y^2)/(13 a^2) + 349/11))/(11/19 – y/(8 a))^2, -x^2/a^2 – x/a – (5 y)/(2 a) – y^2/a^2 – 16/9, -(127 x^2)/(21 a^2) – ((-(118 y)/(23 a) – 47/7) x)/a – (559 y)/(15 a) – (173 y^2)/(22 a^2) – 847/19, -(127 x^2)/(21 a^2) – (((118 y)/(23 a) + 47/7) x)/a – (559 y)/(15 a) – (173 y^2)/(22 a^2) – 847/19)>=0

… and an egg:

min(1/5 – sin(16 p sqrt(x^2/(a^2 (1 – y/(10 a))^2) + (9 y^2)/(16 a^2)) (1 – 1/10 (1 – sqrt(x^2/(a^2 (1 – y/(10 a))^2) + (9 y^2)/(16 a^2))) cos(12 tan^(-1)(x/(a (1 – y/(10 a))), (3 y)/(4 a))))), -x^2/(a^2 (1 – y/(10 a))^2) – (9 y^2)/(16 a^2) + 1)>=0

In Poland, Easter Monday is Śmigus-dyngus, in which boys throw water over girls they like and spank them with pussy willow branches. Traditionally, Wikipedia says, “Boys would sneak into girls’ homes at daybreak on Easter Monday and throw containers of water over them while they were still in bed. After all the water had been thrown, the screaming girls would often be dragged to a nearby river or pond for another drenching. Sometimes a girl would be carried out, still in her bed, before both bed and girl were thrown into the water together. Particularly attractive girls could expect to be soaked repeatedly during the day.”

(Thanks, Danesh and Wade.)

Podcast Episode 336: A Gruesome Cure for Consumption

In the 19th century, some New England communities grew so desperate to help victims of tuberculosis that they resorted to a macabre practice: digging up dead relatives and ritually burning their organs. In this week’s episode of the Futility Closet podcast we’ll examine the causes of this bizarre belief and review some unsettling examples.

We’ll also consider some fighting cyclists and puzzle over Freddie Mercury’s stamp.

See full show notes …

The Endless House

In 1924 architect Frederick Kiesler proposed a house fashioned as a continuous shell rather than an assembly of columns and beams:

The house is not a dome, but an enclosure which rises along the floor uninterrupted into walls and ceilings. Thus the safety of the house does not depend on underground footings but on a strict coordination of all enclosures of the house into one structural unit. Even the window areas are not standardized in size or shape, but are, rather, large and varied in their transparency and translucency, and form part of the continuous flow of the shell.

Storage space is provided between double shells in the walls, radiant heating is built into the floor, and there are no separate bathrooms, as bathing is done in the individual living quarters.

“While the concept of the house does not advocate a ‘return to nature’, it certainly does encourage a more natural way of living, and a greater independence from our constantly increasing automative way of life.”

(Via Ulrich Conrads, The Architecture of Fantasy: Utopian Building and Planning in Modern Times, 1962.)

Podcast Episode 334: Eugene Bullard

Eugene Bullard ran away from home in 1907 to seek his fortune in a more racially accepting Europe. There he led a life of staggering accomplishment, becoming by turns a prizefighter, a combat pilot, a nightclub impresario, and a spy. In this week’s episode of the Futility Closet podcast we’ll tell Bullard’s impressive story, which won him resounding praise in his adopted France.

We’ll also accidentally go to Canada and puzzle over a deadly omission.

See full show notes …


In 1927 Albert Einstein sent a photograph of himself to his friend Cornelia Wolf. He inscribed these lines:

Wherever I go and wherever I stay,
There’s always a picture of me on display.
On top of the desk, or out in the hall,
Tied round a neck, or hung on the wall.

Women and men, they play a strange game,
Asking, beseeching: “Please sign your name.”
From the erudite fellow they brook not a quibble,
But firmly insist on a piece of his scribble.

Sometimes, surrounded by all this good cheer,
I’m puzzled by some of the things that I hear,
And wonder, my mind for a moment not hazy,
If I and not they could really be crazy.

A Promising Paradox

The dean of a university is searching for a new chair for his chemistry department. He offers the job to one candidate, who says she’ll accept only if she can hire three new faculty members. The dean makes a commitment to support these hires.

Such arrangements are common, but that’s troubling: By making a promise, the dean has made a future act obligatory for himself, and that changes its moral status. What if the promised act would otherwise have been wrong? In this case, what if the funding for these three hires ought otherwise properly to have gone to another department?

“The fact that our moral system permits agents to dictate in this manner the moral status of their future actions seems an astonishing power to build into a moral system,” writes University of Arizona philosopher Holly M. Smith. “It is especially troubling when one notes that agents apparently can exploit promises in order to legitimize otherwise objectionable courses of action. What would we say, for example, about a moral system in which an agent may render A obligatory by simply declaring, ‘My doing A next week will be, by virtue of this declaration, morally obligatory’?”

(Holly M. Smith, “A Paradox of Promising,” Philosophical Review 106:2 [April 1997], 153-196.)

One Nation, Indivisible

The second professor of mathematics in the American colonies suggested reckoning coins, weights, and measures in base 8.

Arguing that ordinary arithmetic had already become “mysterious to Women and Youths and often troublesome to the best Artists,” the Rev. Hugh Jones of the College of William and Mary wrote that his proposal was “only to divide every integer in each species into eight equal parts, and every part again into 8 real or imaginary particles, as far as is necessary. For tho’ all nations count universally by tens (originally occasioned by the number of digits on both hands) yet 8 is a far more complete and commodious number; since it is divisible into halves, quarters, and half quarters (or units) without a fraction, of which subdivision ten is uncapable.”

Successive powers of 8 would be called ers, ests, thousets, millets, and billets; cash, casher, and cashest would be used in counting money, ounce, ouncer, and ouncest in weighing, and yard, yarder, and yardest in measuring distance (so “352 yardest” would signify 3 × 82 + 5 × 8 + 2 yards).

Jones pressed this system zealously, arguing that “Arithmetic by Octaves seems most agreeable to the Nature of Things, and therefore may be called Natural Arithmetic in Opposition to that now in Use, by Decades; which may be esteemed Artificial Arithmetic.” But he seems to have had no illusions about its prospects, acknowledging that “there seems no Probability that this will be soon, if ever, universally complied with.”

(H.R. Phalen, “Hugh Jones and Octave Computation,” American Mathematical Monthly 56:7 [August-September 1949), 461-465.)

Podcast Episode 329: The Cock Lane Ghost

In 1759, ghostly rappings started up in the house of a parish clerk in London. In the months that followed they would incite a scandal against one man, an accusation from beyond the grave. In this week’s episode of the Futility Closet podcast we’ll tell the story of the Cock Lane ghost, an enduring portrait of superstition and justice.

We’ll also see what you can get hit with at a sporting event and puzzle over some portentous soccer fields.

See full show notes …