The One-Seventh Ellipse

The decimal expansion of 1/7 is 0.142857142857 …, a repeating decimal. Arrange the six repeating digits into overlapping ordered pairs, like so:

(1, 4), (4, 2), (2, 8), (8, 5), (5, 7) (7, 1),

and, remarkably, all six lie on an ellipse:

19x2 + 36yx + 41y2 – 333x – 531y + 1638 = 0

one-seventh ellipse #1

Even more remarkably, if we take the digits two at a time:

(14, 28), (42, 85), (28, 57), (85, 71), (57, 14), (71, 42),

these points also lie on an ellipse:

-165104x2 + 160804yx + 8385498x – 41651y2 – 3836349y – 7999600 = 0

one-seventh ellipse #2

That’s from David Wells, The Penguin Dictionary of Curious and Interesting Numbers (1986). Victor Hugo wrote, “Mankind is not a circle with a single center but an ellipse with two focal points, of which facts are one and ideas the other.”

Overdone Bacon

Exponents of the theory that Francis Bacon wrote the plays attributed to Shakespeare have gone to sometimes elaborate lengths to find messages hidden in the plays. American physician Orville Ward Owen even invented a “cipher wheel” that could pass the texts under his eyes at various speeds as he looked for hidden meanings.

He didn’t find many supporters. Even Owen’s friend Frederick Mann wrote, “We are asked to believe that such peerless creations as Hamlet, The Tempest, and Romeo and Juliet were not prime productions of the transcendent genius who wrote them, but were subsidiary devices which Bacon designed for the purpose of concealing the cipher therein.”

In his 1910 book Bacon Is Shake-speare, Sir Edwin Durning-Lawrence argues that the long word honorificabilitudinitatibus in Love’s Labour’s Lost is really an anagram:

These plays, F. Bacon’s offspring, are preserved for the world.

“It surpasses the wit of man,” he wrote, to produce another sensible anagram from the long word, and he offered a hundred guineas to anyone who could do it. A Mr. Beevor of St. Albans rather promptly sent him this:

Be off, F. Bacon, the actor has entered and is playing.

Durning-Lawrence was taken aback, but he was a good sport: He paid Beevor his money.

(From John Michell, Who Wrote Shakespeare?, 1996.)

In a Word

n. violent shaking

n. a severe shaking

A lion attacks David Livingstone, 1843:

Starting, and looking half round, I saw the lion just in the act of springing upon me. I was upon a little height; he caught my shoulder as he sprang, and we both came to the ground below together. Growling horribly close to my ear, he shook me as a terrier dog does a rat. The shock produces a stupor similar to that which seems to be felt by a mouse after the first shake of the cat. It caused a sort of dreaminess, in which there was no sense of pain nor feeling of terror, though quite conscious of all that was happening. It was like what patients partially under the influence of chloroform describe, who see all the operation, but feel not the knife. This singular condition was not the result of any mental process. The shake annihilated fear, and allowed no sense of horror in looking round at the beast. This peculiar state is probably produced in all animals killed by carnivora; and if so, is a merciful provision by our benevolent Creator for lessening the pain of death.

The lion left him to attack his companions, who eventually dispatched it. Livingstone could never afterward raise his left arm above his shoulder; when asked by a group of sympathetic friends what he had been thinking during the attack, he said, “I was thinking, with a feeling of disinterested curiosity, which part of me the lion would eat first.”

Podcast Episode 108: The Greenwich Time Lady

As recently as 1939, a London woman made her living by setting her watch precisely at the Greenwich observatory and “carrying the time” to her customers in the city. In this week’s episode of the Futility Closet podcast we’ll meet Ruth Belville, London’s last time carrier, who conducted her strange occupation for 50 years.

We’ll also sample the colorful history of bicycle races and puzzle over a stymied prizewinner.

Sources for our feature on Ruth Belville:

David Rooney, Ruth Belville: The Greenwich Time Lady, 2008.

Ian R. Bartky, Selling the True Time, 2000.

Patricia Fara, “Modest Heroines of Time and Space,” Nature, Oct. 30, 2008.

Stephen Battersby, “The Lady Who Sold Time,” New Scientist, Feb. 25, 2006.

Carlene E. Stephens, “Ruth Belville: The Greenwich Time Lady,” Technology and Culture 51:1 (January 2010), 248-249.

Michael R. Matthews, Colin Gauld, and Arthur Stinner, “The Pendulum: Its Place in Science, Culture and Pedagogy,” in Michael R. Matthews, Colin F. Gauld, and Arthur Stinner, eds., The Pendulum: Scientific, Historical, Philosophical and Educational Perspectives, 2005.

Listener mail:

Eric Niiler, “Tour de France: Top 10 Ways the Race Has Changed,” Seeker, June 29, 2013.

Julian Barnes, “The Hardest Test: Drugs and the Tour de France,” New Yorker, Aug. 21, 2000.

Race Across America.

Wikipedia, “Race Across America” (accessed June 3, 2016).

Wikipedia, “Trans Am Bike Race” (accessed June 3, 2016).

Neil Beltchenko, “2014 Trans Am Race,” Bikepackers Magazine, June 6, 2014.

Trans Am Bike Race 2016.

This week’s lateral thinking puzzle was contributed by listener Tommy Honton, who sent this corroborating link (warning — this spoils the puzzle).

You can listen using the player above, download this episode directly, or subscribe on iTunes or Google Play Music or via the RSS feed at

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and all contributions are greatly appreciated. You can change or cancel your pledge at any time, and we’ve set up some rewards to help thank you for your support. You can also make a one-time donation on the Support Us page of the Futility Closet website.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at Thanks for listening!

A Guest Appearance

The Fibonacci numbers are the ones in this sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Each number is the sum of the two that precede it. But now, interestingly:

\displaystyle  \mathrm{arctan} \left ( \frac{1}{1} \right ) = \mathrm{arctan} \left ( \frac{1}{2} \right ) + \mathrm{arctan} \left ( \frac{1}{3} \right )\\  \mathrm{arctan} \left ( \frac{1}{3} \right ) = \mathrm{arctan} \left ( \frac{1}{5} \right ) + \mathrm{arctan} \left ( \frac{1}{8} \right )\\  \mathrm{arctan} \left ( \frac{1}{8} \right ) = \mathrm{arctan} \left ( \frac{1}{13} \right ) + \mathrm{arctan} \left ( \frac{1}{21} \right )\\  \mathrm{arctan} \left ( \frac{1}{21} \right ) = \mathrm{arctan} \left ( \frac{1}{34} \right ) + \mathrm{arctan} \left ( \frac{1}{55} \right )\\

“And so on!” writes James Tanton in Mathematics Galore! (2012). “The first relation, for instance, states that a line of slope 1/2 stacked with a line of slope 1/3 gives a line of slope 1. (Can you prove the relations?)”

(Ko Hayashi, “Fibonacci Numbers and the Arctangent Function,” Mathematics Magazine 76:3 [June 2003], 215.)

Vulture Picnic

For her 2009 work In Ictu Oculi (“In the Twinkling of an Eye”), artist Greta Alfaro spread a table outside the Spanish village of Fitero and filmed a feast among 40 vultures.

“It was not easy to get them to jump on the table,” she told the Translocal Institute for Contemporary Art. “I had to wait for one week, setting the table every morning and unsetting it at dusk. Vultures have extraordinary eyesight, and if one of them notices that there is food, it will draw circles in the air to let the others know. They approached the scene every day, but either my presence or the presence of the table prevented them from getting closer.”

“I think that it is important today to reflect on the impermanence of almost everything, and on the fact that life cannot be controlled.”

No Waiting

In 1892 … a law firm in the American West came up with the idea of a divorce papers vending machine. For a while, at least, legal divorce papers were items that could be bought from a vending machine in Corinne, Utah. A purchaser could insert $2.50 in coins, pull a lever on the side of the machine, and pick up his papers from a delivery drawer that popped open like a cash register drawer. Those papers were then taken to the local law firm — whose name was printed on the form — where the names of the divorcing couple were written in and witnessed.

— Kerry Segrave, Vending Machines: An American Social History, 2002


“Stopping by Euclid’s Proof of the Infinitude of Primes,” by Presbyterian College mathematician Brian D. Beasley, “with apologies to Robert Frost”:

Whose proof this is I think I know.
I can’t improve upon it, though;
You will not see me trying here
To offer up a better show.

His demonstration is quite clear:
For contradiction, take the mere
n primes (no more), then multiply;
Add one to that … the end is near.

In vain one seeks a prime to try
To split this number — thus, a lie!
The first assumption was a leap;
Instead, the primes will reach the sky.

This proof is lovely, sharp, and deep,
But I have promises to keep,
And tests to grade before I sleep,
And tests to grade before I sleep.

(From Mathematics Magazine 78:2 [April 2005], 171.)

Notes and Measures

Howard Shapiro chose an unusual way to present his paper “Fluorescent Dyes for Differential Counts by Flow Cytometry” at the 1977 meeting of the Histochemical Society — he sang it:

Blood cells are classified by cell and nuclear shape and size
And texture, and affinity for different types of dyes,
And almost all of these parameters can quickly be
Precisely measured by techniques of flow cytometry.

It’s hard to fix a cell suspension rapidly and stain
With several fluorochromes, and this procedure, while it plain-
Ly furnishes the data which one needs to classify,
May fade away, and newer, simpler, methods never dye. …

The full paper, 76 verses with figures and sheet music, is here.