Short Work

A simple proof that \sqrt{2} is irrational:

Assume that it’s rational. Then \sqrt{2} = p/q and p2 = 2q2. But in the latter equation, the left side must have an even number of prime factors and the right an odd number. That’s a contradiction, so our assumption must be wrong.

In the paper below, Manchester Polytechnic mathematician T.J. Randall credits this “marvellous” proof to Philip J. Davis and Reuben Hersh in their 1982 book The Mathematical Experience, but I can’t find it there.

(T.J. Randall, “67.45 \sqrt{2} Revisited,” Mathematical Gazette 67:442 [December 1983], 302-303.)

12/20/2023 UPDATE: Reader Hans Havermann finds the proof mentioned in Stuart Hollingdale’s Makers of Mathematics (1989), after the more familiar proof based on parity of p and q. Hollingdale writes that the alternative proof “can be traced back to the Classical period.”


In 1967, psychiatrists Thomas Holmes and Richard Rahe asked 5,000 medical patients about recent difficult events in their lives. They found a positive correlation with illness. The sum of the “life change units” you’ve amassed in the past year, they said, gives a rough estimate of the effect on your health:

Death of a spouse: 100
Divorce: 73
Marital separation: 65
Imprisonment: 63
Death of a close family member: 63
Personal injury or illness: 53
Marriage: 50
Dismissal from work: 47
Marital reconciliation: 45
Retirement: 45
Change in health of family member: 44
Pregnancy: 40
Sexual difficulties: 39
Gain a new family member: 39
Business readjustment: 39
Change in financial state: 38
Death of a close friend: 37
Change to different line of work: 36
Change in frequency of arguments: 35
Major mortgage: 32
Foreclosure of mortgage or loan: 30
Change in responsibilities at work: 29
Child leaving home: 29
Trouble with in-laws: 29
Outstanding personal achievement: 28
Spouse starts or stops work: 26
Beginning or end of school: 26
Change in living conditions: 25
Revision of personal habits: 24
Trouble with boss: 23
Change in working hours or conditions: 20
Change in residence: 20
Change in schools: 20
Change in recreation: 19
Change in church activities: 19
Change in social activities: 18
Minor mortgage or loan: 17
Change in sleeping habits: 16
Change in number of family reunions: 15
Change in eating habits: 15
Vacation: 13
Major holiday: 12
Minor violation of law: 11

A score of 300+ means you’re at risk of illness.

A score of 150-299 means your risk of illness is moderate (but 30 percent lower than the foregoing group).

A score of less than 150 means your risk of stress-related illness is slight.

Action at a Distance

On April 1, 1976, English astronomer Patrick Moore announced a unique opportunity on BBC Radio 2. At 9:47 a.m., he said, Pluto would pass behind Jupiter, and the combined gravitational forces of the two bodies would noticeably reduce gravity on Earth. If listeners jumped in the air during the conjunction, they’d feel a floating sensation.

Shortly after the appointed moment, the BBC began to receive hundreds of phone calls from listeners who’d confirmed the effect, including “a woman who said that she and 11 friends had been wafted from their chairs and orbited gently around the room.”

None of them had noted the date.

Dunbar’s Number

In the 1990s, after studying the relation between primate brain size and social groups, British anthropologist Robin Dunbar proposed that human beings can comfortably maintain about 150 stable relationships — relationships in which one knows all the other members and how they relate to one another. Informally, he said, this is “the number of people you would not feel embarrassed about joining uninvited for a drink if you happened to bump into them in a bar.”

Notably, in a 2018 article for the Financial Times, Dunbar added that we maintain an inner core of about five people with whom we spend about 40 percent of our social time and 10 more with whom we spend another 20 percent. “In other words, about two-thirds of our total social effort is devoted to just 15 people.”

Smell Test

In an 1899 lecture, University of Wyoming chemistry professor Emory Edmund Slosson announced that he wanted to see how rapidly a certain odor diffused through the air and asked students to raise their hands as soon as they could smell it. Then he poured distilled water over cotton, explaining that it was a chemical with a distinctly unusual odor. Within 15 seconds most of the students in the front row had raised their hands, and within a minute three quarters of the class had.

Similarly, University of California food scientist Michael O’Mahoney told a British TV and radio audience in 1978 that a certain sound frequency would induce the perception of an outdoorsy smell. This suggestion and some audiovisual prompts led hundreds of people to write in claiming they’d smelled something when the frequency was broadcast.

In The Scent of Desire (2009), Brown University psychologist Rachel Herz writes, “This media stunt shows how simply being told that an odor is ‘there’ can convince you that you are smelling something with all of its full-blown consequences.”

12/13/2023 UPDATE: It seems likely that Slosson is Edwin Emery Slosson — possibly Herz mistook his name. (Thanks, Charlotte.)

Higher Things

Winston Churchill published a surprising essay in March 1942: “Are There Men on the Moon?”:

I am not sufficiently conceited to think that my sun is the only one with a family of planets and, therefore, that our little earth is unique. Once we admit that the other stars probably also have planets, at any rate a goodly proportion of them, it is more than likely that a large fraction of these will be the right size to keep on their surface water and, possibly, an atmosphere of some sort; and, furthermore, at the proper distance from their parent sun, to maintain a suitable temperature. Do they house living creatures, or even plants? The answer to this question may never be known.

“[T]he odds are enormous that there must be immense numbers which possess planets whose circumstances would not render life impossible,” he concluded. “I, for one, am not so immensely impressed by the success we are making of our civilization here that I am prepared to think we are the only spot in this immense universe which contains living, thinking creatures, or that we are the highest type of mental and physical development which has ever appeared in the vast compass of space and time.”

(Winston Churchill, “Are There Men on the Moon?”, Sunday Dispatch, March 8, 1942.)

Man of the Hour

Abbreviating litre with a lowercase L can be confusing, as the character can be mistaken for the digit 1. But usually the International System of Units permits a capital letter only when a unit is named after a person.

So, in 1978, University of Waterloo chemist Kenneth Woolner announced in a schoolteachers’ newsletter that the litre had been named for Claude Émile Jean-Baptiste Litre, a fictional French scientist who had proposed a unit of volume measurement before his death in 1778.

Woolner had intended the claim only as an April Fools’ hoax, but the point was made. Today the U.S. National Institute of Standards and Technology recommends abbreviating litre with an uppercase L.

The Machine
Image: Wikimedia Commons

The first life-size obstetrical mannequin was invented by French midwife Angélique du Coudray, who was using it to demonstrate the process of childbirth as early as 1756:

I announced that I would gladly give my advice to poor women who needed it. … I took the tack of making my lessons palpable by having them maneuver in front of me on a machine I constructed for this purpose, and which represented the pelvis of a woman, the womb, its opening, its ligaments, the conduit called the vagina, the bladder and rectum intestine.

The upholstered model included a womb and an extractable baby doll with which her students could practice. The skin and soft organs were made of flesh-colored linen and leather stuffed with padding, and some of the bones were assembled from real skeletons, though wood and wicker later took their place.

“The model is meant mostly for maneuvers that, as others confirm, allow her students to gain confidence, be ‘encouraged, and succeed perfectly,'” writes Nina Rattner Gelbart in The King’s Midwife (1998). “Delivering babies from every conceivable position and presentation will prepare her students for all eventualities. … This machine, as the midwife’s followers will continue to testify, makes an ‘impression that can never be erased,’ ‘an advantage all the more essential because this class of surgeons and these women [of the countryside] do not have the resource of reading … [so] these daily continual maneuvers … [must be] vividly impressed on their senses.'”

Brown Study

In 2016, after three months of study, a team of academics and market researchers determined that the most unattractive color in the world is this one, Pantone 448 C.

The project was seeking to design the most unappealing package possible for Australian cigarette packets. “We didn’t want to create attractive, aspirational packaging designed to win customers,” market researcher Victoria Parr told the Sydney Morning Herald. “Instead our role was to help our client reduce demand, with the ultimate aim to minimize use of the product.”

A thousand smokers decided that “drab dark brown” packages would have the poorest appeal, promising low-quality cigarettes that caused maximum harm. They associated the winning color with dirt, tar, and death, and assigned no positive adjectives.

In announcing the results, the Australian Department of Health referred to the color as “olive green” — until the olive industry objected.