“The Pythagorean Curiosity”

waterhouse pythagorean curiosity

Here’s the item I mentioned in Episode 99 of the podcast — New York City engineer John Waterhouse published it in July 1899. It’s not a proof of the Pythagorean theorem, as I’d thought, but rather a related curiosity. It made a splash at the time — the Proceedings of the American Society of Civil Engineers said it “interested instructors of geometry all over the country, bringing many letters of commendation to him from prominent teachers.” Listener Colin Beveridge has been immensely helpful in devising the diagram above and making sense of Waterhouse’s proof as it appears on page 252 of Elisha Scott Loomis’ 1940 book The Pythagorean Proposition. Click the diagram to enlarge it a bit further.

  1. Red squares BN = AI + CE — Pythagoras’s theorem
  2. Blue triangles AEH, CDN, BMI are all equal in area to ABC, reasoning via X and Y and base sides.
  3. Green angles GHI and IBM are equal and green triangle GHI is congruent to IBM (side angle side), so IG = IK = IM. IH′K is congruent to IHK as angle HIK = angle HIG and the adjacent sides correspond. This means G and K are the same distance from the line HH′, so GK is parallel to HI. Similarly, DE is parallel to PF and MN is parallel to LO.
  4. GK = 4HI, because TU=HI, TG = AH (HTG congruent to EAH) and UK = UG (symmetry). Similarly, PF = 4DE. Dark blue triangles IVK and LWM are equal, so WM = VK. Similarly, OX = QD (dark green triangles PQD and NXO are congruent). Also, WX=MJ and XN=NJ, so M and N are the midpoints of WJ and XJ. That makes WX=2MN, so LO = 4MN.
  5. Each of the trapezia we just looked at (HIKG, OLMN and PFED) have five times the area of ABC.
  6. The areas of orange squares MK and NP are together five times EG. This is because:
    • the square on MI is (the square on MY) + (the square on IY) = (AC2) + (2AB)2 = 4AB2 + AC2.
    • the square on ND is (the square on NZ) + (the square on DZ) = (AB2) + (2AC)2 = 4AC2 + AB2
    • the sum of these is 5(AB2 + AC2) = 5BC2, and BC = HE.
  7. A′S = A′T, so A′SAT is a square and the bisector of angle B′A′C′ passes through A. However, the bisectors of angle A′B′C′ and A′C′B′ do not pass through B and C (resp.) [Colin says Waterhouse’s reasoning for this is not immediately clear.]
  8. Square LO = square GK + square FP, as LO = 4AC, GK = 4AB and FP = 4BC.
  9. [We’re not quite sure what Waterhouse means by “etc. etc.” — perhaps that one could continue to build squares and triangles outward forever.]


Boonville, Calif., has a dwindling language all its own. “Boontling” grew up as a jargon among residents of Anderson Valley around the turn of the 20th century. It includes more than a thousand words and phrases but is dying out among the small population. A brief glossary:

applehead – a young girl
belhoon – a dollar
Bill Nunn – syrup
boshin’ – deer hunting
bucky walter – a pay telephone
can-kicky – angry
dicking – cheating at cards
forbes – a half dollar
glow worm – a lantern
greeley – a newspaper or reporter
harpin’ tidrick – a lengthy discussion
high pockets – a wealthy person
killing snake – working very hard
pearlin – light rain
skee – whiskey
tobe – tobacco
walter – a telephone
zeese – coffee

“A few of us try to keep our skills sharp on the teleef [telephone],” resident Bobby Glover told the San Francisco Chronicle in 2001. “We’re adding new words now that the old-timers are gone.”

Thanks to the efforts of a number of researchers, the jargon has been pretty well documented now — the Chronicle even managed to translate “Old Mother Hubbard”:

The old dame piked for the chigrel nook for gorms for her bahl belljeemer
The gorms had shied, the nook was strung, and the bahl belljeemer had neemer.

Bedtime Stories


In his 1948 book The Lost Art of Profanity, Burges Johnson quotes from an opinion by a Judge Hammond of the Supreme Judicial Court of Massachusetts:

The Watch and Ward Society of Boston years ago brought charges against a certain magazine for printing obscene matter, and my old friend the late Kendall Banning was forced to defend the publication. He felt sure that he could make a case, and during the train ride to Boston he had a sudden idea, and began jotting down such nursery rhymes as he could recall. Then he crossed out significant words and substituted asterisks. In court he asked permission to read these rhymes. They later appeared in a privately printed brochure which aroused delight or horror, according to the state of the reader’s mind. A mere sampling will serve here:

A dillar, a dollar
A ten-o’clock scholar,
What makes you *** so soon?
You used to *** at ten o’clock,
But now you *** at noon.

Jack and Jill went up the hill
To *******
Jack fell down and broke his ***
And Jill came tumbling after.

Johnson says that the courtroom broke out in laughter and that Banning had made his point.

04/15/2016 UPDATE: Related (thanks, Mate):

In a Word

n. an insulted or offended person

In Gelett Burgess’ 1911 novel Find the Woman, a truck driver blocks the way of a parade organized by a society to ban profanity. He is addressed by Dr. Hopbottom, the society’s head:

See here, you slack-salted transubstantiated interdigital germarium, you rantipole sacrosciatic rock-barnacle you, if you give me any of your caprantipolene paragastrular megalopteric jacitation, I’ll make a lamellibranchiate gymnomixine parabolic lepidopteroid out of you! What diacritical right has a binominal oxypendactile advoutrous holoblastic rhizopod like you got with your trinoctial ustilaginous Westphalian holocaust blocking up the teleostean way for, anyway! If you give me any more of your lunarian, snortomaniac hyperbolic pylorectomy, I’ll skive you into a megalopteric diatomeriferous auxospore! You queasy Zoroastrian son of a helicopteric hypotrachelium, you, shut your logarithmic epicycloidal mouth! You let this monopolitan macrocosmic helciform procession go by and wait right here in the anagological street. And no more of your hedonistic primordial supervirescence, you rectangular quillet-eating, vice-presidential amoeboid, either!

The truck driver apologizes: “I see a plain, sea-faring man has no show with a doctor when it comes to exhibiting language in public. … If this here society what’s running this here procession can turn out graduates of the noble art of profanity like you are, I want to say this: Give me the pledge, and I’ll sign it.”

Corner Reflectors

Image: Wikimedia Commons

An arrangement of three mutually perpendicular planes, like those in the corner of a cube, have a pleasing property: They’ll reflect a ray of light back in the direction that it came from. This happy fact is exploited in a variety of technologies, from laser resonators to radar reflectors; the taillights on cars and bicycles contain arrays of tiny corner reflectors.

“A more dramatic application is to reflect laser rays from the Moon, where many such devices have been in place since the 1969 Apollo mission, which sent men to the Moon for the first time,” note mathematicians Juan A. Acebrón and Renato Spigler. “Among other things, the Earth-Moon distance can be measured by firing a laser beam from the Earth to the Moon, and measuring the travel time it takes for the beam to reflect back. This has allowed an estimate of the distance to within an accuracy of 3 cm.”

(Juan A. Acebrón and Renato Spigler, “The Magic Mirror Property of the Cube Corner,” Mathematics Magazine 78:4 [October 2005], 308-311.)

Notable Nightmares


Half sleepless night again — an entirely disgusting dream, about men using flesh and bones, hands of children especially, for fuel — being out of wood and coals. I took a piece to put on someones fire, and found it the side of an animals face, with the jaw and teeth in it.

— John Ruskin, Brantwood Diary, Oct. 29, 1877

I had a dream last night. An amputated head had been stuck on to a man’s trunk, making him look like a drunken actor. The head began to talk. I was terrified and knocked over my folding screen in trying to push a Russian in front of me against the furious creature’s onslaught.

— August Strindberg, Inferno, 1897

The nightmares returned — one terrible one in February 1896 about a tramp, seen holding over a well ‘washing, but with a kind of amused tenderness, an object that I thought was a rabbit, but I presently saw that it was a small deformed hairy child, with a curious lower jaw, very shallow: over the face it had a kind of horny carapace … made of some material resembling pottery. I was disgusted at this but went on, and it grew dark: I heard behind me an odd sound, and turning round saw this horrible creature only a foot or two high, walking complacently after me, with its limbs involved in ugly and shapeless clothes, made, it seemed to me, of oakum, or some more distressing material. The horror of it exceeded all belief.’

— A.C. Benson, quoted in David Newsome, On the Edge of Paradise, 1980

Spring Showers


This is clever — in 1893 Texas inventor Martin Everhart patented a clock-winding mechanism that’s driven by rainwater. The water fills a tank in the attic and then drops through a pipe into a pail in the clock. The pail is balanced with a counterweight, so it falls and rises continuously, accepting a new measure of water at the top and discharging it at the bottom. This motion winds the clock.

I guess the whole thing would stop eventually in a drought, but the clock can be wound by hand if necessary.