the square __ the hypotenuse

One of the choirs I sing in started practicing the chorus of the Major-General’s song from Gilbert and Sullivan’s The Pirates of Penzance, very slowly, starting with 

with many cheerful facts about the square of the hypotenuse.

Hang on, shouldn’t that be on the hypotenuse?  At least, that’s what I’ve always thought it was.

Apparently not. The two books on Gilbert and Sullivan’s operas I have both give of, and the video I linked above has it. But, in general, of and on seem to be interchangeable, with a recent preference for of

There doesn’t seem to be an original Greek form of the theorem, whether formulated by Pythagoras or someone else. If there is a difference, it’s that the square on the hypotenuse is an actual square on an actual side of an actual triangle, and the square of the hypotenuse is a mathematical function of the length of that side. To the ancient Greeks, γεωμετρία (geometria) was literally about measuring the earth.

If you are a singer, use what you conductor provides or tells you. If you are a maths teacher, use what’s in your textbook. If you are anyone else, choose one and don’t worry about it.


10 thoughts on “the square __ the hypotenuse

  1. Always saw/heard/learned it with “of”, as in “the square of the hypotenuse is equal to the sum of the squares of the opposite sides.” And it was conceived in algebraic/arithmetic terms, “square” being the result of multiplying a term by itself rather than naming the geometric figure, and the terms being lengths of the respective sides of the triangle. The Wikipedia definition is more verbose in its reference to the area of the geometric square, but at heart it is a simple c**2 = a**2 + b**2 equation.

    A geometric representation may be used in classes to illustrate the theorem to make the derivation clearer.

    Liked by 1 person

  2. I have a random memory from childhood of my father telling a joke about an Indigenous American man who had three wives who sat on the hides of deer, antelope and hippopotamus. In time each gave birth, the first two to one child each and the third to twins, thereby showing …
    I didn’t understand it at the time, and I’m not sure why I remember him telling it. I have encountered it since. As with the actual theorem, versions of the joke use ‘on’ on ‘of’. Some versions have the three wives in tipis, so it must be ‘of’. (‘in’ just wouldn’t work.)
    (In any case, it’s necessary to ignore the fact that hippopotamuses/i are not native to America.)


  3. I assume you didn’t post the punchline because it presumably contains a term for indigenous Americans that is now considered offensive and is the target of several proposed place renamings in the US.

    Liked by 1 person

    • Partly because I didn’t think it was necessary to labour the punchline and partly because I am aware of the controversy over the word. I was pondering whether to use or mention it. The problem with renaming is what do you do to historical references, eg the 1960 Winter Olympics?
      The word for ‘woman/wife’ in the Indigenous language of the Sydney area followed a similar path. First recorded as diyin, dyin or djin (possibly a retroflex consonant), it became ‘gin’, but remained very informal/slang/offensive even at the best of times, and no places were named using it. It was rare and I can’t remember the last time I heard it, if only because I don’t associate with the kinds of people who would be likely to use it.


  4. (I had to Google the 1980 Winter Olympics to get the reference.)

    Place names changing over time is an issue with greater impact than just political correctness. For example, St. Petersburg changed to Leningrad and then back to St. Petersburg, but Shostakovich’s 7th Symphony is still called the “Leningrad” symphony because of its association with that time and the circumstances thereof. But if the name of the site of the Olympic event changes, the references to it will probably change right along with it.

    “gin” would be homophonous with the Greek combining form “gyn-“, which signifies womanhood or femaleness; if I had heard someone use the term I would have assumed that was the origin. It also sounds similar to “djinn” (genie), FWIW.

    Liked by 1 person

    • There is an old story about a man who was born in St Petersburg, went to school in Petrograd, got married and lived in Leningrad and died in St Petersburg – and lived in one apartment in his life!


  5. Speaking of “gyn-“, I had a Greek language course in college where the instructor mentioned the word “gynecologist” and pronounced it with a soft “g”, saying, “Most gynecologists mispronounce their profession.” I was gratified to hear that because I had always felt that the “g” in “gynecologist” should be soft, not hard, to be consistent with all other English words with Greek origins (through Latin).

    Have you observed Koreans pronouncing words like “ginseng” and “gingko” with a soft initial g as well? And that seems wrong to me. I presume Korean speakers of English dogmatically follow the g-softening rule regardless of whether it is truly applicable or not, and it should apply only to words with Latin or Greek origin,


  6. I think the reverse; terms that came into English from (classical) Greek at a later time may have tended to use the hard sound. It probably has little to do with actual (modern) Greek pronunciation, which has some interesting quirks of its own.

    This could also explain why words like “kleptomania” are spelled with a k instead of a c.

    Liked by 1 person

    • I will stand by my statement that the pronunciation of vowels in most languages tends to drift from hard to soft. Obviously the pronunciation of loanwords is messy, depending on the pronunciation in the source language through history to the time of the borrowing, the circumstances of the borrowing and the comparative phonology of the two languages.


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