Out of the odd words we use in mathematics, I have found “hypotenuse” one of the most curious. Various etymology references mention it comes from the words

hypo- (“under”, as in **hypo**dermic needle)

and

teinein (“to stretch”, as in **ten**don).

But why “under” exactly? While one can place the right triangle so the hypotenuse is “under” the other two sides, it often is oriented like so:

It turns out that the word comes from Euclid, Book I, Proposition 47.

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

In Greek the relevant bit is:

ἡ την ορθην γωνιαν ὑποτεινουσα πλευρα

hê tên orthên gônian hypoteinousa pleura (transliterated)

the line subtending the right angle (translated)

Here’s a close-up of the diagram that goes with the proposition:

Someone later took the sentence (which made perfect sense in context with the diagram, where the hypotenuse is flipped to be on the bottom) and took one word (subtended = hypoteinousa = hypotenuse) to stand for the longest side of a right triangle.

That someone (according to the Oxford English Dictionary) was Thomas Digges in the book *A geometrical practise named Pantometria* (which according to Thomas he wrote based on notes from his father Leonard Digges). Both were mathematicians and surveyors; Leonard is a contender to the title of “inventor of the telescope” (!).

Here’s a diagram from the work (source):

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Filed under: History, Mathematics

midday, on August 7, 2014 at 2:31 pm said:original diagram appreciated!