## On the Word “Hypotenuse”

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 hypodermic needle)
and
teinein (“to stretch”, as in tendon).

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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### One Response

1. original diagram appreciated!