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.