Heron's tricky result can be reproduced on considering the familiar 2D right angled triangle. If h is the height, c the hypotenuse and a the base, then we know by Pythagoras that h = sqrt((c^2) - (a^2)). So if c is 5 and a is 4, then h is sqrt(25 - 16), which is 3. So far, so familiar.

But what if c is still 5, but a is longer, say 7? Then h is sqrt(25 - 49) = the square root of -14.

What would ancient geometers have made of that? Possibly much the same as the other commentator (see box below), namely that:

while c can be as long as one wishes, it cannot be smaller than the minimum needed to connect the other two sides of the triangle (for the case they're the same line, the height is zero).

If the base is longer than the hypotenuse then the latter can never connect up with the side which is perpendicular to the base, ie the height. Not only does such a "hypotenuse" no longer deserve that term, but more fundamentally it could have been considered in violation of Euclid's first two axioms:

1) To draw a straight line from any point to any point
2) To produce [extend] a finite straight line continuously in a straight line
(as worded in Wikipedia).

If c isn't long enough at present, then we've just got to extend it till it is, even if it means changing the given side ratios. Maybe a result incorporating the square root of 14 or of 63 (even if without a minus sign) would have been interpreted as some kind of clue to how much correction the error called for.

Chris G

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