> Jon Marshall wrote: > > > > Beating a dead horse, not even the accepted inclusive definition of > > trapezoid saves us from degenerate cases in the integration rule. If > > one or more of the endpoints of a subinterval of the partition is a > > zero of the function, you get a triangle or a horizontal line for the > > representative area. The benefits of an inclusive definition > > generally seem to be for mathematical or organizational elegance, > > whereas an exclusive definition can afford quick, concise > > communication. Both are worthy goals in their own situations. > > And for particularly quick, concise communication one can use words like > nondegenerate, nonrectangular, etc., if really needed. After all > degenerate cases are often nonproblematic. > > Floor van Lamoen. > I have a feeling, which I think I'll elevate into a conjecture, that in fact the exclusive definitions traditionally given for geometrical figures have never really been used. I'd welcome some assistance in proving or disproving this.
Consider for instance, the old theorem that the midpoints of the edges of a quadrilateral are the vertices of a parallelogram. Has anyone ever seen this stated in print in the form "... are the vertices of a parallelogram, rhombus, rectangle or square"? I must have seen this theorem in a hundred places, many of which will have given the exclusive definitions, but I'm sure that if I'd ever seen it stated like this, I'd still remember it.
No - there's been a tacit understanding that the terms are used inclusively in theorems, and exclusively for classification and description; just as in practice nobody calls a square table rectangular, but nevertheless everyone understands that statements about all rectangular tables are intended to include square ones.
Let me put it like this - a square is indeed a rectangle, but it isn't called one, because it's understood that objects are always assigned to the highest place of the classification to which they belong.
I hope the historians can provide illuminating evidence for or against this hypothesis, in the form of quotations from early geometers giving not merely the definitions they gave, but also examples showing how they actually used the corresponding terms. My guess is that when lip-service has been paid to the exclusivity of the definitions, it's usually been in a parenthetical way, as perhaps in
"a quadrilateral whose diagonals are equal and bisect each other is a rectangle (or perhaps a square)".
I don't mean that actual parentheses were always present, but that the authors used language showing that they really thought of "rectangular" (say) as including "square", no matter what the form of the definition they'd given.