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Topic: Trapezoid definition
Replies: 26   Last Post: Oct 7, 2004 11:51 PM

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John Conway

Posts: 2,238
Registered: 12/3/04
Re: The trapezoid-triangle-line rule?
Posted: Aug 10, 2000 11:36 AM
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On Thu, 10 Aug 2000, Floor van Lamoen wrote:

> 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.

John Conway





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