"Strong typing vs. strong testing" [OT]

[Arnaud Delobelle]

Steven D'Aprano <steve-REMOVE-THIS at cybersource.com.au> writes:

> On Wed, 13 Oct 2010 21:52:54 +0100, Arnaud Delobelle wrote:
>> 
>> Given two circles with radii r1 and r2, circumferences C1 and C2, one is
>> obviously the scaled-up version of the other, therefore the ratio of
>> their circumferences is equal to the ratio of their radii:
>
> That's exactly the sort of thing Peter Nilsson was talking about when he 
> said "Most attempts by students collapse because they assume the formula 
> in advance". It might be "obvious" to you that the two circles are merely 
> scaled up versions of each other, but that is equivalent to assuming that 
> the ratio of the circumference to radius is a constant. Well, yes, it is 
> (at least under Euclidean geometry), but assuming it is a constant 
> doesn't allow you to prove it is a constant -- that's circular reasoning, 
> if you excuse the pun.

There is no circular reasoning.  Read on to find out why.

A circle is, by definition, the locus of points equidistant from a given
point (called its centre), and this constant distance is what we call
its radius.

Let's have two circles with the same centre and radii r1 and r2.  Let's
scale up (from the centre) the first one by a factor r2/r1.  Because all
the points the first circle are r1 units of length away from the centre,
all the points on the scaled up version are r1*r2/r1 = r2 units of
length from the centre.  So the scaled up version of the first circle
*is* the second circle.

I'll let you solve the case when the centres are distinct.

-- 
Arnaud