"Strong typing vs. strong testing" [OT]
steve at REMOVE-THIS-cybersource.com.au
Wed Oct 13 21:06:02 CEST 2010
On Wed, 13 Oct 2010 15:07:07 +0100, Tim Bradshaw wrote:
> On 2010-10-13 14:20:30 +0100, Steven D'Aprano said:
>> ncorrect -- it's not necessarily so that the ratio of the circumference
>> to the radius of a circle is always the same number. It could have
>> turned out that different circles had different ratios.
> But pi is much more basic than that, I think.
Well yes it is, but how did anyone *know* that it was? How did anyone
even know that there was a constant pi = 3.1415... ? It's not like it was
inscribed on the side of some mountain in letters of fire 100 ft high,
and even if it were, why should we believe it?
The context of my comment was the statement that there is no need to
prove that C = 2πr because that's the definition of pi. That may be how
pi was first defined, but the Greeks didn't just *decide* that the ratio
C/r was a constant, they discovered it. They constructed a pair of
regular polygons with n sides, the circle inscribing one polygon and in
turn being inscribed by the second, and observed that as n approached
infinity two things happened: the inner and outer polygons both became
infinitesimally close to the circle, and the ratio of the perimeter of
either polygon to twice the radius approached the same constant.
By modern standards it wasn't *quite* vigorous -- the Greeks hadn't
invented calculus and limits, and so had to do things the hard way -- but
nevertheless it was an inspired proof. I call it a proof rather than a
definition because, prior to this, nobody knew that there was such a
number as pi, let alone what it's value was.
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