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Re: Numbers



I think what we have here is an argument over semantics. To me, you have
supported my view that Arch was NOT using Calculus, but laying the framework
for it. In your own provided quotes, "In fact his technique is similar to
the concept used in modern calculus for the same kind of problem..." and
"..in a sense, Archimedes was already there."

Note. His technique is "similar" to a concept in modern Calc. It doesn't say
it is, it's similar.
Note. "...in a sense,..." means just that. Arch was using much more advanced
mathematical operations as had been previously agreed. However, where in
Netz' writings is it a direct, or even close to a direct, correlation to
actual Calculus. I still hold that Arch deserves alot of credit, but getting
credit for using calculus for his "similar" techniques and "in a sense"
succeeding? I think not.

Again, this type of concluding is 'similar' to saying that Galileo used
Newton's Law of Universal Gravitation and Big Al's Special Relativity. He
did lay the groundwork for both, but did he actually use them or deserve
credit for them?

sigh... Let's agree to disagree on these conclusions and share a beer. I'll
go grab one now...

Daryl L. Taylor, Fizzix Guy
PAEMST '96
Internet Educator of the Year '03
Williamstown HS & Engineering Academy, Rowan University NJ
www.DarylScience.com <http://www.DarylScience.com>
609.330.9571

This email prepared and transmitted using 100% recycled electrons!



-----Original Message-----
From: Forum for Physics Educators [mailto:PHYS-L@lists.nau.edu]On Behalf
Of John Cockman
Sent: Tuesday, November 04, 2003 3:44 PM
To: PHYS-L@lists.nau.edu
Subject: Re: Numbers


My reference is from a BBC interview with a few transcribers of the
Palimpsest:

http://www.bbc.co.uk/science/horizon/2001/archimedestrans.shtml

"...Archimedes was trying to work out the volume of an unusual shape by
dividing it into an infinite number of slices. Archimedes had drawn a
diagram of a triangular prism. Inside this he drew a circular wedge. This
was the volume that he wanted to calculate. He then drew a second curve
inside the wedge. Modern mathematicians already understood that Archimedes
had used some very complex ideas to work out that a slice through the wedge
equals a slice through the curve times a slice through the prism divided by
a slice through the rectangle. But what no-one knew was how Archimedes had
added up an infinite number of these slices to work out the volume of the
wedge. The frustration was that the lines explaining how he had done this
appeared in Heiberg's translation merely as a row of dots. These vital lines
were missing, but then, with the help of the very latest images of the
palimpsest, Reviel Netz went back to study the manuscript again...

Reviel realised that Archimedes had come up with a set of rules for dealing
with infinity. He'd worked out a system for calculating the value of each
slice and then adding up an infinite number of them... What was clear was
that Archimedes had made a huge step towards the understanding of
infinity...

The new finding in the wedge theorem reveals not only that Archimedes was
confident in dealing with infinity, but also that his use of infinite slices
to calculate a volume was far more sophisticated than anyone had realised.
In fact his technique is similar to the concept used in modern calculus for
the same kind of problem...

We always knew that Archimedes was making a step in the direction leading to
modern calculus. What we have found right now is that, in a sense,
Archimedes was already there. He already did develop a special tool with
which you can sum up infinitely many objects in measure of volume."


This seems to be beyond the method of exhaustion which places a circle
between two polygons.

John