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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



----- Original Message -----
From: "Daryl L. Taylor" <Daryl@DARYLSCIENCE.COM>
To: <PHYS-L@lists.nau.edu>
Sent: Tuesday, November 04, 2003 12:14 PM
Subject: Re: Numbers


Ah, but as his own writings show, he did, indeed, use the method of
exhaustion to calculate the area of a circle. He stopped his calculations
of
a 'unit' circle, d=1, with a 96-sided polygon both inscribed (lower limit)
and circumscribed (upper limit). He was actually calculating the area of
that circle. It just happens to be PI. If this isn't the method of
exhaustion, I don't know what is. (Unless it's teaching my 9th period
class...)

He then used the same methods for volumes. I just can't see this as saying
it's calculus. It's one of the foundations of it, but...

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 11:49 AM
To: PHYS-L@lists.nau.edu
Subject: Re: Numbers


By calculus, I mean his method of using the sum of an infinite number of
objects to calculate a volume, not the method of exhaustion used to
calculate pi.

John

----- Original Message -----
From: "Daryl L. Taylor" <Daryl@DARYLSCIENCE.COM>
To: <PHYS-L@lists.nau.edu>
Sent: Tuesday, November 04, 2003 10:51 AM
Subject: Re: Numbers


I have to argue a little with John's statement below... Archimedes
didn't
use Calculus. With all due respect to John and PBS... He used a
calculation
method called "Method of Exhaustion" common to the 'ancient' Greeks. I
point
you to a simple explanation at
http://www.ima.umn.edu/~arnold/graphics.html.
Even Reviel Netz's Nov 01 2002 Science mag article does not imply that
Archimedes 'used' calculus. The truly astonishing thing is that
Archimedes
knew of and attempted to use the concept of infinity. Conventional
wisdom
states the Greeks shunned infinity and never dealt with it. Apparently
Eureka-Man did.

However, to take the leap to the statement that "Archimedes used
Calculus"
is akin to saying that Newton used Quantum Mechanics. Each, Archimedes
and
Newton, used methods, created conclusions, and laid important
foundations
that were needed for future developments, yes. However, as Newton was
fond
of saying, "...it is because I have stood on the shoulders of giants."
(Or
was that David?...)

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!