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 NCB Deposit  # 17

Tom Richmond
Western Kentucky University
1 Big Red Way
Bowling Green, KY 42101

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Animations of Two Classics:

Derivation of the Formula for the Area of a Circle
and the
Pythagorean Theorem

From ( I, 47) of the Elements.
Euclid's (I, 47) "Elements"

Area of Circle animation

Pythagorean Theorem Animation

Ratdolt Edition of "Elements"

From the first page of the
first printed edition of Euclid's
Elements, Venice,1482.  Scholars
name this the Ratdolt edition in
honor of the printer and publisher.
It is not known how the figures
were printed in the text.

Before leaving the above images, we invite the viewer to consider the following:  In the upper right hand corner is the ( I, 47 ) proof of the Pythagorean Theorem from one of the world's oldest hand written copies of Euclid's Elements.  To the above  right are figures taken from the first printed edition.  Now, on the left, we see the contribution of animation, the gift of our generation, to these famous mathematical concepts.

The circle, ellipse, parabola and hyperbola are "sections" of  a cone.
Conic Sections Illustration

Recall the study of cones dates to Apollnius (262-190 BC) and other early Greeks.

Hyperbola animation

Links for the Circle

For a sampler of Richmond's other work, see
<  >

For a Mathematica® reference, see
<  >

For a Maple® source, open
<  >
and type "circle" or "conic" in the search box.

Links for the Pythagorean Theorem

For an interactive proof, see  >

This Java applet was written by Jim Morey.  It won the grand prize in Sun Microsystem's Java programming contest in 1995. 

Printed References for the Circle.

Beckmann, Petr,  A History of. St. Martin's Press, 1971, p. 18.

Jennings, George A., Modern Geometry with Applications, Springer-Verlag,  1994, p. 20.

Printed References for the Pythagorean Theorem

Swetz, Frank J. and T. I. Kao,  Was Pythagoras Chinese?  NCTM, Reston, VA, 1977.

Van der Waerden, B. L.,  Geometry and Algebra in Ancient Civilizations,  Springer, 1983.

Sierpinski, Waclaw,  Pythagorean Triangles, Scripta Mathematica Studies Number Nine. (This is a translation by A. Sharma of a work by the well known mathematician. Yeshiva University, New York, 1962.

Tom and Bettina Richmond have collaborated on several projects.  He is a topologist and she is an algebraist.  See the following publications:
Tom Richmond and Aaron Young, Instant Insanity II, College Mathematics Journal, Vol. 44, no. 4 (Sept. 2013) 265-272.
The Equal Area Zones Property,  American Mathematical Monthly, Vol.100, No. 5  (May 1993) 475-477.
Metric Spaces in Which All Triangles are Degenerate,  American Mathematical Monthly, Vol.104, no. 8 (Oct. 1997) 713-719.
Characterizing Power Functions by Volumes of Revolution,  College Mathematics Journal, Vol. 29, no. 1  (Jan. 1998)  40-41.
A Discrete Transition to Advanced Mathematics (textbook), Brooks/Cole Series in Advanced Mathematics,  Summer, 2003.

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