Michael Woltermann Mathematics Department Washington and Jefferson College Washington PA 153014801 Triumph der Mathematik 100 Great Problems of Elementary Mathematics By Heinrich D ID: 426733
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Slide1
What is it? Is it interesting?
Michael Woltermann Mathematics Department Washington and Jefferson College Washington, PA 15301-4801Slide2
Triumph der Mathematik
100 Great Problems of Elementary MathematicsBy Heinrich Dö
rrieSlide3
Some Background
Heinrich DörriePh. D. Georg-August-Universität Göttingen 1898
Dissertation
Das quadratische Reziprozitätsgesetz im quadratischen Zahlkörper mit der Klassenzahl 1.
Advisor David Hilbert
Triumph der MathematikGerman editions 1932, 1940
Dover (English) edition 1965http://www.washjeff.edu/users/MWoltermann/Dorrie/DorrieContents.htmSlide4
From the Preface
For a long time, I (H. Dörrie) have considered it a necessary and appealing task to write a book of celebrated problems of elementary mathematics, their origins, and above all brief, clear and understandable solutions to them. … The present work contains many pearls of mathematics from Gauss, Euler, Steiner and others.
So then, let this book do its part to awaken and spread interest and pleasure in mathematical thought.Slide5
From a Review at Amazon.com
The selection of problems is outstanding and lives up to the book's original title. The proofs are concise, clever, elegant, often extremely difficult and not particularly enlightening. To say that this book requires a background in college math is like saying that playing chess requires a background in how to move the pieces; it also requires a lot of thought and, preferably, a lot of experience.Slide6
From M.W. (spring 2010)
A lot of things have changed since 1965. For example, terminology has changed, people are not as knowledgeable about some areas of mathematics (especially geometry) as they once were, but more knowledgeable about others (e.g. calculus).
A straightforward translation would not necessarily shed more light on the problems in this book. What was required was in some cases more (or less) mathematical background, current terminology and notation to bring
Triumph der Mathematik
into the twenty first century.Slide7
55. The Curvature of Conic Sections
Determine the curvature of a conic section.
Let the conic section be c
e its eccentricity,
p half the
latus
rectum, q = 1-e2.
An equation for c is
qx
2
+y
2
-2px = 0.
Let n be the length of the normal from a point P on the conic to its axis. Then
The radius of the circle of curvature is
Slide8
ParabolaSlide9
EllipseSlide10
HyperbolaSlide11Slide12
Is it interesting?
Conics by Keith Kendig, MAA 2005Goal is to see conics in a unified way.But n cubed over p squared doesn’t
appear
New Geometric Constructions to Determine the Radius of Curvature of Conics at any Point
by
Jiménez
and Granero, 2007Their approach is based on a “recently found property of conic sections”.
They cite a 1999 article in
Computer Aided Geometric Design,
No. 16.
It’s fun to implement with
Geometer’s Sketchpad
,
Geogebra
, etc. Slide13
The Calculus Approach
The curvature at (x,y) isWith y2=2px-qx
2
,
Slide14
Any