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American Mathematical Monthly - December 2002

December 2002


Mina Rees and the Funding of the Mathematical Sciences
by Amy E. Shell-Gellasch
aa7423@usma.edu

Federal support for research in the mathematical sciences in the United States was essentially developed during the Second World War and the years immediately following. Through her work on the Applied Mathematics Panel during the war, mathematician Mina Rees (1902-1997) was instrumental in facilitating scientific research for the war effort. As Head of the Mathematical Sciences Division and Deputy Science Director of the Office of Naval Research following the war, she helped revolutionize the form and scope of federal support of scientific research, and guided much of the development of early computers. She was dedicated to creating an environment in the United States in which the sciences, and mathematics in particular, grew and flourished.

 

Minimal Perimeter Triangles
by Mark Levi
levi@math.psu.edu

Let K be an arbitrary closed convex curve in the plane. Of all the triangles circumscribed around K, let ABC be the one of least perimeter, and let A' be the point of tangency of the side BC with K. Similarly we define the tangency points B' and C'd . Then the lines AA' , BB' , and CC' are concurrent (i.e., they meet at a point). Moreover, the perpendiculars to the sides of the triangle at the points A' , B', and C' are concurrent as well. This latter property is much more general: it holds for triangles minimizing “most” other functions, such as the area, the sum of the squares of the sides, etc. I had first discovered these facts by an argument based on physics. This physical argument (along with a rigorous proof) is given in the article.

 

Tangents and Subtangents Used to Calculate Areas
by Tom M. Apostol and Mamikon A. Mnatsakanian
apostol@caltech.edu, mamikon@caltech.edu

A recent paper by the authors (Amer. Math. Monthly, June/July 2002) showed that subtangents can be used in practice to draw tangent lines, and it revealed that subtangents provide an unexpected connection between the tractrix and exponential curves. That paper also contains a rigorous justification of a geometric method of tangent sweeps developed by Mamikon Mnatsakanian that can be used to calculate area of many classical plane regions. This paper gives an alternate treatment for power functions, again using tangent sweeps and properties of subtangents. It finds (without calculus) the area of a parabolic segment, a hyperbolic segment, and all generalized ssegments arising from arbitrary real power functions. It also relates subtangents and the mean-value theorem for integrals.

 

Problems and Solutions
 

Notes

Equilateral Triangles and Triangles
by Richard P. Jerrard and John E. Wetzel
jerrard@math.uiuc.edu, j-wetzel@uiuc.
 

An Easy Proof of Hurwitz’s Theorem
by Manuel Benito and J. Javier Escribano
mbenit8@palmera.pntic.mec.es, jesbriba@boj.pntic.mec.es
 

A New Short Proof of Kneser’s Conjecture
by Joshua E. Greene
jgreene@hmc.edu
 

Reviews

In Code: A Mathematical Journey
by Sarah Flannery with David Flannery
Reviewed by Marion Cohen
mathwoman199436@aol.com
 

Dummy View - NOT TO BE DELETED