This discussion has shown how to partition figures such as rectangles, parallelograms, triangles, and regular hexagons, as well as a special case of semicircles, in such a way that Euclid's approach to proving the Pythagorean Theorem remains intact. Although the fact that the Pythagorean cuts exist for rectilinear figures is not really surprising, we have never seen these explicitly shown before. The interested reader is invited to further explore Pythagorean cuts for the general case of semicircles.
Euclid's original proof of the Theorem of Pythagoras is among the topics that appear in several of our courses for mathematics teaching majors, including the upper-division writing course and a senior capstone course in geometry. Many students are surprised to learn that the Pythagorean relationship is geometric, rather than algebraic, in nature, and can be proved purely geometrically. A next step might be to have students explore the notion of finding Pythagorean cuts for the rectangle, parallelogram, and triangle. This could begin with exploration, perhaps using a software tool such as Geometer's Sketchpad or GeoGebra, and lead to conjecture resulting in proof. While some students may get further into this than others, we hope that all students would come away with a better understanding and appreciation of the powerful mathematics that was developed so long ago.
Euclid of Alexandria, Elements (1952). Sir Thomas L. Heath, translator. In Great Books of the Western World, University of Chicago Press, Chicago.
Project Mathematica: Theorem of Pythagoras (1988). Tom Apostol, project director. California Institute of Technology, Pasadena. Available via http://youtu.be/wYp2hWhC8LY
Euclid’s Elements. David Joyce’s website, Clark University, Worcester, Massachusetts.
http://aleph0.clarku.edu/~djoyce/java/elements/elements.html
Marty Bonsangue is Professor of Mathematics at California State University, Fullerton. In 2011 he was named the Outstanding Professor at Cal State Fullerton. He remains deeply devoted to Euclid.
Harris Shultz is Professor of Mathematics Emeritus at California State University, Fullerton. In 1988 he was named Outstanding Professor at California State University, Fullerton and in 1989 he was one of two recipients of the Trustees' Outstanding Professor Award for the then nineteen campuses of The California State University.