By Andrew Freda, posted July 20 –

You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight.—Abraham Lincoln (Henry Ketcham, The Life of Abraham Lincoln [1901])

Should we make time for Euclid in our Geometry classrooms? Yes! When I teach Geometry, the first nontextbook unit I use is always “Geometry and Euclid” (and I encourage everyone to visit a wonderful website, which has all of Euclid’s Elements).

Our textbook refers to point, line, and plane as “undefined terms of Geometry.” This accepted practice is far from helpful. Euclid defines a point as “that which has no part,” a line as “breadthless length,” and a plane as “that which has length and breadth only.” Are these helpful to our students?

At first glance, students cannot make much sense of these, but invariably one brave student will define a point as a “really small” dot. This sounds much better until other students raise the question about whether we can use a “really small” ruler to measure the distance across the “really small” dot, putting us in the position of using a segment to measure a point. This paradox brings us back to Euclid, but now students are in a better position to grapple with the idea of a point as something that has no dimension (i.e., “no part”). The line then opens us to the first dimension, and the plane extends us into the two-dimensional world, where we will spend most of our time in Geometry.

I find these discussions great fun because I get to see students moving from a concrete, nonmathematical understanding (a point is a dot on a page) to the strange, wonderful, and abstract world of mathematical entities. It is gratifying to see students arguing and exploring ideas that they previously took for granted.

Class discussion of the Postulates usually goes smoothly until we reach the fourth postulate, which tells us that all right angles are equal. Of course they are, so I ask my students why we would need a postulate for something we all know to be true. Could it be questioned?

Gentle prodding invariably leads the class to consider lines of longitude, which form right angles with the equator but then converge at the North and South Poles. A quick classroom demonstration that students enjoy involves drawing a right angle on two uninflated balloons and then seeing what happens when one balloon is filled with air. We can then see that all right angles are not equal, but if we restrict ourselves to “flat space,” then (and only then) we can say that all right angles are equal. This postulate is followed by one of the most famous statements of mathematics, the “parallel postulate,” which has a long and rich history and leads to terrific discussions. (Readers might share their insights about the “parallel postulate” by commenting on this post or submitting Reader Reflections to mt@nctm.org.)

The last stage is to move to Euclid’s Proposition 1, at which point I ask students to justify every step of the creation of an equilateral triangle. I call this “proof without tears” because, unlike the dreaded “two-column proofs,” students genuinely enjoy hunting through the Definitions, Common Notions, and Postulates to find the justifications for each and every step of Euclid’s 1.1.

I always get a quiet thrill from students’ responses when I ask them to consider how many students in how many places in how many languages over how many years have been asked to learn these same steps from Euclid—there is nothing else in the high school curriculum that has such deep and far-reaching roots. We then discover that we are also able to create a perpendicular, a perpendicular bisector, an angle bisector, and a rhombus, all with this simple diagram.

“Oh, my . . . that is like Shakespeare!” a student said to me one year after we had discussed Euclid’s Proposition 1. I replied that the poet Edna St. Vincent Millay was quite right: “Euclid alone has looked on Beauty bare.”

ANDREW FREDA, afreda@deerfield.edu, teaches at Deerfield Academy in Deerfield, Massachusetts. He has spoken and written on a variety topics, including the importance of visual representations in algebra, teaching Geometry through stand-alone “units,” ways to understand probability through the use of a dice game, the use of fractals in the high school curriculum, and the role of CAS in the mathematics classroom.