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Euclid’s Elements

“I’ve learned more in one afternoon with Euclid than I did in all my years of high school math!”  Thus exclaimed Kim, a humanities teacher from an all-girls Catholic high school in Louisiana, after the geometry session of the Institute’s first Academic Retreat for Teachers.  Kim had been very skeptical about the quadrivial (or mathematical) portion of the Retreat, especially when she learned she might be called upon to demonstrate one of three propositions from Book I of The Elements.  But when she saw how clearly Euclid explained how to bisect an angle, the reason that the area of a triangle is equal to half of the base times the height, and the celebrated Pythagorean theorem, she was delighted.

Euclid’s Elements was the geometry textbook for students from the third century B.C. through the eighteenth century.  All educated people knew it.  John Adams poured over it with his son during his evenings spent in Europe on diplomatic missions. Thomas Jefferson once wrote to Adams:  "I have given up newspapers in exchange for Tacitus and Thucydides, for Newton and Euclid; and I find myself much the happier." 

Not many textbooks can claim the attention of the greatest men of their times.  Euclid’s power lies in his ability to present beautiful, amazing geometric proofs and constructions with an order and clarity that puts these truths beyond doubt.  He begins his first book with definitions of shapes and some simple assumptions.  Unlike the supposedly “more rigorous” axioms with frightening names, these beginning points are the sorts of things that everyone would readily grant: “An equilateral triangle is that which has its three sides equal” and “All right angles are equal to one another.”  He then leads the student step-by-step to see very complicated truths and constructions.  By the end of Book I, he has shown students how to take any complicated rectilineal figure, such as a drawing of the Pink Panther made out of straight lines (one group of my students drew this as an example), and turn it into an easily measurable square of equal area. 

Book I, like all the other books, is divided into propositions, which prove one important fact about rectilineal shapes.  Each proposition is neatly, clearly laid out.  Euclid begins by stating what he is going to prove (the enunciation), then tells how to construct the diagram he’ll be working from, then lays out each step in his argument, draws his conclusion, and finally restates the enunciation as a now proven truth.

Students of Euclid do not just read his propositions. The central exercise is demonstrating—going before the class to present an assigned proposition.  The student is expected to present the enunciation word for word, then prove the proposition following Euclid’s guidance, offering support for each step.  He then has to answer questions about the demonstration from the teacher and his fellow students. 

Students generally take some time getting used to the whole process.  Their first propositions tend to be sloppy; they do not see why all the steps are important; they will leave parts of enunciations out; their diagrams will not be well-drawn.  But usually by the middle of the first book, they have begun to see the importance of all the steps, and can identify missing steps in the demonstrations of fellow students.  No better training in basic logical presentation exists, for they are not learning abstract or meaningless syllogistic forms, but they are learning the value of precision, clarity, and completeness in coming to know truth.

As students become adept at demonstration, I will frequently have them close their books, then read the enunciation of the next proposition to them as a challenge: “Can we prove that if two angles and a side of one triangle are equal to those of another, the two triangles are equal?”  Students are also encouraged to suggest alternate ways of demonstrating a proposition.  The class can then consider whether the argument is valid and, if so, why Euclid would have done it his way.  Sometimes someone will ask whether, if the diagram could be drawn differently, Euclid’s proof would still work.  (Dr. D.E. Joyce of Clark University has an excellent website featuring the complete text of the Elements along with movable diagrams!)