This blog post shares how a classical approach to Geometry, using Euclid's Elements, written around 300 B.C., integrates art and logic.

About ten years ago I decided to try my hand at reading through Euclid’s Elements. I had been teaching Geometry for over fifteen years. It was time to get to know this founder of a discipline I enjoyed. The effort did not last long. After about twenty minutes I had given up. What was the purpose of those propositions, the arcane drawings, the seemingly useless conclusions? Fast forward to now and you will find Euclid is an essential part of my Geometry class. Why the switch? Suffice it to say that I have found Euclid to be a rich resource for studying mathematics and science.

Euclid is a parallel text in the FCS Geometry curriculum. While we still employ a standard textbook, Euclid provides the theoretical basis for attacking algebraic problems in the standard text. Sometimes we dive right into a proof. Other times we look for patterns in geometrical figures using a dynamic computer program named Geogebra. This enables students to form intuitions and ideas before engaging the proposition. Regardless of the order, students work on their own to prove the propositions and then reason with each other about why their proof is correct or where they struggled to complete it.

Let me list a few of the benefits I have found using Euclid:

• Euclid demands logic from the students. Logic classes are good. Euclid gives the opportunity to put Faith Christian School’s Logic classes to work by necessitating airtight arguments where every assertion must be supported.
• Euclid enables the class to grow as a learning community as students reason together.  
• Euclid demonstrates logical continuity. For instance, like a good novel, Book 1 leads to the climax of what we all know as the Pythagorean Theorem. Using Euclid, the students are not merely given a formula to memorize and employ but can trace the journey on which Euclid has led the reader. 
• Euclid changed the perspective on mathematics. Euclid bases his work, not on precise measurements, but on comparison and proportion. As one formerly trained in nuclear engineering, I value precision, but examining math from Euclid’s perspective enables one to climb above the trees to see the landscape of geometry.
• It is tactile. In every textbook I have used since I started teaching geometry thirty years ago, compass and straightedge have been an unconnected “add-on” to the curriculum. In contrast, from Proposition 1, Euclid embeds the use of compass and straightedge as an organic part of many of his proofs. This resonates with Faith Christian School students who are well-trained in art.
• Euclid gives students a chance to see, appreciate, and enjoy genius. As a teacher, it is a joy to watch the faces of the students light up when they realize Euclid’s brilliance in using proof by contradiction (reductio ad absurdum) to prove a proposition. He gives away the game, only to win in the end.
• Euclid sets the table for the work of other scientists we study. Ptolemy, Copernicus, Kepler, Newton, and Huygens (the list could continue) all show an immersion in the concepts of Euclid that facilitated their work.