- Maths and Sciences
The following post, written loosely in the style of Euclid's Elements, employs the structure of Elements to explain why and how we study this ancient work at The Saint Constantine School.
- A student of mathematics ought to come away from class better able to reason logically, think critically, and apply problem-solving techniques to new situations.
- An effective math education will expose students to mathematical proofs and teach students to ask why formulas and theorems are true, not to merely memorize them.
- In order to understand who we are and how we ought to proceed with regard to any subject, we must understand from where we come.
The Elements of Euclid is a book worthy of study.
The story from Stobaeus goes like this: One day while Euclid was teaching, a young man asked him what could be gained from the study of geometry. His response: To command his slave to give the young man a coin “if he must profit from what he learns,” and to get on with the lesson.
The ancient Greeks are credited with the founding of modern mathematics, for they not only asked “What is true?” but “How can I be certain that this is so?” The attainment of understanding was the ultimate end and its own reward. Much of their mathematical work has been preserved through the centuries in numerous translations of Euclid’s Elements (https://mathcs.clarku.edu/~djoyce/elements/bookI/bookI.html). Compiled around 300 BC by Euclid of Alexandria, Elements served as the geometry textbook of the Western world for over 2000 years. It begins with a set of definitions and ten axioms called “postulates and common notions,” and throughout thirteen books, proceeds to systematically deduce the fundamental results of plane and solid geometry along with many results now considered to be the property of algebra and number theory. Book I culminates in a proof of the Pythagorean theorem. Book II contains geometric justifications for various algebraic identities and a construction for dividing a line into the golden ratio. Book VII contains the Euclidean algorithm for determining the greatest common factor of two positive integers, which is still used today to ensure the security of internet communications, and Book IX contains “Euclid’s Theorem,” which states that there are infinitely many prime numbers, a fact that also plays a role in modern cryptography. Book XIII proves that there are only five Platonic solids, and the list of notable results continues.
Elements is a treasure trove of ancient mathematics and still contains the bulk of results taught in high school geometry courses today. A close reading requires one to understand mathematical proof and teaches students the importance of justifying their inferences at every step. It exposes students to the axiomatic method used by the ancient Greeks that laid the foundation for the rigor demanded by modern mathematics. A study of Elements lays bare our mathematical roots that we may understand the subject not as rote computation, but as a search for truth and order. Postulates 2 and 3 therefore confirm the value of its study.
Elements is highly appropriate in the context of a great books education.
The Saint Constantine School is at its heart a great books program. By exposing students to vast and varied classical texts from many different cultures and time periods, students learn to ask the questions that lead to the cultivation of virtue and wisdom. Elements is one of the greatest texts in all mathematics owing to its contribution to the field’s development and to its longevity, so it cannot be overlooked in the context of a great books education. Reading Elements alongside Plato and Aristotle reveals the role played by mathematical objects and reasoning in the Greek ideas that founded western civilization. A study of Elements is therefore beneficial and appropriate within a great books program [Proposition 1; Postulate 3].
Elements can be effectively merged into a modern curriculum to provide students with a taste of ancient Greek thought as well as a robust problem-solving experience in geometry.
Eventual abandonment of the study of Euclid in most western secondary schools came on the heels of the mathematical discoveries of the 19th and 20th centuries that highlighted the incompleteness of Euclid’s axioms and introduced a host of new, non-Euclidean geometries and applications. Euclid was also criticized for being unnecessarily difficult to read and for requiring students to spend an inordinate amount of time deciphering proofs – some of which are notably inferior to their modern counterparts – rather than thinking creatively and learning problem solving techniques.
At Saint Constantine we do not shy away from difficult texts. The question is always “Is this work worthy of being read? Does it contribute to an understanding of the human soul and its purpose?” rather than “Is this difficult to understand?” Many ideas worthy of rumination require effort to comprehend at first, and the time spent in their contemplation might indeed mean less time spent developing “more practical” skills, but we maintain that among the most practical of skills is that of sustained contemplation. A study of Elements reveals that much of the content is the same as that taught in modern courses and that the propositions and proofs found therein are really quite accessible to high school freshmen under the proper guidance. We love to see our students realizing that they can do “hard” things.
I have come up with a balanced approach to incorporating Euclid’s Elements into a standard geometry curriculum. In the first semester we study Book I in detail, which already contains a good deal of the material in a modern geometry course. Some propositions we study carefully and others we read only to obtain an idea of the truth they present but rely on intuition or more succinct proofs for justification. We learn the structure of a valid proof from Proclus and use this as a model to diagram Euclid’s proofs in two-column form for clarity, using modern terminology and notation to make each proof more concise. Every student is assigned one proposition to study in detail and present to the class. I supplement each lesson with additional problem-solving activities so that we see the relationship between Euclid’s propositions and more modern material. After Book I we rely less on Euclid, though we still study select definitions and propositions from books II through VI as we focus on analytic geometry, trigonometry, and other topics not contained within Elements. Students get a feel for the manner of thinking that lay the foundation for 2000+ years of mathematical development as well as see how these concepts are applied in modern mathematics.
We conclude with a proof by contradiction: Suppose it were not possible to merge Elements into a high school geometry curriculum in a way that simultaneously provided students with a taste of ancient Greek and a robust problem-solving experience. Then our high school math curriculum would not exist. Clearly it does exist, so the proposition is proved.
We invite your children to come experience it.