Introduction to Euclid's Elements

This lesson introduces the ancient Greek mathematician Euclid and the study of his treatise called Elements.


There are two authors in Stepanov's recommended short canon that he did not cover in his "Civilizing the Barbarians" course. One of these two authors is Euclid, who was an ancient Greek mathematician. Euclid is considered to be the "father of geometry" and is known for his treatise "Elements," which established the foundations of geometry. Euclid is one of the greatest mathematicians of antiquity and one of the most influential mathematicians in the history of mathematics.

Not much is known about Euclid's life. It is believed that he was younger than Plato but older than Archimedes, and thus lived around 300 BC. He probably received his early mathematical education in Athens from the pupils of Plato. It is also known that he taught and founded a school at Alexandria under Ptolemy I, who reigned from 306 to 283 BC. Robert Maynard Hutchins provides a short biography of Euclid in Volume 11 of the Great Books of the Western World series in which he states that, "Euclid's great work, the 13 books of the "Elements," must have become a classic soon after publication. From the time of Archimedes they are constantly referred to and used as a basic text-book." Other works by Euclid include "Data," "On Divisions," "Phaenomena," "Optics," and "Porisms." These works cover a variety of mathematical topics, such as optics, spherical astronomy, the properties of figures, and the nature of ratios. Many other works by Euclid are now lost.

An excellent introductory lecture on Euclid's "Elements" is available from Mr. William C. Michael, who is the founding headmaster of the Classical Liberal Arts Academy at The mission of Mr. Michael's academy is to research, restore, and teach the education and culture enjoyed by wise men and saints throughout history. If you enjoy Dr. Michael's style, you can find many more of his lectures on classical education and classical texts on his YouTube channel and on his website,

Mr. Michael's lecture begins by introducing future students of Euclid's "Elements" to the study of classical geometry. Mr. Michael clarifies that when studying classical geometry, the focus is not on physical shapes and sizes, but on perfect imaginary forms that may not even exist in the visible world. He continues by stating that these forms are based on definitions, axioms, and postulates that are used in proofs. He emphasizes that this is a philosophical study and that the goal is to understand the ideas of magnitudes at rest. He teaches with a Catholic worldview and shares how, in the created world, God used geometrical ideas and images to create things. Consequently, he also notes that this study is important for understanding why God chose to use certain figures, such as the sphere, in the design of the earth and other celestial bodies. Thus, there is a theological perspective that can be taken on this work, too.

Mr. Michael's lecture also puts the study of classical geometry and Euclid's "Elements" in context with the Quadrivium, which is the four mathematical arts of arithmetic, geometry, music, and astronomy. "Elements" is a fundamental text in the study of the Quadrivium because it is a comprehensive treatment of the principles of geometry. Its 13 books cover everything from basic definitions and postulates to theorems and proofs of geometric figures and solids. The work has been studied for over 2,000 years and continues to be studied today.

But why should we, in the 21st century, bother with such an ancient text? Because of its enduring value as a source of mathematical knowledge and wisdom. The study of "Elements" is not just about geometry. It is about understanding the underlying structure of the world around us. It is also about learning to think logically and systematically. In his lecture, Mr. Michael states that Plato believed that studying geometry raises one's thoughts to the eternal and helps create a spirit of philosophy. He goes on to explain that studying classical geometry is beneficial for personal spiritual and intellectual growth and happiness and is recommended by the wise men of history.

Mr. Michael believes that the key to success in classical geometry is to start by memorizing the definitions provided in Euclid's "Elements." Mr. Michael understands that this may be a difficult task and requires a significant amount of work, but he is confident that his students have the ability to accomplish it. He encourages students of classical geometry to invest the time and effort necessary to memorize all 23 definitions in Book 1 of "Elements," and to be willing to make sacrifices in order to achieve this goal. Mr. Michael believes that by committing these definitions to memory, students will have a strong foundation on which to build their understanding of the subject. He also believes that memorization is an important step in achieving mastery of any subject.

Watch Mr. Michael's "Classical Geometry, Lesson 01" lecture: 

Here are five other useful links in the study of Euclid's "Elements:"