Book I – The Fundamentals of Geometry: Theories of Triangles, Parallels, and Area

- A straight edge and a compass are the only instruments used in Book I of Euclid’s Elements to prove its propositions.
- Propositions are the results which are obtained by reasoning with the definitions and postulates set forth in the beginning of the book.
- “A problem is a proposition in which something is proposed to be done.” A solution shows how the problem may be solved by using a straight edge and a compass. A demonstration proves that the solution solves the problem.
- “A theorem is a proposition in which the truth of a principle is asserted. The principle must be deduced from axioms and definitions or other truths previously and independently established.”
- A problem is similar to a postulate. A theorem is similar to an axiom. A postulate is a problem whose solution is affirmed. An axiom is a theorem whose truth is granted without demonstration. A corollary is an inference deduced from a proposition. A scholium is an observation of a proposition “not containing an inference of sufficient importance to entitle it a corollary. A lemma is a proposition used to establish a more important proposition. A hypothesis is a condition assumed. A construction is the change made to an original figure during the demonstration. Q.E.D means “which was to be demonstrated.”
- In Book I, Euclid demonstrates 48 propositions, including a proof for the Pythagorean Theorem.

I am mathematically inclined, and have studied Euclid’s elements intensively in previous years, so this was all very familiar to me and a delight to read over again. The following quote has been attributed to Euclid: “The laws of nature are but the mathematical thoughts of God.” There is a particular kind of beauty in mathematics because of the patterns and perfect symmetry of it. But unless you are working in a science, technology, engineering, or mathematics industry, these propositions are utterly useless. However, I believe that Euclid’s methodology is instructive to anyone in search of truth. Euclid begins with fundamental principles that are self-evident and indisputable. From these few axioms, he is able to construct ever more profound truths.

I remember reading a story about Abraham Lincoln’s relationship with Euclid’s Elements. While he was in law school, he was deeply concerned that he did not know what it meant to definitively prove something. Because of this concern, he devoted himself to the study of Euclid’s Elements until he could demonstrate all of the propositions in the book on sight. It is important to note that Euclid’s methodology has practical applications beyond mere mathematics. Like Plato’s Statesman, the exercise of one’s reasoning – i.e. recognizing patterns and similarities, and using logic to form ideas with ever greater complexity – is sometimes more important than whether you actually attain the goal of your investigation.