Well here it is, my first post on Euclid. I have been teaching high school geometry for years and while Euclid comes up frequently, it generally does so in passing. So there seemed nothing more natural to me than to pick up Euclid for the first time and form a discussion group with a couple of my coworkers. Our first meeting about Euclid will be next Friday where we plan to discuss Euclid’s axioms and definitions during the long 2 hour ride to Fresno state for a meeting of geometry teachers. I am not certain how long we will read Euclid together, but I suppose that we will at least make it through the first few books. I suspect that I will frequently sound critical of Euclid as The Elements have had many hundreds of years of criticism as I look back with my modern perspective and compare Euclid with what has evolved since.
I have a few rules for reading a classic such as the Elements. The first is that I will try to refrain from reading commentaries before I have read it all the way through. This means that I may occasionally disagree with orthodox opinions and may even make some mistakes based upon my ignorance of the past. Nonetheless, I like to have an uncolored picture first before I read what others have said. Besides, I am already familiar with quite a bit of commentary merely tangentially. I have enjoyed many years musing casually about geometry as I stand in front of my students and present the material that I have been hired to convey. I plan to post my honest reactions to material, which of course may change as my understanding expands. While I am not new to geometry, I realize that there is still a great deal that I don’t know. I will return to the commentary on my second reading of Euclid.
I think that the part of Euclid that I am the most familiar with and will have the most to say about are his axioms, postulates, and definitions. It is the foundation that will color the rest of the book. It is the part of geometry that as an educator I have thought the most about. I did however, download a copy of John Playfair’s out of print book on geometry, which has been influential on the way educators present geometry today. Specifically, in the case of the parallel postulate….
Another thing that I do when I begin reading a book is consider its structure. Which I intend to discuss in this post.
About The Elements:
I purchased a handsome addition of the first 6 books produced by Oliver Byrne. This is a beautiful copy complete with color coded diagrams to make reading the elements easier for younger readers. I lent this copy to a co-worker, she chose that edition over my budget Dover copy with small print and extensive commentary by Sir Thomas Heath -which as I mentioned earlier I will try to avoid for the time being. First, as I understand it the Elements is a textbook containing a compilation of all of the known mathematical facts of the time organized and presented so that the reader is invited to ponder each fact. The book doesn’t have exercises the way modern textbooks do. It is simply an organized list of facts. Each fact is given as an exercise, following each exercise, the solution of the author is given. So, the book appears to be organized thus: it begins with a discussion regarding the fundamental assumptions and definitions of geometry. Then it proceeds with an organized pattern of assertions for the reader to resolve. Each assertion is intended for the reader to resolve logically. The logical resolution or justification of each assertion is what we call a proof. Immediately following each assertion the student a solution is presented by the author, so that the student may compare it with their solution or master the author’s process. The assertions follow a logical pattern building of building on each other. In many ways Euclid’s book is simply an exercise in pure logic. As I understand it, for many years universities in Europe used Euclid as a final polishing touch towards the end of a scholars education meant to refine and enlarge the scholar’s skills in logic. Abraham Lincoln is said incorporated Euclid’s book in his quest for self-improvement. As a lawyer he reportedly spent his evening proving Euclid’s assertions to sharpen his capacity to make compelling arguments, while many of his peers would spend their evenings drinking. This habit he reportedly maintained as president while on the road. Euclid’s book was a life-long friend. I suppose that the use of Euclid to refine one’s intellect and the lack of redundant exercises in arithmetic or practical application have been part of the wide and enduring appeal of the book. In a similar vein while living in Stockton, California- where I first decided to major in mathematics I came across a appeal in the online resources of the University of the Pacific to minor in mathematics as a way to hone one's intellect, particularly for majors in law. Among textbooks, it seems a little odd in this way. It’s organization is simple and devoid of many flourishes and types of exercises that mark most modern texts. It is an exercise in pure intellect. Nonetheless, I must say of my experience with mathematics in general that my study of math had a greater refining influence on my intellect than any other type of course I took, and I found it the most challenging as well.
In closing, I want to make one last note. When people talk of Euclid, they usually refer to the first few books of his Thirteen Books of The Elements. His treatise also includes other topics in mathematics, namely number theory. One thing that prejudiced me against reading Euclid in the past was the fact that Euclid appears a long time before the invention of Algebra and this makes some of his arguments seem cumbersome and unnecessarily rhetorical compared to more modern streamlined explanations employing modern notation, etc. While I hadn't read Euclid, I had seen a couple proofs that seemed to me could be proven with much less work. Nonetheless, I am now at a place in my life where the so-called classics are beginning to appeal to my mind.
I hope to be done discussing the axioms and definitions by the end of the first week and then be on to the first book. I hope that my discussing the material online and with my colleagues will lend greater enlightenment throughout the process. I hope I will pick up a few more friends online in the process that will add their own comments and enrich the discussion.
