Reinforcing Education for Human & Science Diplomacy
Euclidean Geometry Tutorial
Euclidean Geometry Tutorial

Euclidean Geometry Tutorial

This tutorial covers the Geometry in Euclid’s Elements. Euclid ‘s famous text was “the” book for the study of Geometry until the 19th century. It has been studied by a host of intellectual greats. His systematic approach to Geometry is not a only a tremendous study in how to think and reason, but it became the paradigm that later philosophers would attempt to follow in setting up their own systems of thought. There is really no other mathematical text that rivals its impact on intellectual history.  This tutorial is highly recommended not only for its tremendous historical value, but also as a fine addition to the Geometry-starved Saxon program. The only fault I see with the Saxon programs is its meager treatment of Geometric proofs. Along with Saxon, most modern math texts are downplaying Geometric proofs because they are teaching to the SAT and it does not require proofs. But after going through the magnificent proofs of Euclid , you will see why his work is truly a mathematical classic.  In order to enroll in the second year of the Great Books Tutorial, all students must take my Euclidean Geometry course.  The course requires 5-10 hours a week of preparation and should be seen as a college level mathematics course.  The Elements comprise the greater part of the freshman year mathematics program at St. John’s College and my tutorial is closely modeled after that course. How does this course compare to a conventional Euclidean Geometry course?  Conventional high school Geometry texts study proofs for the first semester of the year and then study the application of Algebra to Geometry that was developed by Descartes (Cartesian) during the second semester.  Though based on Euclid, the conventional study of proofs is usually much less thorough than the proofs in Euclid.  Cartesian Geometry is on the SAT and should be studied as a separate subject if your child will be taking that quiz.  Saxon eliminates a separate year for Geometry entirely and just studies Cartesian Geometry in conjunction with Algebra and Advanced Math.The text for the tutorial is Euclid’s Elements, from Green Lion Press. You might also find “The Bones” helpful during your study as well.  You can find both books, The Elementsand The Bones, at Amazon at discount.  If you are a Greek student, rather than using the Green Lion edition, you might consider the Fitzpatrick editions which have the Greek text included as well-  Volume IVolume II and Volume III.  If you would like a colorized version see the Byrne edition, but please do not use this version for class. 
You can also find the complete text of Euclid  here.  You can see the manipulative possibilities of the online text by clicking here. (Requires Java and Active X)   After studying the Elements, if you would like to see the application of Euclid to the study of Astronomy, I recommend studying Euclid’s Phaenomena and thenPtolemy’s Almagest,

I recommend you download a free demo version or a license of Geometer’s Sketchpad.  Having this software will allow you to make your own versions of the Geometric figures on the computer.  Sketchpad is $10, but you can find a number of fine apps by searching “geometry sketchpad” in the app store. 


Euclid’s Elements form one of the most beautiful and influential works of science in the history of humankind. Its beauty lies in its logical development of geometry and other branches of mathematics. It has influenced all branches of science but none so much as mathematics and the exact sciences. The Elements have been studied 24 centuries in many languages starting, of course, in the original Greek, then in Arabic, Latin, and many modern languages.

I’m creating this version of Euclid’s Elements for a couple of reasons. The main one is to rekindle an interest in the Elements, and the web is a great way to do that. Another reason is to show how Java applets can be used to illustrate geometry. That also helps to bring the Elements alive.

The text of all 13 Books is complete, and all of the figures are illustrated using the Geometry Applet, even those in the last three books on solid geometry that are three-dimensional. I still have a lot to write in the guide sections and that will keep me busy for quite a while.

This edition of Euclid’s Elements uses a Java applet called the Geometry Applet to illustrate the diagrams. If you enable Java on your browser, then you’ll be able to dynamically change the diagrams. In order to see how, please read Using the Geometry Applet before moving on to the Table of Contents.