Course: Euclidean Geometry - Books One - Six

Course description:

This video series consists of a series of 95 recorded videos (accessible online) to accompany the famous 2,000+ year old Euclidean geometry classical textbook.  The videos were recorded during the 2021-2022 school year.  The class consisted of Traditional Catholic high school students of various ages studying the first six of the thirteen books of Euclid together as a group.  It was a fun, challenging, and rewarding experience.  The students did very well, for the most part.   Although your student enrolled in this recorded course will not have the “live experience,” yet we think the videos will still make them feel like they are there, and the benefits in clear thinking will be much the same.   You will find a link below to some sample class recordings below.

The modest fee for this course not only gives you access to the class videos, but also entitles y
ou and/or your student to call us throughout the school year with any questions in understanding the material.   Please do not miss exposing your students to Euclidean classical geometry!    If they are mature enough, we encourage you to do it.   Please read on...

What is Euclidean Geometry?

It is hard to say enough positive and admiring remarks about Euclidean geometry, and what it does for the young (or older!) mind.  There is a reason Euclidean geometry has been studied by just about every great mind throughout the centuries, including St. Thomas himself.   Until our modern era, when men mentioned geometry, the word "Euclid" instantly came to their minds.

Let us begin with the name.   It is called Euclidean geometry because it was compiled by an ancient Greek geometer named Euclid (known as the "father of geometry") some 2,500 years ago!   Euclid actually gave his book the general name The Elements.  This is because besides a simply superb coverage of geometry, it also includes a treatment of arithmetic and other wonderful topics such as ratios and proportions.  

Why should I enroll my student in this course?

The plain truth is that there is simply no course we know of – whether in mathematics or logic – which is more fundamental or effective at forming the mind in the habit of careful and logical reasoning.   In fact, I can honestly say I myself had never truly experienced a true science until I took this course as an adult (36 years old) as a freshman at Thomas Aquinas College.   At that time, I already had a bachelor’s degree in computer science, to obtain which I had studied many math and science courses.   I also say this because, having taught Euclidean geometry to our all five of our children, my wife and I saw firsthand the change in the clarity of their thinking.  The mind greatly matures and expands in this class.  It is true that the student walks away afterwards with a large set of beautifully-done geometric truths.  But more important, the student learns by example how a true science proceeds, step by step with certainty, always being able to defend the current steps by retracing back to certain starting points.  This is a very, very rare thing in the sciences.

If you would like to see The Elements for yourself, you can download or view the entire thirteen books (of which, we will only be studying the first six in this course):  

       Online version:

        Pdf document for download:

Few parents realize this, but The Elements was the standard upper school geometry text in the United States until the early 1900’s.  Until that time, education had been considered first and foremost for the sake itself – truth is simply beautiful, and it is human to form the mind with good intellectual habits.  Afterwards however, with a definite change in philosophy, education came to be considered primarily for making money, solving problems, and making products, etc.  This is tragic.   Of course, we all want our children to succeed in the world and have good jobs after graduation.  But experience shows that the temporal needs are automatically taken care of if the mind is first formed and sharpened.

In our opinion, most modern 'geometry' courses turn out not to be the science of geometry at all, but rather, practice in manipulation skills and problem solving, such as, "Determine the length of this line.  What is the area of that quadrilateral?  How much paint would be required to cover this shape?”   While there is much to be said for having these skills, Euclidean geometry does not solve such particular problems.  For, since as Aristotle shows, there can be no science when dealing with particulars.  Rather, Euclidean geometry is instead concerned with universal truth (e.g. “Prove that ALL triangles have their three interior angles equal to two right angles.”)   This difference in focus has tremendous and lifelong effects on the mind of the student.

What is required for this course?

The primary things required are a zeal for truth and a willingness to work.  This is not an easy course.  It requires supportive parents who want the very best education for their student, and a student who is truly willing to work.  The work involves preparing for class: reading, studying, and memorizing the structure of the careful and beautiful proofs, and then demonstrating those proofs to teacher / parent.  This course does take some adjustment for our modern minds.  But the pain pays off in continued dividends the rest of the student’s life.

On the practical side, you will need the following:
  • The link to the 95 online video recordings.   You will be given this link after paying the course fee.  In the meantime, feel free to watch these four sample class recordings
  • A printed copy of The ElementsWe recommend the Green Lion Press edition (with ISBN #  978-1888009187) which you can purchase through us, or easily find online by searching that ISBN number on Ebay, Amazon, and others.

Is my student mature enough for this course?

Recommended ages are 15 – 110.  (I first took Euclid at 36 years old and I assure you this is good for adults, too!)    That said, our children studied Euclid when they were 13, but this is because we knew each of them was mature enough and ready.  We started them early because we wanted them to be well-prepared to think straight with solid mathematical principles before moving on to modern algebra, trigonometry, and so on.   That strategy paid off well, but even if your student has already done algebra, no worries!   Doing Euclid anytime is eminently worthwhile.   It is humbling and mind-forming like no other course we know of.

What is the time commitment for the student per week?

Again, Euclid is a challenge – but a challenge which will form your child’s mind like nothing else.  You have our promise on that, as long as your student really tries.  But it is hard work.  It requires six hours of work per week:  four hours of class time and at least 2 hours of private preparation outside of class (40 minutes for class session).  We know that is a lot!  But consider: this class serves as both mathematics and logic class.  Euclidean Geometry is definitely not your typical class.   If your student is not disciplined and willing to prepare well for the classes, he should not take the class – it will be frustrating for him, and he would probably be better off with a modern geometry course.

How should the student prepare for each class session?  

For each proof assigned for that day’s class session, the student should:

  • Memorize the annunciation (the thing being proved / constructed) exactly;
  • For the body of the proof, the student should generally strive to memorize the main sequence of steps, in the student's own way - without being slavish (e.g. memorizing each particular word and the exact letters in the book).  In fact, when your student demonstrates the proposition for you (without the aid of the book), he should use other letters than are in the book.  If the student really understands the proof, it does not matter which letters are used, and this should not trip him up. 
  • If the proof is short, the student could indeed memorize the exact words in the body of the proof.  But for longer proofs, this is not possible, and would probably drive the student nuts.

For example, even though proof 1.1 is short (and its body could be memorized exactly), let us pretend it were a long proof.  First then, as with all proofs (long and short), the student should memorize the annunciation exactly, viz.  "To construct an equilateral triangle on a given finite straight line"    Then, for the body, the student would memorize the main sequence of steps in the student's own informal way, something like this: 

"Well, first I draw a straight line.  Then, using that line as a radius, with one of its endpoints as a center, I draw a circle.  Now I draw another circle using that same radius, except this time, I use the other endpoint as the center for this second circle.   To the point where the circles intersect, I draw a line from each end of the original straight line.  End of construction.   Here is the proof:  I know with certainty (science) that I have just constructed an equilateral triangle because all three lines are radii of equal circles, and thus equal.  And they form a figure.  But this is the very definition of an equilateral triangle!"

In other words, the student has true science only when he really sees how the steps fit together to make a lock-solid proof.  A common pitfall however, is for students to slavishly memorize the exact words of the proof as it is in the book.   This technique rarely works.  It is much better to concentrate on recognizing the outline of the main steps, and then letting the mind fill in the details from there.  

What are the recorded classes like?

Again, see the above-mentioned link to the sample classes.  For each class session, the tutor asks a random student to do the next proof.  That student then strives to annunciate the proof exactly as it is in the book, and then demonstrates the body of the proof on the electronic whiteboard, talking it through out loud as he writes / draws.  Again, the student should use other letters than those in the book.  This is critical, because otherwise the student might just “parrot” the proof, with no real understanding. 

After the student does the proof, the real fun begins.  We get to talk about the proof, marvel at how clearly and beautifully Euclid shows us the science.  We sometimes also asked the student questions if either the tutor or the other students got the feeling the student “did not really get it.”  All of this was to help the student and the class, all in a spirt of Catholic charity.   If the student is humble, he will admit if he is lost and appreciate help, rather than trying to act like he gets it. 

Who was the tutor for these videos?

I am Dean Loew, the founder of Angelic Doctor Academy (, a Traditional Catholic school.  I am a graduate of Thomas Aquinas College ( in Santa Paula, California and a homeschooling father for over about 25 years.  Besides having been myself tutored in Euclidean geometry as a student at Thomas Aquinas College, I have since tutored this class four times in the past, with our own four boys, our daughter, as well as about fifteen others.

I use the word ‘tutor’ because in the Thomas Aquinas College tradition, the true teachers are the great minds such as Euclid, and we are here merely to assist the student in learning directly from that great mind.

We hope you will consider having your child watch these recorded videos.   He will learn the proper method of acquriing science.    Remember, the course fee entitles you to call us with any questions your student has as he watches the videos and works his way through this course - the same great book which, in part, formed St. Thomas Aquinas himself.  

Course fee:

$ 100

Course Length (in weeks):


Book(s) used:

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Textbook: Euclidean Geometry, Books One - Six; Video Series


Author: Euclid


Publisher's Description:

Green Lion Press has prepared a new one-volume edition of T.L. Heath's translation of the thirteen books of Euclid's Elements. In keeping with Green Lion's design commitment, diagrams have been placed on every spread for convenient reference while working through the proofs; running heads on every page indicate both Euclid's book number and proposition numbers for that page; and adequate space for notes is allowed between propositions and around diagrams. The all-new index has built into it a glossary of Euclid's Greek terms.

Heath's translation has stood the test of time, and, as one done by a renowned scholar of ancient mathematics, it can be relied upon not to have inadvertantly introduced modern concepts or nomenclature.

Green Lion Press has excised the voluminous historical and scholarly commentary that swells the Dover edition to three volumes and impedes classroom use of the original text. The single volume is not only more convenient, but less expensive as well.

7 x 10", 527 pages, including a new index and glossary of Euclid's Greek terms. Publication date, August 2002.

ISBN: 9781637940891

Publisher's Description:

ISBN: 9781637941980

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