I am an instructor at a mid-sized American university, preparing to teach a two-quarter geometry course for junior and senior math majors. My plan is to use Hartshorne's "Geometry: Euclid and Beyond," which is designed to be read in parallel with Euclid's elements.

I would like the pace of the class to be a little more relaxed than the pace Hartshorne sets (for example, it appears chapter 1 takes Hartshorne three weeks to cover, and this involves reading the first four books of the Elements.) In particular, I am looking for problems that could be used as homework or in-class group work to solidify the students understanding of Euclid.

  1. Are there other companions to Euclid's Elements that are more leisurely paced?

  2. Do you know of a bank of geometry problems that contains problems appropriate for someone working through the Elements?

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    $\begingroup$ Not the question you are asking, but as I have just taught such a course I would recommend not using Hartshorne, and instead using Venema's "Foundations of Geometry". I think Hartshorne's book is incredibly nice, but most students just do not care about history that much. Venema makes them do a lot of interesting, rigorous, axiomatic geometry. His axiom system is much less cumbersome (taking the real numbers for granted). $\endgroup$ Dec 14, 2017 at 14:53
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    $\begingroup$ Another option is Petrunin's notes, but these seem pretty hard for my students. $\endgroup$ Dec 14, 2017 at 14:53
  • $\begingroup$ Euclid's Elements with Exercises by Kathryn Goulding (seems to be aimed at a younger audience, at first glance) $\endgroup$ Feb 1, 2019 at 23:23
  • $\begingroup$ euclidea.xyz sciencevsmagic.net/geo $\endgroup$ Nov 14, 2019 at 22:55

2 Answers 2


A1/ I'm not sure why you would need a companion to Euclid. Euclid is already a textbook and you can progress through it as leisurely as you like...

I prefer Sir Thomas Heath's version. But the two most popular works back in the 19th century (when Euclid teaching was at its height in the UK) were Todhunter's The Elements of Euclid for Schools and Colleges and Potts' Euclid's Elements of Geometry which are more adapted for teaching.

Back in the day the sequence usually followed was BOOK I, BOOK III (excl. 35-37), BOOK II (plus III.35-37). Then a selection from BOOK IV as problems in geometrical drawing. BOOK V on theory of proportions was usually substituted with a treatment in algebra. Then BOOK VI and finally a limited selection of propositions from BOOK XI and XII.

If you're only doing the first two books then Euclid Books I,II by Charles Dodgson (aka Lewis Carroll) is great.

A2/ There are several old collections of geometry problems or "riders" linked to progress through Euclid.

Todhunter includes exercises in his work and he separately published the answers in Key to Exercises in Euclid.

You might like to consult Smith, J Hamblin, Riders in Euclud (I, II, III, IV & VI).

There are quite a few other books dedicated solely to exercises in BOOKS I and II, eg Deakin, Rupert Rider Papers on Euclid (BOOKS I & II).

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    $\begingroup$ Presumably OP wants to teach an axiom system which is actually complete. Euclid, while very good, does leave some things out. For instance, there is no assurance (from his axioms) that a ray starting inside a circle must intersect that circle. Hartshorne's book fixes these issues. $\endgroup$ Apr 18, 2018 at 11:10
  • $\begingroup$ Heath's version is great. You might also want to have a look at this version of the first six books, which is quite lovely. $\endgroup$ Apr 18, 2018 at 13:56
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    $\begingroup$ Thank you, these riders are very helpful. RE: why I need a companion: "In particular, I am looking for problems that could be used as homework or in-class group work to solidify the students understanding of Euclid." It's true Euclid is not a complete axiom system, but I'll address that in the second quarter. $\endgroup$ Apr 18, 2018 at 14:34

Euclid the game is a set of challenges built in Geogebra. It allows the student to progress through the elements gaining additional tools as they go or solve each using only straight edge and compasses. It is accessible and enjoyable. I have used it with students from the age of 12. I don't know whether you were looking for something more formal.


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