Euclid's Constructions:
A Program for Geometric Algebra

Euclid's Elements may be seen as two threads intertwined: geometric algebra and the platonic solids. For a program on geometric algebra, we recommend these 19 constructions:

Book I. Props. 1, 3, 5, 8, 9, 10, 11, 22, 23, 31, 42, 44*, 45*, 46
This sequence of 14 constructions from Book I includes all constructions needed for the constructions of Book II. Those marked with * are part of Euclid's Geometric Algebra thread.
Book II. Props. 11*, 14*
These are fundamental for the Golden Thread, basic to the revolutionary influence of the Elements in the Renaissance.
Book VI. Props. 28*, 29*, 30*
More cases of quadratic equations.

Note: Each constructive proposition in Euclid's Elements consists of three parts:

  • the "setting out" or statement of what is to be done,
  • the actual construction, and
  • the proof that the construction does what was to be done.
We are recommending, in this minimal program, only the actual constructions. These are shown step-by-step, with animations, in the VCE.

According to Heath, Prop. 44 of Book I "will always remain one of the most impressive in all geometry..."

Propositions 44, 45, and 46 of Book I are the basic constructions for the Geometric Algebra Thread, which is one key to the pedagogic success of Euclid's work as a math textbook throughout history.

For a more substantial program, we reccomend the following stages:

  • Stage 1: Book I. Props. 1,2,3,9,10,11,12,22,23,31,46 (11)
    Construction parts only. When another construction is used, fill in all the steps of the subroutine, and its subroutines, and so on. These are all 11 of the "drafting constructions" of Book I. The 3 "area constructions" are listed in Stage 5 below.
  • Stage 2: Book I. Props. 1,9,12,31 (4)
    Proof parts only, of these constructions. Also study the commentaries summarized by Heath for these propositions.
  • Stage 3: Dependency trees for all constructions of Book I (14)
    Some, but not all, dependencies are indicated by Euclid in his text.
  • Stage 4: Book I. Props. 32,47 (2)
    Proofs, these are theorems, not constructions. Which is more important, Prop. 47 or Prop. 48? Wanting more of this? Try Props. 4,26,27,29,34,48 of Book I
  • Stage 5. Book II. Props. 11,14 (2)
    Construction and proof parts. Dependency trees.
  • Stage 6: Area constructions, Props. 42,44,45 of Book I (3)
    Construction parts only.
  • Stage 7: Areas and geometric algebra, Props. 35 to 45 incl. of Book I (11)
    These are the "locus-theorems" of Euclid, see the comments of Heath.
  • Stage 8. More theorems of Book II.
  • Stage 9: Book III.
  • Stage 10: Book IV.
  • Stage 11: Selections from Books V and VI.
  • Stage 12: Solid geometry from Books XI, XII, XIII.
This program proposal is under construction! The mapping from stages into grades (of K-12) is up to you. Stage 1 should begin in grade 4 or 5 at the latest. Stages 2-5 should be appropriate for Grade 6 or 7. Stages 6-8, geometric algebra, should precede the introduction of symbolic algebra, usually in grade 7 or 8. Stages 9-10 are the climax of the Pythagorean tradition, and the golden thread. Stage 11 introduces ratios and irrational numbers, and my be studied concurrently with high school algebra. Stage 12, solid geometry, is traditional for grades 9-12. In short, stages 1-8 should be completed by the end of grade 8 if possible.

Please let me know if you try any of this!

Revised 12 July 1997 by Ralph Abraham