A study of the actual constructions and proofs contained in the 13 books of Euclid’s Elements

I recently purchased a copy of Euclid’s Elements from Amazon published by Green Lion Press.  In reading some of the proofs, I recognized many of them, but noticed peculiarities in Euclid’s style.  For Example, he often uses a first person tense in a proof.  “I would propose that…”  A more significant difference is that he often makes statements without reasons.  For example, he may state that two triangles are congruent with out stating the congruence shortcut used (SSS, SAS, ASA, SAA).  This happens a lot so it helps to have a good feel for geometry when working through a Euclidean proof.

Therefore,I though it might be useful to show the Euclidean proof followed by a more modern two-column proof.  These could be assigned as classroom activities in order to give more practice in proof writing while bringing the original writings of Euclid from 300 BC into your geometry curriculum.

Here we go!

The Postulates:  In order to have a proof of anything, you must start somewhere.  In Euclid’s Elements, we start with the five postulates.  These are five statements that must be accepted without proof;

1.  You can draw a straight line from and point to any point

2. You can produce a finite straight line continuously in a straight line

3. You can describe a circle with any. center and distance (radius)

4.  all Right angles are equal to one another.

5. If a straight line falling on two straight lines (two lines cut by a transverse line) make the interior angles less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.  (This implies parallel lines.  I have heard that Euclid attempted to prove this with the other four postulates, but was unable to do so.  So he made it the fifth postulate.)

Proposition 1


On a given finite straight line to construct an equilateral triangle

Several early propositions are actually constructions.  I chose not to convert constructions into two-column proofs for this post.  This is an example.

Let AB be the given straight line.

Thus it is required to construct an equilateral triangle on the straight line AB.

With the canter A and distance AB, let the circle ACD be described; again, with center B and distance BA, let the circle BCE be described..  and from point C, in which the circles cut one another, to the points A, B let the strainght line s CA, CB be joined.

Now since the post A is the center of the circle CDB, AC is equal to AB.

Again, since the the point B is the center of the circle CAE, BC is equal to BA. 

But CA was also proved to be equal to AB, therefore each of the straight lines CA, CB is equal to AB.

And things which are equal to the same thing are equal to one another; therefore CA is equal to CB.

Therefore the three straight lines CA, AB, and BC are equal to one another.

Therefore the triangle ABC is equilateral and it was constructed on the given finite line AB.

Being what it was required to do.


Note: QED (quod erat demonstrandum) means “which was to demonstrated” and follows a completed proof, QEF (quod erat faciendum) means “which was to be done” and follows a completed construction.

Proposition 2

I am reading through Euclid’s Elements for the first time