Quite Elegantly Done

Euclid’s Elements – written in approximately 300 BCE at the famed library of Alexandria, Egypt, the 13 books (scrolls) of the Elements have long been considered the basis of Euclidian Geometry.  I am endeavoring to modernize Euclid’s proofs into more common two-column proofs which include the statements and reasons of the proof, instead of just the statements, which often appear in Euclid’s work.  I will also comment on statements such as ‘2 right angles” and point out that he is referring to a 180 degree angle.  The 13 books contain 407 proofs ending with QED and 59 constructions ending with QEF.  I will endeavor to analyze every one of these proofs and constructions and convert the proofs into modern two-column proofs. I am using the Green Lion translation of Euclid’s Elements which was published in 2017 along with the book “The Elements of Euclid, the first six books” by Oliver Burne which was published in 1847 and is color coded for easier understanding (although it can be hard the read as the letter s is often represented by the letter f. Ex an ifofceles triangle?) It may take me a year of work, but you can follow my progress and comment on my progress and point out my mistakes.  I am sure I will become a much more knowledgeable Geometry teacher and mathematician as will you.  Let’s go back to 300BCE and get started.

In my translation of the Elements, I 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 thought 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.

Note:  I am not doing proofs to describe constructions, but I will include them here.

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

As I mentioned above, Several early propositions are actually constructions.  Later on there are many mor, which I am excited to explore.  Book five has fifteen constructions and book seven has ten.  I chose not to convert constructions into two-column proofs for this post.  Proposition 1 describes how to construct an equilateral triangle which is helpful if you need a 60 degree angle.  I am imagining a lesson where you have the students construct a 30-60-90 triangle and measure the sides..

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.

QEF

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.

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