On a given straight line to construct an equilateral triangle.

The beauty of this proof is what first captured my imagination. Having very little idea of what Euclid was about, I read the introduction and some of the axioms. Then I turned to the first proposition. The pure, unadulterated beauty of it hit me like a punch in the face. You have a line running from point A to point B. That’s all you have. How will you create an equilateral triangle, using that line as one of the sides and only with your trusty compass and straightedge?

My drawing and notes from the first proposition.

The drawing of the proof itself was beautiful to me, as was the simplicity of it’s solution. Remember that Euclid always states the obvious. You could draw a shorter version of this proof by drawing two short, intersecting arcs with your compass at point C. But go ahead and draw the circles. You need them to understand. And even if you think you understand how it works or intuitively understand it, take the time to read the proof. Remember we’re here not to find shortcuts to drawing triangles but to think philosophically and artistically about proof and demonstration.

Here is the step-by-step method:

1. First set your compass to the width of AB.
2. Describe circle BCD with A as its center.
3. Describe circle ACE with B as its center.
4. Mark one of the intersection points of those two circles as point C.
5. Connect AC and BC with a straight lines.
6. Triangle ABC is equilateral.

Why does it work?

Because AC and AB are radii of circle BCD, they must be equal. And because BA and BC are radii of circle ACE, they must be equal. And if AB = AC, and AB = BC, then AC=BC, according to Euclid’s first axiom: “If one thing is equal to another and that to a third, the first and third are also equal.”

And a triangle with three equal sides must be equilateral.

"Being what was required to do."

I love the Heath translation of Euclid's closing remark. Every proposition, when proven, ends with “Being what was required to do.”

It’s as if Euclid was saying, “Imagine that there was such a thing as a perfect circle. And imagine we could draw a perfectly straight line, ending in two points that would be true points, that is having no height or length or breadth. Can you imagine such things?”

“Yes, Euclid, we can,” we might answer back. “In fact, we’ve been imagining such shapes all of our lives, though we’re not sure why.”

“Good,” Euclid would say. “So if you had a straight line and could set a perfect compass to its endpoints and draw perfect circles, you could create a perfect, equilateral triangle.”

Perhaps you and I would frown a bit at this point and say, “Euclid, are you saying that if we had these perfect things, which we do not have, and if we could set them perfectly and draw perfectly, which we cannot, we could draw a perfect triangle, which in reality cannot be drawn?”

Euclid would beam with pleasure. “EXACTLY!”

If you and I then asked Euclid why we would bother or even care about this, I think he would say, “Because isn’t it perfectly lovely to think about these things and to see them in your mind and know their perfect symmetry and perfect rules?”

And you and I would either “get” that or we would not.

What are these shapes and lines and points? And why do we think on them from an early age and hold them so dearly in our hearts that we require they be taught to every child in elementary school? Learning shapes is one of our first lessons. Why do we think on what we do not have? Why do we imagine and adore a perfection that does not exist? Why would this book be written and why would it be handed down through the ages?

This is what I'm asking. My quest is not to draw shapes but to understand where the structure comes from and why we think about them at all.

rlp

NOTE: Here is a wonderful and legal .pdf version of all 13 books of Euclid, for those who would like to see for themselves without buying a book.