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I am drawing a $19 \times 19$ grid on my desk. For aesthetic purposes, I don't want to use a ruler. Rather, I want to use Euclidean theorems to 'prove' to myself that such and such line meets at a right angle.

I have already marked out the four points that determine the edges of the roughly square rectangle that will contain the grid. I imagine there is some chapter of the elements that would contain a proof that two lines are perpendicular and at right angles to one another.

I imagine myself being able to apply that theorem to each intersection on the grid, piecemeal, in order to 'grow' it, starting at an arbitrary edge, or perhaps starting at all four and working toward the center.

Of course, this is all for fun, and because I love the thinking style of the Elements. But how would I use the book to do that. What process would I use for 'proving' $90^\circ$ perpendicularity, taking each intersection in turn, like an automaton?

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Before you prove anything, you have to have a method of drawing. What do you want to use instead of the ruler? A Peaucellier linkage? That was way after Euclid's times. – Phira Dec 1 '11 at 17:45
I have to admit that I am seriously toying with making a youtube clip on "How to draw a straight line without a ruler using Euclid's elements" where the book is used as a ruler. – Phira Dec 1 '11 at 17:46
(I'm assuming you are using a straight edge, a.k.a. an unmarked ruler, to aid in the drawing of straight lines.) I can't see you needing anything outside of the first book. Note that the Elements contains many instructions for producing an object with desired properties, and also a proof that they work, so that as long as you follow the instructions, you will produce the desired object without any further argument needed. Assuming, that is, that you can draw a perfectly straight line, a perfect circle, etc. – Arthur Fischer Dec 1 '11 at 18:16
@Phira I'm using a knotted cord, ancient Mesoamerican-style. – ixtmixilix Dec 1 '11 at 21:13
i'm not actually sure this requires euclid's elements. it seems to me that i just needed the parallel postulate. i couldn't write a proof of that, though. not yet anyway. i'll have to put it in my list of things to consider. – ixtmixilix Feb 3 '12 at 2:18

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