The study of irrational numbers is thought to have begun with the application of the Pythagorean theorem to the diagonal of a square whose side is 1, resulting in a diagonal whose length is √2. By following the implications of this result to their logical conclusions, the side of the square is shown to be both odd and even, a proposition which would lead, if not to shipwreck, then surely to migraine in any rational Pythagorean. The discovery of irrationals, or what Euclid calls incommensurables, lead to a re-casting of geometric thought, which in turn produced Euclid's gargantuan Book X. The book contains 116 of the 450 Euclidean propositions and is veiled in a similar opacity as I described in my post on Book V.
The Euclidean Books of Lines, as I call them,—Books, V, VII, VIII, IX, and the beginning of X—use straight lines to represent number and magnitude. It is a simple enough system from which our contemporary use of x, y, etc. was developed, designed to steer clear of assigning any specific values to the formulas. For those of us who love the simple things in life, circles, triangles, rhombi, etc., the system of lines can feel more like an army of tiny little sabres slowly bleeding one to death. Take, for instance, Proposition x.10: To find two straight lines incommensurable, the one in length only, and the other in square also, with an assigned straight line. The traditional diagram for this proposition is pictured below, five straight lines of ambiguous length, standing in for the measures and magnitudes. I get intellectual brain freeze when I stare at these diagrams. I understand them, even crave them, but they make me hurt for the pleasure. Below the traditional diagram is an image of my sketched proof which I think is an accurate portrait of how my mind deals with these problems. I assign value and build the square, both of which go against the Euclidean grain.
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On other fronts, Travis Becker from Twinrocker Handmade Paper sent me a sample making of paper for the deluxe edition of Interstices & Intersections. He was trying to make a paper using cotton rag and abaca fibers that would approximate a linen and cotton paper I made with Mina Takahashi last year. The results were beautiful. Yesterday I proofed a variety of plates to test line quality and paper stretch and the sheets performed perfectly. In a few weeks Travis will begin work on the 800 sheets required for the deluxe copies.
The traditional diagram for Proposition x.10, using straight lines to represent number and magnitude.