A Formal System For Euclid's Elements, Jeremy Avigad, Edward Dean, and John Mumma. Review of Symbolic Logic, Vol. 2, No. 4, 2009.

Abstract. We present a formal system, E, which provides a faithful model of the proofs in Euclidâ€™s Elements, including the use of diagrammatic reasoning.

Diagrammatic languages are a perennial favorite discussion topic here, and Euclid's proofs constitute one of the oldest diagrammatic languages around. And yet for hundreds of years (at least since Leibniz) people have argued about whether or not the diagrams are really part of a formal system of reasoning, or whether they are simply visual aids hanging on the side of the true proof. The latter position is the one that Hilbert and Tarski took as well when they gave formal axiomatic systems for geometry.

But was this necessary, or just a contingent fact of the logical machinery available to them? Avigad and his coauthors show the former point of view also works, and that you can do it with very basic proof theory (there's little here unfamiliar to anyone who has read Pierce's book). Yet it sheds a lot of light on how the diagrams in the Elements work, in part because of their very careful analysis of how to read the diagrams -- that is, what conclusion a diagram really licenses you to draw, and which ones are accidents of the specific figure on the page. How they consider these issues is a good model for anyone designing their own visual programming languages.