**BEST STUDENT PAPER OF DIAGRAMS 2020**

**A Diagram of Choice: The Curious Case of Wallis’s Attempted Proof of the Parallel Postulate and the Axiom of Choice**

*ValÃ©rie Lynn Therrien*

**Abstract.**

Wallisâ€™s attempted proof of Euclidâ€™s Parallel Postulate is an important but oft neglected event leading to the discovery of non-Euclidean geometries. Our aim here is to show Wallisâ€™s own reliance on three non-constructive diagrammatic inferences that are not (fully) explicit in his own supplement to Euclidâ€™s axioms. Namely, there is i- an implicit assumption concerning the possibility of motion; ii- an implicit assumption about the continuous nature of space and time; and iii- an explicit assumption about the existence of similar triangles which conceals an appeal to a combinatoric principle of reasoning which is tantamount to appealing to the Axiom of Choice.