We present a formal system, E , which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.
Though pictures are often used to present mathematical arguments, they are not typically thought to be an acceptable means for presenting mathematical arguments rigorously. With respect to the proofs in the Elements in particular, the received view is that Euclid's reliance on geometric diagrams undermines his efforts to develop a gap-free deductive theory.… (More)
In his writings on Hilbert's foundational work, Paul Bernays sometimes contrasts Euclid's geometrical method in the Elements with Hilbert's in Foundations of Geometry, identifying two features that distinguished the latter from the former (notably, in  and in the introduction to ). First, Hilbert's theory is abstract. Though the primitives of the… (More)
For more than two millennia, Euclid's Elements set the standard for rigorous mathematical reasoning. The reasoning practice the text embodied is essentially diagrammatic, and this aspect of it has been captured formally in a logical system termed Eu [2, 3]. In this paper, we review empirical and theoretical works in mathematical cognition and the psychology… (More)
'Now, in calm weather, to swim in the open ocean is as easy to the practised swimmer as to ride in a spring-carriage ashore. But the awful lonesomeness is intolerable. The intense concentration of self in the middle of such a heartless immensity, my God! who can tell it? Mark, how when sailors in a dead calm bathe in the open sea—mark how closely they hug… (More)
* Widely fluctuating numbers of deer apparently characterize mule deer populations. Numbers have varied considerably in historic times, primarily in response to climatic fluctuations, habitat change, and market hunting. Apparently mule deer were common but not abundant prior to European American settlement of the West. They were abundant in the late 1800s… (More)