This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is necessary (...) for this to work. Case-branching occurs when a construction renders a diagram un-representative. The roles of diagrams in reductio arguments, and of objection in probing a demonstration, are discussed. (shrink)

ABSTRACT In this article I argue that the methodology in the Johannes Kepler´s work is guided by two principles of Pythagorean nature, they are: sameness is made known by sameness, and harmony arises from establishing a limit to what is unlimited. By way of hypothesis, I present a general outline, which includes those principles, and I indicate how they are present in Kepler's investigations. I examine two particular cases in the light of that schema.