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Euclid and Geometry: Articles exploring Euclid’s definitions, postulates, common notions, constructions, proofs and the foundations of geometric reasoning.
Euclid’s Definitions Before we can understand the first definitions in Euclid’s Elements, we need to establish a frame of orientation. Geometry begins when the mind is able to distinguish position, relation, extent, boundary and form against a larger spatial field. Geometry does not begin only with circles and triangles drawn on paper. It begins with…
In the previous articles, we explored how Euclid sets a geometric world in motion through his axioms – or common notions as they are referred to in ‘The Elements’ – and his postulates. Once these foundations are in place, something remarkable becomes possible: form can be explored in a two-dimensional realm on a flat surface…
The conditions under which geometry becomes possible Introduction In the previous discussion, we examined the axioms of Euclid as general principles governing equality and comparison. These statements stabilise the field of reasoning. They establish the conditions under which relationships between quantities can be recognised and maintained, allowing a proof to proceed with continuity and coherence….
Introduction Having seen how axioms function at a broader level, both within mathematics and by analogy within human reasoning, we now return to them with a more focused lens. In the introduction to Euclid’s work in the previous post, the structure of geometry was set out as a whole, showing how axioms form the foundation…
