Explore Propositions VI to VIII of Book 1 in Euclid's Elements, examining the first instances of proofs by contradiction in mathematics. Delve into the logical foundations of geometry, questioning the suitability of Euclid's work for modern times. Analyze each proposition in detail, including the proof that equal angles in a triangle are subtended by equal sides, the impossibility of constructing certain equal line segments, and the congruence of triangles with equal sides and bases. Critically evaluate the logical issues present in these proofs and consider alternative foundations for modern geometry. Gain insights into the historical development of mathematical reasoning and its relevance to contemporary mathematical practice.
Euclid Book 1 Props VI-VIII - A Foundation for Geometry - Sociology and Pure Maths - N J Wildberger
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Overview
Syllabus
Intro
Elements Book 1 Prop 6 - If two angles of a triangle are equal, then the sides subtending the equal angles will be equal.
Elements Book 1 Prop 7 - On the same Right Line cannot be constructed two Right Lines equal to two other Right Lines at different points on the same side, and having the same Ends which the first Right Line has.
Elements Book 1 Prop 8 - If two Triangles have two Sides of the one equal to two Sides of the other, each to each, and the Bases equal, then the Angles contained under the equal Sides will be equal.
Logical Issues
Q: If Euclid's Elements are not really a proper logical foundation for geometry - then what is?
Taught by
Insights into Mathematics
