Euclid and van Hiele
Chapter 2
Euclid, van Hiele levels, and GSP
Euclid and van Hiele
Euclid was historically the first person to have achieved van Hiele level 4. He was the pioneer of the axiomatic approach as elucidated in his Elements. His ingenuity and prophetic insight in the recognition of the necessity to formulate the statement on parallel lines as an axiom (the fifth postulate) are witnessed by the failure of attempts in the subsequent two millennia to prove the fifth postulate, only ending with the invention of non-Euclidean geometry in the nineteenth century. His careful avoidance of the use of the fifth postulate in the early propositions in Book I shows his awareness of how changes in axioms affect the resulting geometry (van Hiele’s descriptor 4.2). This helps isolate certain geometric propositions which form a core of all geometries (absolute geometry) regardless of the fifth postulate.
Yet as a master teacher Euclid taught in language that beginners understand. The definitions in his books are formulated in ways that make sense to people without prior knowledge of geometry (with van Hiele level 0). For example, his very first few definitions in Book I, far from being technical definitions, convey the essence of points, line, and surfaces in terms understandable by beginners.
Euclid, however, did not let his students stay long at van Hiele 0. He lifts his readers up, by teaching them how to reason and organize knowledge. His formulation of the postulates and the common notions are important organizing principles of basic knowledge. Three of the first five postulates are basic methods of constructions. The common notions are basic recognition and reasoning principles. Learners of geometry should quickly get into van Hiele level 1 as soon as possible. The development of the propositions in the several books gradually bring readers into van Hiele levels 2 and 3.
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