Exploring Archimedes' Quadrature of Parabola with GeoGebra Snapshots

摘要

This snapshot offers methods of solving the quadrature of parabola, the area of the region (parabolic segment) bounded by the parabola and a chord, using Archimedes' ideas of infinite sums and limits. In the first section, I provide the background of this problem along with relevant terminology and Propositions (1, 3, 19) from the Works of Archimedes. The second section illustrates an exploration of this problem based on snapshots used in GeoGebra technology, a dynamic geometry software (DGS) that intertwines algebra, geometry, and spreadsheets environments. The rationale for using a DGS in the exploration of the quadrature problem is founded in the view of experimental mathematics (Borwein and Bailey 2003; Borwein 2005; Sinclair 2008) in which the role of technology manifests in: "(1) Gaining insight and intuition, (2) Discovering new patterns and relationships, (3) Graphing to expose math principles, (4) Testing and especially falsifying conjectures, (5) Exploring a possible result to see if it merits formal proof, (6) Suggesting approaches for formal proof, (7) Computing replacing lengthy hand derivations, (8) Confirming analytically derived results" (Borwein 2005, p. 76). The role of GeoGebra manifested in the testing and especially falsifying conjectures in an attempt to test various mathematical ideas, which were reflected in the third section. GeoGebra's transformative role as a dynamic modeling mindtool (Jonassen 1996) had a crucial impact in setting the stage for justification of these mathematical ideas. In that sense, the snapshots used in the second section have been a useful asset in the discovery of new patterns and in the setting of the stage for formal proofs, which are presented in the third section.