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Abu Kamil al-Khwarizmi algebraic altitude angle anthyphairesis appears application of areas approximations Arabic Aristaeus Book century Chapter chords circle circumscribed commentary concept of DEMR consider construction cube decagon defining pentagon DEMR development of DEMR diagram diameter discussed divided in EMR division dodecahedron edge Elements equal equation Euclid Euclid-Heath Euclid's Elements Eudoxus example extreme and mean Fibonacci Figure Francesca Gandz geometrical given gives gnomon golden number Greek mathematics Heath hedron Hippasus Hypsicles icosahedron incommensurability indicated inscribed involving DEMR irrational Kepler Knorr larger segment lemma mathematicians mean ratio mentioned method Note obtain original Paccioli Pappus parallelogram pentagon pentagram Plato polyhedra problem Proclus proof proportion Proposition Proposition 58 Ptolemy Pythagorean Theorem radius rectangle references regular pentagon relationship result right triangle S(AB S(AC S(CB says Scholium Section side and diagonal solution sphere square statement Subsection Theaetetus theory Timaeus tion various Waerden