This book is suitable for a course or seminar at the undergraduate level, though it will interest graduate students and professors as well. The prerequisites are only elementary geometry and algebra. This book describes the evolution of geometric ideas in the abstract world of geometric objects. This is not a packaged summary, but instead a journey of creative discovery that will inspire both students and teachers.

This is an overview of the history of ancient Greek geometry. It highlights the major Greek geometers and their work. We also explore and trace the history of many specific topics of interest.
•The evolution of the Pythagorean theorem.
•Formulas to generate integer Pythagorean triples.
•Attempts to solve the classical problems of trisecting an angle, squaring a circle, and doubling a cube.
•Various mechanical "neusis" methods that were used to solve the three classical problems.
•The determination of what is a valid proof.
•The discovery of the conic sections (parabola, hyperbola, and ellipse), and how they solved the classical problems.
•The evolution of the method of exhaustion, leading to integration techniques.
•The discovery of pi.
•The discovery of other useful esoteric curves, including the quadratrix, spiral, conchoid, and cissoid.
•The evolution of spherical geometry, and trigonometry.
•The birth of projective geometry.
•And much more.

Instead of being a chronology of events, this book gives a chronology of people in a sequence of biographies focused on their contributions to geometry. So, there is little about their lives other than their works. This makes the flow of Greek discoveries between individuals and the evolution of geometry more transparent. We begin with the pre-Satics (including Thales and Pythagoras) who started the millennium of Greek geometry. We then continue with 13 geometers who preceded Euclid (including Hippates, Plato, Theaetetus, and Eudoxus) culminating with Euclid and his Elements of Geometry.

After Euclid, a new age of geometry began with Archimedes who was followed by 10 geometers (including Apollonius, Ptolemy, and Pappus). During this period, Archimedes discovered the constant pi, new semi-regular polyhedrons, and new methods of integration. And new curves were discovered such as Archimedes' spiral, Nicomedes' conchoid, and Diocles' cissoid. Also, Apollonius' Conics, Ptolemy's Almagest on astronomy, and Pappus' Collection were written. We give detailed descriptions of these mathematical masterpieces and their main results.

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