Geometry: Euclid and Beyond

by Robin Hartshorne

In recent years, I have been teaching a junior-senior-level course on the classi­ cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa­ rately. The remainder of the book is an exploration of questions that arise natu­ rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks.

Genres: Mathematics, Geometry, Science, Nonfiction, Textbooks

Kindle Edition

First published January 1, 2000

Reviews

[Parsa]

The first chapter examines Euclid chapter by chapter, hinting at the gaps in his proofs(starting from the first proposition, how do we know that the two circles actually meet? or the fourth proposition, how do we justify the method of superposition?)
The second chapter offers a modern look on Euclid, using Hilbert's axioms. Some of these axioms are modern re-interpretations of Euclid's axioms, while some were not mentioned in Euclid, and some are actually theorems in Euclid(SAS is taken as an axiom in Hilbert's system) Using these axioms, we can recover Euclid, with some reinterpretations, and some alternative proofs.

[Diem · Rating 5 out of 5]

Not really fair to say I read this. I used the intro and first chapter for guidance through the first four books of Euclid's "Elements". This would be a rigorous and interesting text book for a geometry student. I can see why it is considered one of the best of its class.