See our review of the first edition. There are three major changes in the new edition:

In Part I, a whole new chapter has been added to cover Arabic Mathematics. This was clearly a huge gap in the original account of the history of geometry, so this addition is very welcome.

In Part II, the author has added a very interesting new chapter on Polyhedra, Tesselations, and Symmetry Groups.

The book now contains exercises, many of them with hints or solutions.
There are new illustrations, including a few in color. Of course, many errors have been corrected, and new material has been added to several chapters.
The historical section has been enriched, both by adding new material and by taking history itself more seriously. For example, the presence of chapter 20, on polyhedra, has allowed the author to expand the discussion of polyhedra in the historical portion as well, including a discussion of the Archimedean semiregular polyhedra. Chapter 2, on "the great river civilizations," now includes a discussion of the debate on how to correctly interpret the Plymptom 322 tablet. This is very welcome, both as a way of alerting mathematicians to the fact that many historical issues remain unsettled, and as a way to signal to students that in history things cannot be settled definitively.
The addition of exercises makes the book more suitable for textbook adoption. It contains much more material than one could reasonably cover in one semester, but with careful selection one could construct several very interesting courses based on it.
Overall, this strikes me as a successul revision, significantly improving the book.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College and is the editor of MAA Reviews.