This book is a slightly condensed version of Sir Heath's two volume History of Greek Mathematics (originally published in 1922). In the preface he notes that the two volume set was primarily geared towards classical scholars whereas this set is essentially the same material condensed for the general reader.
I have not seen a copy of History Vol 2 but from looking at the first volume it seems that mostly what was removed is historical recount and interpretation (ie how one mathematician interpreted the work of another). While that commentary is interesting in some cases, all of the graphs and equations contained within the first volume are reproduced here so I would generally recommend this over the two volume set.
And the material that is here is very high quality and comprehensive in scope, covering everything from the pre-Pythagorean era to Plato and Euclid and beyond to Diophantus and algebra. There is also a decent amount of material on Egyptian and Arabic mathematicians in antiquity.
Highly recommended.