"... there is the general reader who has not lost interest in the studies of his youth and would wish to know how it came about that a Greek of the name Euclid wrote a textbook which, in an almost literal translation, was used in schools and in the universities of this country as the one recognized basis of instruction in elementary geometry, and on which generations of Senior Wranglers no less than average mortals were brought up, asking nothing better, until some fifty years ago."
(Ooff! People don't write sentences like that any more!)
He explains that half the book deals with Euclid and his precursors, and then expresses the hope that readers will want to know more: "And who, having got so far, will not wish to know what heights were scaled by Euclid's successors...?" Indeed!
It's not really a criticism of a book first published in 1931 to say that it is old-fashioned. Of course it is. As the preface makes clear, the first publication of the Rhind papyrus, with its revelations about Egyptian mathematics, happened in the 1920s, after the History was written and before the Manual. Crucial discoveries about the mathematics of ancient Mesopotamia were being made as Heath was writing. And much has happened among scholars of Greek mathematics during the last 70 years.
A particularly noticeable feature of Heath's approach is his exclusive concentration on the content of the major texts. There is very little here about the cultural context in which Greek mathematics was done, very little about the transmission (and reliability!) of the texts, very little about everyday mathematics in ancient cultures.
So while it doesn't count as a criticism, it is still important to say the book is old-fashioned, as a warning to its current readers. By all means use Heath's book, particularly if you would like a quick introduction to the more important mathematical texts. But look for newer books (for example, S. Cuomo's Ancient Mathematics, which despite its title is about Greek, Hellenistic, and Roman mathematics) to fill in the rest of the picture.
Fernando Q. Gouvêa is Professor of Mathematics at Colby College and the co-author, with William Berlinghoff, of Math through the Ages.