The story so far . After the defeat of his generals Datis and Artaphernes at Marathon in 490, Darius intended to invade Greece again, but was distracted by a revolt in Egypt in 486, during which year he died. His son by Atossa, Xerxes, succeeded him and crushed the revolt in 485. Xerxes spent four years preparing his expedition against Greece, the first act being the digging of a canal through the Athos peninsula in 483 (7.22). Late in 481, envoys were sent to demand ‘earth and water’ from the northern Greek states down to Boeotia (46.4n.). The army mustered in Cappadocia, and marched to Sardis, whence in spring 480 it began the expedition; the fleet collected at Abydos (7.20–40). H. gives a total of 5,283,220 men (7.186.2), a fantastic exaggeration no doubt, but indicative of the vast scale of the force. On the way, roads and bridges were constructed, and the Hellespont spanned by pontoons at Abydos (7.33–7). Progress was measured, partly because of the sheer numbers involved, and partly because Xerxes wanted to be able to use the crops in northern Greece to help feed his troops (7.50.4). Army and fleet advanced in contact with each other so as to coordinate their actions (7.236.2), but at the head of the Thermaic gulf in Macedonia, the land route diverged from the coast and they separated, reuniting at Aphetae on the Gulf of Pagasae, where the fleet is waiting at the start of book 8. H. does not tell us enough to be certain which route or (more likely) routes the army took. See map for possible solutions.

This book represents an expansion of the author's lecture notes for a course in Geometry, given in the second year of the Cambridge Mathematical Tripos. Geometry tends to be a neglected part of many undergraduate mathematics courses, despite the recent history of both mathematics and theoretical physics being marked by the continuing importance of geometrical ideas. When an undergraduate geometry course is given, it is often in a form which covers various assorted topics, without necessarily having an underlying theme or philosophy — the author has in the past given such courses himself. One of the aims in this volume has been to set the well-known classical two-dimensional geometries, Euclidean, spherical and hyperbolic, in a more general context, so that certain geometrical themes run throughout the book. The geometries come equipped with well-behaved distance functions, which in turn give rise to curvature of the space. The curved spaces in the title of this book will nearly always be two-dimensional, but this still enables us to study such basic geometrical ideas as geodesics, curvature and topology, and to understand how these ideas are interlinked. The classical examples will act both as an introduction to, and examples of, the more general theory of curved spaces studied later in the book, as represented by embedded surfaces in Euclidean 3-space, and more generally by abstract surfaces with Riemannian metrics.

The author has tried to make this text as self-contained as possible, although the reader will find it very helpful to have been exposed to first courses in Analysis, Algebra, and Complex Variables beforehand. The course is intended to act as a link between these basic undergraduate courses, and more theoretical geometrical theories, as represented say by courses on Riemann Surfaces, Differential Manifolds, Algebraic Topology or Riemannian Geometry. As such, the book is not intended to be another text on Differential Geometry, of which there are many good ones in the literature, but has rather different aims. For books on differential geometry, the author can recommend three in particular, which he has consulted when writing this volume, namely [5], [8] and [9]. The author has also not attempted to put the geometry he describes into a historical perspective, as for instance is done in [8].

MEDES AND PERSIANS

‘Darius the king says: this is the kingdom which I hold: from the Scythians who are beyond Sogdiana to Ethiopia, from Sind to Sardis’. Xerxes inherited from his father an empire that stretched from the Asia Minor coast to India and from the Caucasus to the Persian Gulf, and included Egypt. It far surpassed anything the Near East had seen before, and would not be surpassed in size until the Roman empire.

One unusual feature of this empire is that, despite the fact that it was the successor to the Elamite, Babylonian and Assyrian empires, which made much use of at least nominally ‘historical’ texts recording the deeds of their kings, the Persian empire has left us very little of the kind. There is only one document that can be described as a historical account of specific events, Darius' great inscription at Bisitun (DB = Brosius no. 44), which recounts his crushing of the revolts that greeted his accession to power. Other royal inscriptions list the peoples of the empire, describe the building of great palaces and outline royal ideology, but they do not concern themselves with specific events. Again, apart from the carving that accompanies DB, Achaemenid art does not use representations of individual events. Records were kept of battles, acts of benevolence towards the King etc., but these would have been on perishable material and have not survived (cf. 85.3 and n.).

In this edition I have had two intentions especially in mind: to try to bring to life for the reader the Achaemenid empire, and to offer a good deal of help with the grammatical aspects of the text. The first intention responds to a growing interest in Greece's relationships with the Ancient Near East, and will I hope prevent the commentary and its readers from taking too Hellenocentric a view of Herodotus' account. That Herodotus makes a strong distinction between ‘Greeks’ and ‘Persians’ is an idea that is slowly being revised, as the complexity of his presentation is more and more explored. The second intention responds to my experience at the JACT Greek Summer School, held annually now at Bryanston School, in Dorset. I am very grateful to my various students there not only for making it clearer to me what is required in a modern commentary on a classical text, but also for permitting me to try out on them earlier drafts of the commentary.

Although a new text of Herodotus, based on fresh study of the MSS and a consideration of the linguistic problems involved in constituting such a text, is much to be desired, the text offered here is not the result of a new inspection of the MSS, but aims to be an accessible and readable text.