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.

Introduction

While the notion of worlds beyond our Earth is ancient, the specific idea of planets orbiting distant stars is relatively new. Over two millennia ago Epicurus stated “there are infinite worlds both like and unlike this world of ours,” but he was not speaking of Earth-like planets orbiting Sun-like stars. Indeed, planets orbiting stars would have been a meaningless issue to the Greeks, as the Sun was not recognized as a star, nor the Earth as a planet (Chapter 1).

One of the earliest and most eloquent spokespersons for what is now called astrobiology, and among the first to grasp the implications of the Sun being a star and the Earth a planet, was the mystical Roman Catholic monk Giordano Bruno. In On the Infinite Universe and Worlds (1584) he wrote:

Bruno then concludes with the revolutionary slogan:

Bruno's reward for this prescience and for other heresies was condemnation by the Church, followed by immolation in a public square in Rome in 1600.

Introduction

Prokaryotic microorganisms were the only form of life for at least 80 percent of our evolutionary history (Schopf and Packer, 1987). Multicellular organisms including plants, animals and fungi evolved a mere 0.5–1.0 Ga from single-celled eukaryotic ancestors. Geologists and paleontologists debate the age of life on this planet and when the major microbial lineages first diverged (see Chapter 12 for details). Cyanobacterium-like fossils suggest that life emerged at least 3.45 Ga (Schopf et al., 2002; Schopf and Packer, 1987), but the biogenic origins of these structures are contested (Brasier et al., 2002; Section 12.2.1). The chemical record documents prokaryotic metabolisms that may have existed 3.47–3.85 Ga (Mojzsis et al., 1996) and eukaryotic biosignatures that may be as old as 2.7 Gyr (Brocks et al., 1999). Yet, these are still imprecise interpretations (some might be more recent microbial contamination) and do not set absolute limits on the possible origins of life on Earth. Early periods of heavy bombardment between 4.1 and 3.8 Ga might constrain when life first appeared on Earth, although microorganisms living off chemical energy at kilometer depths could have survived even the largest impact events.

By the standards of multicellular plants and animals, single-cell organisms look relatively simple (Patterson and Sogin, 1993), yet they transformed the atmosphere, the waters, the surface, and the subsurface of the Earth.

Introduction

Picture a future triumph in robotic space exploration: in a complex mission to Mars, a sample has been collected from the martian subsurface near a newly discovered hydrothermally active site at 30°N latitude. Ten years in the detailed planning and execution, the mission's Earth return capsule with its sample canister has landed in the Utah desert. The sample is now under extensive analysis and testing in an ultra-clean containment facility – and initial observations have shown positive indications that it contains life. Only after later testing is completed, checked, rechecked, and repeated is it shown unequivocally that the lifeform contained within the sample is a soil bacterium common to the dirt of an old Soviet launch facility in Baikonur, Kazakhstan, and which has apparently been alive on Mars since a spacecraft crash-landed there in 1972 …

Or picture, as did novelist Michael Crichton (1969) in the very year of the first lunar sample-return mission (Apollo 11), a spacecraft returning to Earth containing a dangerous extraterrestrial organism – The Andromeda Strain – not related in any way to Earth-life and operating by rules scarcely understood even after hundreds of humans have met their grisly demise …

Once you have those events in mind, you are developing a feel for what planetary protection might be, and what it is meant to prevent.