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Most people feel that the times in which they live are special: the best of times, the worst of times, and sometimes both together. Historically, such feelings have been particularly acute around years with lots of zeros at the end. So a sceptic, noting the approach of the year 2000 in western cultures, may be wary of books with titles like The Sixth Extinction, whose concluding words are: "For each of the Big Five [episodes of mass extinction seen in the fossil record] there are theories of what caused them, some of them compelling, but none proven. For The Sixth Extinction, however, we do know the culprit. We are."

After all, the partially informed sceptic may ask, is not extinction the fate of essentially all species that ever lived? Are not origination and extinction the Yin and Yang of evolution? Is it not true that new species exploit the opportunities created by the disappearance of earlier ones, in a self-amplifying process of relay and replacement that has carried us from the simple unicellular organisms at the dawn of biological history to the abundant diversity of living forms that enrich the earth today? Given all this, some have argued that we should not be disturbed by current extinctions, but should view them simply as part of life's rich tapestry, a way-station, like others in the fossil record, on a natural journey to a different but interesting future.

The great strength of Richard Leakey and Roger Lewin's book that sets it above many other recent books on the same broad theme is that it deals very clearly with such questions. And it leaves no doubt that, millennial resonances or not, we truly are living in a singular time. The sixth extinction of the title will be unlike all previous extinction events, having no parallel in the history of life on earth.

The book begins with an account of the sweep of the evolution of life on earth, with particular focus on the Big Five episodes of mass extinction seen in the fossil records. These roughly correspond to the ends of the conventionally defined Ordovician epoch (440 million years ago, or mya), the Devonian (365 mya), the Permian (225 mya), the Triassic (210 mya), and the Cretaceous or end of the age of dinosaurs (65 mya). In each of these great convulsions, two-thirds or more of all extant species were extinguished. The End-Permian episode, which is taken as marking the end of the Palaeozoic or "old era of animals", saw the vanishing of 95 per cent of marine animal species. Leakey and Lewin give a vivid survey of current ideas about the environmental events that may have caused the Big Five. But the jury is still out, and we do not yet have a verdict. Asteroid impacts, changing climate, effects of continental drift and other ideas still contend. The truth could yet involve combinations of causes.

Leakey and Lewin also note that the Big Five are interspersed with a spread of lesser events, on ever finer time scales; a fully detailed record of extinction events over the sweep of geological time arguably has a fractal structure. There is no agreed explanation for this, either.

Against this background, there follow chapters on the evolution of Homo sapiens, on the concept of "progress" in evolution, and on the nature of diversity. Most readers of this review will be familiar with the brilliant essays of Stephen Jay Gould, one of whose abiding themes is the accidental nature of evolutionary history and the concomitant absence of any evidence for progress or advance. I agree with Leakey and Lewin when they acknowledge the role of chance in evolution, but differ from Gould in reading the evolutionary story as one of increasing complexity. They do this partly in general terms, but mainly by focusing on neurological complexity or, more baldly, brain size in animals. Gould may be more politically correct to deny humans any evolutionarily special status, but Leakey and Lewin differ in seeing progressive complexification as an evolutionary theme, culminating in "the undeniable superiority of the human brain I in this sense, then, humans are the pinnacle of evolution, the highest expression of biological computation". They approvingly quote E. O. Wilson: "Let us not pretend to deny in our philosophy what we know in our heart to be true." Personally, I think the sentiment in Wilson's seductive quote is dangerous; in this case, however, I think heart and head are in unison.

The Sixth Extinction next gives an excellent overview of current ecological thinking about how communities of plants and animals function, and how they cope with natural or human-created disturbance. We see how wistful older notions of the "balance of nature" have given way to a recognition of the dynamic character of most ecosystems, with ceaseless fluctuations in the abundance of particular species driven by external environmental events and/or by internal interactions among populations (which can cause cycles or even dynamical chaos).

The book's many strands are woven together in its final chapters, which deal with the eponymous Sixth Extinction.

The main trouble with all discussions of current and likely future extinction rates is that we know so little about the creatures we share the world with. As Leakey and Lewin make clear, we do not know to within an order of magnitude how many species there are on earth today. Estimates of the true total range from three to 100 million or more. My own guess would be at the low end, in the three to eight million range. The number of these species that have been named and recorded is smaller, around 1.7 million. Even this smaller number is uncertain to within 10 per cent or so because we lack centralised databases for most groups, and especially for the most diverse groups like insects or nematode worms. Many of the recorded names are synonyms - the same species unwittingly identified under two or more separate names by different people in different places at different times - so the actual number recorded may be 1.4 million or fewer.

Such ignorance is astonishing. In large part, it reflects ill-understood vagaries of intellectual fashion. Systematic study of the earth's richness of living things comes comparatively late in the history of science. The starting date may be taken as 1758, when Linnaeus published his Systema Naturae, recognising some 9,000 species of plants and animals. This is almost a full century after Newton, building on two millennia of systematic studies of the heavens, had given us a basic understanding of the laws of motion.

Whatever the historical reasons for our lack of knowledge about the numbers of species alive today, one obvious consequence is that estimates of the numbers becoming extinct is even more uncertain. Leakey and Lewin cite guesses ranging from 17,000 to 100,000 or more species going extinct each year. These are derived by pyramiding conjectures about the consequences of rates of tropical deforestation on already uncertain estimates of the numbers of living species.

As a sometime theoretical physicist, I find these estimates a bit exasperating. Although our ignorance of species numbers makes it impossible to be accurate about absolute numbers of extinctions, there is an alternative approach that enables us to draw fairly precise conclusions about current extinction rates relative to the average rates seen over the sweep of geological time. To see how this method works, consider first the question, how many of the plant and animal species ever to have lived are alive today? Leakey and Lewin quote an estimate of 0.1 per cent, deduced by dividing a guess that 30 million are alive today by someone's wild guess that 30 billion species have existed in total. But in the next paragraph, they cite the estimate by David Raup of the University of Chicago that the average lifetime of a species in the fossil record, from origination to extinction, is roughly four million years. This is a much more secure figure than guesses at species totals, being based on detailed studies of particular species, of various kinds, in the fossil record. As discussed by Leakey and Lewin, the Cambrian explosion in diversity of multicellular organisms, which occurred a bit less than 600 million years ago, may for most practical purposes be taken as a starting date. With an average "species lifetime" of four million years, there thus have been about 600/4 or 150 "generations" of species. On this basis, we would estimate that extant species represent a little less than 1 per cent of those ever to have lived. This assumes that the total numbers of species have been roughly constant over this interval of 600 million years or so. In fact, the fossil record shows a very roughly linear rise in total species abundance, albeit with many interruptions and fluctuations. As we lie at the high end of the rise, we should probably double the 1/150 estimate, to conclude that the species alive today represent 1 to 2 per cent of those ever to have graced the planet.

Admittedly this kind of estimate has its own faults. For one thing, species' lifetimes from origination to extinction vary greatly, both within and between groups, so there is much variation about the four million year average. Some think this figure is a bit high, although for insects - the most numerous group today - it could be low. For another thing, such species' lifetimes derive from a fossil record that in total includes only 250,000 species from the much larger total that must have existed. Moreover, 95 per cent of these species are shallow-water organisms, which contrasts with today's species, more than half of which are terrestrial insects. Even so, we shall now see how this "species lifetime" approach enables us to say quite sharp things about current and likely future extinction rates.

The International Union for the Conservation of Nature has for the past 20 years or more published Red Data Books, recording extinctions and also identifying species that are "endangered" or "vulnerable" according to specific criteria. These latter categories find applications in, for example, the Convention governing International Trade in Endangered Species (Cites). The list of extinct species at first sight looks reassuringly small: a little more than 200 species of vertebrates over the past two centuries or so. A second look reveals its inadequacies: only 61 species of insects certified extinct, and not one of these from the tropics. The rigorous standards needed for certification of extinction essentially exclude all but the most charismatic and best-studied groups of organisms. The furries and featheries, mammals and birds, receive vastly more attention than other groups. This, however, enables us to draw interesting conclusions. Among all birds and mammals, on average roughly one species has been certified extinct each year over the past century. There are about 4,000 species of mammals and 9,000 of birds, making a total of 13,000 species. We can turn this around, to say that if roughly one in 10,000 species of birds and mammals goes extinct each year, then today's life-expectancy of an average bird or mammal species is around 10,000 years. This may sound a long time, but it is 400 times shorter than the four-million-year average seen in the fossil record.

Even among the comparatively well-studied birds and mammals, certified extinctions are known to be an underestimate. Many tropical birds have not been seen for 20, 40, 80 years but have not been subject to the rigorous scrutiny needed for a certified extinction. If birds and mammals are representative of species more generally - and there is no reason to doubt it - then extinction rates over the past century have run around 1,000 times higher than the background rate in the fossil record.

What about coming centuries? Here we cannot escape the need for indirect estimates. Until recently, all such assessments were based on the ecologists' so-called species-area rule. This rule, which is grounded both on empirical evidence and some theoretical understanding, tells us that if we compare the number of species of beetles or plants or whatever on two islands in an archipelago, one of which is ten times bigger than the other, then in general we will find twice as many species on the bigger island. A rather shaky extension of this rule suggests that if 1 per cent of the tropical forest is destroyed each year, then about 1/400 of the resident species will thereby be committed to eventual extinction. Again, we can invert this, reinterpreting it to say that the present life expectancy of a typical such species has shortened to around 400 years.

Two alternative approaches are too recent for Leakey and Lewin's book. One, by myself and colleagues, uses estimates of the average rates at which species in better studied groups (birds, mammals, palm trees) are "climbing the ladder" of IUCN categories of threat, from unendangered to vulnerable to endangered to extinct. These rates are low, but nevertheless suggest the average species in these groups may reach the top of the ladder and fall off into oblivion in around 100 to 400 years or so. Georgina Mace at the London Zoo has used more precise studies of extinction probabilities, based on groups of interest to Cites, to estimate how long before half the species in each of ten vertebrate taxa will be extinct (3, 4, 3 orders or families of reptiles, birds, mammals, respectively). The numbers range from 100 to 1,000 years, averaging around 300 to 400 years for the mammals and birds. All three methods, each different from the other, agree in suggesting a shortening of species life-expectancies from the previous century's 10,000 years to a few hundred years.

It is important to realise that we can say quite precise things about the dramatic recent acceleration in extinction rates. Too much conservation literature gives a misleadingly woolly impression, by focusing on absolute numbers, rather than on relative rates.

In short, over the past century extinctions have been occurring at rates about 1,000 times higher than the average background in the fossil record. And the coming century will see these rates notch up by a further factor of more than ten. Such extinction rates are fully comparable with those associated with the Big Five mass extinctions in the geological record.

As Leakey and Lewin conclude by emphasising, this sixth wave of extinction, on whose breaking tip we stand, differs from the previous Big Five. Although details are uncertain, they were all caused by environmental changes of one kind or another. The sixth extinction is unambiguously caused by the activities of a single other species, whose numbers and associated impacts have grown to rival the scale of the natural processes that built and maintain the biosphere as a place where life can flourish. Homo sapiens sequesters around 40 per cent of all net global primary productivity on land, and about one tenth of primary productivity in the seas.

These facts distance the coming sixth extinction from its five predecessors, and undercut any comparisons with their aftermaths. Maybe we will lurch into some form of sustainable future, in a world like that of the cult movie Blade Runner, a teeming technological culture on a biological impoverished planet. Perhaps the current and growing perturbations to the balance of the biosphere - greenhouse effects and ozone holes - portend a grimmer future.

It may even be that the typical evolutionary fate of inhabited planets, if other inhabited planets indeed exist, is for one species to gain a technological mastery that runs disastrously ahead of its wisdom in the application of such mastery. We do not know enough to answer any of these questions.

Sir Robert May is chief scientific adviser to the Government, on leave from his Royal Society professorship in the department of zoology, University of Oxford.