Fatal attacks, however, have decreased. Forty years ago more than half of all attack victims died; today more than four out of five victims survive. Improvements in communications and emergency medical care have, of course, saved many lives, but Peter Klimley believes that in several cases the sharks have simply changed their minds. “Can you imagine?” he said. “These sharks are seizing people and holding them to make this decision. They strike and hold and release.”

At the moment science accepts about 400 species of sharks, but the number changes as new species are discovered. Of all known species, only four attack human beings with any frequency: bull sharks, tiger sharks, oceanic whitetips, and great whites.

The old adage is true: A swimmer has a better chance of being struck by lightning than killed by a shark. And around the world many, many more people die every year from bee stings, snakebites, falling off ladders, or drowning in bathtubs than from shark attack … none of which, to be sure, detracts from the ghastly, visceral horror of being eaten by a huge fish, but all of which should give some comfort to the recreational swimmer.

In Australia, between 1876 and 1999, 52 attacks by great whites were recorded, and of them 27 were fatal. In the Mediterranean Sea since 1900 there have been 23 reliably recorded encounters with great whites, including one in 1909 in which the remains of two adults and a child were found inside a single 15-foot-long female shark caught off Augusta, Sicily.

Curiously, there has been relatively little progress over the past 50 years in the development of shark repellents. Dyes have been tried; so have chemicals and bubble curtains. The current state of the art uses electricity. So far, though, nothing has been proved to discourage a hungry great white in full attack.

We went to South Africa in the southern winter because of a phenomenon long known to locals but only recently deemed significant by researchers: In wintertime great whites gather in large numbers in a few bays in a little pocket between the Cape of Good Hope and Danger Point.

“The warm Agulhas Current sweeps down the east coast of Africa, out of the Indian Ocean,” Andre Hartman explained, “and meets the cold Benguela Current that flows up the west coast. Where the currents mix, there’s a temperate zone in False Bay, Gansbaai, and Kleinbaai, and sharks gather there in the winter. The temperature seems to suit them.”

So does the abundance of yearling South African fur seals, which swarm over rocky islands in the bays and provide a smorgasbord for the sharks.

We drove from Cape Town to False Bay, so named perhaps because early explorers who rounded the Cape of Good Hope thought that their next landfall would be India—until they bumped into the eastern shore of the bay, 20 miles away. Here Robert Lawrence and Chris Fallows run a modest cage-diving operation. For roughly a hundred dollars the young men will take visitors and scientists a few miles offshore and show them great white sharks from their choice of shelters: a small boat or an even smaller cage. They promised to show us behavior that, till now, we all had believed was the stuff of extreme rarity, if not outright legend.