On the swells of the Sea of Cortez, everything looks like a whale. But the suggestive shapes usually turn out to be whitecaps or a cloud’s shadow. Lulled by disappointment, the rocking boat and general monotony, I drift into torpor. Then, less than half a mile away, a series of unmistakable spouts erupts, and bursts of exhalation carry across the water.

The BIP XII, a trawler from Mexico’s Center for Biological Research, changes course and chugs toward a group of about 25 sperm whales—adult females, juveniles and suckling calves up to 2 years old. The calves and juveniles are 15 to 20 feet long, and some of the larger females are more than 30 feet from head to tail (a male would be almost twice as long). We approach one that appears to be sleeping, its rumpled back and bulging head rolling with the waves. It snorts awake and swims off as its companions drift away from us in loose pairs and trios. We trail after one of the pairs, a female and calf. The two idle along, nudging each other and blowing mist. Then the female surges forward. The huge muscles of her flanks go taut as she arches her back and heaves out her tail. Water cascades off her broad tail flukes, and she dives. The calf follows, Leviathan in miniature, its flukes aloft as it slides into the sea.

The other whales start to dive and our boat slows to a stop. The 12 of us aboard, a mix of biologists and crew members, gather at the rail to await the whales’ return. Five minutes turns into ten, then fifteen. Still they do not surface. We have a schedule to keep and so must motor on.

The life of a sperm whale remains largely a mystery. The animals spend most of their time at great depths, diving more than 6,000 feet in pursuit of prey and staying down for more than an hour. They are the largest toothed whales (a few filter-feeders, like the blue whale, are larger) and can grow to more than 60 feet long; their brains are larger than those of any other animal on earth. But even after decades...