RUNNING THE AMAZON <i> by Joe Kane (Alfred A. Knopf: $19.95; 278 pp.) </i>

Adventurer Joe Kane tells the story of the first expedition to run the entire length of the Earth’s longest river, 4,200 miles from high in the Andes Mountains almost at the Pacific Coast across Brazil to the Atlantic.

In 1985, ten people--nine men and a woman from various countries--set out on the journey, brought together through several sponsors and for various reasons: scientific interest, filmic possibilities, reckless enthusiasm or habitual addiction to adventure. The group’s first challenge was finding the source of the river, a goal that has obsessed explorers from the time of the Spaniards. Kane weaves the history and legend of the South American interior into his text, as well as comments on the current situation (“In the four centuries since Pizarro’s butchery the Colca valley has regained a delicate equilibrium, though it is not nearly as productive as it once was. Now there is also a new joker in the deck, the $900 million Majes project, which, if ever completed, will pump Andean water westward over the mountains into Peru’s coastal plain”).

Their starting point located, the travelers set out in kayaks, and the trip begins in earnest. The Polish participant “each day prepared himself as if for combat. . . . He carefully applied a combination of lotions and sunscreens that looked exactly like war paint.” And combat it is. As the river becomes increasingly wild, the group breaks into those who run the rapids and those who elect to portage. Each day’s run is a dramatic tale: “Though it was midday, no sun reached the water. The river itself was a mess--fast, mud brown, roiling from the rain and still rising, studded with boulders that towered over our little fleet. This was the Acomamba Abyss.” Along the way there are the inevitable near-disasters, discomforts and triumphs. Kane carries off a terrific adventure story, a torrent of stories from the first dusty road to the final champagne drunk at the Atlantic.