''As a mathematician I did know whatever the medical examiner said would be undercounting it,'' said Dr. Steven Whitman, who was director of epidemiology for Chicago's Department of Public Health then. For each heat-related death, he said, the medical examiner must confirm one of three factors: a body temperature of 105 degrees or more, a high environmental temperature at the death scene, or decomposition in a body seen alive just before the heat wave.

But since autopsies are not performed on all bodies, not all heat-related deaths are recorded. In addition, fatal strokes and heart attacks attended by physicians are generally not counted as heat-related deaths, even if they probably would not have occurred without the heat.

For a more accurate picture, Dr. Whitman compared the number of deaths that month, around 3,000, with the expected number in an average month, 2,200, attributing the difference, or ''excess deaths,'' to the heat.

Other deadly heat waves in the United States occurred in New York City in 1972, when 891 died; in Los Angeles in 1955, when 946 died; and across the Midwest in 1980, when hundreds were killed.

Perhaps surprisingly, cities that suffer most in heat waves are not those in the hottest zones.

''A heat wave is a very relative factor,'' said Dr. Laurence S. Kalkstein, a climatologist at the University of Delaware Center for Climatic Research. ''What makes a heat wave in Duluth is not what makes a heat wave in Dallas. We respond to the normal weather conditions of wherever we live.''

A heat wave, as he defines it, is a variation from normal. Areas with irregular but intense variability, like New York or Chicago, where the temperature can vary by 20 degrees or more in a few days, fare worse than places like Phoenix or Miami, where the weather is warmer but more constant.

''In New York we once calculated a threshold of 93 degrees for a few days, whereas in Phoenix it might be 112,'' Dr. Kalkstein said.