Persons exposed to high temperature, or to equivalent environmental factors, have quantifiable reactions, such as reducing the resistance to both heat and moisture flow in skin tissues and clothing needed to maintain thermal equilibrium. The one-to-one relationship between this resistance in the walking person and temperature, with the other factors neutral, is the basis for the apparent temperature scale and the derived heat index. When this approach is taken to assess the thermal environment for a still person exposed to heat in still air, there is a zone of ambient conditions in which there are three solutions to the heat-balance equation. Extraordinary thermal stress occurs, depending slightly on other conditions, at ambient temperatures near 41°C, especially at high humidity, because of the difficulty in carrying sweat vapor from the person when free convection is minimal. This anomaly is examined for a range of ambient vapor pressures and extra radiation. The rapid rise in heat stress when ambient temperature just exceeds body temperature in still conditions may explain the severity of some observed distress.