Abstract: A thermostat is described for controlling a furnace for a hot air, hot water, or steam heating system which delivers heat to a heated space via a delivery system. The thermostat causes the system to conserve energy by limiting the "burn" or on-time cycle to a system-specific interval during which the heat exchanger of the furnace operates in its linear region. At the end of this interval (designated a "MAX.sub.-- ON" interval), the burn cycle is terminated, but delivery of heat to the heated space continues for a secondary-delivery interval during which the furnace's blower (or other delivery means) continues operation. The secondary-delivery interval ends when the residual heat has been extracted from the furnace and delivery system. The system then remains off until the next burn cycle begins. The duration of the off-time interval is such that heat input to the heated space and heat outflow to the ambient from the heated space are kept in equilibrium.