My Honeywell MagicStat thermostat indicates 'system on', but my furnace is not on and the current temperature in my house is 10 cooler than what I have programmed it to be (during the heating season). I have found turning the thermostat to 'off' and then back to 'auto' gets the furnace to start, but the problem persists the next time the temperature falls below the programmed setting

What brand furnace do you have? Look in the bottom door for a flashing light.