This section is from the book "Experimental Cookery From The Chemical And Physical Standpoint", by Belle Lowe. Also available from Amazon: Experimental cookery.
The freezing point of a liquid cannot be defined as the temperature at which the liquid becomes a solid, for supercooled liquids are cooled below the freezing point. The freezing point and the melting point are identical, so that the freezing point may be defined as the temperature at which the solid melts. The freezing point of a liquid may also be defined as the temperature at which the solid and liquid are in equilibrium. Here equilibrium means the temperature at which any proportion of solid and liquid can exist without change, that is, no solid melts and no liquid freezes. For water, this temperature is 0°C. Of course, equilibrium can exist for a long period of time only if the solid and liquid are completely insulated or if the temperature of the surroundings is at the freezing point of the liquid. If the temperature of the surroundings changes slightly above 0°C. so that heat is absorbed, some of the solid melts. If the temperature of the surroundings is below 0°C, so that heat is withdrawn from the mixture, the liquid freezes. After all the liquid is frozen, the temperature of the ice may drop below 0°C, just as the temperature of the ground or a stone may assume the temperature of the surroundings.
If ice and water that are not at the same temperature are mixed, they are not in equilibrium, since ice and water are in equilibrium only at the freezing point or 0°C. If the temperature of the ice is 0°C. and the temperature of the water is 20°C. some of the ice melts. In melting, each gram of ice absorbs 80 calories of heat, and the temperature of the water is lowered. If sufficient ice is added to the water, and the temperature of the surroundings does not influence the mixture, ice melts until the temperature of the water reaches 0°C. and equilibrium is established. If the quantity of ice added to the water is not sufficient to lower the temperature of the water to 0°C, the temperature of the water is lowered as long as the ice melts and provided the surroundings do not influence the temperature. After the ice is all melted the temperature of the water cannot be lowered except from the surroundings.