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Alcoholic proof is a measure of how much ethanol there is in an alcoholic drink. It is commonly used in the United States, where it is about double the percentage of alcohol by volume (ABV or Alc/Vol).
Where it comes from
This system was introduced in the 18th century. It might have happened earlier, when spirits were tested with gunpowder: a mixture of water and alcohol proved itself when it could be poured on a small amount of gunpowder and still light up the wet powder. If the powder did not ignite, the mixture had too much water in it and the proof was considered low. This process led to the nickname firewater, given by Native Americans.
A "proven" solution was defined as 100 degrees proof (100°). People have found out that this takes 57.15% ethanol. This value is still used as the British definition. A simpler ratio to remember is seven to four: 70° proof is 40% alcohol by volume.
A hydrometer can be used to measure the precise proof of a spirit. This test has gone through many formal changes.
British proof spirits
In Britain, this replaced the Sikes hydrometer system (based on proof spirit) which was used since 1816, although officially the Customs and Excise Act of 1952 defined "spirits of proof strength" (or proof spirits):
- "Spirits shall be deemed to be at proof if the volume of the ethyl alcohol contained therein made up to the volume of the spirits with distilled water has a weight equal to that of twelve-thirteenths of a volume of distilled water equal to the volume of the spirits, the volume of each liquid being computed as at fifty-one degrees Fahrenheit."
In the definition of the United States, the proof number is twice the percentage of the alcohol content measured by volume at a temperature of 60°F (15.5°C). Therefore "80 proof" is 40% alcohol by volume (most of the other 60% is water). If an 150 proof beverage is mixed half-and-half with water, the drink is 75 proof.
US Federal regulation (CFR 27 5.37 Alcohol Content) requires that liquor labels state the percentage alcohol by volume (sometimes abbreviated ABV). The regulations permit (but do not require) a statement of the degrees proof as long as it is right next to the percentage alcohol by volume. 
Alcohol during production
Alcohol is produced by yeast during the process of fermentation. The other product of fermentation is carbon dioxide, which is the gas that can make beer bottles explode or blow their tops off. The amount of alcohol in the finished liquid depends on how much sugar there was at the beginning for the yeast to convert into alcohol. In beer, the alcohol is generally 3% to 12% (6 to 24 proof) and usually about 4% to 6% (8 to 12 proof). Depending on the strain of yeast, wines top out at about 14% to 16% (28 to 32 proof), because that is the point in the fermentation process where the alcohol concentration denatures the yeast. Since the 1990s, a few alcohol-tolerant 'superyeast' strains have become commercially available, which can ferment up to 20%. 
Very few microorganisms can live in alcoholic solutions. The main three are yeast, Brettanomyces, and Acetobacter. In what is essentially disinfection, yeast keeps multiplying as long as there is sugar to "eat", gradually increasing the alcoholic content of the solution and killing off all other microorganisms, and eventually themselves. There are "fortified" wines with a higher alcohol concentration than that because stronger alcohol has been mixed with them.
Stronger liquors are distilled after fermentation is complete to separate the alcoholic liquid from the remains of the grain, fruit, or whatever it was made from. The idea of distillation is that a mixture of liquids is heated, the one with the lowest boiling point will evaporate (or "boil off") first, and then the one with the next lowest boiling point, and so on. The catch is that water and alcohol form a mixture (called an azeotrope) that has a lower boiling point than either one of them, so what distills off first is that mixture that is 95% alcohol and 5% water. Thus a distilled liquor cannot be stronger than 95% (190 proof); there are other techniques for separating liquids that can produce 100% ethanol (or "absolute alcohol"), but they are used only for scientific or industrial purposes. 100% ethanol does not stay 100% for very long, because it is hygroscopic and absorbs water out of the atmosphere.