Short answers, pulled from the story.
The inch is a unit of length in the British Imperial and United States customary systems of measurement. It is equal to one-twelfth of a foot. Since the adoption of the international yard in the 1950s and 1960s, the inch has been defined as exactly 25.4 mm, which is exactly 2.54 cm.
The word inch was an early English borrowing from the Latin uncia, meaning one-twelfth, Roman inch, or Roman ounce. It is cognate with the word ounce, which was reborrowed later through Anglo-Norman. In many European languages the word for inch instead comes from the word for thumb, because a man's thumb is about an inch wide.
After 1066 the inch was defined as equal to three barleycorns laid end to end. A statute of Edward II of England in 1324 set it as three grains of barley, dry and round, placed end to end, lengthwise. The Welsh Laws of Hywel Dda, from the first half of the 10th century, similarly held that three lengths of a barleycorn is the inch.
The two inches differed because they came from different definitions. The U.S. inch was effectively 25.4000508 mm, referenced at 68 degrees Fahrenheit, based on the 1866 conversion factor of one metre equal to 39.37 inches. The UK inch was 25.399977 mm, referenced at 62 degrees Fahrenheit, defined against the Imperial Standard Yard.
The British Standards Institution adopted an inch of exactly 25.4 mm in 1930, and the American Standards Association followed in 1933. The international yard of exactly 0.9144 metres, which fixed the inch at 25.4 mm, was adopted by the United States on the 1st of July 1959 and by the United Kingdom in 1963, effective the 1st of January 1964.
The US survey inch came from the older metre-based definition the United States retained for surveying, creating a difference of about one inch per mile from the standard inch. In 2020 the National Institute of Standards and Technology announced that the U.S. survey foot would be phased out on the 1st of January 2023 and replaced by the international foot of 0.3048 metres exactly.
The inch is commonly used in the United States, Canada, and the United Kingdom, where it is a primary unit for road signs since the 1st of October 1995. Worldwide it measures display screens such as televisions and computer monitors, and it specifies the diameter of vehicle wheel rims and the inner diameter of tyres in tyre codes.