What is the problem?
Watches are machines, they behave in completely deterministic ways. The key elements of a watch are a power source (mainspring), the gear train, an governor mechanism (escapement and balance), and an output (the hands). The gears are not arbitrary, they have to have specific ratios to each other in motion. For example, there are 60 seconds in a minute. So the 4th wheel, that carries the seconds hand, has to go around 60 times exactly for each single revolution of the center wheel, which carries the minute hand.
The exact speed of these wheels, what we think of as the watch's accuracy, is a function of the escapement. But the wheels must turn at the appropriate rate relative to each other for minutes and seconds to be even meaningful. Gear ratios dictate this relationship.
Sometimes, when a watch is running fine, but reading incorrectly, the degree of error can tell you exactly where the problem is. You don't even have to see the watch.
This watch gained 22 minutes per 24 hours. There are 64 teeth on the center wheel (typical of American pocketwatches). The passing of each center wheel tooth against the pinion of its neighbor, Mr. 3rd wheel, represents 1/64th of a minute. 1/64th of a minute error per hour (each turn of the center wheel) adds up to 22.5 minutes error in 24 hours.
Inspection of the center wheel reveals that I missed a slightly bent tooth. Each time this part of the wheel came around, it was not enough to stop the watch, but it did pop over to the next tooth, skipping exactly one, per hour. The problem is visible in the above image.
The lesson is not to get lazy about checking each wheel, especially on a watch like this that did have damage to the train to begin with.
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