In the 1870's, most of the games in the US would have been American four-ball pool, which was played with two balls and two cue balls, and the score depended on how you bounced your cue ball off the other balls. First person to get to 61 points won the game.
15-ball pool was just coming into popularity at that time, and you scored by sinking a ball into a pocket, getting the number of points that was printed on that particular ball (1-15). Another game was three-ball pool, also called Straight Rail - this was the same as the English/French game Carombole - you made points by bouncing your cue ball off the third ball. There are no pockets in Straight Rail pool.
In Europe, you'd probably be playing either the three-ball game, or something similar to four-ball billiards only with no pockets. There were many varieties of the game, sometimes with each player taking only one shot per turn and others with a successful player continuing to shoot until they missed. Snooker, an English game, was invented in 1875 and is similar to 15-ball pool with the exception that the point of the game is to leave your opponent's cue ball in the worst possible location to make a shot (being "snookered").
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