We use the phrase “to see which way the cat jumps” in a figurative sense as meaning to notice how events are shaping, so as to be able to act accordingly.
It is the general assumption that the saying came from the game of tipcat, which boys have been playing since the sixteenth century at least.
In this game the “cat” is a short stick of wood, about an inch and a half or so in diameter and five or six inches long, each end tapered from the center like the frustum of a cone.
In playing the game the “cat” is struck on one of its tapered ends with sufficient force to cause it to spring into the air and is then knocked away by the player. To strike the “cat” while in the air the player must observe closely the direction of its spring.
But figurative expressions do not often arise from boys’ games; they are more likely to achieve permanence from the sports of men.
For this reason it seems to me that the source of the expression is more likely to have been the same ancient sport which produced the expressions, “room to swing a cat,” and “to let the old cat die.” That is, the “cat” in this instance was probably tied within a leather sack which in turn, hung from a tree, was used as a target in archery.
The sportsman had literally to watch how the cat jumped in order to be able to hit the target.