To see the most important implication of Galileo’s rule, let’s look at what happens if we add consecutive odd numbers. After one unit of time, the ball has traveled one unit of distance. After the next unit of time the ball has traveled another three units of distance, for a total of 1 + 3 = 4 units traveled since the motion started. After the third unit of time, the total becomes 1 + 3 + 5 = 9 units of distance. Notice the pattern: the numbers 1, 4, and 9 are the squares of consecutive integers—12 = 1, 22 = 4, 32 = 9. So Galileo’s odd-number rule seems to be implying that the total distance
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