Well, using Math to see how far the Moon is from the Earth could be hard. Also, he saw that the Moon is moving in a "circle" instead of flying off, so the Earth did have a pull on the Moon so that it didn't fly away. Newton had to use Math in order to be able to identify what it's now called gravity. There's a lot that the Moon has to offer, there were no computers back then, so there's was no calculator and everything had to be done by hand.
