How is the Law of Universal Gravitation different from the understanding of the motion before Newton theorized it?

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The soviet union beat the us to the first successful launch, with the october 4, 1957, orbiting of sputnik 1, and later beat the us to have the first human in earth orbit, yuri gagarin, on april 12, 1961. the "race" peaked with the july 20, 1969, us landing of the first humans on the moon with apollo 11