Yes, Newton conjectured that the same type of force, which is gravitational in nature, acts between the Earth and both the apple and the moon.
Did Newton believe that the force between the Earth and the moon was the same as the force between the Earth and the apple?
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Yes, Newton believed that the force between the Earth and the Moon was governed by the same fundamental force of gravity as the force between the Earth and an apple. He formulated the law of universal gravitation to describe this force acting between any two masses in the universe.