How did scientists measure the mass and weight of planets and sun in such olden times?
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Mass is the amount of “stuff” in something, and is the same no matter where in the universe it is, while weight on earth is mass times the acceleration due to gravity on the surface of the earth (g = 9.8 m/s^(2)). On the moon you have the same mass but different weight, due to the different acceleration due to gravity on the surface of the moon. It’s better to speak only of the mass of different celestial bodies.
“Olden times” for measuring the mass of the earth was 1798: the Cavendish experiment. This was essentially 70 years after Newton died. The page https://earthscience.stackexchange.com/questions/88/how-is-the-mass-of-the-earth-determined discusses how such measurements were made.
The first measurement of the mass of Earth was done by instead measuring the gravitational force between lead spheres. The mass of these lead spheres as well as the distance between them was known, and so the mass of Earth could be calculated based on the distance to its center and the strength of local gravity.
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