At the beginning,how and using which principles scientists determined the mass and distance from sun of various planets?
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For the distances, between planets, see my answer to the next question, Historical knowledge of Distance of Earth from Sun Once you know the distance and the apparent (angular) size you get the radius. Apparent size was known only for the Sun and Moon before telescopes came in use. But with a telescope, you see a planet as a disk, and can measure the angular size of this disk.
Mass is trickier. The size of a planet can be measured with a telescope and knowing the distance. Then you make some guess about the density. This gives some estimate. It is as good as your guess of the density.
For planets having satellites the situation is better. You know the orbit of a satellite, and can compute the force of gravitational attraction. Using Newton's law of attraction, you find the mass. This is about all you could do in the era before the space craft. See
A more subtle method involves the perturbation by a planet of the orbits of other planets. For example when Neptune's existence was calculated (before it was discovered), this calculation came with a rough estimate of its mass. This was long before its satellites were discovered.
(EDIT: See the very interesting reference in the comment of Franz Lemmermeyer, where the results obtained with this method are described.)
The difficult case is Mercury and Venus. The mass of Venus was reliably measured only after a space craft was sent to it. Observing the motion of the spacecraft near the Venus, you can compute the mass using the same Newton's Law.
A remarkable story is related with Sirius. It has a satellite (also a star, but invisible: too small). This star orbits Sirius and perturbs Sirius's orbit. From this perturbation, the mass of the satellite could be found. So in the beginning of 20th century we had a paradoxial situation: we knew the mass of the invisible satellite of Sirius (9 light years away!) but did not know the mass of our closest neighbor (Venus).
Alexandre Eremenko
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The masses of Mercury and Venus were well known around the middle of the 19th century, and quite likely a lot earlier. – May 20 '17 at 06:33
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@Franz Lemmermeyer: Would you explain then how they were found? – Alexandre Eremenko May 20 '17 at 09:16
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The same way Neptune and Uranus were discovered, by computing their gravitational action on other planets, in this case Venus and Earth. – May 20 '17 at 10:54
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Gravitational action of Venus on Earth is too small for evaluation of the mass of Venus. For Mercury it is even worse. – Alexandre Eremenko May 20 '17 at 13:26
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google for books from the 19th century with the keywords Masse and Venus. – May 20 '17 at 13:36
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If you already did this, perhaps you share your best result? – Alexandre Eremenko May 20 '17 at 13:47
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https://books.google.de/books?id=EN0aAAAAYAAJ&pg=PA588&dq=masse+venus&hl=en&sa=X&ved=0ahUKEwjVgIT73v7TAhXEwBQKHeP5CU0Q6AEIIzAA#v=onepage&q=masse%20venus&f=false – May 20 '17 at 15:04
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See https://archive.org/details/naturwissenschaf71892deut p.~83 for a discussion in which Leverrier is credited with these estimates. – May 20 '17 at 15:11
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@Franz Lemmermeyer: Thanks for the interesting reference. It confirms what I wrote in my answer. This method gives some estimate, but not very accurate. – Alexandre Eremenko May 20 '17 at 16:05
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You may be interested in First observation that the movement of a planet or asteroid in its orbit was affected by another planet? I can move it here if you think it's better suited. – uhoh Feb 28 '19 at 05:21
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In case of the Earth's mass, it was measured by William Cavendish through his experiment with a spring pendulum (like Coulomb did for the determination of the Coulomb's Constant).
D. LaRocque
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