Back in the 1600's how did they calculate the radius of the earth and the sun and the distance to the sun?

Answer 1

They have used the paralax method since about that time. This is measuring the arc of items in space with known distances. They have known diamater of the earths rotation 186,000,000 for a long time. Knowing that it is possible to measure the distance of astronomical items using one second of angle in the difference of an items location in the sky measured six months apart.
B

Answer 2

Early 1600 is when Galileo started using the telescope for astronomy. Calculations done before that were based on naked-eye observations (large quadrants and lots of observations to average out the errors).

John T has already described how Erastothenes had determined the circumference of Earth by noting that the sun fully lit the bottom of a well in Alexandria (Egypt) while still 'tilted' by 7 degrees in Syrene, 5000 stadia directly to the South. (One stadium is approximately 0.1 statute mile or 0.16 km).

A simple "rule of three" gave him an answer that was within 1% of the true value (if 7 degrees represent 5000 stadia, how many stadia are in 360 degrees).

Thanks to Kepler who had worked out a relation between a planet's period and its distance from the Sun, astronomers (at least, those who believed in a system centred on the Sun) knew the relative distances: for example, they knew that Jupiter's mean orbital radius was 5.2 times the mean orbital radius of Earth.

The relationship is P^2 = d^3 where P is the planet's orbital period (in years) and d is the mean orbital distance (in Astronomical Units, where 1 AU is the average Sun-Earth distance). But they did not know the true length of an astronomical unit.

Early astronomers (as early as Aristarchus of Samos) tried to use parallax to determine the distance to the Moon and to the Sun. View the Moon from two positions on Earth (a known distance apart) and calculate the triangle using the difference in the Moon's position in the sky. The angle is too small for accurate results without a telescope.

Aristarchus and Hipparchus tried a method using the time interval between Moon phases to determine the distance to the Sun. If the Sun is relatively close, the time interval will be different than if the Sun is far away. Unfortunately, their method was based on the Moon's orbit being circular. We now know it is not.

With telescopes (after 1600), the distance to the Moon was determined with better accuracy using the parallax method.

An accurate baseline for the Solar system was not determined until 1761 using a transit of Venus in front of the Sun (using equations developed by Halley).

Answer 3

The tribesmen of Oolongatanga walked around the world, counting footsteps. The French mathematician Le Boof Balogne commissioned this amazing feat, and then applied the formula relating diameter to radius to find Earth's radius. For the distance to the sun, the tribesmen were paid to stare at the sun and log the point at which blindness occurred. The formula of intensity to distance gave a rough estimate of distance.

Answer 4

Actually, the ancient Greeks figured out the radius of the earth.
One fellow noted that on noon on June 21, one city had columns with no shadow, directly below the sun, and the other showed an angle of 7 degrees. Thus, by figuring out the distance between the cities - he paid a man to pace it out! he could work out the size of the Earth.

I'm not exactly sure of how the distance to the Sun was computed.

Answer 5

They did it the same way we do it now. They used mathematics and formulas that they developed to assimilate all the information. We didn't develop these formulas, they did.