If someone asked me to figure out the distance to (say) Mars, and also it’s size and mass, how exactly could I do this using just the tools that would have been available 100 years ago.
What’s the step by step process to calculate these things?
For distance, it’s easy. Parallax. You take a measurement of the angle to planet C from position A, another measurement from position B, and then you have the distance from A to B, and the angles BAC and ABC, and so you can solve ACB and distances AC and BC.
It’s size, once you know the distance, is easy to calculate by measuring the apparent disk at magnification X.
Once you know the size you can guess the mass by plugging in hypothetical materials. But if something is orbiting the body, you can calculate the mass by observing the orbit of the orbiting body, and figuring out how high the orbit is and how fast, and therefore the effect of gravity on the orbiting body.
Once you know Kepler’s Laws then it’s easy enough to work out the size of the different orbits relative to the size of the Earth’s. The tricky part is then measuring an absolute distance to anything further than the Moon.
For a long time, transits of Venus were the best method, essentially using the backdrop of the Sun’s disk to measure the planet’s parallax in front of it. There are practical complications with the method, however, and even at the start of the 20th century the accuracy was still, IIRC, on the order of 1% or so.
I think there were then a couple of relatively close asteroid passages which brought something close enough to accurately measure the parallax without a transit. However, even then it had to wait until after WWII and the advent of radar ranging of planets - basically bouncing a radar beam off Venus - for the era of really high precision absolute solar system distance measurements to start.
By 1900 measuring the apparent diameters of the planets was straightforward and hence using the distance measurements to derive sizes.
For masses, as Lemur866 explained, you can use their moons to derive masses of those that have them.
Another handle was gravitational perturbations - their minor effects on each others’ orbits. But that was pretty imprecise. For Mercury and Venus, people still had to pretty much guess they had a similar density to the Earth and Mars and estimate from there. Good mass measurements were significant science targets of the first flybys of the inner planets.
Getting Kepler’s laws is still a big step–even bigger if you can’t pull out a measuring rod and must instead rely on triangulated observations to derive them.
The main 17th century problem in determining the movement of the planets was the movement of the Earth itself, which was still not precisely known. You need to know the details of the Earth’s movement first before you can use Earthbound observation to calculate the position of the other planets.
Kepler solved this problem in an ingenious way. The period of Mars–nearly 687 days–was known quite precisely from thousands of years of observation. So he compared Brahe’s observations of Mars from Earth that were 687 days apart–theoretically Mars was in the exact same place for each of these observations–and used these to work out the initial details of Earth’s exact position relative to the sun in terms of Mars’ fixed distance to the Sun. When he exhaused one set of periodic observations for Mars, he worked the process in reverse–checking Mars’ position on observations made exactly one earth-year apart, when now the Earth-Sun distance was fixed–to refine the position details on Mars.
By bouncing back and forth between the two sets of observations Kepler was able to work out the relative positions of the two planets quite precisely. And since he had precise detail about Earth’s position, the position calculations for the other planets would be much easier to find. And, of course, all these measurements would still be relative to the unknown but fixed baseline of the AU.