#### When and how were relative distances to the planets first measured?

I understand that the absolute distance to a planet can be measured using earth-baseline (e.g., diurnal) parallax, and that the first reasonably accurate such measurement was made for Mars by Cassini (and his assistant Richer) in 1672, and then, famously (with some modifications) for Venus by Halley, Cook, and others and others during its 1769 transit; but when were earth-baseline *relative* distance measurements first made for each of the planets?

Interpretation of such measurements requires assumptions about the nature of the orbits of planets, but some of the most compelling evidence for the validity of these assumptions is Kepler's third law of planetary motion, the persuasiveness of which itself rests on observations of relative distances. In fact, all of my texts say, in effect, that the third law "fit" the observed pattern of orbital period and relative distance.

Where did these distance measurements come from? What relative distance measurements were available to Kepler in 1619? When was earth-orbit-baseline parallax used to make these measurements for each of the planets?

The relative distances to the planets is fixed immediately by Copernican model, and this is what makes heliocentrism ten thousand times better than geocentrism, even without any known physical cause for the orbits.

The relative distances are fixed from the radius of the epicycle — the epicycle transfers Earth's orbit onto the planet, and the ratio of the epicycle radius (not the angular extent, which also includes the planet's motion along the deferent) to the deferent size in the Copernican interpretation directly gives the ratio of the Earth's orbit to the planet's orbit. The relative size of Venus and Mercury's orbit, relative to the Earth's distance from the sun, is given by the maximum in angle they get away from the sun.

This is not surprising, because the epicycle radius is giving you the parallax from the point of view of the Earth's orbit of the different planets. Once you know the absolute size of Earth's orbit, you know the distance to everything else, which is why the Earth's orbit is called the "Astronomical Unit".

This means that just Brahe's observations are sufficient to fix the entire solar system size except for the absolute scale of the Astronomical unit. The location of all the planets in 3 dimensions is completely determined from the assumption that the Earth's orbit is shared between all of them. The fact that the epicycles all are given by a one-year orbital period for the Earth is Baysian-wise extremely compelling evidence for heliocentrism without anything further to say.

This is why it is not correct to say that geocentrists were somehow justified, or had any valid points, or were anything other than the dimwitted reactionaries that they were. This includes Ptolmey, who buried the heliocentric work of Appolonius for political reasons, although even the most casual astronomer of the era was aware that heliocentrism was correct.

Given the ratio of people trying to explain planets motion to ordinary people just minding their business, one understands why geocentrism remained a good day to day point of view for so long.

Yeah, that's not a good explanation for the bias among astronomers. The real reason is that those day-to-day people needed a philosopher to justify them owning slaves and amusing themselves by torturing people in an arena, and Aristotle was their go-to guy, or else Roman Gods or whatnot, and heliocentrism showed these to be stupid. Further, when Christianity replaced Aristotlism, it had an even more stupid cosmology (although not so stupid ethics), and this meant nobody except astronomers had any interest in astronomical truth. But instead of sticking to their guns, they fudged.

I'm still not clear how this worked: Were the distances measured using parallax (or some other technique) or can they be calculated from their epicycles and the assumption that the orbits are in fact circular (in which case I need help visualizing: a diagram would be great!)?

There is nothing much to visualize--- imagine jupiter standing still in space, at a distance of it's orbit (this is a reasonable approximation), then every year it wobbles back and forth, and the angular wobble in the sky is the parallax due to Earth's orbit, and the angle subtended by the wobble is equal (to first order) to the ratio of Earth's orbit to Jupiter's orbit. The only correction is because jupiter is moving a little along its orbit too, but the parallax motion of the planet due to the orbit of the Earth is another name for the epicycle radius.

Remember, what I'm asking is a historical question. So I want to confirm: is that in fact how Copernicus arrived at his ratios (0,3763:0,7193:1.000:1,5198:5.2192:9.1743); and Kepler (via Brahe) at his (0.389:0.724:1.000:1.523:5.200:9.510)?

I didn't read it, and I don't know the exact ratios in the two works, but yes, Copernicus had all the ratios of the distances correct to 5% for sure. I would rather give this credit to Aristarchus and Appolonius in ancient times, they knew the relative distances to the planets.

You cannot praise Tycho Brahe and then condemn all geocentrist astronomers as dimwitted, as he was one of them and did not even believe in the Earth's daily rotation.

Brahe was half-heliocentrist in that he believed the inner planets travel around the sun, and he was open-minded enough to take accurate measurements. That he was dimwitted enough to hold on to geocentrist views doesn't diminish his achievement, and I am sure had he been alive to see Kepler's orbit reconstruction, he would have changed his mind

Ron: Brahe believed that the Earth could not move at all as it was too heavy. I suspect it is more likely that seeing Kepler's calculations he would have thought the Sun moved around the Earth in a daily circle and annual ellipse and the other planets around the sun in ellipses.

This is reasonable, it's a coodinate transformation minus the parallax and abberation effects of the Earth's motion on the distant stars, things which came much later, and minus Newtonian gravity, which came a little later. What's dimwitted is that the epicycles of the outer planets are independent. As long as you believe the relative distances are as in Copernicus's model, you could reasonably believe back then the Earth is fixed. But it is ridiculous to ascribe the yearly epicycle motion to the planets, rather than to one thing with a yearly period, either the sun or the Earth.

So all these guys used the method you describe historically? And they were calculated in the 100s BC?