I was reading some literature and I found that long before the actual distances between other planets and Earth or distance between Sun and Earth were known, physicists had calculated the ratios between these distances. Can anybody tell me the technique used at that time to measure these ratio? This must have been done before 1650.
[Physics] How were the ratios of distances between planets and the Sun first calculated
distancegeometryhistoryobservational-astronomysolar system
Related Solutions
The other parts, other than the inverse square, were clear already before Newton, or at least were easy to guess. That the force of gravity is proportionality to mass of a small object responding to the field of another comes from Galileo's observation of the universal acceleration of free fall. If the acceleration is constant, the force is proportional to the mass. By Newton's third law, the force is equal and opposite on the two objects, so you can conclude that it should be proportional to the second mass too.
The model which gives you this is if you assume that everything is made from some kind of universal atom, and this atom feels an inverse square attraction of some magnitude. If you sum over all the pairwise attractions in two bodies, you get an attraction which is proportional to the number of atoms in body one times the number of atoms in body two.
So the only part that was not determined by simple considerations like this was the falloff rate. I should point out that if you look at two sources of a scalar field, and look at the force, it is always proportional to $g_1$ times $g_2$, where $g_1$ and $g_2$ are the propensity of each source to make a field by itself. Further, if you put two noninteracting sources next to each other, this g is additive, if the field is noninteracting, essentially for the reasons described above--- the independent attractions are independent. So that the proportionality to an additive body constant you multiply over the two bodies is clear. That for gravity, the g is the mass, this was established by Galileo.
More mathemtically
Let's call the force law between the objects $F(m_1,m_2,r)$. We know that if we put the body m_1 in free fall, the acceleration doesn't depend on the mass, so
$$ F(m_1,m_2,r) = m_1 G(m_2,r) $$
So that the mass will cancel in Newton's law to give a universal acceleration. This gives you the relation
$$ F( a m_1 , m_2, r ) = a F(m_1,m_2,r) $$
We know that if we put body 2 in free fall, the same cancellation happens, but we also know Newton's third law: $F(m_1,m_2,r)= F(m_2,m_1,r)$ so that
$$ F( m_1, a m_2, r) = a F(m_1,m_2,r) $$
So you now write
$$ F( m_1 \times 1 , m_2 \times 1, r) = m_1 F( 1, m_2\times 1 , r) = m_1 m_2 F(1,1,r) $$
And this tells you that the force is proportional to the masses times a function of r. The form of the function is undetermined.
An independent argument for the scaling is that if you consider the object m_1 as composed of two nearby independent objects of mass $m_1/2$, then
$$ F(m_1/2 , m_2 , r) + F(m_1/2 , m_2 , r) = F(m_1,m_2,r)$$
Then the same conclusion follows.
These types of scaling arguments are second nature by now, and they are automatically done by matching units. So if you have a force per unit mass, the force between two massive particles must be per unit mass 1 and per unit mass 2.
This general argument fails for direct three-body forces, where the force between 3 bodies is not decomposable as a sum of forces between the pairs bodies individually. There are no macroscopic examples, since the pairwise additivity is true for linear fields, but the force between nucleons has a 3-body component.
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.
Best Answer
The relative distances of the earth, sun and moon were determined by Aristarchus. See my summary here. By measuring the size of the earth (as e.g. Eratosthenes did) these can be turned into absolute distances.
Once heliocentrism was introduced the planetary distances could be determined as follows:
Distance from Venus (or Mercury) to the sun: continually measure the angle VES; when it is at a maximum the angle EVS will be right, and we know ES so we can find VS. (Since Venus and Mercury move much faster than the earth, the earth can be considered stationary for the purposes of this demonstration.)
Distance from an outer planet P to the sun. Note when P is in opposition, i.e., when SEP is a straight line. Then wait for the earth and planet to move until the angle SE'P' becomes a right angle. Since we know the orbital times of E and P we know the angles ESE' and PSP' (assuming the orbits to be circles centred at the sun). The angle P'SE' follows, and we already know angle SE'P' and length ES so we can compute SP'.