No assumptions can be made. For example we do not know the shape of the Earth or its distance to the moon or the size of the moon, because none of those distances or sizes have ever been measured, only calculated. To me, it seems like an algebraic problem that is missing a fundamental value.
EDIT: Sorry I should have explained, the reason I mention assuptions, is because whenever I try to find out how it was discovered or measured in which ever version you find, Someone seems to already KNOW a value that doesn´t seem to be able to be calculated. I guess the real question should have been how to calculate the size and distance from scratch without ASSUMING anything, only by MEASURING, (we have modern computing technology available and access to all current mathematical knowledge) but what would be the logical order of measurements and calculations in order to solve the problem?
Neglecting the eccentricity of the Earths orbit we can pretend that the Sun is always at a constant distance from the Earth. The length of the year can be expressed in terms of the Sun's distance from the Earth $R $ and it's mass $M$:
$$\text{year} = \frac{2\pi}{\sqrt{MG}} R^{\frac{3}{2}}$$
where $G$ is the gravitational constant. We then need another measurement to eliminate $M$, e.g. the expression for the magnitude of the tidal force due to the Sun's gravity is:
$$F_{\text{tidal}} = \frac{2 M G r}{R^3}$$
where $r$ is the length of the component in the direction of the Sun of the vector pointing from the Earth's center to a point on the surface where we want to evaluate the tidal force.