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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?

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    I very much disagree with the person who down-voted this question. Determination of the distance to the Sun (likewise the Moon$ was a very challenging issue (as was determining the radius of the Earth). the ingredient the OP may be missing is you can measure parallax by, for example, viewing objects from the bottom of separated wells. But that requires synchronizing in time, a technical challenge in ancient times. – 2017-01-13
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    The contraction it's means "it is." The possessive for an inanimate object becomes its, backwards of usage for, say, people. Just a quirk because English does not have many types of accent signs, compared with French for example. – 2017-01-13
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    **"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."** Well, this is making 3 assertions, The first one, we do not know the shape of the earth because it must be measured? How do you measure a shape? And if you say we don't measure the radius of the earth because we didn't use ruler, we also don't use ruler for much smaller things, like 1 KM distances one might use a [teodolite](https://en.wikipedia.org/wiki/Theodolite). Is that measuring or calculating? – 2017-01-13
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    You may find an answer here: How do you measure the distance between Earth and the Sun? http://curious.astro.cornell.edu/physics/41-our-solar-system/the-earth/orbit/87-how-do-you-measure-the-distance-between-earth-and-the-sun-intermediate – 2017-01-13
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    "none of those distances or sizes have ever been measured, only calculated" That's a heck of a restriction if we can't use things we know just because they were never measure. And it makes the problem futile because we won't *measure* the distance to the sun; "only" calculate it. Are you discounting these measures because the haven't been measured, or because we are going back to first principals and need to derive them if we want to use them? – 2017-01-14
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    @MarkFischler and even the wells idea from Eratosthenes assumed sphericity, his goal was actually to calculate distances based on the assumtion of sphericity, using the same method (assuming sphericity, it is possible to calculate the radius of anything, even a house brick). – 2017-01-16
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    @Santropedro I would accept a theodolite as measuring, however there is Geodetic surveying (which assumes sphericity) and Plane surveying (which assumes a flat level plane) distances of KM perhaps even up to hundreds of KM could probably be measured and confirmed fairly accurately and easily, perhaps this might be a good starting point? – 2017-01-16
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    Related: http://physics.stackexchange.com/questions/5192 – 2017-01-16
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    @fleablood yes, back to the very beginning, derive all values from at least some real measurement without making assumptions – 2017-01-16
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    Well, then we can determine the earth is spherical via observation at sample points. We can determine the size of the earth via observing angles of the sun and stars at different locations. We can replicate Kepler and Newton and Galileo and determine their results etc. – 2017-01-16
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    @GeoffDutton I'm willing to help you! But we need you to write more text in your question with more examples of things you consider measurement vs calculation. Do you accept the earth is approximately spherical? Do you think it's flat? Note that we have taken pictures of the earth from satellites and from the moon. – 2017-01-16
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    @fleablood My beliefs are unimportant, if we assume sphericity using sticks and shadows we get millions of miles to the Sun if we assume a dead flat plane we get a very close Sun, and all the variations in between based on the shape of the Earth and location of the measurements taken, so the question is I suppose in principal, is assumption necessary to calculate the distance to the Sun or can it be done from a set of confirmable measurements? this is a maths question and nothing else. – 2017-01-19
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    @Santropedro purely for the purposes of this question I cannot accept any assumption, unless it can be measured accurately (relatively, depending on the size and conditions of the measurements taken and used to calculate) – 2017-01-19

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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.

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    "No assumptions can be made." hence, we don't know tidal force, gravitational constant, the mechanics of gravity and orbits, etc. – 2017-01-14
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    Furthermore the main influence on tides is the Moon, not the Sun. – 2017-01-14
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    I appreciate the answer but before using equations we would need values for all the necessary parts required to calculate any of the other parts, for example M, G, R that´s why I wondered if it could be done from measurements or at least a set of measurements that could be confirmed – 2017-01-16
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    I think you are very ambiguous about what you consider available to us or not. I'd find it perfect acceptable to say "get a library card. Go to the library. Read up on the history of astrophysics. Replicate all the results on your own." – 2017-01-16
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    @GeoffDutton Yes, but note that with the two measurement results you can eliminate $MG$. This elimination process is at the heart of all measurements in physics. Note also that the fact that we use different units with incompatible dimensions when in modern physics they can be given the same units (e.g. natural units). So, even the familiar equations of physics that are taught in high school contain hidden baggage in the form of the units that are actually expressible in terms of other supposedly incompatible units by using $\hbar$, $c$, and $G$ as conversion factors. – 2017-01-16
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    @fleablood it seems astronomers based all their calculations on the result of an experiment which was devised in order to calculate the dimensions of a sphere... http://astronomy.stackexchange.com/questions/6274/how-was-the-earth-sun-distance-originally-calculated the obvious assumptions here are that the elevation and longitude of both locations were identical, and they in fact existed on a sphere, that´s why the no assumptions caveat comes in, is there an alternative method using measurments only as a starting point. – 2017-01-17
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    " that´s why the no assumptions caveat comes in" Um... why?... What exactly are you trying to do? They didn't assume the world was a sphere. They observed it seemed most likely and then verified with experiments. It seems you are very ambiguous about what to accept and not. – 2017-01-17
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    nobody had observed a sphere, many people had pondered it but nobody had seen it, or measured the inverse square curve, therfore it was an assumption, it´s fine if there is no way to do it without making assumptions, but I wanted to know if it is at all possible or do we have to make assumptions and hope they turn out to be correct later? – 2017-01-19