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#1




False hypothesis for proof of axiom
The axiom is that the Earth orbits the Sun along with the other planets while turning to the Sun each day as a function of daily rotation.
The original geocentric astronomers founded their hypotheses on the observation that the Sun passed directly through the constellations while the planets displayed direct/retrograde motions with the Earth as stationary and central. In a moving Earth/Sun centred system, the Sun's motion around the Earth daily is due to the rotation of the Earth hence the axiom is proved. The orbital motion of the Earth is proved by an alternative hypothesis based on the observation that stars, in a lineofsight transition, move from left to right of the Sun. Now the false hypothesis and the basis for a base/step mathematical induction  "That the fixed stars being at rest, the periodic times of the five primary planets, and (whether of the sun about the earth, or) of the earth about the sun, are in the sesquiplicate proportion of their mean distances from the sun.... for the periodic times are the same, and the dimensions of the orbits are the same, whether the sun revolves about the earth, or the earth about the sun." Newton There is no geocentric/heliocentric equivalency as the "sun about the earth" refers to daily rotation and can only be answered in rotational terms so it cannot be equated with the Earth's orbital motion around the central Sun.. 
#2




False hypothesis for proof of axiom
Imagine trying to make sense of outside motions while being on a carousel yet this is exactly what celestial sphere proponents did by giving a North/South/East/West (RA/Dec) framework priority 
https://www.youtube.com/watch?v=BYy0EQBnqHI The foundation of astrophysics is based on a notion that the Sun about the Earth and the Earth about the Sun are equivalent in terms of observations however the Sun around the Earth is taken up by daily rotation whereas the Sun through the constellations is the closest to the Earth's orbital motion. It is no wonder people over the last 20 years in this newsgroup haven't a clue how to deal with Huygen's geocentric/heliocentric equivalency that is related to the Equation of Time even though it is flawed  "Here take notice, that the Sun or the Earth passes the 12 signs or makes an entire revolution in the ecliptic in 365 days, 5 hours 49 min. or there about, and that those days, reckoned from noon to noon, are of different lengths; as is known to all that are versed in Astronomy. Now between the longest and the shortest of those days, a day may be taken of such a length, as 365 such days, 5. hours &c. (the same numbers as before) make up, or are equal to that revolution: And this is called the Equal or Mean day, according to which the watches are to be set; and therefore the hour or minute showed by the watches, though they be perfectly just and equal, must needs differ almost continually from those that are showed by the Sun, or are reckoned according to its motion. But this difference is regular, and is otherwise called the Equation of Time.." Huygens While entertaining in its own way, most if not all of you imitate Newton who made things up as he went along but, unlike the nuisances here, he did it in a very careful way  more like the actions of a politicians filling in deficiencies and gaps while appearing to affirm astronomical or timekeeping principles, even flawed ones. I have seen the ground where mathematicians can work productively and with great creativity and that is what is so dismaying, they would rather lean on old familiar junk like solar vs sidereal time than act to clean up a mess they created. 
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