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1. If the Cheshire-Cat is alluding to a very special kind of cheese, which we can be fairly sure of, then the chain of associations will lead us from there to Moon (not to Sun), because I have heard the full moon looks like a big round cheese.

But the cat is an animal which certainly corresponds to Sun, which can be 'proven' for instance by the constellation Leo which once stood at high summer, the prime instant for Sun. Even earlier 'the Cat' announced the beginning of summer, some 10,000 years ago:

 

This picture is from Hamlet's Mill (where it has been copied from Robert Eisler's, The Royal Art of Astrology). The precession of the equinoxes, as it is called, moves the starry sky dome slowly around so that for instance spring equinox gradually moves from left to right in this picture.

We have a great 'nut' which needs to be 'tightened', viz. where Rogo in day 409 is located in the skyscape of time. To do it we must start by standing back and take a broad overview of the situation.

As to the Cheshire-Cat I think Carroll used the double associations - Cat made it clear it was about Sun, but 'Cheeseshire' made it equally clear that it was about Moon. Sun does not rule during the whole cycle of the year - Moon takes over at some point:

"Hamiora Pio once spoke as follows to the writer: 'Friend! Let me tell of the offspring of Tangaroa-akiukiu, whose two daughters were Hine-raumati (the Summer Maid - personified form of summer) and Hine-takurua (the Winter Maid - personification of winter), both of whom where taken to wife by the sun ...

Now, these women had different homes. Hine-takurua lived with her elder Tangaroa (a sea being - origin and personified form of fish). Her labours were connected with Tangaroa - that is, with fish. Hine-raumati dwelt on land, where she cultivated food products, and attended to the taking of game and forest products, all such things connected with Tane."

(Elsdon Best, The Astronomical Knowledge of the Maori.)