Certainly his wife, George, from a younger generation—and probably more accustomed to using printed blanks—always used a circular layout. And that is the layout that dominates throughout the automatic script. Even when the spirits were taking control, their scrawls are almost always based on circular forms.
Yet certain features of the system lend themselves readily to a square format. For now I’ll limit myself to the two most fundamental and practical.
The first and most obvious one is that a standard chessboard, with eight squares on each side, has a perimeter of twenty-eight squares. Yeats actually states that "the individual phases are alternately primary and antithetical" (AVB 88), and here we have the alternating squares of black and white (and, as with Yeats’s comment, this does lead to the slightly odd situation of Phases 1 and 15 being the same).
It’s probably best to put the square on point, so that it can be seen in the same orientation as normal presentation of the Wheel in A Vision.
The second feature, that is probably the most interesting in some respects, although one which the Yeatses themselves don’t seem to have used, is the way that the square format helps the mapping of the twenty-eight phases onto the twelve months or signs of the Zodiac.
When a circle is mapped onto a larger circle, the divisions scale up naturally, but when a square is mapped onto a larger square the sides transfer automatically but new squares appear at the corners, as the corner squares map on to each side and the new corner square. It makes more sense when you see it.
While 28 does not map easily onto to 12, if you place another row of squares along each side of the chessboard so that it is ten squares on each side, it has a perimeter of 36 squares, easily mapped onto a twelvefold scheme, with three squares for each twelfth. The division of the months into three periods of roughly ten days each may seem a little arbitrary, though it has good calendrical precedent as far back as ancient Egypt.
The diagram here takes the correspondences sketched out in "The Completed Symbol", Section VI, with March aligned with the Full Moon and September aligned with the New Moon and both phases and calendar proceeding anti-clockwise. Whereas Yeats comments "There is no reason why March, June, etc., should have one Phase, all others three; it is classification not symbolism" (AVB 196), here it is simply logical that the corner or cardinal phases should have a month or sign apiece.
It is also worth remembering that the Zodiac has long been divided into 36 decanates, divisions of 10˚ each, so that each sign has three sectors.
The correspondences here are those sketched out applied to the Zodiac rather than the months, so with Aries aligned with the Full Moon and Libra aligned with the New Moon, though Yeats actually places "the Ides of March, at the full moon in March" with "the Vernal Equinox, symbolical of the first degree of Aries, the first day of our symbolical or ideal year", so we'll need to look at the fine tuning beyond the broad principle shortly.
While we are about it, it is also worth considering that the central space is made up of four squares, surrounded by twelve squares, then twenty and then twenty-eight. The four and the twelve have relatively obvious applications (the inner Zodiac here is the one that corresponds with the phases during the afterlife, see AVB 223).
We'll come back to these boards again soon, since there are dimensions that touch on chess, both regular and Enochian, tarot and the I Ching, all of which are worth exploring in greater depth.
*Update: 3 August 2022. The reference appears in Fagan's "The Origin of Horoscope Form", 12/61 American Astrology:
Before beginning a delineation of an astrological chart we must fully comprehend the meaning of the horoscope form itself and its interpretation. The circular diagram is popular in the west today; but strange to say, it is a comparative innovation, notwithstanding the fact that the dome of the heavens is obviously circular. Up to the 19th century the square of rectangular shaped horoscope form was the vogue, as it is still the vogue in India. In his letters to the present writer, the Irish poet, W.B. Yeats, the Nobel prize winner for literature (1923), always used the square-shaped horoscope form. Why did the square-shaped form persist for so many centuries before it was gradually supplanted by the circular design, and how did it originate? The glib answer is, of course, that it was easier to draw. But is that the only reason?
Cyril Fagan, Heliacal Phenomena, page 19