“Squaring the circle” is an expression that I run into frequently, whether in a Hermetic/esoteric or a Masonic connection. Often the first image that pops into people’s minds is this image supposedly of Hermes Trismegistus.
Recently I was reading the book Quadrivrium and ‘squaring the circle’ is mentioned four times, but quite different from the above. According to the authors, a ‘circled square’ (or ‘squared circle’) are a circle and a square with the same area. So the square would certainly be smaller than in the image above, more like this:
Or like in Leonardo da Vinci’s “Vitruvian Man”
You can read all about the mathematical approach on Wikipedia.
Later I was rereading a book by René Guénon (Symbols Of Sacred Science). He also mentions ‘squaring the circle’ a couple of times and with Guénon you can expect a different approach. Let us see.
In this connection we should note the important nuance between the two forms, circular and square, of the figure of the triple precinct: they refer respectively to the symbolism of the Terrestrial Paradise and the Celestial Jerusalem, as we have explained in one of our works. In fact, there is always analogy and correspondence between the beginning and the end of any cycle; but at the end, the circle is replaced by the square, and this indicates the realization of what the Hermeticists designated symbolically as the ‘squaring of the circle’. (p. 77)
Not too far before this quote Guénon spoke of building symbolism. Many traditional buildings have a dome placed on a square base, heaven and earth united in the building. In a note to the quote above, Guénon says:
The ‘squaring’ cannot be achieved in ‘becoming’, or in the movement of the cycle, because it expresses the fixation resulting from the ‘passage to the limit’, and since all cyclic movement is properly speaking indefinite, the limit cannot be reached successively and analytically by traversing all the points corresponding to each moment of the development of manifestation. (p. 77)
So is the square refers to the ‘terrestrial’ and the circle to the ‘celestial’, it is more a matter of ‘circling the square’ as you live in the terrestrial, make yourself a circle ‘of the same size’ in order to forward to the celestial.