(I know this is an old thread, but this is something I've discussed frequently)
It is the geometry of "straight-edge and compass" - the square is just a short cut, as there are SE&C functions to draw a line perpendicular to another line.
The two "organic" divisions of the circle are the square, and the hexagon - half of which is the equilateral triangle. The angle of square to triangle is 90:60, reducing to 3:2 - the perfect fifth. In number of sides, it's 4:3, the perfect fourth:
The square and triangle are also the squares and trines of astrology, which also figure heavily in Raymond Lully's "Art".
In the alchemical diagram posted after, The point below the dragon's belly is 1, the Monad/centrum. The circle is split for 2. the triangle is 3, the square is 4. That is all you need to complete the "decad" (1-4) itself becomes a symbol of the decad (1-10) (references to the tetraktys):
I first found the following image in Georg Klet's 'Cabala Chymica' (1606), basically a reworking of the image posted above:
But it was used again in a publication of Johann Grashoff's 'Cabalae Verior Descriptio' (1680), with the following caption on the opposite page (rough translation):
'This is to observe that everything that Master/Magician [Artifex] learns or wants to start in this high art. Let the Balance, the right-hand line [right angle, the square in his left hand], and the circle [compass in his right hand] be well observed; so there is nothing unequal, nothing crooked, nothing except the cube, that is, everything is to be brought into one, from one.' (the points of the hexagon are the same points used to construct the cube)
This is something else I found, from a 1496 publication of "Philosophia pauperum" (pseudo-Albertus Magnus):
There you have the decad (1-10), the tools of geometry. in the middle is the world of the four elements.. the "World" (observible universe).. And 12 hours of day, and 12 hours of night (which happens to be the angles of the hexagon bisected, and those resulting 12 bisected again). (Wisdom 11:20-22 But you have disposed all things by measure and number and weight. For great strength is always present with you; who can resist the might of your arm? Indeed, before you the whole universe is like a grain from a balance,[f] or a drop of morning dew come down upon the earth.)
These are how the universe are created, ex nihilo. The monad, the decad, the 4 elements, the 7 powers/planets, geometry, sound. Only then does the actual "work" of physical creation begin. It "unfolds" from (or contracts to) a point, one of no dimensions (1 point), then the line, which is one dimension (2 points); then the triangle (shape), two dimensions (3 points); and then the tetrahedron (body), three dimensions (4 points). That's the Greek version anyways. Kabbalah has it point-line-circle-sphere.. And i've seen it point-line-square-cube as well.
This is something i whipped up quick awhile ago dealing with the same question of the square and compass. The bottom right picture is Abraham von Franckenburg's "
Key to the House of David". It illustrates most of the points I made above.
This is some "basic" geometry and arithmetic (quadvirium) that the authors of these old texts presume their readers are familiar with.