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The famous Square and Compass symbol has meanings within Masonry which relate to their ideals, but here I am just looking at the design as a stand alone reality theory. The labels show how the square and compass are good emblems of public physics vs private physics. Using a compass to draw a circle involves sticking the point into the surface to create a fixed point of reference, around which circles of different sizes can be drawn according to how far the compass is opened. Right away there is private preference being expressed.
The design on the compass and its circular joint at the top have a celestial connotation, of solar dominion and perhaps lunar modulation of light and dark, with the vines or leaves extending downward. The three nested circles can be a syzygy of heavenly bodies, or an indication of the power of comparison. Setting the compass allows a circle to be drawn, but the ability to set the compass again and draw a precisely gauged circle of a different size makes the compass into a machine of Algebraic substitution through time. The drawing of the circles is a mechanical process which exposes ratio and reference.
The ruled square is, by contrast, geometric and static. It is used as a public index, and therefore the more rigid the better. While the compass has decorative elements, the square is both literal and abstract. It is literally a ruler, and can be used as such, but a ruler does not literally represent anything except the idea of pure measurement.

”Le baiser de l’hôtel de ville” (The Kiss) by Robert Doisneau