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Introduction
Whilst no doubt most frequenters of “Physics Forums” will be familiar with Nicolaus Copernicus as the scientist who advanced the (at the time) radical thesis that the earth revolved around the sun rather than vice versa, a perhaps less well-known aspect of his work is the “nuts and bolts” geometry underlying his ground-breaking treatise: “De Revolutionibus Orbium Coelestium” (On the Revolutions of the Heavenly Spheres).  In this article (and perhaps others to follow), we analyse “Copernican geometry” in light of its stated intent:
Because the proofs which we shall use in almost the entire work deal with straight lines and arcs, with plane and spherical triangles, and because Euclid’s Elements, although they clear up much of this, do not have what is here most required, namely how to find the sides from the angles and the angles from the sides, since the angle does not measure the subtending straight line – just as the line does not measure the angle – but the arc does...

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Greg Bernhardt