The harmony of the spheres was a once popular idea about the geometry of the universe.
Read about historical scientific conceptions
Read historical examples of alternative conceptions
The spheres
The spheres originally referred to a nested sequence of crystalline spheres centred on the earth, and supposedly carrying heavenly bodies around the earth. So this was based on a 'geocentric' model The seven 'planets' or wandering stars (at the time this meant Moon, Sun, Mercury, Venus, Mars, Jupiter and Saturn) were each embedded within discrete spheres, and the fixed stars all shared one outermost sphere. Supposedly, the movement of the outer sphere due to a prime mover (God, or some agent of God) to make the stars revolve around the earth would be transmitted to the next outermost sphere, and so forth, so everything revolved around the earth.
Harmony and number
The idea of a 'harmony' of sphere was due to an analogy with music where it had been found that strings arranged with lengths in certain simple ratios * could played to give pleasing melodies. Sounds from certain combinations of different strings would blend to give chords that were heard as aesthetically pleasing. [* Note: the strings need to be of the same material and under the same tension for the simple ratio to produce the required combinations.]
The discoveries about these mathematical patterns and music had been known since the school of Pythagoras (6th Century BCE) who saw geometry and mathematics to be inherent in the world. Followers of Pythagoras assumed that the heavens had been designed to be harmonious, such that the movement of the heavenly bodies would each be associated with a tone that would collectively blend harmoniously. The music of the spheres was therefore the sound made (literally, or analogously) by the cosmos. However, followers of Pythagoras put a great fire at the centre of the Cosmos, rather than the Earth as in the Ptolemaic model that dominated for many centuries.
The Copernican revolution – a different kind of hamrony
Although Nicolaus Copernicus is seen as a revolutionary who argued for a sun centred cosmos, in which the earth orbited the sun along with Mercury, Venus, Mars, Saturn and Jupiter, he was very influenced by the kinds of ideas about the inherent mathematical structure of the world as proposed by Pythagorean.
So, Copernicus did not give up the ancient idea that anything that moved in the heavens had to move in perfect circles (even if that did not fit observations unless arbitrary combinations of circles were used). He also thought that the distances of the planets from the Sun was dictated by mathematical patterns.
Read about heavenly circular motion
In particular he suggested that the system would be based on arrangement of the five so-called platonic solids that had regular faces and so high symmetry: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. He envisaged that there was a nested system of these shapes such that when they were sequentially placed one within another, they would define the orbits of the different planets.
Although these geometric figures were assumed to define the geometry of the system, Copernicus retained the idea of spheres for locating his heavenly bodies: in English his great work, 'De revolutionibus orbium coelestium' is known as On the Revolutions of the Heavenly Spheres.
