How Copernicus figured out that it took 7 parts of water and 1 part earth until the entire Earth was submersed in water

Question

In chapter 3 of "The Revolutions of the Heavenly Spheres" where Copernicus defines "how the Earth forms a single sphere with water", the Edward Rosen translation of the book states this.

For they do not realize that the water cannot be even seven times greater and still leave any part of the land dry, unless earth as a whole vacated the center of gravity and yielded that position to water, as if the latter were heavier than itself For, spheres are to each other as the cubes of their diameters. Therefore, if Earth were the eighth part to seven parts of water, earth's diameter could not be greater than the distance from [their joint] center to the circumference of the waters. For context, "they" is referring to Aristotelians which according to Copernicus, believe that there is ten times more water than land.

My main lack of understanding comes from how Copernicus uses the statement "spheres are to each other as the cubes of their diameters." To prove that "if Earth were the eighth part to seven parts of water, earth's diameter could not be greater than the distance from [their joint] center to the circumference of the waters." Thereby showing that a 1:7 ratio of land to water is the threshold where water submerges the entire Earth.

Answer 1

I am not able to give a complete answer to your question and admit that I do not really understand what Copernicus is saying, but give this as background information: The notion that the proportion of water to earth in the cosmos is 7 : 1 goes back to Averoes (Ibn Rushd) in his Shorter Commentary to Aristotle’s Meteorology, with decidedly specious arguments. It seems that Copernicus accepted this bit of mediaeval speculation. To my recollection, Aristotle himself has nothing to say on this matter.