# How are the elevations of the poles proportionate to distances of earth that have been traversed according to Copernicus

In chapter 2 of "The Revolutions of the Heavenly Spheres" where Copernicus maintains that the Earth is spherical, the Edward Rosen translation of the book states that

the elevations of the poles have the same ratio everywhere to the portions of the earth that have been traversed.

I'd like to know what Copernicus is referring to in the aforementioned piece of text and how it provides evidence for the spherical shape of the Earth.

• Sounds to me like the "distance of Earth that have been traversed" is referring to the distance from the equator, which in terms of latitude always equals the altitude of the pole in the sky. E.g. on the equator the North pole is at 0 degrees altitude, or at 30 degrees latitude it is 30 degrees above the horizon. – Dean Jun 29 '17 at 16:42
• @Dean On the equator, the north pole is not at 0 degrees altitude from the horizon. Due to the curvature of the Earth, it would be below the horizon and have a negative altitude. – zephyr Jun 29 '17 at 16:56
• @zephyr I think by definition the north pole is at 0 degrees altitude on the equator, regardless of if you can see it or not. Otherwise the opposite to this where the north pole is overhead at the north pole cannot hold true, mathematically speaking anyway. – Dean Jun 29 '17 at 17:03
• @Dean That's absolutely not right. Your horizon is defined based on how far you can see at your given location and height. For example the distance to your horizon (regardless of obstructing buildings, mountains, etc.) is given by $d=\sqrt{h^2+2R_Eh}$. The location of the north pole has nothing to do with what you can see from a particular location at a particular height. It makes no sense for the north pole to be on your horizon at the Equator. What if you grew two inches? Your statement about the opposite not being true then also makes no sense. I see no connection between the two ideas. – zephyr Jun 29 '17 at 17:09
• @Dean Don't get me wrong, the north pole will be very near the horizon while at the Equator, but it won't be exactly at the horizon. That's just the nature of the geometry you're looking at. We might call it zero for convenience, but my point is that it isn't exactly zero, mathematically. – zephyr Jun 29 '17 at 17:35

You can see the evidence of this in star trail photographs. The stars will always appear to rotate about the celestial poles. This image shows the star trails above Ecuador, at a latitude of about 1$^{\circ}$. You can clearly see that the stars are rolling directly over head; with the North and South celestial poles located on the horizon to the right and left, respectively.