To pursue this study, it is necessary to obtain geometrical arguments, in which we can believe without doubting. I will make then a figure of the earth, and I will place E in the centre of it, and I will describe another circle from the north of it to the south, and draw a straight line from the Arctic Pole to the Antarctic Pole through the earth and through the orbit of the earth, and place A at the top of the firmament, and B in the north pole of the circle, and C down below, and D in the south pole.
Therefore, whosoever being in position E, should take the astrolabe in his hand (for with it will be obtained full certain knowledge of this matter), and placing his face along the middle line of the astrolabe which he holds suspended by a thread from his thumb, and beholding the Arctic Pole through the two holes of its two surfaces, would find that pole level with the earth; and if you travel three score six and two-thirds of a mile from E to B and then place the astrolabe opposite the Arctic Pole, and look through it as you did before, you would find it to be six degrees in height over the earth
Again, if you move another three score six and two-third miles from that towards B, and place the astrolabe opposite the same pole, and look as before, you will find two degrees in height overhead, and so on, always, from E to B, for every three score six and two-third miles until one would reach B, one would find the same pole increasing in height by one degree. The amount of all those miles put together in accordance with the amount of the three hundred and sixty degrees which are in the circumference of the sphere of the earth, make 24,000 miles, which is the circumference measurement, including the orbit of the water and of the land. And the alkoterra, i.e., the diameter of the earth, is eight thousand miles, and, accordingly, it is four thousand miles to the centre of the earth, and for every mile of those to the centre there ought to be three and one seventh miles of circumference.