different places, and that it was recognised that, if a traveller were to go far enough north, he would find the pole to coincide with the zenith, whereas by going south he would reach a region (not very far beyond the limits of actual Greek travel) where the pole would be on the horizon and the equator consequently pass through the zenith; in regions still farther south the north pole would be permanently invisible, and the south pole would appear above the horizon.
Further, if in the figure h e k w represents the horizon, meeting the equator q e r w in the east and west points e w, and the meridian h q z p k in the south and north points 
Fig. 15.—The equator, the horizon, and the meridian. h and k, z being the zenith and p the pole, then it is easily seen that q z is equal to p k, the height of the pole above the horizon. Any celestial body, therefore, the distance of which from the equator towards the north (declination) is less than p k, will cross the meridian to the south of the zenith, whereas if its declination be greater than p k, it will cross to the north of the zenith. Now the greatest distance of the sun from the equator is equal to the angle between the ecliptic and the equator, or about 2312°. Consequently at places at which the height of the pole is less than 2312° the sun will, during part of the year, cast shadows at midday towards the south. This was known actually to be the case not very far south of Alexandria. It was similarly recognised that on the other side of the equator there must be a region in which the sun ordinarily cast shadows towards the south, but occasionally towards the north. These two regions are the torrid zones of modern geographers.
Again, if the distance of the sun from the equator is 2312°, its distance from the pole is 6612°; therefore in regions so far north that the height p k of the north pole
