They would have endeavoured to determine where the pole appears to be raised exactly thirty degrees above the horizon. But the effect of refraction being to raise every celestial object above its true position, they would have supposed the pole to be raised thirty degrees, when in reality it was less raised than this. In other words, they would have supposed they were in latitude 30°, when, in reality, they were in some lower latitude, for the pole of the heavens rises higher and higher above the horizon as we pass to higher and higher latitudes. Thus they would set their station somewhat to the south of latitude 30°, instead of to the north, as when they were supposed to have used the shadow method. Here again we can find how far they would set it south of that latitude. Using the Greenwich refraction table (which is the same as Bessel's), we find that they would have made a much greater error than when using the other method, simply because they would be observing a body at an elevation of about thirty degrees only, whereas in taking the sun's mid-day altitude in spring or autumn, they would be observing a body at twice as great an elevation. The error would be, in fact, in this case, about 1 mile 1512 yards.
It seems not at all unlikely that astronomers, so skilful and ingenious as the builders of the pyramid manifestly were, would have employed both methods. In that case they would certainly have obtained widely discrepant results, rough as their means and methods must unquestionably have been, compared with modern instruments and methods. The exact determination from the shadow plan would have set them 1125 yards to the north of the true latitude; while the exact determination from the Pole-star method would have set them 1 mile 1512 yards south of the true latitude. Whether they would thus have been led to detect the effect of atmospheric refraction on celestial bodies high above the horizon may be open to question. But certainly they would have recognized the action of some cause or other, rendering one or other method, or both methods, unsatisfactory If so, and we can scarcely doubt that this would actually happen (for certainly they would recognize the theoretical justice of both methods, and we can hardly imagine that having two available methods, they would limit their operations to one method only), they would scarcely see any better way of proceeding than to take a position intermediate between the two which they had thus obtained. Such a position would lie almost exactly 1072 yards south of true latitude 30° north.