To set the centre of the pyramid's future base in latitude 30°, two methods could be used, both already to some degree considered--the shadow method, and the Pole-star method. If at noon, at the season when the sun rose due east and set due west, an upright A C were found to throw a shadow C D, so proportioned to A C that A C D would be one-half of an equal-sided triangle, then, theoretically, the point where this upright was placed would be in latitude 30°. As a matter of fact it would not be, because the air, by bending the sun's rays, throws the sun apparently somewhat above his true position. Apart from this, at the time of true spring or autumn, the sun does not seem to rise due east, or set due west, for he is raised above the horizon by atmospheric refraction, before he has really reached it in the morning, and he remains raised above it after he has really passed below--understanding the word "really" to relate to his actual geometrical direction. Thus, at true spring and autumn, the sun rises slightly to the north of east, and sets slightly to the north of west. The atmospheric refraction is indeed so marked, as respects these parts of the sun's apparent course, that it must have been quickly recognized. Probably, however, it would be regarded as a peculiarity only affecting the sun when close to the horizon, and would be (correctly) associated with his apparent change of shape when so situated. Astronomers would be prevented in this way from using the sun's horizontal position at any season to guide them with respect to the cardinal points, but they would still consider the sun, when raised high above the horizon, as a suitable astronomical index (so to speak), and would have no idea that even at a height of sixty degrees above the horizon, or seen as in direction D A, Fig. 1, he is seen appreciably above his true position. Adopting this method--the shadow method--to fix the latitude of the pyramid's base, they would conceive the sun was sixty degrees above the horizon at noon, at true spring or autumn, when in reality he was somewhat below that elevation. Or, in other words, they would conceive they were in latitude 30° north, when in reality they were farther north (the mid-day sun at any season sinking lower and lower as we travel farther and farther north). The actual amount by which, supposing their observations exact, they would thus set this station north of its proper position, would depend on the refractive qualities of the air in Egypt. But although there is some slight difference in this respect between Egypt and Greenwich, it is but small; and we can determine from the Greenwich refraction tables, within a very slight limit of error, the amount by which the architects of the great pyramid would have set the centre or the base north of latitude 30°, if they had trusted solely to the shadow method. The distance would have been as nearly as possible 1125 yards, or say three furlongs.