Both these objects have been attained with wonderful accuracy. The orientation of the Great Pyramid is correct, and the centre of its base corresponds with the thirtieth degree of north latitude within a slight error which was inevitable, if, as is probable, the Egyptian astronomers were unacquainted with the effect of atmospheric refraction in raising the apparent above the true place of celestial bodies, or had formed an insufficient estimate of its amount. The centre of the base is 2328 yards south of the real thirtieth parallel of latitude, which is 944 yards north of the position which would have been deduced from the pole-star method, and 3459 yards south of that from the shadow method, by astronomers ignorant of the effect of refraction. The shadow method could never have been so reliable as the polar method, and it is certain therefore _à priori_ that the latter must have been adopted either wholly or principally, and this conclusion is confirmed by the internal construction of the pyramid itself, which is shown by the subjoined vertical section. [Illustration: PYRAMID] The tunnel A B C is bored for a distance of 350 feet underground through the solid rock, and is inclined at an angle pointing directly to what was then the pole-star, Alpha Draconis, at its lower culmination. As there is no bright star at the true pole, its position is ascertained by taking the point half-way between the highest and lowest positions of the conspicuous star nearest to it, and which therefore revolves in the smallest circle about it. This star is not always the same on account of the precession of the equinoxes, and Alpha Draconis supplied the place of the present pole-star about 3440 B.C., and practically for several centuries before and after that date. Now the underground tunnel is bored exactly at the angle of 26° 17´ to the horizon, at which Alpha Draconis would shine down it at its lower culmination when 3° 42´ from the pole; and the ascending passage and grand gallery are inclined at the same angle in an opposite direction, so that the image of the star reflected from a plane mirror or from water at B, would be seen on the southern meridian line by an observer in the grand gallery, while another very conspicuous star in the southern hemisphere, Alpha Centauri, would at that period shine directly down it. The passages therefore would have the double effect, 1st, of enabling the builders to orient the base and lower layers of the pyramid up to the king's chamber in a perfectly true north and south line; 2nd, of making the grand gallery the equivalent of an equatorially-mounted telescope of a modern observatory, by which the transit of heavenly bodies in a considerable section of the sky comprising the equatorial and zodiacal regions, across the meridian, and therefore at their highest elevations, could be observed by the naked eye with great accuracy.