and least when nearest the point vertically overhead. This effect is, indeed, so marked on bodies close to the horizon that if the astronomers of the pyramid times had observed the sun, moon, and stars attentively when so placed, they could not have failed to discover the peculiarity. Probably, however, though they noted the time of rising and setting of the celestial bodies, they only made instrumental observations upon them when these bodies were high in the heavens. Thus they remained ignorant of the refractive powers of the air.[18] Now, if they had determined the position of the thirtieth parallel of latitude by observations of the noonday sun (in spring or autumn), then since, owing to refraction, they would have judged the sun to be higher than he really was, it follows that they would have supposed the latitude of any station from which they observed to be lower than it really was. For the lower the latitude the higher is the noonday sun at any given season. Thus, when really in latitude 30° they would have supposed themselves in a latitude lower than 30°, and would have travelled a little further north to find the proper place, as they would have supposed, for erecting the great pyramid. On the other hand, if they determined the place from observations of the movements of stars near the pole of the heavens, they would make an error of a precisely opposite nature. For the higher the latitude the higher is the pole of the heavens; and refraction, therefore, which apparently raises the pole of the heavens, gives to a station the appearance of being in a higher latitude than it really is, so that the observer would consider he was in latitude 30 north when in reality somewhat south of that latitude. We have only then to inquire whether the great pyramid was set north or south of latitude 30°, to ascertain whether the pyramid architects observed the noonday sun or circumpolar stars to determine their latitude; always assuming (as we reasonably may) that those architects did propose to set the pyramid in that particular latitude, and that they were able to make very accurate observations of the apparent positions of the celestial bodies, but that they were not acquainted with the refractive effects of the atmosphere. The answer comes in no doubtful terms. The centre of the great pyramid's base lies about one mile and a third south of the thirtieth parallel of latitude; and from this position the pole of the heavens, as raised by refraction, would appear to be very near indeed to the required position. In fact, if the pyramid had been set about half a mile still farther south the pole would have seemed just right.
Of course, such an explanation as I have here suggested appears altogether heretical to the pyramidalists. According to them the pyramid architects knew perfectly well where the true thirtieth parallel lay, and knew also all that modern science has discovered about refraction; but set the pyramid south of the true parallel and north of the position where refraction would just have made the apparent elevation of the pole correct, simply in order that the pyramid might correspond as nearly as possible to each of two conditions, whereof both could not be fulfilled at once. The pyramid would indeed, they say, have been set even more closely midway between the true and the apparent parallels of 30° north, but that the Jeezeh hill on which it is set does not afford a rock foundation any farther north. 'So very close,' says Professor Smyth, 'was the great pyramid placed to the northern brink of its hill, that the edges of the cliff might have broken off under the terrible pressure had not the builders banked up there most firmly the immense mounds of rubbish which came from their work, and which Strabo looked so particularly for 1800 years ago, but could not find. Here they were, however, and still are, utilised in enabling the great pyramid to stand on the very utmost verge of its commanding hill, within the limits of the two required latitudes, as well as over the centre of the land's physical and radial formation, and at the same time on the sure and proverbially wise foundation of rock.'
The next circumstance to be noted in the position of the great pyramid (as of all the pyramids) is that the sides are carefully oriented. This, like the approximation to a particular latitude, must be regarded as an astronomical rather than a geographical relation. The accuracy with which the orientation has been effected will serve to show how far the builders had mastered the methods of astronomical observation by which orientation was to be secured. The problem was not so simple as might be supposed by those who are not acquainted with the way in which the cardinal points are correctly determined. By solar observations, or rather by the observations of shadows cast by vertical shafts before and after noon, the direction of the meridian, or north and south line, can theoretically be ascertained. But probably in this case, as in determining the latitude, the builders took the stars for their guide. The pole of the heavens would mark the true north; and equally the pole-star, when below or above the pole, would give the true north, but, of course, most conveniently when below the pole. Nor is it difficult to see how the builders would make use of the pole-star for this purpose. From the middle of the northern side of the intended base they would bore a slant passage tending always from the position of the pole-star at its lower meridional passage, that star at each successive return to that position serving to direct their progress; while its small range, east and west of the pole, would enable them most accurately to determine the star's true mid-point below the pole; that is, the true north. When they had thus obtained a slant tunnel pointing truly to the meridian, and had carried it down to a point nearly below the middle of the proposed square base, they could, from the middle of the base, bore vertically downwards, until by rough calculation they were near the lower end of the slant tunnel; or both tunnels could be made at the same time. Then a subterranean chamber would be opened out from the slant tunnel. The vertical boring, which need not be wider than necessary to allow a plumb-line to be suspended down it, would enable the architects to determine the point vertically below the point of suspension. The slant tunnel would give the direction of the true north, either from that point or from a point at some known small distance east or west of that point.[19] Thus, a line from some ascertained point near the mouth of the vertical boring to the mouth of the slant tunnel would lie due north and south, and serve as the required guide for the orientation of the pyramid's base. If this base extended beyond the opening of the slant tunnel, then, by continuing this tunnelling through the base tiers of the pyramid, the means would be obtained of correcting the orientation.
This, I say, would be the course naturally suggested to astronomical architects who had determined the latitude in the manner described above. It may even be described as the only very accurate method available before the telescope had been invented. So that if the accuracy of the orientation appears to be greater than could be obtained by the shadow method, the natural inference, even in the absence of corroborative evidence, would be that the stellar method, and no other, had been employed. Now, in 1779, Nouet, by refined observations, found the error of orientation measured by less than 20 minutes of arc, corresponding roughly to a displacement of the corners by about 37-1⁄2 inches from their true position, as supposed to be determined from the centre; or to a displacement of a southern corner by 53 inches on an east and west line from a point due south of the corresponding northern corner. This error, for a base length of 9140 inches, would not be serious, being only one inch in about five yards (when estimated in the second way). Yet the result is not quite worthy of the praise given to it by Professor Smyth. He himself, however, by much more exact observations, with an excellent altazimuth, reduced the alleged error from 20 minutes to only 4-1⁄2, or to 9⁄40ths of its formerly supposed value. This made the total displacement of a southern corner from the true meridian through the corresponding northern corner, almost exactly one foot, or one inch in about twenty-one yards—a degree of accuracy rendering it practically certain that some stellar method was used in orienting the base.
Now there is a slanting tunnel occupying precisely the position of the tunnel which should, according to this view, have been formed in order accurately to orient the pyramid's base, assuming that the time of the building of the pyramid corresponded with one of the epochs when the star Alpha Draconis was distant 3° 42' from the pole of the heavens. In other words, there is a slant tunnel directed northwards and upwards from a point deep down below the middle of the pyramid's base, and inclined 26° 17' to the horizon, the elevation of Alpha Draconis at its lower culmination when 3° 42' from the pole. The last epoch when the star was thus placed was circiter 2160 B.C.; the epoch next before that was 3440 B.C. Between these two we should have to choose, on the hypothesis that the slant tunnel was really directed to that star when the foundations of the pyramid were laid. For the next epoch before the earlier of the two named was about 28,000 B.C., and the pyramid's date cannot have been more remote than 4000 B.C.
