Having their base properly oriented, and being about to erect the building itself, the architects would certainly not have closed the mouth of the slant tunnel pointing northwards, but would have carried the passage onwards through the basement layers of the edifice, until these had reached the height corresponding to the place where the prolongation of the passage would meet the slanting north face of the building. I incline to think that at this place they would not be content to allow the north face to remain in steps, but would fit in casing stones (not necessarily those which would eventually form the slant surface of the pyramid, but more probably slanted so as to be perpendicular to the axis of the ascending passage.) They would probably cut a square aperture through such slant stones corresponding to the size of the passage elsewhere, so as to make the four surfaces of the passage perfectly plane from its greatest depth below the base of the pyramid to its aperture, close to the surface to be formed eventually by the casing stones of the pyramid itself.
Now, in this part of his work, the astronomical architect could scarcely fail to take into account the circumstance that the inclined passage, however convenient as bearing upon a bright star near the pole when that star was due north, was, nevertheless, not coincident in direction with the true polar axis of the celestial sphere. I cannot but think he would in some way mark the position of their true polar axis. And the natural way of marking it would be to indicate where the passage of his Pole-star _above_ the pole ceased to be visible through the slant tube. In other words he would mark where a line from the middle of the lowest face of the inclined passage to the middle of the upper edge of the mouth was inclined by twice the angle 3° 42' to the axis of the passage. To an eye placed on the optical axis of the passage, at this distance from the mouth the middle of the upper edge of the mouth would (_quam proximé_) show the place of the true pole of the heavens. It certainly is a singular coincidence that at the part of the tube where this condition would be fulfilled, there is a peculiarity in the construction of the entrance passage, which has been indeed otherwise explained, but I shall leave the reader to determine whether the other explanation is altogether a likely one. The feature is described by Smyth as "a most singular portion of the passage--viz., a place where two adjacent wall-joints, similar, too, on either side of the passage, were vertical or nearly so; while every other wall-joint, both above and below, was _rectangular_ to the length of the passage, and, therefore, largely _inclined_ to the vertical." Now I take the mean of Smyth's determinations of the transverse height of the entrance passage as 47.23 inches (the extreme values are 47.14 and 47.32), and I find that, from a point on the floor of the entrance passage, this transverse height would subtend an angle of 7° 24' (the range of Alpha Draconis in altitude when on the meridian) at a distance 363.65 inches from the transverse mouth of the passage. Taking this distance from Smyth's scale in Plate xvii. of his work on the pyramid ("Our Inheritance in the Great Pyramid"), I find that, if measured along the base of the entrance passage from the lowest edge of the vertical stone, it falls exactly upon the spot where he has marked in the probable outline of the uncased pyramid, while, if measured from the upper edge of the same stone, it falls just about as far within the outline of the cased pyramid as we should expect the outer edge of a sloped end stone to the tunnel to have lain.