be high above the horizon
even when exactly under the pole. No star so far from the pole as to
pass close to the horizon would be of use even for the work of
orientation, while for the work of obtaining the latitude it would be
absolutely essential that a star close to the pole should be used.

A line along the feet of the uprights would run north-and-south. But the
very object for which the great astronomical edifice was being raised,
was that the north-and-south line amongst others should be indicated by
more perfect methods.

Now at this stage of proceedings, what could be more perfect as a method
of obtaining the true bearing of the pole than to dig a tubular hole
into the solid rock, along which tube the Pole-star at its lower
culmination should be visible? Perfect stability would be thus insured
for this fundamental direction line. It would be easy to obtain the
direction with great accuracy, even though at first starting the borings
were not quite correctly made. And the further the boring was continued
downwards towards the south the greater the accuracy of the direction
line thus obtained. Of course there could be no question whatever in
such underground boring, of the advantage of taking the lower passage of
the Pole-star, not the upper. For a line directly from the star at its
upper passage would slant downwards at an angle of more than thirty
degrees from the horizon, while a line directly from the star at its
lower passage would slant downwards at an angle of less than thirty
degrees; and the smaller this angle the less would be the length, and
the less the depth of the boring required for any given horizontal
range.

Besides perfect stability, a boring through the solid rock would present
another most important advantage over any other method of orienting the
base of the pyramid. In the case of an inclined direction line above the
level of the horizontal base, there would be the difficulty of
determining the precise position of points under the raised line; for
manifest difficulties would arise in letting fall plumb-lines from
various points along the optical axis of a raised tubing. But nothing
could be simpler than the plan by which the horizontal line
corresponding to the underground tube could be determined. All that
would be necessary would be to allow the tube to terminate in a
tolerably large open space; and from a point in the base vertically
above this, to let fall a plumb-line through a fine