ity of question that the shadow method was not the method on
which sole or chief reliance was placed, though this method must have
been known to the builders of the pyramid. It does not, however, prove
that the star method was the only method followed. A distance of 944
yards is so small in a matter of this sort that we might fairly enough
assume that the position of the base was determined by the Pole-star
method. If, however, we supposed the builders of the pyramid to have
been exceedingly skilful in applying the methods available to them, we
might not unreasonably conclude from the position of the pyramid's base
that they used both the shadow method and the Pole-star method, but
that, recognizing the superiority of the latter, they gave greater
weight to the result of employing this method. Supposing, for instance,
they applied the Pole-star method three times as often as the shadow
method, and took the mean of all the results thus obtained, then the
deduced position would lie three times as far from the northern position
obtained by the shadow method as from the southern position obtained by
the Pole-star method. In this case their result, if correctly deduced,
would have been only about 156 yards north of the actual present
position of the centre of the base.

It is impossible, however, to place the least reliance on any
calculation like that made in the last few lines. By _a posteriori_
reasoning such as this one can prove almost anything about the pyramids.
For observe, though presented as _a priori_ reasoning, it is in reality
not so, being based on the observed fact, that the true position lies
more than three times as far from the northerly limit as from the
southern one. Now, if in any other way, not open to exception, we knew
that the builders of the pyramid used both the sun method and the star
method, with perfect observational accuracy, but without knowledge of
the laws of atmospheric refraction, we could infer from the observed
position the precise relative weights they attached to the two methods.
But it is altogether unsafe, or, to speak plainly, it is in the logical
sense a perfectly vicious manner of reasoning, to ascertain first such
relative weights on an assumption of this kind, and having so found
them, to assert that the relation thus detected is a probable one in
itself, and that since, when assumed, it accounts precisely for the
observed position of the pyramid, therefore the pyramid was posite