g from
the effects of atmospheric refraction. They would proceed probably
somewhat in this wise:--Using any kind of direction lines, they would
take the altitude of their Polar star (1) when passing immediately under
the pole, and (2) when passing immediately above the pole. The mean of
the altitudes thus obtained would be the altitude of the true pole of
the heavens. Now, atmospheric refraction affects the stars in the same
way that it affects the sun, and the nearer a star is to the horizon,
the more it is raised by atmospheric refraction. The Pole-star in both
its positions--that is when passing below the pole, and when passing
above that point--is raised by refraction, rather more when below than
when above; but the estimated position of the pole itself, raised by
about the mean of these two effects, is in effect raised almost exactly
as much as it would be if it were itself directly observed (that is, if
a star occupied the pole itself, instead of merely circling close round
the pole). We may then simplify matters by leaving out of consideration
at present all questions of the actual Pole-star in the time of the
pyramid builders, and simply considering how far they would have set the
pyramid's base in error, if they had determined their latitude by
observing a star occupying the position of the true pole of the heavens.

They would have endeavoured to determine where the pole appears to be
raised exactly thirty degrees above the horizon. But the effect of
refraction being to raise every celestial object above its true
position, they would have supposed the pole to be raised thirty degrees,
when in reality it was less raised than this. In other words, they would
have supposed they were in latitude 30 deg., when, in reality, they were in
some lower latitude, for the pole of the heavens rises higher and higher
above the horizon as we pass to higher and higher latitudes. Thus they
would set their station somewhat to the south of latitude 30 deg., instead
of to the north, as when they were supposed to have used the shadow
method. Here again we can find how far they would set it south of that
latitude. Using the Greenwich refraction table (which is the same as
Bessel's), we find that they would have made a much greater error than
when using the other method, simply because they would be observing a
body at an elevation of about thirty degrees only, whereas in taking the
sun's mid-day altitude in spring or autumn, they wou