tude where the true pole of the heavens
is one-third of the way from the horizon to the point overhead (the
zenith), and where the noon sun at true spring or autumn (when the sun
rises almost exactly in the east, and sets almost exactly in the west)
is two-thirds of the way from the horizon to the point overhead. In an
observatory set exactly in this position, some of the calculations or
geometrical constructions, as the case may be, involved in astronomical
problems, are considerably simplified. The first problem in Euclid, for
example, by which a triangle of three equal sides is made, affords the
means of drawing the proper angle at which the mid-day sun in spring or
autumn is raised above the horizon, and at which the pole of the heavens
is removed from the point overhead. Relations depending on this angle
are also more readily calculated, for the very same reason, in fact,
that the angle itself is more readily drawn. And though the builders of
the great pyramid must have been advanced far beyond the stage at which
any difficulty in dealing directly with other angles would be involved,
yet they would perceive the great advantage of having one among the
angles entering into their problems thus conveniently chosen. In our
time, when by the use of logarithmic and other tables, all calculations
are greatly simplified, and when also astronomers have learned to
recognize that no possible choice of latitude would simplify their
labours (unless an observatory could be set up at the North Pole itself,
which would be in other respects inconvenient), matters of this sort are
no longer worth considering, but to the mathematicians who planned the
great pyramid they would have possessed extreme importance.

[Illustration: Fig. 1.]

To set the centre of the pyramid's future base in latitude 30 deg., two
methods could be used, both already to some degree considered--the
shadow method, and the Pole-star method. If at noon, at the season when
the sun rose due east and set due west, an upright A C were found to
throw a shadow C D, so proportioned to A C that A C D would be one-half
of an equal-sided triangle, then, theoretically, the point where this
upright was placed would be in latitude 30 deg.. As a matter of fact it
would not be, because the air, by bending the sun's rays, throws the sun
apparently somewhat above his true position. Apart from this, at the
time of true spring or autumn, the sun does not seem to rise due east,
or s