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The way I look at this, and I am not sure I am right, is best described by
thinking of a cross section of the earth drawn as a circle and as a ellipse
with common centers.  The circle would represent a spherical earth and the
ellipse an ellipsoidal earth.  At the north pole and at the equator, tangents
to the circle and to the ellipse are parallel to each other.  Lines normal to
those tangents all pass through the common center.  The normals from the
poles intersect those from the equators at right angles giving the poles a
latitude of 90 degrees.  The normals are the orientation of plumb bobs at the
equator and at the poles.

A line drawn outward from the center of the earth half way between the
equator and the north pole strikes the circle at a point where a tangent to
the circle would be at right angles to that line.  A pumb bob at that
location would be parallel to that line from the center of the earth.  The
location would be 45 degrees north.

The same line from the center of the earth would not strike the ellipse at
the point where a normal to a tangent line would be 45 degrees to the
equator.  That point would be just a little bit farther south.  (I can not do
the math, but maybe for the slightly flattened earth 11'32").  The plumb bob
at the location called 45 degrees north latitude on the ellipse would not
pass through the center of the earth, but would be parallel to a line from
the center of the earth at 45 degrees to the equator.

Bill Murdoch

   

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