Euclid defined an angle as an 'inclination'. 1 Unfortunately, Euclid or his translators
forgot to tell us what an inclination is, but, in the context of structure, an inclination is
a static attribute. 2 Euclid also circumscribed angles to a plane. Relativists have
altered both of these attributes to suit their arguments and end up chasing their tails.
Contemporary mathematicians perceive an angle as ‘the amount of rotation’ 3 and
talk of angles lying on spheres. 4 Of course, the loophole for the ‘curved angle’
proposal is to measure the angle between two tangents, both of which are straight
and lie on the same plane. More troublesome is to reconcile the modern dynamic
with Euclid’s static notion of angle. The relativistic result is not a photograph, but
rather a movie. The ET has to wait until the misguided relativist moves a line
sufficiently to call it an angle. If the video recorder breaks down in the middle of the
experiment, the ET returns to his planet without ever knowing what an earthly angle
is. This definition summarily outlaws the 0° angle.

The relativistic notions create insurmountable semantic problems. Among the most
notorious are the claims that there are triangles that conform to the surface of
saddles and spheres. In the religion of relativity, this amusing notion results in
triangles whose internal angles add up to more or less than 180°. If a triangle is a
geometry that has three angles and an angle is contingent on a plane, Lobachevski
and Riemann-like ‘geometries’ do not qualify as triangles. They lack two of the most
important properties of this fundamental figure: flatness and angles. First, the two
tangents that relativists use to measure angles on spheres violate their ‘rotation’
definition. Then, if these tangents are alleged to meet at a point, relativists must
define what they mean by touch. Do the tangents share a common point or are the
points ‘meeting’ side by side? Are these points spherical, circular, or hexagonal?
But let’s concede whatever relativists wish to assume. A tangent makes contact with
a sphere at a single point. We can rely on two tangents lying on an imaginary plane
to make an angle. These two tangents cannot and do not meet a third tangent that
touches the sphere at any other point. The most relativists can construct with this
hocus pocus maneuver is one angle, not three simultaneously. However, when they
argue that the internal ‘angles’ of  a Riemann ‘triangle’ add up to more than 180° they
are inadvertently doing just that. The alleged Riemann triangle is a 3D figure viewed
in 2D. That’s why the alleged angles add up to more than 180°.

Finally, assuming the tangents have a point in common, this point by itself does not
qualify as an angle. Therefore, the tangents are run above parallel
The next point after the common point on each tangent does not coincide with the
curved line lying flat against the sphere. Since relativists allege that there are infinite
points between any two, the point we choose makes a difference in the angle. In
order to have an angle, both the ‘rotation’ and ‘tangent’ definitions require two lines.
As soon as the lines diverge along the surface of the sphere, we can have no angle.
[In science we don’t define a line. We point to a line and say line.]


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