The contemporary definition of parallel can be synthesized as 'coplanar lines that
don't intersect'. This definition is nothing but a rewording of Euclid's definition # 23,
which states:
"Parallel straight lines are straight lines which lie in the same plane and, if produced
indefinitely in diametrical directions, do not meet in either direction."
Euclid's definition was intended exclusively for plane geometry. Since the
development of hyperbolic (Lobachevsky) and elliptic (Riemann) geometry the
essence of Euclid's definition -- lines that do not meet' --was extended to lines lying
on solids. This has generated a host of interesting semantic problems. Of course, a
prosecutor must define words to suit his theory. The task of the jury is to verify that
the definition is not irrational to begin with and is used consistently throughout trial. I
will show that Euclid's definition is irrational and that the one extrapolated by
relativists is irrational and used inconsistently throughout the trial. None of them
belong in Geometry.
The reader should by now be able to recognize some of the problems with
definitions. First, Euclid's definition is a actually a proof. Indeed, for centuries
mathematicians have tried to 'prove' Euclid's 'parallel' postulate, which is essentially
a rewording of his 23rd definition. Meanwhile, the ET has to wait until Euclid extends
his line indefinitely to understand what parallel means. Euclid's parallel lines do not
constitute a pattern that we can synthesize in a static photograph. We must film his
parallels frame by frame. Euclid's movie is still in production. Moreover, Euclid has
modified the initial assumption if he forcefully extends two lines to prove his
definition. The devil's advocate only wanted to know whether the two line segments
before us were parallel. Relativists have so distorted his definition that not even
Euclid would recognize it today. According to one version, parallel ultimately
depends on the perception of the observer. Some relativists contend that railroad
tracks that remain ideally parallel throughout their entire extent and that never touch
each other are NOT parallel if from the observer’s perspective the tracks appear to
meet in the horizon! This is a good example of the inadequacy of making definitions
contingent upon proofs and observers.
So where does this leave the relativistic definition, which was extrapolated from
Euclid's imperfect attempt? In plane (or Euclidean) geometry, parallel and
equidistant lines are synonymous. Irrespective of how far Euclid extends them, it
seems reasonable to conclude that they will not only never meet but also maintain
the same distance from edge to edge throughout. Not so in hyperbolic and elliptic
geometry. The reason for this is that relativists have changed the rules. Relativists
insist that the concept known as 'straight line' should conform to the physical
medium it rests on. (This, I believe, is the first documented instance in history where
man physically bends a concept.) The way they 'prove' this to novices is by drawing
parallel lines on a paper and molding the paper into a saddle. If the experiment was
performed correctly, the two lines should diverge, which shows that despite not
being equidistant they are nevertheless parallel (i.e., they do not meet).
What relativists have overlooked in their 'empirical' definition is that it relies on the
concept 'straight line'. It turns out that a straight line stands on its own and does not
need a plane or a solid as a backdrop. A straight line has its own definition and
cannot be changed retroactively to suit the prosecutor's theory. The definition of
straight line makes no provision for a backdrop although it clearly implies a plane.
Likewise, the various contemporary definitions of parallel fail to make provisions for
a solid. Hence, relativists can define parallel as they wish, but if their definition
invokes the term 'straight line', they cannot flex this definition in their experiment
with the paper. The definition of straight line doesn't change either because it lies on
a cube or on a sphere. A straight line that lies on the face of a cube remains just as
straight as it does when it forms a tangent with the surface of a sphere. The way we
'prove' this in Physics is by removing the solid on which relativists drew their lines
so as not to distract them. We discover when we do this that what the relativist
actually did was to modify the definition of the concept straight line retroactively the
moment he bent the physical paper.
If, on the other hand, relativists insist that they are referring to a line that conforms
to the surface of a sphere, they are describing a curve. This proposal is now three
elements removed from Euclid's original hypothesis. The lines are neither coplanar
nor straight, and cannot be extended indefinitely. Whether a great circle is regarded
as a series of points or an itinerary, it has a beginning location and an ending one. If
our stroll around the world begins at the Eiffel Tower, it must end at the Eiffel Tower
minus one step. If relativists want to go around again, their tireless walk does not
make the great circle any longer. Nevertheless, they would be talking about motion
and not about a standalone geometric figure called a curve. So if Euclid's definition
has flaws, by conserving its essence -- intersection -- and by further changing the
rules, relativists have not come up with a better mouse trap. It would have been less
embarrassing for relativists to apologize and fix the vase after Euclid dropped it than
to try to convince the owner that the scattered pieces constitute a new type of art.
Therefore, I will revamp Euclid's erroneous definition and have parallel designate in
Physics what most people understand for parallel in everyday life: equidistant lines.
This does not mean that we will measure from edge to edge to prove whether the
definition holds its ground, but the opposite: that IF two edges are equidistant
throughout, they will henceforth be known as parallels for the remainder of the
presentation.
This definition summarily excludes lines which are allegedly made of points and
curves because neither can meet the orthogonal 'facing' requirement. In the case of
the 'point' line we would in addition be constructing parallelism point by point, which
results in a proof rather than a definition. This notion of parallel is also predicated on
edges as opposed to lines. Euclid's infinite line is not a geometric figure, but a
pattern, an unfinished object (Sec. xxx and xxx). Note that there is no provision for a
backdrop: the edges do not have to belong to the same plane or solid. Again,
whether this definition is useful to mathematicians is irrelevant. This is what a set of
parallel lines is in Physics.
1. Equidistant straight edges throughout the length of the shorter one, where the
distance is regarded in the direction in which the edge faces.
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Copyright © by Nila Gaede 2008