Transposition from Geometrical to Mechanical Principles.  www.rolfkeppler.de 

Impressum 
When we transpose the geometrical principles under consideration to mechanics, we have to deal with material surfaces and angles instead of geometric lines which bound figures on paper; the principles are the same, for geometry treats of the relations of form of material things.
Beginning
of the Air Line. Starting from Naples Dock, goini South alon the Gulf Coast.
Upper View, looking East; lower view looking West
p.93
When we carry our demonstrations
from the realm of principles to principles applied, we do no violence to
our conclusion. Such transposition is necessary to demonstrate the truth
about the material form of the earth, or the character of its tangible
contour.
The principles of right angles
constitute the basis of mechanics as well as of geometry.
Every mechanic considers right
angles; he works with the square. The laws of these relations of form must
be obeyed alike by the builder, engineer, and surveyor.
If we place two mechanic's squares
in line, the perpendicular arms will be parallel, and may be closely fitted
together. It also follows that if we place the perpendicular arms together,
the horizontal blades will form a straight line; the result are identical.
Two rectangular metal plates accurately trimmed and placed edge to edge
will force the other sides to form a straight line, as in the accompanying
diagram.
If two plates joined will form
a straight line, it follows that in the adjustment of three plates, the
result would be the same; the same result would be invariable for every
subsequent adjustment.
Ten thousand plates joined would
form a continuous rectiline.
A straight line would be forced
because of the fact that at every junction the surfaces joined would be
parallel, with the horizontals at right angles; consequently the horizontals
would necessarily be in line, Flagstones upon pavements running for miles,
are illustrations of the principles involved.
p.94
If such squares laid in a straight
line will join accurately the contiguous surfaces, it follows that if the
surfaces were joined accurately, the pavement would extend in the
same directionin a straight line.
There is no possible escape from this conclusion.
We may further illustrate this
principle by reference to the survey of railroad tangents, which often
extend for miles in a straight line, as related to the right and the left.
These tangents are surveyed by
means of the transit instrument, a small telescope mounted in a horizontal
axis, and made to revolve perpendicularly. The instrument may be taken
one mile from a given point, and adjusted so that the signal staff is coincidental
with the perpendicular crosshair.
Fourmile R.R. Tangent Surveyed by Revolving Transit on Horizontal, Rihtangled AxisPerpendicular View
As the instrument has a fixed axis
at right angles with the line of vision, or horizontal axis of the tube,
if the tube be revolved on its rightangled axis so as to point in the
opposite direction, another staff, two miles from the first, may be placed
exactly in line with the first staff and the line of collimation extending
through the tube.
In the accompanying diagram, presenting
a view of the ground from the vertical, A is the point of the first staff;
B, the telescope, CD, its axis at right angles with the line X; E is the
point of the second staff, and Y is the new line extended from B, by revolving
the telescope perpendicularly on its axis CD.
If the axis be true, the lines
X and Y will form a continuous
p.95
straight line two miles in length,
its extension being dependent upon a rightangled axis only three inches
from either side of the line!
If such long distances can be connected
together by so short a rightangled axis, it follows that if surfaces of
rectangles of greater adjusting leverage are placed in conjunction, they
are capable of extending absolutely straight lines.