lowering of the false bottom. As long as the exposed sand
above this false bottom had cohesion enough to prevent the collapse of
the "centering," this arch could have been loaded with safety up to the
limits of the compressive strength of the sand.

To quote again from Mr. Goodrich:

"Furthermore, the author's reason for bisecting the angle between
the vertical and the angle of repose of the material, when he
undertakes to determine the thickness of key, is not obvious. This
assumption is shown to be absurd when carried to either limit, for
when the angle of repose equals zero, as is the case with water,
this method would give a definite thickness of key, while there can
be absolutely no arch action possible in such a case; and, when the
angle of repose is 90 deg., as may be assumed in the case of rock, this
method would give an infinite thickness of key, which is again seen
to be absurd."

Mr. Goodrich assumes that water or liquid has an angle of repose equal
to zero, which is true, but the writer's assumptions applied only to
solid material, and the liquid gives an essentially different condition
of pressure, as shown by a careful reading of the paper. In solid rock
Mr. Goodrich assumes an angle of repose equal to 90 deg., for which there is
no authority; that is, solid rock has no known angle of repose. In order
to carry these assumptions to a definite conclusion, we must assume for
that material with an angle of repose of 90 deg. some solid material which
has weight but no thrust, such as blocks of ice piled vertically. In
this case Mr. Goodrich can readily see that there will be no arching
action over the structure, and that the required thickness of key would
be infinite. As to the other case, it is somewhat difficult to conceive
of a solid with an angle of repose of zero; aqueous material does not
fulfill this condition, as it is either a liquid or a combination of
water and solid material. The best illustration, perhaps, would be to
assume a material composed of iron filings, into which had been driven a
powerful magnet, so that the iron filings would be drawn horizontally in
one direction. It is easy to conceive, then, that in tunneling through
this material there would be no necessity for holding up the roof; the
definite thickness of key given, as being at the point of intersection
of two 45 deg. angles, would be merely a precautionary measure, and would
no