7

7. The first waves from a very distant earthquake come to us
directly through the Earth. The observed speeds of transmission
are the greater, in general, the more nearly the earthquake
origin is exactly on the opposite side of the Earth from the
observer; that is, the speeds of transmission are greater the
nearer the center of the Earth the waves pass. Now, we know
that the speeds are functions of the rigidity and density of
the materials traversed. The observed speeds require for their
explanation, so far as we can now see, that the rigidity of the
Earth"s central volume be much greater than that of steel, and
the rigidity of the Earth"s outer strata considerably less than
that of steel. Wiechert has shown that a core of radius 4,900
km. whose rigidity is somewhat greater than that of steel and
whose average density is 8.3, overlaid by an outer stony shell
of thickness 1,500 km. and average density 3.2, would satisfy
the observed facts as to the average density of the Earth, as
to the speeds of earthquake waves, as to the flattening of the
Earth,--assuming the concentric strata to be homogeneous in
themselves,--and as to the relative strengths of gravity at the
Poles and at the Equator. The dividing line, 1,500 km. below
the surface--1,600 km. would be just one fourth of the way from
the surface to the center--places a little over half the volume
in the outer shell and a little less than half in the core.
Wiechert did not mean that there must be a sudden change of
density at the depth of 1,500 km., with uniform density 8.3
below that surface and uniform density 3.2 above that surface.
The change of density is probably fairly continuous. It was
necessary in such a preliminary investigation to simplify the
assumptions. The observational data are not yet sufficiently
accurate to let us say what the law of increase in density and
rigidity is as we pass from the surface to the center.


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To obey reason is, then, the first duty, at the root of all others, and

itself resting directly upon the relation between liberty and reason;
in a sense, to remain reasonable is the sole duty
To obey reason is, then, the first duty, at the root of all others, and
itself resting directly upon the relation between liberty and reason;
in a sense, to remain reasonable is the sole duty. But it assumes
special forms amid the diversity of human relations. He first considers
the relations wherein we stand to ourselves and the corresponding
duties. That there should be any such duties is at first sight strange,
seeing we belong to ourselves; but this is not the same as having
complete power over ourselves. Possessing liberty, we must not abdicate
it by yielding to passions, and treat ourselves as if there were
nothing in us that merits respect. We are to distinguish between what
is peculiar to each of us, and what we share with humanity. Individual
peculiarities are things indifferent, but the liberty and intelligence
that constitute us persons, rather than individuals, demand to be
respected even by ourselves. There is an obligation of self-respect
imposed upon us as moral persons that was not established, and is not
to be destroyed, by us. As special cases of this respect of the moral
person in us, he cites (1) the duty of _self-control_ against anger or
melancholy, not for their pernicious consequences, but as trenching
upon the moral dignity of liberty and intelligence; (2) the duty of
_prudence_, meaning providence in all things, which regulates courage,
enjoins temperance, is, as the ancients said, the mother of all the
virtues,--in short, the government of liberty by reason; (3)
_veracity_; (4) duty towards the _body_; (5) duty of _perfecting_ (and
not merely keeping intact) the intelligence, liberty, and sensibility
that constitute us moral beings.


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It would not be difficult to increase greatly the number of the

given illustrations of unsolved questions relating directly to
the natural numbers
It would not be difficult to increase greatly the number of the
given illustrations of unsolved questions relating directly to
the natural numbers. In fact, the well-known greater Fermat
theorem is a question of this type, which does not appear more
important intrinsically than many others but has received
unusual attention in recent years on account of a very large
prize offered for its solution. In view of the fact that those
who have become interested in this theorem often experience
difficulty in finding the desired information in any English
publication, we proceed to give some details about this theorem
and the offered prize. The following is a free translation of a
part of the announcement made in regard to this prize by the
Konigliche Gesellschaft der Wissenschaften, Gottingen, Germany:


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Let me illustrate by a supposititious example

Let me illustrate by a supposititious example. A nation has an
expenditure of $100,000,000 a year. It raises the sum by
taxation of some sort and thus lives within its means. But
$100,000,000 is the interest on a much larger sum, let us say
$2,500,000,000. If instead of paying out a hundred million year
by year for expenses, we capitalize it, we may have immediately
at hand a sum twenty-five times as great. The interest on this
sum is the same as the annual expense account. Let us then
borrow $2,500,000,000 on which the interest charges are
$100,000,000 a year. But while paying these charges the nation
has the principal to live on for a generation. Half of it will
meet current expenses for a dozen years, and the other half is
at once available for public purposes, for dockyards, for
wharves, for fortresses, for public buildings and, above all,
for the ever-growing demands of military conscription and of
naval power. Meanwhile the nation is not standing still. In
these twelve years the progress of invention and of commerce
may have doubled the national income. There is then still
another $100,000,000 yearly to be added to the sum available
for running expenses. This again can be capitalized, another
$2,500,000,000 can be borrowed, not all at once perhaps, but
with due regard to the exigencies of banking and the temper of
the people. With repeated borrowings the rate of taxation
rises. Living on the principal sets a new fashion in
expenditure. The same fashion extends throughout the body
politic. Individuals, corporations, municipalities all live on
their principal.


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