Evening star. [volume] (Washington, D.C.) 1854-1972, 1929-02-03

Then move the pointer UP to the point
P passing Y horizontal lines. The point
P is then described without ambiguity
by the numbers x Y (co-ordinates). If
one had used, instead of ruled milli-
meter paper, q piece which had been
stretched or deformed the same de-
termination could still be carried out;
but in this case the lines passed would
no longer be horizontals or verticals or
even straight lines. The same point
would then of course yield different
numbers, but the possibility Of deter
mining a point by means of two num-
bers (Gaussian co-ordinates) still re-
mains. Moreover, II p and Y are two
points which lie very close to one an
other then their co-ordinates only diff
fer very slightly.
When point can be described by two
numbers in this way we speak of a
wo-dimensional continuum (surface).
Explains Riemannian Metric.
Now consider two neighboring points
P, @ on the surface and a little way
off another pair of points p. @. What
does it mean to say that the distance
Pc Is equal to the distance Pc? This
statement only has clear meaning
when we have a small measuring rod
which We can take from one pair of
points to the other and If the result
of the comparison is independent of
the particular measuring rod selected.
If this Is so, the magnitudes of the
tracts Pc, Pc can be compared. II a
continuum Is of this kind we say it
has a metric. OF course, the distance
of the two points Pc must depend on
the coordinate differences ok, DY)
But the form of this dependence is
not known priori. If it Is of the form:
ds'=g DIX Fl 'dxdy + g''dy'
Then it is called Riemannian met
ric.
If it Is possible to choose the co-or-
dinateS so that this expression takes
the form: ds'=dx + dy' (Pythagoras's
theorem). then the continuum is Eu-
clidean IA plane).
Thus it Is clear that the Euclidean
continuum is special case of the
Riemannian. Inversely, the Keenan,
MAN continuum is metric continuum
which is Eclidean in infinitely small re-
gions, but not in finite regions. The
quantities, g', gf, ga. describe the
metrical properties of the surface, that
is the metrical field.
BY making use of empirically known
pIOpeTtleS Oh space, especially CIle law
of the propagation of light, 1t is pos
sible to show that the space-time
continuum has Riemannian metric.
The quantities gu, etc, appertaining
to it determine not only the metric
of the continuum, but also the gravita-
tional field The law governing the
gravitational field is found in answer
to the question: Which are the simplest
mathematical laws to which the metric
chat is the g'i, etc. can be subjected?
The answer was given by the discovery
of the field laws of gravitation, which
have proved themselves more accurate
than the Newtonian law. This rough
outline is intended only to give gen.
eral idea of the sense in which have
spoken of the speculative methods
OF the general theory of relatively.
Expanding the Theory.
This theory having brought together
the metric and gravitation would have
been completely satisfactory If the
world had only gravitation fields and
no electro magnetic fields. Now 1t is
true that the latter can be included
within the general theory of relatively
by taking over and appropriately mod:-