
If now a series of stationary posts are placed in space in straight line,
along the positions of which the moving body travels, the intuitional concept
of time arises from the succession of events realized by its progress forward
from station to station.  That concept is not applicable in the same way to a
reverse motion of the body.  This is due to the fact that in the finite
universe of conflicting forces and never-ending movement, progress in any
direction involves frictions, wearing out, change, death.  In an infinite
universe of non-complex and hence indestructible beings, time becomes
converted into eternity, disappears as a necessary concept.  This would
certainly be true if our observer, the moving object and the posts were alike
incorruptible and changeless.  In such a case time would have no
meaning.  Its reality is in the eternal restlessness of things.  Our
observer's mind functions with finite memory, in a finite body, which
develops and wears out and dies, so that for him, observing the retrograde
motion of the traveling and corruptible objects, as it reaches, in
succession, the same station posts previously passed reverse order, those
posts are seen in comparison with their images recalled in memory, and with
such changes, greater or less, as the interval will have brought in the
moving body, the station posts, and the observer's mortal self.  Thus for us
the time concept, though so closely related to that of space, can have a
forward direction only.  For mortals, that which has happened, can never
happen again, except in reminiscence, and then it is a different
happening.  Four-dimensionalism may involve a different view.  For it, all
events, past, present and future, may be coincidentally existent, developed
for the individual as the space-time world-line of his travel encounters
them.


TIME, IS CONCEIVED AND MEASURED BY US IN TERMS OF MOTION THROUGH SPACE.  The
day is, the concept of the repeated motion of the sun about the earth, or, in
more precise terms, the concept of the sequential motion of the earth about
its axis.  Similarly, the year is the concept of the sequence of the earth's
motion around the sun.  SUCH TIME MEASUREMENTS ARE ESSENTIAL FOR FINITE
BEINGS LIVING IN A CHANGING WORLD OF FINITE PARTS.

In such a world an object, in order to be perceptible to consciousness, must
ENDURE.  The mind cannot perceive it until the eye, by the movement of its
muscles or by the associations of perspective with other muscular movements,
has traveled over it, taking in by complexes, its extensions in three
directions, or until the hand or other member has touched it, traveling over
its surfaces.  Any such sequences of motion take time in space, and, as the
perception is impossible without the motion, THE OBJECT CANNOT EXIST FOR US
IN SPACE WITHOUT THE TIME ELEMENT.

Thus, each of the station posts of our illustration, though deemed at rest,
may be said, in a sense, by its persistent endurance in space, by the quality
of the static energy which constitutes its existence, to be moving along the
time dimension, in the same way as, by the addition of kinetic energy, it
could be given apparent motion in space.  That, at all events, is the view of
four- dimensionalists; and in this sense the space medium may be said to have
four dimensions.  The energy of the enduring post constitutes its travel in
time.  From the viewpoint of the moving observer, who deems himself
stationary, it may be said to be traveling in space also, that is in a
composite space-time.  If it helps anyone to think in that way, no harm is
done; but it is a fair inquiry whether, even granted our qualified abilities
of perception, there is any analogy between travel in space and endurance in
time.  The only readily perceptible relationship is the measurement of
endurance in terms of space motion.

THE SERIOUS MEANING UNDERLYING THE SPACE-TIME CONCEPTION IS THAT WE DO NOT
GET A TRUE IDEA OF THE POSITION IN THE UNIVERSE OF AN EVENT WHEN WE ARE GIVEN
SPACE CO-ORDINATES ONLY.  The complete picture can only be afforded by the
addition of the time co-ordinate.  To say that a battle took place in Paris
is to say that it took place so many miles and in such a direction from New
York but the event is not truly placed until we add the date, that is, a
dimension measured, in days or years, in a number of certain periodic
motions, from another time.  To determine the full time and space
relationship between two objects in space, speaking in orthodox language, one
must know their distance apart inn terms of an agreed length unit, and if one
was in its place before the other, the time between them in terms of a time
unit based on some periodic occurrence.  In the new language we are said to
be ling in a space-time world.  The persistence of objects and the duration
of events are viewed as travel along the time dimension in a sense analogous
to that in which we seem too see objects moving along space dimensions.  This
conception, has never been realized by anybody except in mathematical
symbols.


We shall now endeavor to give an elementary idea of the mathematical
formulation of the space-time world.  This will give some notion of the
foundation upon which rests the space-time conception and, together with it,
the general theory and the new law of gravitation.


The following is a method of determining the distance or length of the line
between two points in a plane, in reference to a Cartesian co-ordinate system
to neither axis of which the line is parallel.  It makes no difference if this
line be curved inasmuch as, for the purposes about to be described, the
selected distance of the two points must be so minute that the length will be
practically the same whether the line be straight or curved.  WE SHALL
DESIGNATE THE MINUTE DISTANCE IN QUESTION AS THE 'DISTANCE' ELEMENT.  From
each end of the distance element are dropped two perpendiculars, one to the
'x' axis of the co-ordinate system and the other to the 'y' axis, extending
the perpendiculars in such way as to form, by their respective intersections,
together with the distance element, the sides of a right angle triangle, of
which the element is the hypotenuse and the respective 'x' and 'y'
co-ordinate differences of the two end points are the remaining sides.  We
then have a method of measuring the distance element by use of the Euclidean
proposition, known as the theorem of Pythagoras, that the square of the
hypotenuse is equal to the sum of the squares of the other two sides.
Designating then the x co-ordinate difference as x and the y co-ordinate
difference as y, we have the result that the square of the distance element
will be equal to

x^2 + y^2


Let us now select different co-ordinate axes, for example, x' and y', and
draw the same perpendiculars as before from the ends of the same distance
element to these new axes, forming again, in the same way, a right angle
triangle, of which the distance element is the hypotenuse.  You may place
your new co-ordinate system at any angle you please in relation to the first
one and still, since the distance element is always the same, you will always
have the same expression for it, though in terms of the new co-ordinate
system, that is to say, the square of the distance element will now equal

x'^2 + y'^2.

Your new right angle triangle may be of an entirely different shape from the
first one, but the same relationship must of course still hold.


If we now desire to place our distance element in 3-dimensional space, that
is to say, in relation to a Cartesian co-ordinate system of 3-dimensions, we
shall merely, in order to obtain the quantitative relations, have to drop our
perpendiculars not only to the x and y axes, but also the z axis, but the
relationship will still turn out the same; and we shall have the result that
the distance element will be expressed by the form

x^2 + y^2 + z^2

or on another co-ordinate system by the like form

x'^2 + y'^2 + Z'^2.


The distance element, thus being always the same and expressible in like
forms of equal value, whatever the stationary Cartesian co-ordinate system
selected may be, is itself a quantity known as an INVARIANT.  As above,
shown in 2-Dimensional space it is formulated in various expressions always
involving two quantities; in 3-Dimensional space in various expressions,
always involving three quantities.  The question immediately suggests itself,
whether the distance element is still an invariant, which may be similarly
expressed in terms of the co-ordinates- of a moving Cartesian system at any
given time, by the Lorentz transformation.  We have seen that, according to
the special theory, the answer must apparently be in the negative.  The
motion, even at any particular instant renders the time and space quantities
variable, and introduces into the equations three new quantities--the lapsed
time--the velocity of the system-- and that of light.


Now the strange variance of time and space quantities may be expressive of a
world of 4-Dimensions.  The non-Euclidean geometry would, in that event
become useful.  If a straight line may no longer be the shortest distance
between two points, and parallel lines may now meet, strange results must
follow.  Minkowski therefore, set out to discover whether he could devise, by
the use of the Lorentz transformation, a formula for the distance element in
the 4-Dimensional world, which would contain 4 terms and be analogous in form
to that above set forth for the 3-Dimensional world, and invariant in
relation to any system, in spite of the apparent disturbance in values caused
by motion.


When he applied the Lorentz equations to this determination of the distance
element, now become the "interval element" in the space-time world, the
result was, at first, an equation which contained, besides the x, y, z
quantities the new quantities, v, the velocity of the system, c, the velocity
of light, and t, the time.  He did find, that when he took the form

x^2 + y^2 + Z^2 - c^2 t^2

and transformed it by the Lorentz equations, the v's cancelled, and it turned
out equal to the form

x'^2 + y'^2 + z'^2 - c'^2 t'^2.


He and Professor Einstein have exhibited the greatest confidence that this
equation expresses the truth of a 4-Dimensional world, the minus quantity,
which is the departure from the analogy of the previous equations,
expressing, in some subtle and incomprehensible way, the inconceivable twist
or curvature of the fourth dimension, time.


In order to make the analogy complete, by rendering the form of the equation
identical with the 3-Dimensional form, these mathematicians first concluded
to adopt c, the velocity of light, as a unit with, therefore, the quantity 1,
which immediately reduces the expression c^2 t^2 to t^2.  The minus sign
before that expression, being a differentiation from the Pythagorean formula,
they then adopted for the expression t, the imaginary expression

sqrt( -1) t,

which immediately transforms the last quantity of the expression to + t^2,
thus giving, with these corrections, the following equation:

x^2 + y^2 + z^2 + t^2 = x'^2 + y'^2 + z'^2 + t'^2.

This merely means that, in adding to the expression the term containing the
quantity t, we are adding as expressive of the 4-Dimensional continuum, an
inconceivable negative quantity.  The above exposition does not cover details
and is merely intended as an elementary guide.


It will be observed, that the attempt at mathematical expression for
4-Dimensional space involves the use of the great Pythagorean theorem, the
backbone of Euclidean geometry, that the time quantity appearing in the
equation finally produced is a negative and has no comprehensible value, that
the reasoning is based largely on analogy, and that the resulting invariant
interval element is a mere expression not represented by any conceivable
reality.  THIS INTERVAL ELEMENT, OF COURSE, CEASES TO BE DISTANCE.  IT IS THE
EXPRESSION FOR THE INTERVAL, NOT BETWEEN TWO POINTS BUT BETWEEN TWO ADJACENT
EVENTS IN THE SPACE-TIME CONTINUUM; and its theoretic extension constitutes a
world-line which, in the space-time world conception, means travel, in a
united space and time, from event to event.  A direct method of the
enlargement of the minute, space-time interval element into a real
world-line has so far not been devised, except by the methods of classic
mathematics.  Only the element is conceived, by reason of its minuteness, and
not very logically, to be susceptible of treatment in accordance with the
Pythagorean proposition, but neither that proposition nor any other classic
formula ought properly to be applicable to a 4-Dimensional world.

