made, it is found that it never remains at rest. Beginning with the
changes of shortest duration, we have a change which takes place every
day, and is therefore called diurnal. In our northern latitudes it is
found that during the six hours from nine o'clock at night until three
in the morning the direction of the magnet remains nearly the same. But
between three and four A.M. it begins to deviate towards the east,
going farther and farther east until about 8 A.M. Then, rather
suddenly, it begins to swing towards the west with a much more rapid
movement, which comes to an end between one and two o'clock in the
afternoon. Then, more slowly, it returns in an easterly direction until
about nine at night, when it becomes once more nearly quiescent.
Happily, the amount of this change is so small that the navigator need
not trouble himself with it. The entire range of movement rarely
amounts to one-quarter of a degree.
It is a curious fact that the amount of the change is twice as great in
June as it is in December. This indicates that it is caused by the
sun's radiation. But how or why this cause should produce such an
effect no one has yet discovered.
Another curious feature is that in the southern hemisphere the
direction of the motion is reversed, although its general character
remains the same. The pointing deviates towards the west in the
morning, then rapidly moves towards the east until about two o'clock,
after which it slowly returns to its original direction.
The dip of the needle goes through a similar cycle of daily changes. In
northern latitudes it is found that at about six in the morning the dip
begins to increase, and continues to do so until noon, after which it
diminishes until seven or eight o'clock in the evening, when it becomes
nearly constant for the rest of the night. In the southern hemisphere
the direction of the movement is reversed.
When the pointing of the needle is compared with the direction of the
moon, it is found that there is a similar change. But, instead of
following the moon in its course, it goes through two periods in a day,
like the tides. When the moon is on the meridian, whether above or
below us, the effect is in one direction, while when it is rising or
setting it is in the opposite direction. In other words, there is a
complete swinging backward and forward twice in a lunar day. It might
be supposed that such an effect would be due to the moon, like the
earth, being a magnet. But were this the case there would be only one
swing back and forth during the passage of the moon from the meridian
until it came back to the meridian again. The effect would be opposite
at the rising and setting of the moon, which we have seen is not the
case. To make the explanation yet more difficult, it is found that, as
in the case of the sun, the change is opposite in the northern and
southern hemispheres and very small at the equator, where, by virtue of
any action that we can conceive of, it ought to be greatest. The
pointing is also found to change with the age of the moon and with the
season of the year. But these motions are too small to be set forth in
the present article.
There is yet another class of changes much wider than these. The
observations recorded since the time of Columbus show that, in the
course of centuries, the variation of the compass, at any one point,
changes very widely. It is well known that in 1490 the needle pointed
east of north in the Mediterranean, as well as in those portions of the
Atlantic which were then navigated. Columbus was therefore much
astonished when, on his first voyage, in mid-ocean, he found that the
deviation was reversed, and was now towards the west. It follows that a
line of no variation then passed through the Atlantic Ocean. But this
line has since been moving towards the east. About 1662 it passed the
meridian of Paris. During the two hundred and forty years which have
since elapsed, it has passed over Central Europe, and now, as we have
already said, passes through European Russia.
The existence of natural magnets composed of iron ore, and their
property of attracting iron and making it magnetic, have been known
from the remotest antiquity. But the question as to who first
discovered the fact that a magnetized needle points north and south,
and applied this discovery to navigation, has given rise to much
discussion. That the property was known to the Chinese about the
beginning of our era seems to be fairly well established, the
statements to that effect being of a kind that could not well have been
invented. Historical evidence of the use of the magnetic needle in
navigation dates from the twelfth century. The earliest compass
consisted simply of a splinter of wood or a piece of straw to which the
magnetized needle was attached, and which was floated in water. A
curious obstacle is said to have interfered with the first uses of this
instrument. Jack is a superstitious fellow, and we may be sure that he
was not less so in former times than he is today. From his point of
view there was something uncanny in so very simple a contrivance as a
floating straw persistently showing him the direction in which he must
sail. It made him very uncomfortable to go to sea under the guidance of
an invisible power. But with him, as with the rest of us, familiarity
breeds contempt, and it did not take more than a generation to show
that much good and no harm came to those who used the magic pointer.
The modern compass, as made in the most approved form for naval and
other large ships, is the liquid one. This does not mean that the card
bearing the needle floats on the liquid, but only that a part of the
force is taken off from the pivot on which it turns, so as to make the
friction as small as possible, and to prevent the oscillation back and
forth which would continually go on if the card were perfectly free to
turn. The compass-card is marked not only with the thirty-two familiar
points of the compass, but is also divided into degrees. In the most
accurate navigation it is probable that very little use of the points
is made, the ship being directed according to the degrees.
A single needle is not relied upon to secure the direction of the card,
the latter being attached to a system of four or even more magnets, all
pointing in the same direction. The compass must have no iron in its
construction or support, because the attraction of that substance on
the needle would be fatal to its performance.
From this cause the use of iron as ship-building material introduced a
difficulty which it was feared would prove very serious. The thousands
of tons of iron in a ship must exert a strong attraction on the
magnetic needle. Another complication is introduced by the fact that
the iron of the ship will always become more or less magnetic, and when
the ship is built of steel, as modern ones are, this magnetism will be
more or less permanent.
We have already said that a magnet has the property of making steel or
iron in its neighborhood into another magnet, with its poles pointing
in the opposite direction. The consequence is that the magnetism of the
earth itself will make iron or steel more or less magnetic. As a ship
is built she thus becomes a great repository of magnetism, the
direction of the force of which will depend upon the position in which
she lay while building. If erected on the bank of an east and west
stream, the north end of the ship will become the north pole of a
magnet and the south end the south pole. Accordingly, when she is
launched and proceeds to sea, the compass points not exactly according
to the magnetism of the earth, but partly according to that of the ship
also.
The methods of obviating this difficulty have exercised the ingenuity
of the ablest physicists from the beginning of iron ship building. One
method is to place in the neighborhood of the compass, but not too near
it, a steel bar magnetized in the opposite direction from that of the
ship, so that the action of the latter shall be neutralized. But a
perfect neutralization cannot be thus effected. It is all the more
difficult to effect it because the magnetism of a ship is liable to
change.
The practical method therefore adopted is called "swinging the ship,"
an operation which passengers on ocean liners may have frequently
noticed when approaching land. The ship is swung around so that her bow
shall point in various directions. At each pointing the direction of
the ship is noticed by sighting on the sun, and also the direction of
the compass itself. In this way the error of the pointing of the
compass as the ship swings around is found for every direction in which
she may be sailing. A table can then be made showing what the pointing,
according to the compass, should be in order that the ship may sail in
any given direction.
This, however, does not wholly avoid the danger. The tables thus made
are good when the ship is on a level keel. If, from any cause whatever,
she heels over to one side, the action will be different. Thus there is
a "heeling error" which must be allowed for. It is supposed to have
been from this source of error not having been sufficiently determined
or appreciated that the lamentable wreck of the United States ship
Huron off the coast of Hatteras occurred some twenty years ago.
X
THE FAIRYLAND OF GEOMETRY
If the reader were asked in what branch of science the imagination is
confined within the strictest limits, he would, I fancy, reply that it
must be that of mathematics. The pursuer of this science deals only
with problems requiring the most exact statements and the most rigorous
reasoning. In all other fields of thought more or less room for play
may be allowed to the imagination, but here it is fettered by iron
rules, expressed in the most rigid logical form, from which no
deviation can be allowed. We are told by philosophers that absolute
certainty is unattainable in all ordinary human affairs, the only field
in which it is reached being that of geometric demonstration.
And yet geometry itself has its fairyland--a land in which the
imagination, while adhering to the forms of the strictest
demonstration, roams farther than it ever did in the dreams of Grimm or
Andersen. One thing which gives this field its strictly mathematical
character is that it was discovered and explored in the search after
something to supply an actual want of mathematical science, and was
incited by this want rather than by any desire to give play to fancy.
Geometricians have always sought to found their science on the most
logical basis possible, and thus have carefully and critically inquired
into its foundations. The new geometry which has thus arisen is of two
closely related yet distinct forms. One of these is called
NON-EUCLIDIAN, because Euclid's axiom of parallels, which we shall
presently explain, is ignored. In the other form space is assumed to
have one or more dimensions in addition to the three to which the space
we actually inhabit is confined. As we go beyond the limits set by
Euclid in adding a fourth dimension to space, this last branch as well
as the other is often designated non-Euclidian. But the more common
term is hypergeometry, which, though belonging more especially to space
of more than three dimensions, is also sometimes applied to any
geometric system which transcends our ordinary ideas.
In all geometric reasoning some propositions are necessarily taken for
granted. These are called axioms, and are commonly regarded as
self-evident. Yet their vital principle is not so much that of being
self-evident as being, from the nature of the case, incapable of
demonstration. Our edifice must have some support to rest upon, and we
take these axioms as its foundation. One example of such a geometric
axiom is that only one straight line can be drawn between two fixed
points; in other words, two straight lines can never intersect in more
than a single point. The axiom with which we are at present concerned
is commonly known as the 11th of Euclid, and may be set forth in the
following way: We have given a straight line, A B, and a point, P, with
another line, C D, passing through it and capable of being turned
around on P. Euclid assumes that this line C D will have one position
in which it will be parallel to A B, that is, a position such that if
the two lines are produced without end, they will never meet. His axiom
is that only one such line can be drawn through P. That is to say, if
we make the slightest possible change in the direction of the line C D,
it will intersect the other line, either in one direction or the other.
The new geometry grew out of the feeling that this proposition ought to
be proved rather than taken as an axiom; in fact, that it could in some
way be derived from the other axioms. Many demonstrations of it were
attempted, but it was always found, on critical examination, that the
proposition itself, or its equivalent, had slyly worked itself in as
part of the base of the reasoning, so that the very thing to be proved
was really taken for granted.
[Illustration with caption: FIG. 1]
This suggested another course of inquiry. If this axiom of parallels
does not follow from the other axioms, then from these latter we may
construct a system of geometry in which the axiom of parallels shall
not be true. This was done by Lobatchewsky and Bolyai, the one a
Russian the other a Hungarian geometer, about 1830.
To show how a result which looks absurd, and is really inconceivable by
us, can be treated as possible in geometry, we must have recourse to
analogy. Suppose a world consisting of a boundless flat plane to be
inhabited by reasoning beings who can move about at pleasure on the
plane, but are not able to turn their heads up or down, or even to see
or think of such terms as above them and below them, and things around
them can be pushed or pulled about in any direction, but cannot be
lifted up. People and things can pass around each other, but cannot
step over anything. These dwellers in "flatland" could construct a
plane geometry which would be exactly like ours in being based on the
axioms of Euclid. Two parallel straight lines would never meet, though
continued indefinitely.
But suppose that the surface on which these beings live, instead of
being an infinitely extended plane, is really the surface of an immense
globe, like the earth on which we live. It needs no knowledge of
geometry, but only an examination of any globular object--an apple, for
example--to show that if we draw a line as straight as possible on a
sphere, and parallel to it draw a small piece of a second line, and
continue this in as straight a line as we can, the two lines will meet
when we proceed in either direction one-quarter of the way around the
sphere. For our "flat-land" people these lines would both be perfectly
straight, because the only curvature would be in the direction
downward, which they could never either perceive or discover. The lines
would also correspond to the definition of straight lines, because any
portion of either contained between two of its points would be the
shortest distance between those points. And yet, if these people should
extend their measures far enough, they would find any two parallel
lines to meet in two points in opposite directions. For all small
spaces the axioms of their geometry would apparently hold good, but
when they came to spaces as immense as the semi-diameter of the earth,
they would find the seemingly absurd result that two parallel lines
would, in the course of thousands of miles, come together. Another
result yet more astonishing would be that, going ahead far enough in a
straight line, they would find that although they had been going
forward all the time in what seemed to them the same direction, they
would at the end of 25,000 miles find themselves once more at their
starting-point.
One form of the modern non-Euclidian geometry assumes that a similar
theorem is true for the space in which our universe is contained.
Although two straight lines, when continued indefinitely, do not appear
to converge even at the immense distances which separate us from the
fixed stars, it is possible that there may be a point at which they
would eventually meet without either line having deviated from its
primitive direction as we understand the case. It would follow that, if
we could start out from the earth and fly through space in a perfectly
straight line with a velocity perhaps millions of times that of light,
we might at length find ourselves approaching the earth from a
direction the opposite of that in which we started. Our straight-line
circle would be complete.
Another result of the theory is that, if it be true, space, though
still unbounded, is not infinite, just as the surface of a sphere,
though without any edge or boundary, has only a limited extent of
surface. Space would then have only a certain volume--a volume which,
though perhaps greater than that of all the atoms in the material
universe, would still be capable of being expressed in cubic miles. If
we imagine our earth to grow larger and larger in every direction
without limit, and with a speed similar to that we have described, so
that to-morrow it was large enough to extend to the nearest fixed
stars, the day after to yet farther stars, and so on, and we, living
upon it, looked out for the result, we should, in time, see the other
side of the earth above us, coming down upon us? as it were. The space
intervening would grow smaller, at last being filled up. The earth
would then be so expanded as to fill all existing space.
This, although to us the most interesting form of the non-Euclidian
geometry, is not the only one. The idea which Lobatchewsky worked out
was that through a point more than one parallel to a given line could
be drawn; that is to say, if through the point P we have already
supposed another line were drawn making ever so small an angle with CD,
this line also would never meet the line AB. It might approach the
latter at first, but would eventually diverge. The two lines AB and CD,
starting parallel, would eventually, perhaps at distances greater than
that of the fixed stars, gradually diverge from each other. This system
does not admit of being shown by analogy so easily as the other, but an
idea of it may be had by supposing that the surface of "flat-land,"
instead of being spherical, is saddle-shaped. Apparently straight
parallel lines drawn upon it would then diverge, as supposed by Bolyai.
We cannot, however, imagine such a surface extended indefinitely
without losing its properties. The analogy is not so clearly marked as
in the other case.
To explain hypergeometry proper we must first set forth what a fourth
dimension of space means, and show how natural the way is by which it
may be approached. We continue our analogy from "flat-land" In this
supposed land let us make a cross--two straight lines intersecting at
right angles. The inhabitants of this land understand the cross
perfectly, and conceive of it just as we do. But let us ask them to
draw a third line, intersecting in the same point, and perpendicular to
both the other lines. They would at once pronounce this absurd and
impossible. It is equally absurd and impossible to us if we require the
third line to be drawn on the paper. But we should reply, "If you allow
us to leave the paper or flat surface, then we can solve the problem by
simply drawing the third line through the paper perpendicular to its
surface."
[Illustration with caption: FIG. 2]
Now, to pursue the analogy, suppose that, after we have drawn three
mutually perpendicular lines, some being from another sphere proposes
to us the drawing of a fourth line through the same point,
perpendicular to all three of the lines already there. We should answer
him in the same way that the inhabitants of "flat-land" answered us:
"The problem is impossible. You cannot draw any such line in space as
we understand it." If our visitor conceived of the fourth dimension, he
would reply to us as we replied to the "flat-land" people: "The problem
is absurd and impossible if you confine your line to space as you
understand it. But for me there is a fourth dimension in space. Draw
your line through that dimension, and the problem will be solved. This
is perfectly simple to me; it is impossible to you solely because your
conceptions do not admit of more than three dimensions."
Supposing the inhabitants of "flat-land" to be intellectual beings as
we are, it would be interesting to them to be told what dwellers of
space in three dimensions could do. Let us pursue the analogy by
showing what dwellers in four dimensions might do. Place a dweller of
"flat-land" inside a circle drawn on his plane, and ask him to step
outside of it without breaking through it. He would go all around, and,
finding every inch of it closed, he would say it was impossible from
the very nature of the conditions. "But," we would reply, "that is
because of your limited conceptions. We can step over it."
"Step over it!" he would exclaim. "I do not know what that means. I can
pass around anything if there is a way open, but I cannot imagine what
you mean by stepping over it."
But we should simply step over the line and reappear on the other side.
So, if we confine a being able to move in a fourth dimension in the
walls of a dungeon of which the sides, the floor, and the ceiling were
all impenetrable, he would step outside of it without touching any part
of the building, just as easily as we could step over a circle drawn on
the plane without touching it. He would simply disappear from our view
like a spirit, and perhaps reappear the next moment outside the prison.
To do this he would only have to make a little excursion in the fourth
dimension.
[Illustration with caption: FIG. 3]
Another curious application of the principle is more purely
geometrical. We have here two triangles, of which the sides and angles
of the one are all equal to corresponding sides and angles of the
other. Euclid takes it for granted that the one triangle can be laid
upon the other so that the two shall fit together. But this cannot be
done unless we lift one up and turn it over. In the geometry of
"flat-land" such a thing as lifting up is inconceivable; the two
triangles could never be fitted together.
[Illustration with caption: FIG 4]
Now let us suppose two pyramids similarly related. All the faces and
angles of the one correspond to the faces and angles of the other. Yet,
lift them about as we please, we could never fit them together. If we
fit the bases together the two will lie on opposite sides, one being
below the other. But the dweller in four dimensions of space will fit
them together without any trouble. By the mere turning over of one he
will convert it into the other without any change whatever in the
relative position of its parts. What he could do with the pyramids he
could also do with one of us if we allowed him to take hold of us and
turn a somersault with us in the fourth dimension. We should then come
back into our natural space, but changed as if we were seen in a
mirror. Everything on us would be changed from right to left, even the
seams in our clothes, and every hair on our head. All this would be
done without, during any of the motion, any change having occurred in
the positions of the parts of the body.
It is very curious that, in these transcendental speculations, the most
rigorous mathematical methods correspond to the most mystical ideas of
the Swedenborgian and other forms of religion. Right around us, but in
a direction which we cannot conceive any more than the inhabitants of
"flat-land" can conceive up and down, there may exist not merely
another universe, but any number of universes. All that physical
science can say against the supposition is that, even if a fourth
dimension exists, there is some law of all the matter with which we are
acquainted which prevents any of it from entering that dimension, so
that, in our natural condition, it must forever remain unknown to us.
Another possibility in space of four dimensions would be that of
turning a hollow sphere, an india-rubber ball, for example, inside out
by simple bending without tearing it. To show the motion in our space
to which this is analogous, let us take a thin, round sheet of
india-rubber, and cut out all the central part, leaving only a narrow
ring round the border. Suppose the outer edge of this ring fastened
down on a table, while we take hold of the inner edge and stretch it
upward and outward over the outer edge until we flatten the whole ring
on the table, upside down, with the inner edge now the outer one. This
motion would be as inconceivable in "flat-land" as turning the ball
inside out is to us.
XI
THE ORGANIZATION OF SCIENTIFIC RESEARCH