But, alas! when this experiment is tried it is found that a south pole instantly develops itself on one side of the break, and a north pole on the other side, so that the two pieces will simply form two magnets, each with its north and south pole. There is no possibility of making a magnet with only one pole.

It was formerly supposed that the central portions of the earth consisted of an immense magnet directed north and south. Although this view is found, for reasons which need not be set forth in detail, to be untenable, it gives us a good general idea of the nature of terrestrial magnetism. One result that follows from the law of poles already mentioned is that the magnetism which seems to belong to the north pole of the earth is what we call south on the magnet, and vice versa.

Careful experiment shows us that the region around every magnet is filled with magnetic force, strongest near the poles of the magnet, but diminishing as the inverse square of the distance from the pole. This force, at each point, acts along a certain line, called a line of force. These lines are very prettily shown by the familiar experiment of placing a sheet of paper over a magnet, and then scattering iron filings on the surface of the paper. It will be noticed that the filings arrange themselves along a series of curved lines, diverging in every direction from each pole, but always pa.s.sing from one pole to the other. It is a universal law that whenever a magnet is brought into a region where this force acts, it is attracted into such a position that it shall have the same direction as the lines of force. Its north pole will take the direction of the curve leading to the south pole of the other magnet, and its south pole the opposite one.

The fact of terrestrial magnetism may be expressed by saying that the s.p.a.ce within and around the whole earth is filled by lines of magnetic force, which we know nothing about until we suspend a magnet so perfectly balanced that it may point in any direction whatever. Then it turns and points in the direction of the lines of force, which may thus be mapped out for all points of the earth.

We commonly say that the pole of the needle points towards the north.

The poets tell us how the needle is true to the pole. Every reader, however, is now familiar with the general fact of a variation of the compa.s.s. On our eastern seaboard, and all the way across the Atlantic, the north pointing of the compa.s.s varies so far to the west that a ship going to Europe and making no allowance for this deviation would find herself making more nearly for the North Cape than for her destination.

The "declination," as it is termed in scientific language, varies from one region of the earth to another. In some places it is towards the west, in others towards the east.

The pointing of the needle in various regions of the world is shown by means of magnetic maps. Such maps are published by the United States Coast Survey, whose experts make a careful study of the magnetic force all over the country. It is found that there is a line running nearly north and south through the Middle States along which there is no variation of the compa.s.s. To the east of it the variation of the north pole of the magnet is west; to the west of it, east. The most rapid changes in the pointing of the needle are towards the northeast and northwest regions. When we travel to the northeastern boundary of Maine the westerly variation has risen to 20 degrees. Towards the northwest the easterly variation continually increases, until, in the northern part of the State of Washington, it amounts to 23 degrees.

When we cross the Atlantic into Europe we find the west variation diminishing until we reach a certain line pa.s.sing through central Russia and western Asia. This is again a line of no variation. Crossing it, the variation is once more towards the east. This direction continues over most of the continent of Asia, but varies in a somewhat irregular manner from one part of the continent to another.

As a general rule, the lines of the earth's magnetic force are not horizontal, and therefore one end or the other of a perfectly suspended magnet will dip below the horizontal position. This is called the "dip of the needle." It is observed by means of a bra.s.s circle, of which the circ.u.mference is marked off in degrees. A magnet is attached to this circle so as to form a diameter, and suspended on a horizontal axis pa.s.sing through the centre of gravity, so that the magnet shall be free to point in the direction indicated by the earth's lines of magnetic force. Armed with this apparatus, scientific travellers and navigators have visited various points of the earth in order to determine the dip.

It is thus found that there is a belt pa.s.sing around the earth near the equator, but sometimes deviating several degrees from it, in which there is no dip; that is to say, the lines of magnetic force are horizontal. Taking any point on this belt and going north, it will be found that the north pole of the magnet gradually tends downward, the dip constantly increasing as we go farther north. In the southern part of the United States the dip is about 60 degrees, and the direction of the needle is nearly perpendicular to the earth's axis. In the northern part of the country, including the region of the Great Lakes, the dip increases to 75 degrees. Noticing that a dip of 90 degrees would mean that the north end of the magnet points straight downward, it follows that it would be more nearly correct to say that, throughout the United States, the magnetic needle points up and down than that it points north and south.

Going yet farther north, we find the dip still increasing, until at a certain point in the arctic regions the north pole of the needle points downward. In this region the compa.s.s is of no use to the traveller or the navigator. The point is called the Magnetic Pole. Its position has been located several times by scientific observers. The best determinations made during the last eighty years agree fairly well in placing it near 70 degrees north lat.i.tude and 97 degrees longitude west from Greenwich. This point is situated on the west sh.o.r.e of the Boothian Peninsula, which is bounded on the south end by McClintock Channel. It is about five hundred miles north of the northwest part of Hudson Bay. There is a corresponding magnetic pole in the Antarctic Ocean, or rather on Victoria Land, nearly south of Australia. Its position has not been so exactly located as in the north, but it is supposed to be at about 74 degrees of south lat.i.tude and 147 degrees of east longitude from Greenwich.

The magnetic poles used to be looked upon as the points towards which the respective ends of the needle were attracted. And, as a matter of fact, the magnetic force is stronger near the poles than elsewhere.

When located in this way by strength of force, it is found that there is a second north pole in northern Siberia. Its location has not, however, been so well determined as in the case of the American pole, and it is not yet satisfactorily shown that there is any one point in Siberia where the direction of the force is exactly downward.

[Ill.u.s.tration with caption: DIP OF THE MAGNETIC NEEDLE IN VARIOUS LAt.i.tUDES. The arrow points show the direction of the north end of the magnetic needle, which dips downward in north lat.i.tudes, while the south end dips in south lat.i.tudes.]

The declination and dip, taken together, show the exact direction of the magnetic force at any place. But in order to complete the statement of the force, one more element must be given--its amount. The intensity of the magnetic force is determined by suspending a magnet in a horizontal position, and then allowing it to oscillate back and forth around the suspension. The stronger the force, the less the time it will take to oscillate. Thus, by carrying a magnet to various parts of the world, the magnetic force can be determined at every point where a proper support for the magnet is obtainable. The intensity thus found is called the horizontal force. This is not really the total force, because the latter depends upon the dip; the greater the dip, the less will be the horizontal force which corresponds to a certain total force. But a very simple computation enables the one to be determined when the value of the other is known. In this way it is found that, as a general rule, the magnetic force is least in the earth's equatorial regions and increases as we approach either of the magnetic poles.

When the most exact observations on the direction of the needle are 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 lat.i.tudes 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 lat.i.tudes 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 pa.s.sage 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 cla.s.s 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 compa.s.s, 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 pa.s.sed through the Atlantic Ocean. But this line has since been moving towards the east. About 1662 it pa.s.sed the meridian of Paris. During the two hundred and forty years which have since elapsed, it has pa.s.sed over Central Europe, and now, as we have already said, pa.s.ses 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 compa.s.s 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 superst.i.tious 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 compa.s.s, 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 compa.s.s-card is marked not only with the thirty-two familiar points of the compa.s.s, 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 compa.s.s 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 compa.s.s 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 compa.s.s, 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 pa.s.sengers 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 compa.s.s itself. In this way the error of the pointing of the compa.s.s 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 compa.s.s, 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 s.p.a.ce is a.s.sumed to have one or more dimensions in addition to the three to which the s.p.a.ce we actually inhabit is confined. As we go beyond the limits set by Euclid in adding a fourth dimension to s.p.a.ce, 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 s.p.a.ce 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, pa.s.sing through it and capable of being turned around on P. Euclid a.s.sumes 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.

[Ill.u.s.tration 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 a.n.a.logy. 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 pa.s.s 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 s.p.a.ces the axioms of their geometry would apparently hold good, but when they came to s.p.a.ces 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 a.s.sumes that a similar theorem is true for the s.p.a.ce 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 s.p.a.ce 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, s.p.a.ce, 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. s.p.a.ce 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 s.p.a.ce intervening would grow smaller, at last being filled up. The earth would then be so expanded as to fill all existing s.p.a.ce.

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 a.n.a.logy 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 a.n.a.logy is not so clearly marked as in the other case.

To explain hypergeometry proper we must first set forth what a fourth dimension of s.p.a.ce means, and show how natural the way is by which it may be approached. We continue our a.n.a.logy 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 p.r.o.nounce 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."

[Ill.u.s.tration with caption: FIG. 2]

Now, to pursue the a.n.a.logy, 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 s.p.a.ce 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 s.p.a.ce as you understand it. But for me there is a fourth dimension in s.p.a.ce. 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 s.p.a.ce in three dimensions could do. Let us pursue the a.n.a.logy 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."