Among those provisions of Nature which seem to us as especially designed for the use of man, none is more striking than the seeming magnetism of the earth. What would our civilization have been if the mariner's compass had never been known? That Columbus could never have crossed the Atlantic is certain; in what generation since his time our continent would have been discovered is doubtful. Did the reader ever reflect what a problem the captain of the finest ocean liner of our day would face if he had to cross the ocean without this little instrument? With the aid of a pilot he gets his ship outside of Sandy Hook without much difficulty. Even later, so long as the sun is visible and the air is clear, he will have some apparatus for sailing by the direction of the sun. But after a few hours clouds cover the sky. From that moment he has not the slightest idea of east, west, north, or south, except so far as he may infer it from the direction in which he notices the wind to blow. For a few hours he may be guided by the wind, provided he is sure he is not going ashore on Long Island. Thus, in time, he feels his way out into the open sea. By day he has some idea of direction with the aid of the sun; by night, when the sky is clear he can steer by the Great Bear, or "Cynosure," the compass of his ancient predecessors on the Mediterranean. But when it is cloudy, if he persists in steaming ahead, he may be running towards the Azores or towards Greenland, or he may be making his way back to New York without knowing it. So, keeping up steam only when sun or star is visible, he at length finds that he is approaching the coast of Ireland. Then he has to grope along much like a blind man with his staff, feeling his way along the edge of a precipice. He can determine the latitude at noon if the sky is clear, and his longitude in the morning or evening in the same conditions. In this way he will get a general idea of his whereabouts. But if he ventures to make headway in a fog, he may find himself on the rocks at any moment. He reaches his haven only after many spells of patient waiting for favoring skies.
The fact that the earth acts like a magnet, that the needle points to the north, has been generally known to navigators for nearly a thousand years, and is said to have been known to the Chinese at a yet earlier period. And yet, to-day, if any professor of physical science is asked to explain the magnetic property of the earth, he will acknowledge his inability to do so to his own satisfaction. Happily this does not hinder us from finding out by what law these forces act, and how they enable us to navigate the ocean. I therefore hope the reader will be interested in a short exposition of the very curious and interesting laws on which the science of magnetism is based, and which are applied in the use of the compass.
The force known as magnetic, on which the compass depends, is different from all other natural forces with which we are familiar. It is very remarkable that iron is the only substance which can become magnetic in any considerable degree. Nickel and one or two other metals have the same property, but in a very slight degree. It is also remarkable that, however powerfully a bar of steel may be magnetized, not the slightest effect of the magnetism can be seen by its action on other than magnetic substances. It is no heavier than before. Its magnetism does not produce the slightest influence upon the human body. No one would know that it was magnetic until something containing iron was brought into its immediate neighborhood; then the attraction is set up. The most important principle of magnetic science is that there are two opposite kinds of magnetism, which are, in a certain sense, contrary in their manifestations. The difference is seen in the behavior of the magnet itself. One particular end points north, and the other end south. What is it that distinguishes these two ends? The answer is that one end has what we call north magnetism, while the other has south magnetism. Every magnetic bar has two poles, one near one end, one near the other. The north pole is drawn towards the north pole of the earth, the south pole towards the south pole, and thus it is that the direction of the magnet is determined. Now, when we bring two magnets near each other we find another curious phenomenon. If the two like poles are brought together, they do not attract but repel each other. But the two opposite poles attract each other. The attraction and repulsion are exactly equal under the same conditions. There is no more attraction than repulsion. If we seal one magnet up in a paper or a box, and then suspend another over the box, the north pole of the one outside will tend to the south pole of the one in the box, and vice versa.
Our next discovery is, that whenever a magnet attracts a piece of iron it makes that iron into a magnet, at least for the time being. In the case of ordinary soft or untempered iron the magnetism disappears instantly when the magnet is removed. But if the magnet be made to attract a piece of hardened steel, the latter will retain the magnetism produced in it and become itself a permanent magnet.
This fact must have been known from the time that the compass came into use. To make this instrument it was necessary to magnetize a small bar or needle by passing a natural magnet over it.
In our times the magnetization is effected by an electric current. The latter has curious magnetic properties; a magnetic needle brought alongside of it will be found placing itself at right angles to the wire bearing the current. On this principle is made the galvanometer for measuring the intensity of a current. Moreover, if a piece of wire is coiled round a bar of steel, and a powerful electric current pass through the coil, the bar will become a magnet.
Another curious property of magnetism is that we cannot develop north magnetism in a bar without developing south magnetism at the same time. If it were otherwise, important consequences would result. A separate north pole of a magnet would, if attached to a floating object and thrown into the ocean, start on a journey towards the north all by itself. A possible method of bringing this result about may suggest itself. Let us take an ordinary bar magnet, with a pole at each end, and break it in the middle; then would not the north end be all ready to start on its voyage north, and the south end to make its way south? 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 passing 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 space 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 compass. On our eastern seaboard, and all the way across the Atlantic, the north pointing of the compass 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 compass. 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 passing 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 brass circle, of which the circumference is marked off in degrees. A magnet is attached to this circle so as to form a diameter, and suspended on a horizontal axis passing 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 passing 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 compass 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 latitude and 97 degrees longitude west from Greenwich. This point is situated on the west shore 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 latitude 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.
[Illustration with caption: DIP OF THE MAGNETIC NEEDLE IN VARIOUS LATITUDES. The arrow points show the direction of the north end of the magnetic needle, which dips downward in north latitudes, while the south end dips in south latitudes.]
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 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.
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.
The claims of scientific research on the public were never more forcibly urged than in Professor Ray Lankester's recent Romanes Lecture before the University of Oxford. Man is here eloquently pictured as Nature's rebel, who, under conditions where his great superior commands "Thou shalt die," replies "I will live." In pursuance of this determination, civilized man has proceeded so far in his interference with the regular course of Nature that he must either go on and acquire firmer control of the conditions, or perish miserably by the vengeance certain to be inflicted on the half-hearted meddler in great affairs. This rebel by every step forward renders himself liable to greater and greater penalties, and so cannot afford to pause or fail in one single step. One of Nature's most powerful agencies in thwarting his determination to live is found in disease-producing parasites. "Where there is one man of first-rate intelligence now employed in gaining knowledge of this agency, there should be a thousand. It should be as much the purpose of civilized nations to protect their citizens in this respect as it is to provide defence against human aggression."
It was no part of the function of the lecturer to devise a plan for carrying on the great war he proposes to wage. The object of the present article is to contribute some suggestions in this direction; with especial reference to conditions in our own country; and no better text can be found for a discourse on the subject than the preceding quotation. In saying that there should be a thousand investigators of disease where there is now one, I believe that Professor Lankester would be the first to admit that this statement was that of an ideal to be aimed at, rather than of an end to be practically reached. Every careful thinker will agree that to gather a body of men, young or old, supply them with laboratories and microscopes, and tell them to investigate disease, would be much like sending out an army without trained leaders to invade an enemy's country.
There is at least one condition of success in this line which is better fulfilled in our own country than in any other; and that is liberality of support on the part of munificent citizens desirous of so employing their wealth as to promote the public good. Combining this instrumentality with the general public spirit of our people, it must be admitted that, with all the disadvantages under which scientific research among us has hitherto labored, there is still no country to which we can look more hopefully than to our own as the field in which the ideal set forth by Professor Lankester is to be pursued. Some thoughts on the question how scientific research may be most effectively promoted in our own country through organized effort may therefore be of interest. Our first step will be to inquire what general lessons are to be learned from the experience of the past.
The first and most important of these lessons is that research has never reached its highest development except at centres where bodies of men engaged in it have been brought together, and stimulated to action by mutual sympathy and support. We must call to mind that, although the beginnings of modern science were laid by such men as Copernicus, Galileo, Leonardo da Vinci, and Torricelli, before the middle of the seventeenth century, unbroken activity and progress date from the foundations of the Academy of Sciences of Paris and the Royal Society of London at that time. The historic fact that the bringing of men together, and their support by an intelligent and interested community, is the first requirement to be kept in view can easily be explained. Effective research involves so intricate a network of problems and considerations that no one engaged in it can fail to profit by the suggestions of kindred spirits, even if less acquainted with the subject than he is himself. Intelligent discussion suggests new ideas and continually carries the mind to a higher level of thought. We must not regard the typical scientific worker, even of the highest class, as one who, having chosen his special field and met with success in cultivating it, has only to be supplied with the facilities he may be supposed to need in order to continue his work in the most efficient way. What we have to deal with is not a fixed and permanent body of learned men, each knowing all about the field of work in which he is engaged, but a changing and growing class, constantly recruited by beginners at the bottom of the scale, and constantly depleted by the old dropping away at the top. No view of the subject is complete which does not embrace the entire activity of the investigator, from the tyro to the leader. The leader himself, unless engaged in the prosecution of some narrow specialty, can rarely be so completely acquainted with his field as not to need information from others. Without this, he is constantly liable to be repeating what has already been better done than he can do it himself, of following lines which are known to lead to no result, and of adopting methods shown by the experience of others not to be the best. Even the books and published researches to which he must have access may be so voluminous that he cannot find time to completely examine them for himself; or they may be inaccessible. All this will make it clear that, with an occasional exception, the best results of research are not to be expected except at centres where large bodies of men are brought into close personal contact.
In addition to the power and facility acquired by frequent discussion with his fellows, the appreciation and support of an intelligent community, to whom the investigator may, from time to time, make known his thoughts and the results of his work, add a most effective stimulus. The greater the number of men of like minds that can be brought together and the larger the community which interests itself in what they are doing, the more rapid will be the advance and the more effective the work carried on. It is thus that London, with its munificently supported institutions, and Paris and Berlin, with their bodies of investigators supported either by the government or by various foundations, have been for more than three centuries the great centres where we find scientific activity most active and most effective. Looking at this undoubted fact, which has asserted itself through so long a period, and which asserts itself today more strongly than ever, the writer conceives that there can be no question as to one proposition. If we aim at the single object of promoting the advance of knowledge in the most effective way, and making our own country the leading one in research, our efforts should be directed towards bringing together as many scientific workers as possible at a single centre, where they can profit in the highest degree by mutual help, support, and sympathy.
In thus strongly setting forth what must seem an indisputable conclusion, the writer does not deny that there are drawbacks to such a policy, as there are to every policy that can be devised aiming at a good result. Nature offers to society no good that she does not accompany by a greater or less measure of evil The only question is whether the good outweighs the evil. In the present case, the seeming evil, whether real or not, is that of centralization. A policy tending in this direction is held to be contrary to the best interests of science in quarters entitled to so much respect that we must inquire into the soundness of the objection.
It would be idle to discuss so extreme a question as whether we shall take all the best scientific investigators of our country from their several seats of learning and attract them to some one point. We know that this cannot be done, even were it granted that success would be productive of great results. The most that can be done is to choose some existing centre of learning, population, wealth, and influence, and do what we can to foster the growth of science at that centre by attracting thither the greatest possible number of scientific investigators, especially of the younger class, and making it possible for them to pursue their researches in the most effective way. This policy would not result in the slightest harm to any institution or community situated elsewhere. It would not be even like building up a university to outrank all the others of our country; because the functions of the new institution, if such should be founded, would in its relations to the country be radically different from those of a university. Its primary object would not be the education of youth, but the increase of knowledge. So far as the interests of any community or of the world at large are concerned, it is quite indifferent where knowledge may be acquired, because, when once acquired and made public, it is free to the world. The drawbacks suffered by other centres would be no greater than those suffered by our Western cities, because all the great departments of the government are situated at a single distant point. Strong arguments could doubtless be made for locating some of these departments in the Far West, in the Mississippi Valley, or in various cities of the Atlantic coast; but every one knows that any local advantages thus gained would be of no importance compared with the loss of that administrative efficiency which is essential to the whole country.
There is, therefore, no real danger from centralization. The actual danger is rather in the opposite direction; that the sentiment against concentrating research will prove to operate too strongly. There is a feeling that it is rather better to leave every investigator where he chances to be at the moment, a feeling which sometimes finds expression in the apothegm that we cannot transplant a genius. That such a proposition should find acceptance affords a striking example of the readiness of men to accept a euphonious phrase without inquiring whether the facts support the doctrine which it enunciates. The fact is that many, perhaps the majority, of the great scientific investigators of this and of former times have done their best work through being transplanted. As soon as the enlightened monarchs of Europe felt the importance of making their capitals great centres of learning, they began to invite eminent men of other countries to their own. Lagrange was an Italian transplanted to Paris, as a member of the Academy of Sciences, after he had shown his powers in his native country. His great contemporary, Euler, was a Swiss, transplanted first to St. Petersburg, then invited by Frederick the Great to become a member of the Berlin Academy, then again attracted to St. Petersburg. Huyghens was transplanted from his native country to Paris. Agassiz was an exotic, brought among us from Switzerland, whose activity during the generation he passed among us was as great and effective as at any time of his life. On the Continent, outside of France, the most eminent professors in the universities have been and still are brought from distant points. So numerous are the cases of which these are examples that it would be more in accord with the facts to claim that it is only by transplanting a genius that we stimulate him to his best work.
Having shown that the best results can be expected only by bringing into contact as many scientific investigators as possible, the next question which arises is that of their relations to one another. It may be asked whether we shall aim at individualism or collectivism. Shall our ideal be an organized system of directors, professors, associates, assistants, fellows; or shall it be a collection of individual workers, each pursuing his own task in the way he deems best, untrammelled by authority?
The reply to this question is that there is in this special case no antagonism between the two ideas. The most effective organization will aim both at the promotion of individual effort, and at subordination and co-operation. It would be a serious error to formulate any general rule by which all cases should be governed. The experience of the past should be our guide, so far as it applies to present and future conditions; but in availing ourselves of it we must remember that conditions are constantly changing, and must adapt our policy to the problems of the future. In doing this, we shall find that different fields of research require very different policies as regards co-operation and subordination. It will be profitable to point out those special differences, because we shall thereby gain a more luminous insight into the problems which now confront the scientific investigator, and better appreciate their variety, and the necessity of different methods of dealing with them.
At one extreme, we have the field of normative science, work in which is of necessity that of the individual mind alone. This embraces pure mathematics and the methods of science in their widest range. The common interests of science require that these methods shall be worked out and formulated for the guidance of investigators generally, and this work is necessarily that of the individual brain.
At the other extreme, we have the great and growing body of sciences of observation. Through the whole nineteenth century, to say nothing of previous centuries, organizations, and even individuals, have been engaged in recording the innumerable phases of the course of nature, hoping to accumulate material that posterity shall be able to utilize for its benefit. We have observations astronomical, meteorological, magnetic, and social, accumulating in constantly increasing volume, the mass of which is so unmanageable with our present organizations that the question might well arise whether almost the whole of it will not have to be consigned to oblivion. Such a conclusion should not be entertained until we have made a vigorous effort to find what pure metal of value can be extracted from the mass of ore. To do this requires the co-operation of minds of various orders, quite akin in their relations to those necessary in a mine or great manufacturing establishment. Laborers whose duties are in a large measure matters of routine must be guided by the skill of a class higher in quality and smaller in number than their own, and these again by the technical knowledge of leaders in research. Between these extremes we have a great variety of systems of co-operation.
There is another feature of modern research the apprehension of which is necessary to the completeness of our view. A cursory survey of the field of science conveys the impression that it embraces only a constantly increasing number of disconnected specialties, in which each cultivator knows little or nothing of what is being done by others. Measured by its bulk, the published mass of scientific research is increasing in a more than geometrical ratio. Not only do the publications of nearly every scientific society increase in number and volume, but new and vigorous societies are constantly organized to add to the sum total. The stately quartos issued from the presses of the leading academies of Europe are, in most cases, to be counted by hundreds. The Philosophical Transactions of the Royal Society already number about two hundred volumes, and the time when the Memoirs of the French Academy of Sciences shall reach the thousand mark does not belong to the very remote future. Besides such large volumes, these and other societies publish smaller ones in a constantly growing number. In addition to the publications of learned societies, there are journals devoted to each scientific specialty, which seem to propagate their species by subdivision in much the same way as some of the lower orders of animal life. Every new publication of the kind is suggested by the wants of a body of specialists, who require a new medium for their researches and communications. The time has already come when we cannot assume that any specialist is acquainted with all that is being done even in his own line. To keep the run of this may well be beyond his own powers; more he can rarely attempt.
What is the science of the future to do when this huge mass outgrows the space that can be found for it in the libraries, and what are we to say of the value of it all? Are all these scientific researches to be classed as really valuable contributions to knowledge, or have we only a pile in which nuggets of gold are here and there to be sought for? One encouraging answer to such a question is that, taking the interests of the world as a whole, scientific investigation has paid for itself in benefits to humanity a thousand times over, and that all that is known to-day is but an insignificant fraction of what Nature has to show us. Apart from this, another feature of the science of our time demands attention. While we cannot hope that the multiplication of specialties will cease, we find that upon the process of differentiation and subdivision is now being superposed a form of evolution, tending towards the general unity of all the sciences, of which some examples may be pointed out.
Biological science, which a generation ago was supposed to be at the antipodes of exact science, is becoming more and more exact, and is cultivated by methods which are developed and taught by mathematicians. Psychophysics—the study of the operations of the mind by physical apparatus of the same general nature as that used by the chemist and physicist—is now an established branch of research. A natural science which, if any comparisons are possible, may outweigh all others in importance to the race, is the rising one of "eugenics,"—the improvement of the human race by controlling the production of its offspring. No better example of the drawbacks which our country suffers as a seat of science can be given than the fact that the beginning of such a science has been possible only at the seat of a larger body of cultivated men than our land has yet been able to bring together. Generations may elapse before the seed sown by Mr. Francis Galton, from which grew the Eugenic Society, shall bear full fruit in the adoption of those individual efforts and social regulations necessary to the propagation of sound and healthy offspring on the part of the human family. But when this comes about, then indeed will Professor Lankester's "rebel against Nature" find his independence acknowledged by the hitherto merciless despot that has decreed punishment for his treason.
This new branch of science from which so much may be expected is the offshoot of another, the rapid growth of which illustrates the rapid invasion of the most important fields of thought by the methods of exact science. It is only a few years since it was remarked of Professor Karl Pearson's mathematical investigations into the laws of heredity, and the biological questions associated with these laws, that he was working almost alone, because the biologists did not understand his mathematics, while the mathematicians were not interested in his biology. Had he not lived at a great centre of active thought, within the sphere of influence of the two great universities of England, it is quite likely that this condition of isolation would have been his to the end. But, one by one, men were found possessing the skill and interest in the subject necessary to unite in his work, which now has not only a journal of its own, but is growing in a way which, though slow, has all the marks of healthy progress towards an end the importance of which has scarcely dawned upon the public mind.
Admitting that an organized association of investigators is of the first necessity to secure the best results in the scientific work of the future, we meet the question of the conditions and auspices under which they are to be brought together. The first thought to strike us at this point may well be that we have, in our great universities, organizations which include most of the leading men now engaged in scientific research, whose personnel and facilities we should utilize. Admitting, as we all do, that there are already too many universities, and that better work would be done by a consolidation of the smaller ones, a natural conclusion is that the end in view will be best reached through existing organizations. But it would be a great mistake to jump at this conclusion without a careful study of the conditions. The brief argument—there are already too many institutions—instead of having more we should strengthen those we have—should not be accepted without examination. Had it been accepted thirty years ago, there are at least two great American universities of to-day which would not have come into being, the means devoted to their support having been divided among others. These are the Johns Hopkins and the University of Chicago. What would have been gained by applying the argument in these cases? The advantage would have been that, instead of 146 so-called universities which appear to-day in the Annual Report of the Bureau of Education, we should have had only 144. The work of these 144 would have been strengthened by an addition, to their resources, represented by the endowments of Baltimore and Chicago, and sufficient to add perhaps one professor to the staff of each. Would the result have been better than it actually has been? Have we not gained anything by allowing the argument to be forgotten in the cases of these two institutions? I do not believe that any who carefully look at the subject will hesitate in answering this question in the affirmative. The essential point is that the Johns Hopkins University did not merely add one to an already overcrowded list, but that it undertook a mission which none of the others was then adequately carrying out. If it did not plant the university idea in American soil, it at least gave it an impetus which has now made it the dominant one in the higher education of almost every state.
The question whether the country at large would have reaped a greater benefit, had the professors of the University of Chicago, with the appliances they now command, been distributed among fifty or a hundred institutions in every quarter of the land, than it has actually reaped from that university, is one which answers itself. Our two youngest universities have attained success, not because two have thus been added to the number of American institutions of learning, but because they had a special mission, required by the advance of the age, for which existing institutions were inadequate.
The conclusion to which these considerations lead is simple. No new institution is needed to pursue work on traditional lines, guided by traditional ideas. But, if a new idea is to be vigorously prosecuted, then a young and vigorous institution, specially organized to put the idea into effect, is necessary. The project of building up in our midst, at the most appropriate point, an organization of leading scientific investigators, for the single purpose of giving a new impetus to American science and, if possible, elevating the thought of the country and of the world to a higher plane, involves a new idea, which can best be realized by an institution organized for the special purpose. While this purpose is quite in line with that of the leading universities, it goes too far beyond them to admit of its complete attainment through their instrumentality. The first object of a university is the training of the growing individual for the highest duties of life. Additions to the mass of knowledge have not been its principal function, nor even an important function in our own country, until a recent time. The primary object of the proposed institution is the advance of knowledge and the opening up of new lines of thought, which, it may be hoped, are to prove of great import to humanity. It does not follow that the function of teaching shall be wholly foreign to its activities. It must take up the best young men at the point where universities leave them, and train them in the arts of thinking and investigating. But this training will be beyond that which any regular university is carrying out.
In pursuing our theme the question next arises as to the special features of the proposed association. The leading requirement is one that cannot be too highly emphasized. How clearly soever the organizers may have in their minds' eye the end in view, they must recognize the fact that it cannot be attained in a day. In every branch of work which is undertaken, there must be a single leader, and he must be the best that the country, perhaps even the world, can produce. The required man is not to be found without careful inquiry; in many branches he may be unattainable for years. When such is the case, wait patiently till he appears. Prudence requires that the fewest possible risks would be taken, and that no leader should be chosen except one of tried experience and world-wide reputation. Yet we should not leave wholly out of sight the success of the Johns Hopkins University in selecting, at its very foundation, young men who were to prove themselves the leaders of the future. This experience may admit of being repeated, if it be carefully borne in mind that young men of promise are to be avoided and young men of performance only to be considered. The performance need not be striking: ex pede Herculem may be possible; but we must be sure of the soundness of our judgment before accepting our Hercules. This requires a master. Clerk-Maxwell, who never left his native island to visit our shores, is entitled to honor as a promoter of American science for seeing the lion's paw in the early efforts of Rowland, for which the latter was unable to find a medium of publication in his own country. It must also be admitted that the task is more serious now than it was then, because, from the constantly increasing specialization of science, it has become difficult for a specialist in one line to ascertain the soundness of work in another. With all the risks that may be involved in the proceeding, it will be quite possible to select an effective body of leaders, young and old, with whom an institution can begin. The wants of these men will be of the most varied kind. One needs scarcely more than a study and library; another must have small pieces of apparatus which he can perhaps design and make for himself. Another may need apparatus and appliances so expensive that only an institution at least as wealthy as an ordinary university would be able to supply them. The apparatus required by others will be very largely human—assistants of every grade, from university graduates of the highest standing down to routine drudges and day-laborers. Workrooms there must be; but it is hardly probable that buildings and laboratories of a highly specialized character will be required at the outset. The best counsel will be necessary at every step, and in this respect the institution must start from simple beginnings and grow slowly. Leaders must be added one by one, each being judged by those who have preceded him before becoming in his turn a member of the body. As the body grows its members must be kept in personal touch, talk together, pull together, and act together.
The writer submits these views to the great body of his fellow-citizens interested in the promotion of American science with the feeling that, though his conclusions may need amendment in details, they rest upon facts of the past and present which have not received the consideration which they merit. What he most strongly urges is that the whole subject of the most efficient method of promoting research upon a higher plane shall be considered with special reference to conditions in our own country; and that the lessons taught by the history and progress of scientific research in all countries shall be fully weighed and discussed by those most interested in making this form of effort a more important feature of our national life. When this is done, he will feel that his purpose in inviting special consideration to his individual views has been in great measure reached.