The Fourth Dimension

Lobatchewsky, Bolyai, and Gauss Before entering on a description of the work of Lobatchewsky and Bolyai it will not be out of place to give a brief account of them, the materials for which are to be found in an article by Franz Schmidt in the forty-second volume of the Mathematische Annalen, and in Engel’s edition of Lobatchewsky.

Lobatchewsky was a man of the most complete and wonderful talents. As a youth he was full of vivacity, carrying his exuberance so far as to fall into serious trouble for hazing a professor, and other freaks. Saved by the good offices of the mathematician Bartels, who appreciated his ability, he managed to restrain himself within the bounds of prudence. Appointed professor at his own University, Kasan, he entered on his duties under the regime of a pietistic reactionary, who surrounded himself with sycophants and hypocrites. Esteeming probably the interests of his pupils as higher than any attempt at a vain resistance, he made himself the tyrant’s right-hand man, doing an incredible amount of teaching and performing the most varied official duties. Amidst all his activities he found time to make important contributions to science. His theory of parallels is most closely connected with his name, but a study of his writings shows that he was a man capable of carrying on mathematics in its main lines of advance, and of a judgment equal to discerning what these lines were. Appointed rector of his University, he died at an advanced age, surrounded by friends, honoured, with the results of his beneficent activity all around him. To him no subject came amiss, from the foundations of geometry to the improvement of the stoves by which the peasants warmed their houses.

He was born in 1793. His scientific work was unnoticed till, in 1867, Houel, the French mathematician, drew attention to its importance.

Johann Bolyai de Bolyai was born in Klausenburg, a town in Transylvania, December 15th, 1802.

His father, Wolfgang Bolyai, a professor in the Reformed College of Maros Vasarhely, retained the ardour in mathematical studies which had made him a chosen companion of Gauss in their early student days at Göttingen.

He found an eager pupil in Johann. He relates that the boy sprang before him like a devil. As soon as he had enunciated a problem the child would give the solution and command him to go on further. As a thirteen-year-old boy his father sometimes sent him to fill his place when incapacitated from taking his classes. The pupils listened to him with more attention than to his father for they found him clearer to understand.

In a letter to Gauss Wolfgang Bolyai writes:—

“My boy is strongly built. He has learned to recognise many constellations, and the ordinary figures of geometry. He makes apt applications of his notions, drawing for instance the positions of the stars with their constellations. Last winter in the country, seeing Jupiter he asked: ‘How is it that we can see him from here as well as from the town? He must be far off.’ And as to three different places to which he had been he asked me to tell him about them in one word. I did not know what he meant, and then he asked me if one was in a line with the other and all in a row, or if they were in a triangle.

“He enjoys cutting paper figures with a pair of scissors, and without my ever having told him about triangles remarked that a right-angled triangle which he had cut out was half of an oblong. I exercise his body with care, he can dig well in the earth with his little hands. The blossom can fall and no fruit left. When he is fifteen I want to send him to you to be your pupil.”

In Johann’s autobiography he says:—

“My father called my attention to the imperfections and gaps in the theory of parallels. He told me he had gained more satisfactory results than his predecessors, but had obtained no perfect and satisfying conclusion. None of his assumptions had the necessary degree of geometrical certainty, although they sufficed to prove the eleventh axiom and appeared acceptable on first sight.

“He begged of me, anxious not without a reason, to hold myself aloof and to shun all investigation on this subject, if I did not wish to live all my life in vain.”

Johann, in the failure of his father to obtain any response from Gauss, in answer to a letter in which he asked the great mathematician to make of his son “an apostle of truth in a far land,” entered the Engineering School at Vienna. He writes from Temesvar, where he was appointed sub-lieutenant September, 1823:—

The discovery of which Johann here speaks was published as an appendix to Wolfgang Bolyai’s Tentamen.

Sending the book to Gauss, Wolfgang writes, after an interruption of eighteen years in his correspondence:—

Wolfgang received no answer from Gauss to this letter, but sending a second copy of the book received the following reply:—

The impression which we receive from Gauss’s inexplicable silence towards his old friend is swept away by this letter. Hence we breathe the clear air of the mountain tops. Gauss would not have failed to perceive the vast significance of his thoughts, sure to be all the greater in their effect on future ages from the want of comprehension of the present. Yet there is not a word or a sign in his writing to claim the thought for himself. He published no single line on the subject. By the measure of what he thus silently relinquishes, by such a measure of a world-transforming thought, we can appreciate his greatness.

It is a long step from Gauss’s serenity to the disturbed and passionate life of Johann Bolyai—he and Galois, the two most interesting figures in the history of mathematics. For Bolyai, the wild soldier, the duellist, fell at odds with the world. It is related of him that he was challenged by thirteen officers of his garrison, a thing not unlikely to happen considering how differently he thought from every one else. He fought them all in succession—making it his only condition that he should be allowed to play on his violin for an interval between meeting each opponent. He disarmed or wounded all his antagonists. It can be easily imagined that a temperament such as his was one not congenial to his military superiors. He was retired in 1833.

His epoch-making discovery awoke no attention. He seems to have conceived the idea that his father had betrayed him in some inexplicable way by his communications with Gauss, and he challenged the excellent Wolfgang to a duel. He passed his life in poverty, many a time, says his biographer, seeking to snatch himself from dissipation and apply himself again to mathematics. But his efforts had no result. He died January 27th, 1860, fallen out with the world and with himself.

Metageometry

The theories which are generally connected with the names of Lobatchewsky and Bolyai bear a singular and curious relation to the subject of higher space.

In order to show what this relation is, I must ask the reader to be at the pains to count carefully the sets of points by which I shall estimate the volumes of certain figures.

No mathematical processes beyond this simple one of counting will be necessary.

Let us suppose we have before us in fig. 19 a plane covered with points at regular intervals, so placed that every four determine a square.

Now it is evident that as four points determine a square, so four squares meet in a point.

Thus, considering a point inside a square as belonging to it, we may say that a point on the corner of a square belongs to it and to three others equally: belongs a quarter of it to each square.

Thus the square ACDE (fig. 21) contains one point, and has four points at the four corners. Since one-fourth of each of these four belongs to the square, the four together count as one point, and the point value of the square is two points—the one inside and the four at the corner make two points belonging to it exclusively.

Now the area of this square is two unit squares, as can be seen by drawing two diagonals in fig. 22.

We also notice that the square in question is equal to the sum of the squares on the sides AB, BC, of the right-angled triangle ABC. Thus we recognise the proposition that the square on the hypothenuse is equal to the sum of the squares on the two sides of a right-angled triangle.

Now suppose we set ourselves the question of determining the whereabouts in the ordered system of points, the end of a line would come when it turned about a point keeping one extremity fixed at the point.

We can solve this problem in a particular case. If we can find a square lying slantwise amongst the dots which is equal to one which goes regularly, we shall know that the two sides are equal, and that the slanting side is equal to the straight-way side. Thus the volume and shape of a figure remaining unchanged will be the test of its having rotated about the point, so that we can say that its side in its first position would turn into its side in the second position.

Now, such a square can be found in the one whose side is five units in length.

In fig. 23, in the square on AB, there are—

The total is 16. There are 9 points in the square on BC.

In the square on AC there are—

or 25 altogether.

Hence we see again that the square on the hypothenuse is equal to the squares on the sides.

Now take the square AFHG, which is larger than the square on AB. It contains 25 points.

making 25 altogether.

If two squares are equal we conclude the sides are equal. Hence, the line AF turning round A would move so that it would after a certain turning coincide with AC.

This is preliminary, but it involves all the mathematical difficulties that will present themselves.

There are two alterations of a body by which its volume is not changed.

One is the one we have just considered, rotation, the other is what is called shear.

Consider a book, or heap of loose pages. They can be slid so that each one slips over the preceding one, and the whole assumes the shape b in fig. 24.

This deformation is not shear alone, but shear accompanied by rotation.

Shear can be considered as produced in another way.

Take the square ABCD (fig. 25), and suppose that it is pulled out from along one of its diagonals both ways, and proportionately compressed along the other diagonal. It will assume the shape in fig. 26.

This compression and expansion along two lines at right angles is what is called shear; it is equivalent to the sliding illustrated above, combined with a turning round.

In pure shear a body is compressed and extended in two directions at right angles to each other, so that its volume remains unchanged.

Now we know that our material bodies resist shear—shear does violence to the internal arrangement of their particles, but they turn as wholes without such internal resistance.

But there is an exception. In a liquid shear and rotation take place equally easily, there is no more resistance against a shear than there is against a rotation.

Now, suppose all bodies were to be reduced to the liquid state, in which they yield to shear and to rotation equally easily, and then were to be reconstructed as solids, but in such a way that shear and rotation had interchanged places.

That is to say, let us suppose that when they had become solids again they would shear without offering any internal resistance, but a rotation would do violence to their internal arrangement.

That is, we should have a world in which shear would have taken the place of rotation.

A shear does not alter the volume of a body: thus an inhabitant living in such a world would look on a body sheared as we look on a body rotated. He would say that it was of the same shape, but had turned a bit round.

Let us imagine a Pythagoras in this world going to work to investigate, as is his wont.

Fig. 27 represents a square unsheared. Fig. 28 represents a square sheared. It is not the figure into which the square in fig. 27 would turn, but the result of shear on some square not drawn. It is a simple slanting placed figure, taken now as we took a simple slanting placed square before. Now, since bodies in this world of shear offer no internal resistance to shearing, and keep their volume when sheared, an inhabitant accustomed to them would not consider that they altered their shape under shear. He would call ACDE as much a square as the square in fig. 27. We will call such figures shear squares. Counting the dots in ACDE, we find—

or a total of 3.

Now, the square on the side AB has 4 points, that on BC has 1 point. Here the shear square on the hypothenuse has not 5 points but 3; it is not the sum of the squares on the sides, but the difference.

This relation always holds. Look at fig. 29.

Shear square on hypothenuse—

Square on one side—which the reader can draw for himself—

and the square on the other side is 1. Hence in this case again the difference is equal to the shear square on the hypothenuse, 9 - 1 = 8.

Thus in a world of shear the square on the hypothenuse would be equal to the difference of the squares on the sides of a right-angled triangle.

In fig. 29 bis another shear square is drawn on which the above relation can be tested.

What now would be the position a line on turning by shear would take up?

We must settle this in the same way as previously with our turning.

Since a body sheared remains the same, we must find two equal bodies, one in the straight way, one in the slanting way, which have the same volume. Then the side of one will by turning become the side of the other, for the two figures are each what the other becomes by a shear turning.

We can solve the problem in a particular case—

In the figure ACDE (fig. 30) there are—

a total of 16.

Now in the square ABGF, there are 16—

Hence the square on AB would, by the shear turning, become the shear square ACDE.

And hence the inhabitant of this world would say that the line AB turned into the line AC. These two lines would be to him two lines of equal length, one turned a little way round from the other.

That is, putting shear in place of rotation, we get a different kind of figure, as the result of the shear rotation, from what we got with our ordinary rotation. And as a consequence we get a position for the end of a line of invariable length when it turns by the shear rotation, different from the position which it would assume on turning by our rotation.

A real material rod in the shear world would, on turning about A, pass from the position AB to the position AC. We say that its length alters when it becomes AC, but this transformation of AB would seem to an inhabitant of the shear world like a turning of AB without altering in length.

If now we suppose a communication of ideas that takes place between one of ourselves and an inhabitant of the shear world, there would evidently be a difference between his views of distance and ours.

We should say that his line AB increased in length in turning to AC. He would say that our line AF (fig. 23) decreased in length in turning to AC. He would think that what we called an equal line was in reality a shorter one.

We should say that a rod turning round would have its extremities in the positions we call at equal distances. So would he—but the positions would be different. He could, like us, appeal to the properties of matter. His rod to him alters as little as ours does to us.

Now, is there any standard to which we could appeal, to say which of the two is right in this argument? There is no standard.

We should say that, with a change of position, the configuration and shape of his objects altered. He would say that the configuration and shape of our objects altered in what we called merely a change of position. Hence distance independent of position is inconceivable, or practically distance is solely a property of matter.

There is no principle to which either party in this controversy could appeal. There is nothing to connect the definition of distance with our ideas rather than with his, except the behaviour of an actual piece of matter.

For the study of the processes which go on in our world the definition of distance given by taking the sum of the squares is of paramount importance to us. But as a question of pure space without making any unnecessary assumptions the shear world is just as possible and just as interesting as our world.

It was the geometry of such conceivable worlds that Lobatchewsky and Bolyai studied.

This kind of geometry has evidently nothing to do directly with four-dimensional space.

But a connection arises in this way. It is evident that, instead of taking a simple shear as I have done, and defining it as that change of the arrangement of the particles of a solid which they will undergo without offering any resistance due to their mutual action, I might take a complex motion, composed of a shear and a rotation together, or some other kind of deformation.

Let us suppose such an alteration picked out and defined as the one which means simple rotation, then the type, according to which all bodies will alter by this rotation, is fixed.

Looking at the movements of this kind, we should say that the objects were altering their shape as well as rotating. But to the inhabitants of that world they would seem to be unaltered, and our figures in their motions would seem to them to alter.

In such a world the features of geometry are different. We have seen one such difference in the case of our illustration of the world of shear, where the square on the hypothenuse was equal to the difference, not the sum, of the squares on the sides.

In our illustration we have the same laws of parallel lines as in our ordinary rotation world, but in general the laws of parallel lines are different.

In one of these worlds of a different constitution of matter through one point there can be two parallels to a given line, in another of them there can be none, that is, although a line be drawn parallel to another it will meet it after a time.

Now it was precisely in this respect of parallels that Lobatchewsky and Bolyai discovered these different worlds. They did not think of them as worlds of matter, but they discovered that space did not necessarily mean that our law of parallels is true. They made the distinction between laws of space and laws of matter, although that is not the form in which they stated their results.

The way in which they were led to these results was the following. Euclid had stated the existence of parallel lines as a postulate—putting frankly this unproved proposition—that one line and only one parallel to a given straight line can be drawn, as a demand, as something that must be assumed. The words of his ninth postulate are these: “If a straight line meeting two other straight lines makes the interior angles on the same side of it equal to two right angles, the two straight lines will never meet.”

The mathematicians of later ages did not like this bald assumption, and not being able to prove the proposition they called it an axiom—the eleventh axiom.

Many attempts were made to prove the axiom; no one doubted of its truth, but no means could be found to demonstrate it. At last an Italian, Sacchieri, unable to find a proof, said: “Let us suppose it not true.” He deduced the results of there being possibly two parallels to one given line through a given point, but feeling the waters too deep for the human reason, he devoted the latter half of his book to disproving what he had assumed in the first part.

Then Bolyai and Lobatchewsky with firm step entered on the forbidden path. There can be no greater evidence of the indomitable nature of the human spirit, or of its manifest destiny to conquer all those limitations which bind it down within the sphere of sense than this grand assertion of Bolyai and Lobatchewsky.

Take a line AB and a point C. We say and see and know that through C can only be drawn one line parallel to AB.

But Bolyai said: “I will draw two.” Let CD be parallel to AB, that is, not meet AB however far produced, and let lines beyond CD also not meet AB; let there be a certain region between CD and CE, in which no line drawn meets AB. CE and CD produced backwards through C will give a similar region on the other side of C.

Nothing so triumphantly, one may almost say so insolently, ignoring of sense had ever been written before. Men had struggled against the limitations of the body, fought them, despised them, conquered them. But no one had ever thought simply as if the body, the bodily eyes, the organs of vision, all this vast experience of space, had never existed. The age-long contest of the soul with the body, the struggle for mastery, had come to a culmination. Bolyai and Lobatchewsky simply thought as if the body was not. The struggle for dominion, the strife and combat of the soul were over; they had mastered, and the Hungarian drew his line.

Can we point out any connection, as in the case of Parmenides, between these speculations and higher space? Can we suppose it was any inner perception by the soul of a motion not known to the senses, which resulted in this theory so free from the bonds of sense? No such supposition appears to be possible.

Practically, however, metageometry had a great influence in bringing the higher space to the front as a working hypothesis. This can be traced to the tendency the mind has to move in the direction of least resistance. The results of the new geometry could not be neglected, the problem of parallels had occupied a place too prominent in the development of mathematical thought for its final solution to be neglected. But this utter independence of all mechanical considerations, this perfect cutting loose from the familiar intuitions, was so difficult that almost any other hypothesis was more easy of acceptance, and when Beltrami showed that the geometry of Lobatchewsky and Bolyai was the geometry of shortest lines drawn on certain curved surfaces, the ordinary definitions of measurement being retained, attention was drawn to the theory of a higher space. An illustration of Beltrami’s theory is furnished by the simple consideration of hypothetical beings living on a spherical surface.

Let ABCD be the equator of a globe, and AP, BP, meridian lines drawn to the pole, P. The lines AB, AP, BP would seem to be perfectly straight to a person moving on the surface of the sphere, and unconscious of its curvature. Now AP and BP both make right angles with AB. Hence they satisfy the definition of parallels. Yet they meet in P. Hence a being living on a spherical surface, and unconscious of its curvature, would find that parallel lines would meet. He would also find that the angles in a triangle were greater than two right angles. In the triangle PAB, for instance, the angles at A and B are right angles, so the three angles of the triangle PAB are greater than two right angles.

Now in one of the systems of metageometry (for after Lobatchewsky had shown the way it was found that other systems were possible besides his) the angles of a triangle are greater than two right angles.

Thus a being on a sphere would form conclusions about his space which are the same as he would form if he lived on a plane, the matter in which had such properties as are presupposed by one of these systems of geometry. Beltrami also discovered a certain surface on which there could be drawn more than one “straight” line through a point which would not meet another given line. I use the word straight as equivalent to the line having the property of giving the shortest path between any two points on it. Hence, without giving up the ordinary methods of measurement, it was possible to find conditions in which a plane being would necessarily have an experience corresponding to Lobatchewsky’s geometry. And by the consideration of a higher space, and a solid curved in such a higher space, it was possible to account for a similar experience in a space of three dimensions.

Now, it is far more easy to conceive of a higher dimensionality to space than to imagine that a rod in rotating does not move so that its end describes a circle. Hence, a logical conception having been found harder than that of a four dimensional space, thought turned to the latter as a simple explanation of the possibilities to which Lobatchewsky had awakened it. Thinkers became accustomed to deal with the geometry of higher space—it was Kant, says Veronese, who first used the expression of “different spaces”—and with familiarity the inevitableness of the conception made itself felt.

From this point it is but a small step to adapt the ordinary mechanical conceptions to a higher spatial existence, and then the recognition of its objective existence could be delayed no longer. Here, too, as in so many cases, it turns out that the order and connection of our ideas is the order and connection of things.

What is the significance of Lobatchewsky’s and Bolyai’s work?

It must be recognised as something totally different from the conception of a higher space; it is applicable to spaces of any number of dimensions. By immersing the conception of distance in matter to which it properly belongs, it promises to be of the greatest aid in analysis for the effective distance of any two particles is the product of complex material conditions and cannot be measured by hard and fast rules. Its ultimate significance is altogether unknown. It is a cutting loose from the bonds of sense, not coincident with the recognition of a higher dimensionality, but indirectly contributory thereto.

Thus, finally, we have come to accept what Plato held in the hollow of his hand; what Aristotle’s doctrine of the relativity of substance implies. The vast universe, too, has its higher, and in recognising it we find that the directing being within us no longer stands inevitably outside our systematic knowledge.