THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE
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:—
“Temesvar, November 3rd, 1823.
“DEAR GOOD FATHER,
“I have so overwhelmingly much to write about my discovery that I know
no other way of checking myself than taking a quarter of a sheet only
to write on. I want an answer to my four-sheet letter.
“I am unbroken in my determination to publish a work on Parallels, as
soon as I have put my material in order and have the means.
“At present I have not made any discovery, but the way I have followed
almost certainly promises me the attainment of my object if any
possibility of it exists.
“I have not got my object yet, but I have produced such stupendous
things that I was overwhelmed myself, and it would be an eternal shame
if they were lost. When you see them you will find that it is so. Now
I can only say that I have made a new world out of nothing. Everything
that I have sent you before is a house of cards in comparison with a
tower. I am convinced that it will be no less to my honour than if I
had already discovered it.”
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:—
“My son is first lieutenant of Engineers and will soon be captain.
He is a fine youth, a good violin player, a skilful fencer, and
brave, but has had many duels, and is wild even for a soldier. Yet he
is distinguished—light in darkness and darkness in light. He is an
impassioned mathematician with extraordinary capacities.... He will
think more of your judgment on his work than that of all Europe.”
Wolfgang received no answer from Gauss to this letter, but sending a
second copy of the book received the following reply:—
“You have rejoiced me, my unforgotten friend, by your letters. I
delayed answering the first because I wanted to wait for the arrival
of the promised little book.
“Now something about your son’s work.
“If I begin with saying that ‘I ought not to praise it,’ you will be
staggered for a moment. But I cannot say anything else. To praise it
is to praise myself, for the path your son has broken in upon and the
results to which he has been led are almost exactly the same as my own
reflections, some of which date from thirty to thirty-five years ago.
“In fact I am astonished to the uttermost. My intention was to let
nothing be known in my lifetime about my own work, of which, for the
rest, but little is committed to writing. Most people have but little
perception of the problem, and I have found very few who took any
interest in the views I expressed to them. To be able to do that one
must first of all have had a real live feeling of what is wanting, and
as to that most men are completely in the dark.
“Still it was my intention to commit everything to writing in the
course of time, so that at least it should not perish with me.
“I am deeply surprised that this task can be spared me, and I am most
of all pleased in this that it is the son of my old friend who has in
so remarkable a manner preceded me.”
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.
[Illustration: Fig. 19.]
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.
[Illustration: Fig. 20.]
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.
[Illustration: Fig. 21.]
[Illustration: Fig. 22.]
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.
[Illustration: Fig. 23.]
In fig. 23, in the square on AB, there are—
9 points interior 9
4 at the corners 1
4 sides with 3 on each side, considered as
1½ on each side, because belonging
equally to two squares 6
The total is 16. There are 9 points in the square on BC.
In the square on AC there are—
24 points inside 24
4 at the corners 1
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.
16 inside 16
16 on the sides, counting as 8
4 on the corners 1
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.
[Illustration: 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.
[Illustration: Fig. 25.] [Illustration: Fig. 26.]
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.
[Illustration: Fig. 27.] [Illustration: Fig. 28.]
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—
2 inside = 2
4 at corners = 1
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.
[Illustration: Fig. 29.]
Shear square on hypothenuse—
7 internal 7
4 at corners 1
—
8
[Illustration: Fig. 29 _bis_.]
Square on one side—which the reader can draw for himself—
4 internal 4
8 on sides 4
4 at corners 1
—
9
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—
[Illustration: Fig. 30.]
In the figure ACDE (fig. 30) there are—
15 inside 15
4 at corners 1
a total of 16.
Now in the square ABGF, there are 16—
9 inside 9
12 on sides 6
4 at corners 1
—
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.
───────────────────────────
C D
───────────────────────────────────
A B
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.
[Illustration: Fig. 32.]
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.
[Illustration: Fig. 33.]
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.