HISTORICAL SKETCH OF THE HYPERSPACE MOVEMENT
Egypt the Birthplace of Geometry--Precursors: NASIR-EDDIN, CHRISTOPH
CLAVIUS, SACCHERI, LAMBERT, LA GRANGE, KANT--Influence of the
_Mecanique Analytique_--The Parallel-Postulate the Root and Substance
of the Non-Euclidean Geometry--The Three Great Periods: The Formative,
Determinative and Elaborative--RIEMANN and the Properties of Analytic
Spaces.
The evolution of the idea of a fourth dimension of space covers a long
period of years. The earliest known record of the beginnings of the
study of space is found in a hieratic papyrus which forms a part of the
Rhind Collection in the British Museum and which has been deciphered by
EISENLOHR. It is believed to be a copy of an older manuscript of date
3400 B. C., and is entitled "_Directions for Knowing All Dark Things_"
The copy is said to have been made by AHMES, an Egyptian priest
between 1700 and 1100 B. C. It begins by giving the dimensions of
barns; then follows the consideration of various rectilineal figures,
circles, pyramids, and the value of pi ([Greek: p]). Although many of
the solutions given in the manuscript have been found to be incorrect
in minor particulars, the fact remains that Egypt is really the
birth-place of geometry. And this fact is buttressed by the knowledge
that THALES, long before he founded the Ionian School which was the
beginning of Greek influence in the study of mathematics, is found
studying geometry and astronomy in Egypt.
The concept of hyperspace began to germinate in the latter part of the
first century, B. C. For it was at this date that GEMINOS of RHODES
(B. C. 70) began to think seriously of the mathematical labyrinth
into which EUCLID'S parallel-postulate most certainly would lead if
an attempt at demonstrating its certitude were made. He recognized
the difficulties which would engage the attention of those who might
venture to delve into the mysterious possibilities of the problem.
There is no doubt, too, but that EUCLID himself was aware, in some
measure at least, of these difficulties; for his own attitude towards
this postulate seems to have been one of noncommittance. It is,
therefore, not strange that the astronomer, PTOLEMY (A. D. 87-165),
should be found seeking to prove the postulate by a consideration of
the possibilities of interstellar triangles. His researches, however,
brought him no relief from the general dissatisfaction which he felt
with respect to the validity of the problem itself.
For nearly one thousand years after the attempts at solving the
postulate by GEMINOS and PTOLEMY, the field of mathematics lay
undisturbed. For it was at this time that there arose a strange
phenomenon, more commonly known as the "Dark Ages," which put an
effectual check to further research or independent investigations.
Mathematicians throughout this long lapse of time were content to
accept EUCLID as the one incontrovertible, unimpeachable authority,
and even such investigations as were made did not have a rebellious
tendence, but were mainly endeavors to substantiate his claims.
Accordingly, it was not until about the first half of the thirteenth
century that any real advance was made. At this time there appeared
an Arab, NASIR-EDDIN (1201-1274) who attempted to make an improvement
on the problem of parallelism. His work on EUCLID was printed in
Rome in 1594 A. D., about three hundred and twenty years after his
demise and was communicated in 1651 by JOHN WALLIS (1616-1703) to the
mathematicians of Oxford University. Although his calculations and
conclusions were respectfully received by the Oxford authorities no
definite results were regarded as accomplished by what he had done.
It is believed, however, that his work reopened speculation upon
the problem and served as a basis, however slight, for the greater
work that was to be done by those who followed him during the next
succeeding eight hundred years.
About twenty years before the printing of the work of NASIR-EDDIN,
CHRISTOPH CLAVIUS (1574) deduced the axiom of parallels from the
assumption that a line whose points are all equidistant from a straight
line is itself straight. In his consideration of the parallel-postulate
he is said to have regarded it as EUCLID'S XIIIth axiom. Later BOLYAI
spoke of it as the XIth and later still, TODHUNTER treated it as the
XIIth. Hence, there does not seem to have been any general unanimity
of opinion as to the exact status of the parallel-postulate, and
especially is this true in view of the uncertainty now known to have
existed in EUCLID'S mind concerning it.
GIROLAMO SACCHERI (1667-1733), a learned Jesuit, born at San Remo,
came next upon the stage. And so important was his work that it will
perpetuate the memory of his name in the history of mathematics. He was
a teacher of grammar in the Jesuit _Collegio di Brera_ where TOMMASO
CEVA, a brother of GIOVANNI, the well-known mathematician, was teacher
of mathematics. His association with the CEVA brothers was especially
beneficial to him. He made use of CEVA'S very ingenious methods in his
first published book, 1693, entitled _Solutions of Six Geometrical
Problems Proposed by Count Roger Ventimiglia_.
A +--------------------------------+ B
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C +--------------------------------+ D
FIG. 1.
SACCHERI attacked the problem of parallels in quite a new way.
Examining a quadrilateral, _ABCD_, in which the angles _A_ and _B_ are
right angles and the sides _AC_ and _BD_ are equal, he determined to
show that the angles _C_ and _D_ are equal. He also sought to prove
that they are either right angles, obtus acute. He undertook to prove
the falsity of the latter two propositions (that they are either
obtuse or acute), leaving as the only possibility that they must be
right angles. In doing so, he found that his assumptions led him into
contradictions which he experienced difficulty in explaining.
His labors in connection with the solution of the problems proposed by
COUNT VENTIMIGLIA, including his work on the question of parallels,
led directly into the field of metageometrical researches, and perhaps
to him as to no other who had preceded him, or at least to him in a
larger degree, belongs the credit for a continued renewal of interest
in that series of investigations which resulted in the formulation of
the non-Euclidean geometry.
The last published work of SACCHERI was a recital of his endeavors at
demonstrating the parallel-postulate. This received the "Imprimatur"
of the Inquisition, July 13, 1733; the Provincial Company of Jesus
took possession of the book for perusal on August 16, 1733; but
unfortunately within two months after it had been reviewed by these
authorities, SACCHERI passed away.
All efforts which had been made prior to the work of SACCHERI were
based upon the assumption that there must be an equivalent postulate
which, if it could be demonstrated, would lead to a direct, positive
proof of EUCLID'S proposition. Although these and all other attempts
at reaching such a proof have signally failed and although it may
correctly be said that the entire history of demonstrations aiming at
the solution of the famous postulate has been one long series of utter
failures, it can be asserted with equal certitude that it has proven
to be one of the most fruitful problems in the history of mathematical
thought. For out of these failures has been built a superstructure
of analytical investigations which surpasses the most sanguine
expectations of those who had labored and failed.
In 1766 JOHN LAMBERT (1728-1777) wrote a paper upon the _Theory
of Parallels_ dated Sept. 5, 1766, first published in 1786, from
the papers left by F. BERNOULLI, which contained the following
assertions:[2]
1. The parallel-axiom needs proof, since it does not hold for geometry
on the surface of the sphere.
2. In order to make intuitive a geometry in which the triangle's sum
is less than two right angles, we need an "imaginary" sphere (the
pseudosphere).
3. In a space in which the triangle's sum is different from two right
angles there is an absolute measure (a natural unit for length).
At this time IMMANUEL KANT (1724-1804), the noted German metaphysician,
was in the midst of his philosophical labors. And it is believed that
it was he who first suggested the idea of different _spaces_. Below is
given a statement taken from his _Prolegomena_[3] which corroborates
this view.
"That complete space (which is itself no longer the boundary of
another space) has three dimensions, and that space in general
cannot have more, is based on the proposition that not more than
three lines can intersect at right angles in one point.... That we
can require a line to be drawn to infinity, a series of changes to
be continued (for example, _spaces_ passed through by motion) in
indefinitum, presupposes a representation of space and time which
can only attach to intuition."
[2] Vide _New York Mathematical Society Bulletin_, Vol. III,
1893-4, p. 79, G. B. HALSTEAD on _Lambert's Non-Euclidean Geometry_.
[3] _Prolegomena_, KANT, p. 37, Trans. by J. P. MAHAFFY and J.
H. BERNARD.
His differentiation between space in general and space which may be
considered as the "boundary of another space" shows, in the light of
the subsequent developments of the mathematical idea of space that
he very fully appreciated the marvelous scope of analytic spaces. His
conception of space, therefore, must have had a profound influence
upon the mathematic thought of the day causing it to undergo a rapid
reconstruction at the hands of geometers who came after him.
Under the masterly influence of LA GRANGE (1736-1813) the idea of
different spaces began to take definite shape and direction; the
geometry of hyperspace began to crystallize; and the field of mathesis
prepared for the growth of a conception the comprehension of which
was destined to be the profoundest undertaking ever attempted by the
human mind. Unlike most great men whom the world learns tardily to
admire, LA GRANGE lived to see his talents and genius fully recognized
by his compeers; for he was the recipient of many honors both from his
countrymen and his admirers in foreign lands. He spent twenty years in
Prussia where he went upon the invitation of FREDERICK the Great who in
the Royal summons referred to himself as the "greatest king in Europe"
and to LA GRANGE as the "greatest mathematician" in Europe. In Prussia
the _Mecanique Analytique_ and a long series of memoirs which were
published in the Berlin and Turin Transactions were produced. LA GRANGE
did not exhibit any marked taste for mathematics until he was 17 years
of age. Soon thereafter he came into possession of a memoir by HALLEY
quite by accident and this so aroused his latent genius that within one
year after he had reviewed HALLEY'S memoir he became an accomplished
mathematician.
He created the calculus of variations, solved most of the problems
proposed by Fermat, adding a number of theorems of his own contrivance;
raised the theory of differential equations to the position of a
science rather than a series of ingenious methods for the solution
of special problems and furnished a solution for the famous
isoperimetrical problem which had baffled the skill of the foremost
mathematicians for nearly half a century. All these stupendous tasks he
performed by the time he reached the age of nineteen.
The _Mecanique Analytique_ is his greatest and most comprehensive
work. In this he established the law of virtual work from which,
by the aid of his calculus of variations, he deduced the whole of
mechanics, including both solids and liquids. It was his object in the
_Analytique_ to show that the whole subject of mechanics is implicitly
embraced in a single principle, and to lay down certain formulae from
which any particular result can be obtained. He frequently made the
assertion that he had, in the _Mecanique Analytique_, transformed
mechanics which he persistently defined as a "geometry of four
dimensions"[4] into a branch of analytics and had shown the so-called
mechanical principles to be the simple results of the calculus.
Hence, there can be no doubt but that LA GRANGE not only completed
the foundation, but provided most of the material in his analyses and
other "abstract results of great generality" which he obtained in his
numerous calculations, for the superstructure subsequently known as the
geometry of hyperspace, and in which the fourth dimensional concept
occupies a very fundamental place.
[4] In 1754 D'ALEMBERT (1717-1783) published an article in the
famous old _Encyclopedia_ edited by DIDEROT and himself on _Dimension_.
In this article the idea of the fourth dimension is dwelt upon at
length. The view which he expressed in this article, of course, served
greatly to popularize the conception among the learned men of the day,
and owing to the close relationship existing between D'ALEMBERT and LA
GRANGE, it is not surprising that the latter should have been very much
enamored of the idea.
It is as if for nearly seventeen hundred years workmen, such as
GEMINOS, of RHODES, PTOLEMY, SACCHERI, NASIR-EDDIN, LAMBERT, CLAVIUS,
and hundred of others who struggled with the problem of parallels,
had made more or less sporadic attempts at the excavation of the land
whereon a marvelously intricate building was to be constructed. There
is no historical evidence to show that any of them ever dreamed that
the results of their labors would be utilized in the manner in which
they have been used. Then came KANT with the wonderfully penetrating
searchlight of his masterful intellect who from the elevation which
he occupied saw that the site had great possibilities, but he had not
the mathematical talent to undertake the work of actual, methodical
construction. Indeed his task was of a different sort. However, he
succeeded in opening the way for LA GRANGE and others who followed
him. LA GRANGE immediately seized upon the idea which for more than
a thousand years had been impinging upon the minds of mathematicians
vainly seeking lodgment and began the elaboration of a plan in
accordance with which minds better skilled in the pragmatic application
of abstract principles than his could complete the work begun.
Unfortunately, on account of his intense devotion and loyalty to the
study of pure mathematics, and when he had reached the summit of
his greatness where he stood "without a rival as the foremost living
mathematician," his health became seriously affected, causing him to
suffer constant attacks of profound melancholia from which he died on
April 10, 1813.
We come now to one of the most remarkable periods in the history of
mental development. During the six hundred years between the birth of
NASIR-EDDIN and the death of LA GRANGE the entire world of mathesis
was being reconstituted. Since there had been gradually going on an
internal process which, when completed, forever would liberate the mind
from the narrow confines of consciousness limited to the three-space,
it is not surprising that we should find, in the mathematical thought
of the time, an absolutely epoch-making departure. The innumerable
attempts at the solution of the parallel-postulate, all failures in the
sense that they did not prove, have intensified greatly the esteem in
which the never-dying elements of EUCLID are held to-day. And despite
the fact that there may come a time when his axioms and conclusions
may be found to be incongruent with the facts of sensuous reality; and
though all of his fundamental conceptions of space in general, his
theorems, propositions and postulates may have to give way before the
searching glare of a deeper knowledge because of some revealed fault,
the perfection of his work in the realm of pure mathematics will remain
forever a master piece demanding the undiminished admiration of mankind.
The parallel-postulate, as stated by EUCLID in his _Elements of
Geometry_, reads as follows:
"If a straight line meet two straight lines so as to make the two
interior angles on the same side of it taken together less than
two right angles, these straight lines being continually produced,
shall at length meet upon that side on which are the angles which
are less than two right angles."
On this postulate hang all the "law and the prophets" of the
non-Euclidean Geometry. In it are the virtual elements of three
possible geometries. Furthermore, it is both the warp and the woof
of the loom of present-day metageometrical researches. It is the
golden egg laid by the god SEB at the beginning of a new life cycle in
psychogenesis. Its progeny are numerous--hyperspaces, sects, straights,
digons, equidistantials, polars, planars, coplanars, invariants,
quaternions, complex variables, groups and many others. A wonderfully
interesting breed, full of meaning and pregnant with the power of final
emancipations for the human intellect!
When the conclusions which were systematically formulated as a result
of the investigations along the lines of hypotheses which controverted
the parallel-postulate were examined it was found that they fell into
three main divisions, namely: the synthetic or hyperbolic; the analytic
or RIEMANNIAN and the elliptic or CAYLEY-KLEIN. These divisions or
groups are based upon the three possibilities which inhere in the
conception taken of the sum of the angles referred to in the above
postulate as to whether it is equal to, greater or less than two right
angles.
The assumption that the angular sum is congruent to a straight angle is
called the Euclidean or parabolic hypothesis and is to be distinguished
from the synthetic or hyperbolic hypothesis established by GAUSS,
LOBACHEVSKI and BOLYAI and which assumes that the angular sum is less
than a straight angle. The elliptic or CAYLEY-KLEIN hypothesis assumes
that the angular sum is greater than a straight angle. LOBACHEVSKI,
however, not satisfied with the statement of the parallel-postulate
as given by EUCLID and which had caused the age-long controversy,
substituted for it the following:
"All straight lines which, in a plane, radiate from a given
point, can, with respect to any other straight line, in the same
plane, be divided into two classes--the intersecting and the
non-intersecting. The boundary line of the one and the other class
is called parallel to the given line."
This is but another way of saying about the same thing that EUCLID had
declared before, and yet, curiously enough it afforded just the liberty
that LOBACHEVSKI needed to enable him to elaborate his theory.
For the purposes of this sketch the field of the development of
non-Euclidean geometry is divided into three periods to be known as:
(1) the _formative_ period in which mathematical thought was being
formulated for the new departure; (2) the _determinative_ period during
which the mathematical ideas were given direction, purpose and a
general tendence; (3) the _elaborative_ period during which the results
of the former periods were elaborated into definite kinds of geometries
and attempts made at popularizing the hypotheses.
The Formative Period
CHARLES FREDERICH GAUSS (1777-1855) by some has been regarded as the
most influential mathematician that figured in the formulation of
the non-Euclidean geometry; but closer examination into his efforts
at investigating the properties of a triangle shows that while his
researches led to the establishment of the theorem that a regular
polygon of seventeen sides (or of any number which is prime, and also
one more than a power of two) can be inscribed, under the Euclidean
restrictions as to means, in a circle, and also that the common
spherical angle on the surface of a sphere is closely connected with
the constitution of the area inclosed thereby, he cannot justly be
designated as the leader of those who formulated the synthetic school.
And this, too, for the simple reason that, as he himself admits in
one of his letters to TAURINUS, he had not "published anything on the
subject." In this same letter he informs TAURINUS that he had pondered
the subject for more than thirty years and expressed the belief that
there could not be any one who had "concerned himself more exhaustively
with this second part (that the sum of the angles of a triangle cannot
be more than 180 degrees)" than he had.
Writing from Göttingen to TAURINUS, November 8, 1824, and commenting
upon the geometric value of the sum of the angles of a triangle, he
says:
"Your presentation of the demonstration that the sum of the angles
of a plane triangle cannot be greater than 180 degrees does,
indeed, leave something to be desired in point of geometrical
precision. But this could be supplied, and there is no doubt
that the impossibility in question admits of the most rigorous
demonstration. But the case is quite different with the second
part, namely, that the sum of the angles cannot be smaller than
180 degrees; this is the real difficulty, the rock upon which
all endeavors are wrecked.... The assumption that the sum of the
three angles is smaller than 180 degrees leads to a new geometry
entirely different from our Euclidean--a geometry which is
throughout consistent with itself, and which I have elaborated
in a manner entirely satisfactory to myself, so that I can solve
every problem in it with the exception of the determining of a
constant which is not _a priori_ obtainable."
It appears from this correspondence that GAUSS had in the privacy
of his own study elaborated a complete non-Euclidean geometry, and
had so thoroughly familiarized himself with its characteristics and
possibilities that the solution of every problem embraced within it was
very clear to him except that of the determination of a constant. He
concluded the above letter by saying:
"All my endeavors to discover contradiction or inconsistencies in
this non-Euclidean geometry have been in vain, and the only thing
in it that conflicts with our reason is the fact that if it were
true there would necessarily exist in space a linear magnitude
quite determinate in itself; yet unknown to us."
Judging from the correspondence between GAUSS and GERLING (1788-1857),
BESSEL (1784-1846), SCHUMACHER and TAURINUS, the nephew of SCHWEIKART,
and that between SCHWEIKART and GERLING, there had grown up a general
dissatisfaction in the minds of mathematicians of this period with
Euclidean geometry and especially the parallel-postulate and its
connotations. BESSEL expresses this general discontent in one of his
letters to GAUSS, dated February 10, 1829, in which he says:
"Through that which LAMBERT said and what SCHWEIKART disclosed
orally, it has become clear to me that our geometry is incomplete,
and should receive a correction, which is hypothetical, and if the
sum of the angles of the plane triangle is equal to 180 degrees,
vanishes."
The opinion of leading mathematicians at this time seems to have been
crystallizing very rapidly. Unconsciously the men of this formative
period were adducing evidence which would give form and tendence to the
developments in the field of mathesis at a later date. They appear to
have been reaching out for that which, ignis fatuus-like, was always
within easy reach, but not quite apprehensible.
A bolder student than GAUSS was FERDINAND CARL SCHWEIKART (1780-1857)
who also has been credited with the founding of the non-Euclidean
geometry. In fact, if judged by the same standards as GAUSS, he would
be called the "father of the geometry of hyperspace"; for he really
published the first treatise on the subject. This was in the nature of
an inclosure which he inserted between the leaves of a book he loaned
to GERLING. He also asked that it be shown to GAUSS that he might give
his judgment as to its merits.
SCHWEIKART'S treatise, dated Marburg, December, 1818, is here quoted in
full:
"There is a two-fold geometry--a geometry in the narrower sense,
the Euclidean, and an astral science of magnitude.
"The triangles of the latter have the peculiarity that the sum of
the three angles is not equal to two right angles.
"This presumed, it can be most rigorously proven: (_a_) That the
sum of the three angles in the triangle is less than two right
angles.
"(_b_) That this sum becomes ever smaller, the more content
the angle incloses. (_c_) That the altitude of an isosceles
right-angled triangle indeed ever increases, the more one
lengthens the side; that it, however, cannot surpass a certain
line which I call the constant."
Squares have consequently the following form:
[Illustration: FIG. 2.]
"If this constant were for us the radius of the earth (so that
every line drawn in the universe, from one fixed star to another,
distant 90° from the first, would be a tangent to the surface of
the earth) it would be infinitely great in comparison with the
spaces which occur in daily life."
The above, being the first published, not printed, treatise on the new
geometry occupies a unique place in the history of higher mathematics.
It gave additional strength to the formative tendencies which
characterized this period and marked SCHWEIKART as a constructive and
original thinker.
The nascent aspects of this stage received a fruitful contribution
when NICOLAI LOBACHEVSKI (1793-1847) created his _Imaginary Geometry_
and JANOS BOLYAI (1802-1860) published as an appendix to his father's
_Tentamen_, his _Science Absolute of Space_. LOBACHEVSKI and BOLYAI
have been called the "Creators of the Non-Euclidean Geometry." And this
appellation seems richly to be deserved by these pioneers. Their work
gave just the impetus most needed to fix the status of the new line
of researches which led to such remarkable discoveries in the more
recent years. The _Imaginary Geometry_ and the _Science Absolute of
Space_ were translated by the French mathematician, J. HOÜEL in 1868
and by him elevated out of their forty-five years of obscurity and
non-effectiveness to a position where they became available for the
mathematical public. To BOLYAI and LOBACHEVSKI, consequently, belong
the honor of starting the movement which resulted in the development of
metageometry and hence that which has proved to be the gateway of a new
mathematical freedom.
GAUSS, SCHWEIKART, LOBACHEVSKI, WOLFGANG and JANOS BOLYAI were the
principal figures of the formative period and the value of their
work with respect to the formulation of principles upon which was
constructed the Temple of Metageometry cannot be overestimated.
The Determinative Period
This period is characterized chiefly by its close relationship to the
theory of surfaces. RIEMANN'S Habilitation Lecture on _The Hypotheses
Which Constitute the Bases of Geometry_ marks the beginning of this
epoch. In this dissertation, RIEMANN not only promulgated the system
upon which GAUSS had spent more than thirty years of his life in
elaborating, for he was a disciple of GAUSS; but he disclosed his own
views with respect to space which he regarded as a particular case
of manifold. His work contains two fundamental concepts, namely, the
_manifold_ and the _measure of curvature_ of a continuous manifold,
possessed of what he called _flatness_ in the smallest parts. The
conception of the measure of curvature is extended by RIEMANN from
surfaces to spaces and a new kind of space, finite, but unbounded, is
shown to be possible. He showed that the dimensions of any space are
determined by the number of measurements necessary to establish the
position of a point in that space. Conceiving, therefore, that space
is a manifold of finite, but unbounded, extension, he established the
fact that the passage from one element of a manifold to another may
be either discrete or continuous and that the manifold is discrete or
continuous according to the manner of passage. Where the manifold
is regarded as discrete two portions of it can be compared, as to
magnitude, by counting; where continuous, by measurement. If the whole
manifold be caused to pass over into another manifold each of its
elements passing through a one-dimensional manifold, a two-dimensional
manifold is thus generated. In this way, a manifold of _n_-dimensions
can be generated. On the other hand, a manifold of _n_-dimensions can
be analyzed into one of one dimension and one of (_n_-1) dimensions.
To RIEMANN, then, is due the credit for first promulgating the
idea that space being a special case of manifold is generable, and
therefore, _finite_. He laid the foundation for the establishment of
a special kind of geometry known as the "elliptic." Space, as viewed
by him, possessed the following properties, viz.: generability,
divisibility, measurability, ponderability, finity and flexity.
These are the six pillars upon which rests the structure of hyperspace
analyses.[5]
[5] Vide _Nature_, Vol. VIII, pp. 14-17; 36, 37 (1873); also
_Mathematical Papers_, pp. 65-71.
_Generability_ is that property of geometric space by virtue of which
it may be generated, or constructed, by the movement of a line, plane,
surface or solid in a direction without itself. _Divisibility_ is that
property of geometric space by virtue of which it may be segmented
or divided into separate parts and superposed, or inserted, upon
or between each other. _Measurability_ is that property by virtue
of which geometric space is determined to be a manifold of either
a positive or negative curvature, also by which its extent may be
measured. _Ponderability_ is that property of geometric space by virtue
of which it may be regarded as a quantity which can be manipulated,
assorted, shelved or otherwise disposed of. _Finity_ is that property
by virtue of which geometric space is limited to the scope of the
individual consciousness of a unodim, a duodim or a tridim and by
virtue of which it is finite in extent. _Flexity_ is that property by
virtue of which geometric space is regarded as possessing curvature,
and in consequence of which progress through it is made in a curved,
rather than a geodetic line, also by virtue of which it may be flexed
without disruption or dilatation.
RIEMANN who thus prepared the way for entrance into a veritable
labyrinth of hyperspaces is, therefore, correctly styled "The father
of metageometry," and the fourth dimension is his eldest born. He
died while but forty years of age and never lived long enough fully
to elaborate his theory with respect to its application to the
measure of curvature of space. This was left for his very energetic
disciple, EUGENIO BELTRAMI (1835-1900) who was born nine years after
RIEMANN and lived thirty-four years longer than he. His labors mark
the characteristic standpoint of the determinative period. BELTRAMI'S
mathematical investigations were devoted mainly to the non-Euclidean
geometry. These led him to the rather remarkable conclusion that the
propositions embodied therein relate to figures lying upon surfaces of
constant negative curvature.
BELTRAMI sought to show that such surfaces partake of the nature of the
pseudosphere, and in doing so, made use of the following illustration:
[Illustration: FIG. 3.]
[Illustration: FIG. 4.]
If the plane figure _aabb_ is made to revolve upon its axis of symmetry
_AB_ the two arcs, _ab_ and _ab_ will describe a pseudospherical
concave-convex surface like that of a solid anchor ring. Above and
below, toward _aa_ and _bb_, the surface will turn outward with
ever-increasing flexure till it becomes perpendicular to the axis
and ends at the edge with one curvature infinite. Or, the half of a
pseudospherical surface may be rolled up into the shape of a champagne
glass, as in Fig. 4. In this way, the two straightest lines of the
pseudospherical surface may be indefinitely produced, giving a kind of
space (pseudospherical) in which the axiom of parallels does not hold
true.
The determinative period marks the most important stage in the
development of non-Euclidean geometry and certainly the most
significant in the evolution of the idea of hyperspaces and multiple
dimensionality. RIEMANN and BELTRAMI are chief among those whose labors
characterize the scope of this period. Their work gave direction and
general outline for later developments and all subsequent researches
along these lines have been conducted in strict conformity with the
principles laid down by these pioneer constructionists. They laid out
the field and designated its confines beyond which no adventurer has
since dared to pass.
The great importance of the work of RIEMANN at this time may be seen
further in the fact that it not only marked the beginning of a new
epoch in geometry; but his pronouncement of the hypothesis that space
is unbounded, though finite, is really the first time in the history
of human thought that expression was ever given to the idea that space
may yet be only of limited extent. Before that time the minds of all
men seemed to have been unanimous in the consideration of space as an
illimitable and infinite quantity.
The Elaborative Period
The elaborative stage includes the work of all those who, working upon
the bases laid down by LOBACHEVSKI, BOLYAI, SCHWEIKART and RIEMANN,
have sought to amplify the conclusions reached by them. Among those
whose investigations have greatly multiplied the applications of
hyperspace conceptions are HOÜEL (1866) and FLYE ST. MARIE (1871)
of France; HELMHOLTZ (1868), FRISCHAUF (1872), KLEIN (1849), and
BALTZER (1877) of Germany; BELTRAMI (1872) of Italy; DE TILLY (1879)
of Belgium; CLIFFORD and CAYLEY (1821) of England; NEWCOMB (1835) and
HALSTEAD of America.
These have been most active in popularizing the subject of
non-Euclidean geometry and incidentally the idea of the fourth
dimension. The great mass of non-professional mathematical readers,
therefore, owe these men an immeasurable debt of gratitude for the work
that they have done in the matter of rendering the conceptions which
constitute the fabric of metageometry understandable and thinkable. A
glance at the bibliography appended at the end of this volume will give
some idea of the enormous amount of labor that has been expended in an
effort to translate the most abstract mathematical principles into a
language that could easily be comprehended by the average intelligent
person.
The characteristic standpoint of this period is the popular
comprehension of the hyperspace concept and the consequent mental
liberation which follows. For there is no doubt but that unheard of
possibilities of thought have been revealed by investigations into
the nature of space. An entirely new world has been opened to view and
only a beginning has been made at the exploration of its extent and
resources.
One of the notable incidents of the early years of this period is the
position taken by FELIX KLEIN who stands in about the same relation
to CAYLEY as BELTRAMI does to RIEMANN, in that he assumed the task of
completing the work of his predecessor. KLEIN held that there are only
two kinds of RIEMANNIAN _space_--the elliptical and the spherical. Or
in other words, that there are only two possible kinds of space in
which the propositions announced by RIEMANN could apply. SOPHUS LIE,
called the "great comparative anatomist of geometric theories," carried
his classifications to a final conclusion in connection with spaces of
all kinds and decided that there are possible only four kinds of three
dimensional spaces.
But whether men with peering, microscopic, histological vision shall
establish the existence of one or many spaces, and regardless of the
mathematic rigor with which they shall demonstrate the self-consistency
of the doctrines which they hold, the fact remains that the hypotheses
thus maintained, while they may be regarded as true descriptions of
the spaces concerned, are, nevertheless, incompatible. All of them
cannot be valid. It will perhaps be found that none of them are
valid, especially objectively so. The only true view, therefore, of
these systems of hyperspaces is that which assigns them to their
rightful place in the infinitely vast world of pure mathesis where
their validity may go unchallenged and their existence unquestioned;
for in that domain of unconfined mentation, in that realm of divine
intuitability, the marvelous wonderland of ideas and notions, one is
not only disinclined to doubt their logical actuality, but is quite
willing to accede their claims.