ESSENTIALS OF THE NON-EUCLIDEAN GEOMETRY
The Non-Euclidean Geometry Concerned with Conceptual
Space Entirely--Outcome of Failures at Solving the
Parallel-Postulate--The Basis of the Non-Euclidean Geometry--Space
Curvature and Manifoldness--Some Elements of the Non-Euclidean
Geometry--Certainty, Necessity and Universality as Bulwarks
of Geometry--Some Consequences of Efforts at Solving the
Parallel-Postulate--The Final Issue of the Non-Euclidean
Geometry--Extended Consciousness.
The term "non-Euclidean" is used to designate any system of geometry
which is not strictly Euclidean in content.
It is interesting to note how the term came to be used. It appears to
have been employed first by GAUSS. He did not strike upon it suddenly,
however, as in the correspondence between him and WACHTER in 1816
he used the designation "anti-Euclidean" and then, later, following
SCHWEIKART, he adopted the latter's terminology and called it "Astral
Geometry." This he found in SCHWEIKART'S first published _treatise_
known by that name and which made its appearance at Marburg in
December, 1818. Finally, in his correspondence with TAURINUS in 1824,
GAUSS first used the expression "non-Euclidean" to designate the system
which he had elaborated and continued to use it in his correspondence
with SCHUMACHER in 1831.
"Non-Legendrean," "semi-Euclidean" and "non-Archimedean" are titles
used by M. DEHN to denote all kinds of geometries which represented
variations from the hypotheses laid down by LEGENDRE, EUCLID and
ARCHIMEDES.
The semi-Euclidean is a system of geometry in which the sum of the
angles of a triangle is said to be equal to two right angles, but
in which one may draw an infinity of parallels to a straight line
through a given point. The non-Euclidean geometry embraces all the
results obtained as a consequence of efforts made at finding a
satisfactory proof of the parallel-postulate and is, therefore, based
upon a conception of space which is at variance with that held by
EUCLID. According to the Ionian school space is an infinite continuum
possessing uniformity throughout its entire extent. The non-Euclideans
maintain that space is not an infinite extension; but a finite though
unbounded manifold capable of being generated by the movement of
a point, line or plane in a direction without itself. It is also
held that space is curved and exists in the shape of a sphere or
pseudosphere and is consequently elliptical.
The inapplicability of EUCLID's parallel-postulate to lines drawn
upon the surface of a sphere suggested the possibility of a space
in which the postulate could apply to all possible surfaces or that
space itself may be spherical in which case the postulate would be
invalidated altogether. Hence, it is quite natural that mathematicians
finding themselves unable to prove the postulate with due mathetic
precision should turn their attention to the conceptually possible.
In this virtual abandonment of the perceptual for the conceptual lies
the fundamental difference between the Euclidean and the non-Euclidean
geometries. It may be said to the credit of the Euclideans that they
have sought to make their geometric conceptions conform as closely as
possible to the actual nature of things in the sensuous world while
at the same time they must have perceived that at best their spatial
notions were only approximations to the sensuous actuality of objects
in space.
On the other hand, non-Euclideans make no pretense at discovering any
congruency between their notions and things as they actually are.
The attitude of the metageometricians in this respect is very aptly
described by CASSIUS JACKSON KEYSER who says:
"He constructs in thought a summitless hierarchy of hyperspaces,
an endless series of orderly worlds, worlds that are possible and
logically actual, and he is content not to know if any of them be
otherwise actual or actualized."[6]
[6] _Mathematics_, by C. J. KEYSER, Adrian Professor of
Mathematics, Columbia University.
The non-Euclidean is, therefore, not concerned about the applicability
of ensembles, notions and propositions to real, perceptual space
conditions. It is sufficient for him to know that his creations
are thinkable. As soon as he can resolve the nebulosity of his
consciousness into the conceptual "star-forms" of definite ideas
and notions, he sits down to the feast which he finds provided by
superfoetated hypotheses fabricated in the deeps of mind and logical
actualities imperturbed and unmindful of the weal of perceptual space
in its homogeneity of form and dimensionality.
Fundamentally, the non-Euclidean geometry is constructed upon the
basis of conceptual space almost entirely. Knowledge of its content is
accordingly derived from a superperceptual representation of relations
and interrelations subsisting between and among notions, ideas,
propositions and magnitudes arising out of a conceptual consideration
thereof. In other words, representations of the non-Euclidean
magnitudes, cannot be said to be strictly perceptual in the same sense
that three-space magnitudes are perceived; for three-space magnitudes
are really sense objects while hyperspace magnitudes are not sense
objects. They are far removed from the sensuous world and in order to
conceive them one must raise his consciousness from the sensuous plane
to the conceptual plane and become aware of a class of perceptions
which are not perceptions in the strict sense of the word, but
superperceptions; because they are representations of concepts rather
than precepts.
Notions of perceptual space are constituted of the triple presentations
arising out of the visual, tactual and motor sensations which are fused
together in their final delivery to the consciousness. The synthesis
of these three sense-deliveries is accomplished by equilibrating
their respective differences and by correcting the perceptions of one
sense by those of another in such a way as to obtain a completely
reliable perception of the object. This is the manner in which the
characteristics of Euclidean space are established.
The characteristics of non-Euclidean space are not arrived at exactly
in this way. Being beyond the scope of the visual, tactile and motor
sense apprehensions, it cannot be said to represent judgments derived
from any consideration or elaboration of the deliveries presented
through these media. Yet, the substance of metageometry, or the science
of the measurement of hyperspaces, may not be regarded as an _a priori_
substructure upon which the system is founded. That is, the conceptual
space of non-Euclidean geometry is not presented to the consciousness
as an _a priori_ notion. On the other hand, the _a posterioristic_
quality of metageometric spaces marks the entire scope of motility of
the notions appertaining thereto.
The notions, therefore, of conceptual space are derivable only from the
perception of concepts, or, otherwise consist of judgments concerning
interconceptual relations. The process of apperception involved in the
recognition of relations which may be methodically determined is much
removed from the primary procedure of perceiving sense-impressions
and fusing them into final deliveries to the consciousness for
conceptualization or the elaboration into concepts or general
notions. It is a procedure which is in every way superconceptual and
extra-sensuous. The metageometrician or analyst in no way relies upon
sense-deliveries for the data of his constructions; for, if he did, he
should, then, be reduced to the necessity of confining his conclusions
to the sphere of motility imposed by the sensible world with the result
that we should be able to verify empirically all his postulations. But,
contrarily, he goes to the extra-sensuous, and there in the realm of
pure conceptuality, he finds the requisite freedom for his theories;
thus, environed by a sort of intellectual anarchism, he pursues
analytical pleasures quite unrestrainedly. The difference between
the two mental processes--that which leads from the sensible world to
conception and that which veers into the fields beyond--is so great
that it is hardly permissible to view the results arrived at in the
outcome of the separate processes as being identical.
To illustrate this difference, let us draw an analogy. The miner
digs the iron ore out of the ground. The iron is separated from the
extraneous material and delivered to the furnaces where the metal is
melted and turned out as pig iron. It is further treated, and steel,
of various grades, cast iron and other kinds of iron are produced. The
treatment of the iron ore up to this stage is similar to the treatment
of sense-impressions by the Thinker. Steel, cast iron, et cetera, are
similar to mental concepts. Later, the steel and other products are
converted into instruments and numerous articles. This represents
the superperceptual process. Trafficking in iron ore products, such
as instruments of precision, watch springs, and the like, represents
a stage still farther removed from the primary treatment of the
ore and is similar to that to which concepts are treated when the
metageometrician manipulates them in the construction of conceptual
space-forms. Perception is the dealing with raw iron ore while
conception is analogous to the production of the finished product.
Superperception would be analogous to the trafficking in the
finished product as such and without any reference to the source
or the preceding processes. Thus the notions and judgments of the
non-Euclidean geometry are arrived at as a result of a triple process
of perception, conception and superperception the latter being merely
superconceived as formal space-notions. But it is obvious that the
more complex the processes by which judgments purporting to relate to
perceptual things are derived the more likely are those judgments to be
at variance with the nature of the things themselves.
In view of the foregoing, the dangers resulting from identifying
the products of the two processes are very obvious indeed. But the
difference between the two procedures is the difference between
Euclidean and non-Euclidean geometries or the difference between
perceptual space notions and conceptual space notions. Hence, it is
not understood just how or why it has occurred to anyone that the two
notions could be made congruent. Magnitudes in perceptual, sensible
space are things apart from those that may be said to exist in
mathematical space or that space whose qualities and properties have no
existence outside of the mind which has conceived them. It is believed
to be quite impossible to approach the study of metageometrical
propositions with a clear, open mind without previously understanding
the fundamental distinctions which exist between them.
It follows, therefore, as a logical conclusion that geometric space of
whatsoever nature is a purely formal construction of the intellect, and
for this reason is completely under the sovereignty of the intellect
however whimsical its demands may be. Being thus the creature of the
intellect, its possibilities are limited only by the limitations of the
intellect itself. Perceptual space, being neither the creature of the
intellect nor necessarily an _a priori_ notion resident in the mental
substructure, but existing entirely independent of the intellect or
its apprehension thereof, cannot be expected to conform to the purely
formal restrictions imposed by the mind except in so far as those
restrictions may be determined by the nature of perceptual space. And
for that matter, it should not be forgotten that, as yet, we have no
means of determining whether or not the testimony of the intellect is
thoroughly credible simply because there is no other standard by which
we may prove its testimony. It is possible to justify the deliveries
of the eye by the sense of touch, or vice versa. It is also possible
to prove all our sense-deliveries by one or the other of the senses.
But we have no such good fortune with the deliveries of the intellect.
We have simply to accept its testimony as final; because we cannot
do any better. But if it were possible to correct the testimony of
the intellect by some other faculty or power which by nature might be
more accurate than the intellect it should be found that the intellect
itself is sadly limited.
The possible curvature of space is a notion which also characterizes
the content of the non-Euclidean geometry. It is upon this notion
that the question of the finity and unboundedness of space, in the
mathematical sense, rests. In the curved space, the straightest line
is a curved line which returns upon itself. Progression eastward
brings one to the west; progression northward brings one to the south,
et cetera. On this view space is finite, but may not be regarded as
possessing boundaries.
Space-curvature, reinforced by the idea that space is also a manifold
is the enabling clause of metageometry and without them the analyst
dares not proceed. Here again, we are led to the confession that
however fantastic these two notions may seem and evidently are, there
is nevertheless to be recognized in them a "dim glimpse" of a veritable
reality--a slight foreshadowing of the revelation of some great kosmic
mystery.
The manifoldness of space is the fiat of analysis. It is the inevitable
outcome of the analyst's method of procedure. His education, training
and view of things in general inhibit his arriving at any other
result and he may be pardoned with good grace for his manufacture of
the space-manifold. For by it perhaps a better appreciation of that
wonderful extension of consciousness in the nature of which is involved
the explanation of the perplexing problems which the manifold and other
metageometrical expedients faintly adumbrate may be gained.
It is pertinent, in the light of the above, to examine into some of the
relative merits of the three formal bulwarks of geometrical knowledge.
These are _certainty_, _necessity_ and _universality_.
Geometric certainty is derived solely from the nature of the premises
upon which it is based. If the premises be contradictory, it is,
of course, defective. But if the premises are non-contradictory
or self-evident, then the certainty of geometric notions and
conclusions is valid. Another consideration of prime importance in
this connection is the _definition_. From it all premises proceed.
Hence, the definition is even more important than the premise; for
it is the persisting determinant of all geometric conclusions while
the premise is dependent upon the limitations of the definition. The
determinative character of the definition has led to its apotheosis;
but this, admittedly, has been necessary in order to give stability
and permanency to the conclusions which followed. But in spite of this
it would appear that the certainty of geometric conclusions is not a
quality to be reckoned as absolute or final.
With the same certainty that it can be said the sum of the angles of
the triangle is equal to two right angles it may be asserted that
that sum is also greater or less than two right angles. Certainty
which is based upon the inherent congruity of definitions, premises
and propositions is an entirely different matter from that certainty
which arises out of the real, abiding validity of a scheme of thought.
But this difference is not lessened by the fact that the latter is
dependent, in a measure, upon the correct systematization of our
spatial experiences by means of methodical processes. Euclidean
geometry, accordingly, is not so certain in its applications as it is
utilitarian; but non-Euclidean geometry is even less certain than the
former and consequently more lacking in its utilitarian possibilities.
The necessity of geometrical determinations is merely the necessity
which inheres in logical inferences or deductions. These may or may
not be valid. Inasmuch as the necessariness of deductions is primarily
based upon the conditional certainty of premises and definitions it
appears that this quality is in no way peculiar to geometry whether
Euclidean or non-Euclidean. In like manner, the universality of
geometric judgments may not properly be regarded as a peculiarity of
geometry; but is explicable upon the basis of the formal character
of the assumptions which underlie it. The chief value, then, of
non-Euclidean geometry seems to abide in the fact that it clarifies our
understanding as to the complex processes by which it is possible to
organize and systematize our spatial experiences for assimilation and
use in other branches of knowledge.
With the above statement of the case of the non-Euclidean geometry
it is now thought permissible to state briefly some of the elements
thereof.[7]
[7] The science of pure mathematics is perhaps indebted to no one in so
great a degree as to GEORGE BRUCE HALSTEAD, formerly of the University
of Texas, whose labors in connection with the popular exposition of
the non-Euclidean geometry have been most untiring and effectual. Vide
_Popular Astronomy_, Vol. VII and VIII, 1900, Dr. G. B. HALSTEAD.
Below will be found some of the elements obtained as a consequence of
efforts made both at proving and disproving the parallel-postulate of
Euclid:
"If two points determine a line it is called a straight."
"If two straights make with a transversal equal alternate angles they
have a common perpendicular."
"A piece of a straight is called a sect."
"If two equal coplanar sects are erected perpendicular to a straight,
if they do not meet, then the sect joining their extremities makes
equal angles with them and is bisected by a perpendicular erected
midway between their feet."
"The sum of the angles of a rectilineal triangle is a straight angle,
in the hypothesis of the right (angle); is greater than a straight
angle in the hypothesis of the obtuse (angle); is less than a straight
angle in the hypothesis of the acute (angle)."
"The hypothesis of right is Euclidean; the hypothesis of the acute is
BOLYAI-LOBACHEVSKIAN; the hypothesis of obtuse is RIEMANNIAN."
"If one straight is parallel to a second the second is parallel to the
first."
"Parallels continually approach each other."
"The perpendiculars erected at the middle point of the sides of a
triangle are all parallel, if two are parallel."
"If the foot of a perpendicular slides on a straight its extremity
describes a curve called an equidistant curve, or an equidistantial."
"An equidistantial will slide on its trace."
"In the hypothesis of the obtuse a straight is of finite size and
returns into itself."
"Two straights always intersect."
"Two straights perpendicular to a third straight intersect at a point
half a straight from the third either way."
"A pole is half a straight from its polar."
"A polar is the locus of coplanar points half a straight from its pole.
Therefore, if the pole of one straight lies on another straight the
pole of this second straight is on the first straight."
"The cross of two straights is the pole of the join of their poles."
"Any two straights inclose a plane figure, a digon."
"Two digons are congruent if their angles are equal."
"The equidistantial is a circle with center at the poles of its basal
straight."
A typical postulate based upon the BOLYAI hypothesis of the acute angle
is the following:
"From any point _P_ drop _PC_, a perpendicular to any given straight
line _AB_. If _D_ move off indefinitely on the ray _CB_, the sect will
approach as limit _PF_ copunctal with _AB_ at infinity.
[Illustration: FIG. 5.]
_PD_ is said to be at _P_ the parallel to _AB_ toward _B_. _PF_ makes
with _PC_ an angle _CPF_ which is called the angle of parallelism for
the perpendicular _PC_. It is less than a right angle by an amount
which is the limit of the deficiency of the triangle _PCD_. On the
other side of _PC_, an equal angle of parallelism gives the parallel
_P_ to _BA_ towards _AM_.[8] Thus at any point there are two parallels
to a straight. A straight has, therefore, two separate points at
infinity."
"Straights through _P_ which make with _PC_ an angle greater than the
angle of parallelism and less than its supplement do not meet the
straight _AB_ at all not even at infinity."
[8] NOTE.--_M_ may be any point on the line _BA_ indefinitely
produced.
The parallel-postulate is stated in the non-Euclidean geometry as
follows:
"If a straight line meeting two straight lines make those angles which
are inward and upon the same side of it less than two right angles the
two straight lines being produced indefinitely will meet each other on
this side where the angles are less than two right angles."
It is stated by MANNING[9] in the following language:
"If two lines are cut by a third and the sum of the interior angles on
the same side of the cutting line is less than two right angles the
line will meet on that side when sufficiently produced."
It is rather significant that in this postulate which is really
a definition of space should be found grounds for such diverse
interpretations as to its nature. Of course, the moment the mind
seeks to understand the infinite by interpreting it in the unmodified
terms of the apparently unchangeable finite it entangles itself into
insurmountable difficulties. As a drowning man grasps after straws so
the mind, immersed in endless abysses of infinity, fails to conduct
itself in a seemly manner; but gasps, struggles and flounders and
is happy if it can, in the depths of its perplexity, discover a way
of logical escape. The pure mathematician has a hankering after the
logically consistent in all his pursuits; to him it is the "Holy
Grail" of his highest aspirations. He seeks it as the devotee seeks
immortality. It is to him a philosopher's stone, the elixir of
perpetual youth, the eternal criterion of all knowledge.
[9] Vide _Non-Euclidean Geometry_, p. 91.
Failures to demonstrate the celebrated postulate of EUCLID led, as a
matter of course, to the substitution of various other postulates more
or less equivalent to it in that each of them may be deduced from the
other without the aid of any new hypothesis.
Among those who sought proof by a restatement of the problem are the
following:
1. PTOLEMY: The internal angles which two parallels make with a
transversal on the same side are supplementary.
2. CLAVIUS: Two parallel straight lines are equidistant.
3. PROCLUS: If a straight line intersects one of two parallels it also
intersects the other.
4. WALLIS: A triangle being given another triangle can be constructed
similar to the given one and of any size whatever.
5. BOLYAI (W.): Through three points not lying on a straight line a
sphere can always be drawn.
6. LORENZ: Through a point between the lines bounding an angle a
straight line can always be drawn which will intersect these two lines.
7. SACCHERI: The sum of the angles of a triangle is equal to two right
angles.
There were, of course, many other statements and substitutions used by
mathematicians in their endeavors satisfactorily to establish the truth
of the parallel-postulate. That their labors should have terminated,
first, by doubting it, then by denying, and finally, by building up a
system of geometries which altogether ignores the postulate is just
what might naturally be expected of these men who have given to the
world the non-Euclidean geometry. In doing what they did many, if
not all of them, were not aware in any measure of the proportions
of the imposing superstructure that would be built upon their
apparent failures. All of them undoubtedly must have sensed the vague
adumbrations forecast by the unfolding mysteries which they sought to
lay bare; all of them must have felt as they executed the early tasks
of those crepuscular days of pure mathematics that the way which they
were traveling would lead to the inner shrine of a higher knowledge
and a wider freedom; they may have been led by divine intuition to
strike out on this new path and yet they could not have known how fully
their dreams would be realized by the mathematicians of the twentieth
century. If so, they were truly gods and mathesis is their kingdom.
The analyst proceeds upon a basis entirely at variance with that which
guides the ordinary investigator in the formulation of his conclusions.
The empirical scientist in arriving at his theories or hypotheses is
governed at all times by the degree of conformity which his postulates
exhibit to the actual phenomena of nature. He endeavors to ascertain
just how far or in what degree his hypothesis is congruent with
things found in nature. If the dissidence is found to predominate he
abandons his theory and makes another statement and again sets out to
determine the degree of conformity. If he then finds that the natural
phenomena agree with his theory he accepts it as for the time being
finally settling the question. In all things he is limited by the
answer which nature gives to his queries. Not so with the exponent of
pure mathematics. For him the truth of hypotheses and postulates is
not dependent upon the fact that physical nature contains phenomena
which answer to them. The sole determining factor for him is whether
or not he is able to state with _rational consistency_ the assumed
first principles and then logically develop their consequences. If he
can do this, that is, if he can state his hypotheses with consistency
and develop their consequences into a logical system of thought, he is
quite satisfied and well pleased with his performances. But the fact
that this is true is of vital significance for all who seek clearly to
understand the essential character of hyperspatiality.
It appears, therefore, that the science of consequences is the radical
essence of pure geometry. The metageometrician enjoys unlimited freedom
in the choice of his postulates and suffers curtailment only when it
comes to the question of consistency. He is at liberty to formulate as
many systems of geometry as the barriers of consistency will permit
and these are practically innumerable. So long then as the laws of
compatibility remain inviolate his multiplication of postulate-systems
may proceed indefinitely. Is it strange then that under conditions
where an investigator has such unbridled liberty he should be found
indulging in mathetic excesses?
KANT held that the axioms of geometry are synthetic judgments _a
priori_; but it appears that in the strictest sense this is not the
case. It depends upon the type of mind which is taken as a standard of
reference. If it be the uncultivated mind, it is certain that to it
the relations expressed by an axiom would never appear spontaneously.
If on the other hand, the standard be that of a cultivated mind it is
also equally certain that to it these relations would be discovered
only after methodical operations. All judgments arrived at as a result
of logical processes should, it seems, be regarded as judgments _a
posteriori_, i.e., the results of empirical operations. Confessedly,
the facts adduced in course of experimentation serve as guides in
choosing among all of the many possible logical conventions; but our
choice remains untrammeled except by the compulsion arising out of
a fear of inconsistency. The real criterion then of all geometries
is neither truth, conformability nor necessity, but consistency and
convenience.
The difficulty with the non-Euclideans resolved itself into the
question as to whether it is more consistent, as well as convenient, to
establish a proof of the postulate by taking advantage of the support
to be found in other postulates or whether, by seeking a demonstration
based upon the deliveries of sense-experience as to the nature of
space and its properties, a still more consistent conclusion might
be reached. They had further perplexity, however, when it came to a
decision as to whether the organic world is produced and maintained
in Euclidean space or in a purely conceptual space which alone can
be apprehended by the mind's powers of representation. Unwilling to
admit the existence of the world in Euclidean space, they turned their
attention to the examination of the properties of another kind of
space so-called which unlike the space of the Ionian school could be
made to answer not only all the purposes of plane and solid figures,
but of spherics as well. And so, the manifold space was invented by
RIEMANN and later underwent some remarkable improvements at the hands
of his disciple, BELTRAMI. But it may be said here, parenthetically,
that the truth of the whole matter is that our world is neither in
Euclidean nor non-Euclidean space, both of which, in the last analysis,
are conceptual abstractions. Although it may not be denied that the
Euclidean space is the more compatible.
The problem of devising a space, if only a very limited portion, in
which could be demonstrated the assumed alternative hypothesis and
its consequences logically developed, occasioned no inconsiderable
concern for the non-Euclidean investigators; but neither LOBACHEVSKI,
BOLYAI nor RIEMANN were to be baffled by the difficulties which they
met. These only cited them to more laborious toil. Having succeeded in
mentally constructing the particular kind of space which was adaptable
to their rigorous mathetic requirements it immediately occurred to
them that all the qualities of the limited space thus devised might
logically be amplified and extended to the entire world of space and
that what is true of figures constructed in the segmented portion
of space which they used for experimental purposes is also true of
figures drawn anywhere in the universe of this space as all lines
drawn in the finite, bounded portion could be extended indefinitely
and all magnitudes similarly treated. From these results, it was
but a single step to the conclusion which followed--that either an
entirely new world of space had been discovered or that our notion of
the space in which the organic world was produced is wholly wrong and
needs revision. But notwithstanding the insurmountable obstacles which
stood in the way of the investigators who made the attempt to discover
the homology which might exist between the characteristics of the
newly fabricated space and the phenomenal world, investigations were
carried forward with almost amazing recklessness and loyalty to the
mathetic spirit until it was discovered that all efforts to trace out
any definite lines of correspondence were futile. Then the policy of
ignoring the question of conformability was adopted and has since been
pursued with unchangeable regularity by the analytical investigator.
Among the results obtained by the non-Euclideans in their profound
researches into the nature of hyperspace are these: 1. It was found
that the angular sum of a triangle, being ordinarily assumed to be a
variable quantity, is either less or greater than two right angles
so that a strictly Euclidean rectangle could not be constructed. 2.
The angle sums of two triangles of equal area are equal. 3. No two
triangles not equal can have the same angles so that similar triangles
are impossible unless they are of the same size. 4. If two equal
perpendiculars are erected to the same line, their distance apart
increases with their length. 5. A line every point of which is equally
distant from a given straight line is a curved line. 6. Any two lines
which do not meet, even at infinity, have one common perpendicular
which measures their minimum distance. 7. _Lines which meet at infinity
are parallel._ But it is apparent that these results have not followed
upon any mathematical consequence of other supporting postulates or
axioms such as would place them on a coördinate basis with those used
as a support for the parallel-postulate; for they are based upon the
envisagement of an entirely new principle of space-perception and
belong to a wholly different set of space qualities.
The final issue then of the non-Euclidean geometry is neither in
the utility of its processes and conclusions nor in the increscent
inclination towards a new outlook upon the world of mathesis; but
resides solely in the possibilities yet to be developed in that vast
domain of analytical thought which it has discovered and opened to
view. To say that it sheds any light upon the nature of the universe
is perhaps to take the radical view; yet it cannot be doubted that the
researches incident to the formulation of the non-Euclidean geometry
have greatly extended the scope of consciousness. Whether the extension
is valid and normal or simply a hypertrophic excrescence of mental
feverishness; whether by virtue of it we shall more closely approach
an understanding of the true nature of the mind of the Infinite, or
shall all fall into insanity, are certainly debatable questions. It
nevertheless appears evident that humanity has gained something of
real, abiding permanence by this new departure. If that something be
merely an extended consciousness or an awakening to the fact that
there are stages of awareness beyond the strictly sensuous, and every
observable evidence points to this, then there has only begun the
process by which the faculty of conscious functioning in this new world
shall become the normal possession of the human species. But this new
world cannot be said to be of mathematical import; for it is doubtful
if mathematical laws such as have been devised up to the present time,
would obtain therein. So that if anything, it must be psychological and
vital.
On this view the worlds of hyperspace inlaid with analytic
manifoldnesses and constant curvatures are but the primal excitants
which will finally awaken in the mind the faculty of awareness in the
new domain of psychological content. Then will come the blooming of
the diurnal flower of the mind's immortality and the outputting of the
organ of consciousness wherewith the infinite stretches of hyperspaces,
the low-lying valleys of reals and imaginaries and the uplifting hills
of finites and infinites shall be divested of their mysteries and
stand out in their unitariness no longer draped in the veil of the
inscrutable and the incomprehensible.
The fourth dimension, regarded by some as a new scope of motion for
objects in space, by others as a new and strange direction of spatial
extent and by others still as the doorway of the temple of exegesis
wherein an explanation may be found for the entire congeries of
mysteries and supermysteries which now perplex the human mind, may also
be said to be the key to the non-Euclidean geometry. But it really
complicates the situation; for one has to be capable of prolonged
abstract thought even to envisage is as a conceptual possibility.
POINCARÉ[10] says: "Any one who should dedicate his life to it could,
perhaps, eventually imagine the fourth dimension," implying thereby
that a lifetime of prolonged abstract thought is necessary to bring the
mind to that point of ecstasy where it could even so much as imagine
this additional dimension. Nevertheless by it (the fourth dimension)
was the non-Euclidean geometry made and without it was not any of the
hyperspaces made that were made. It is the view which geometers have
taken of space in general that has made the fourth dimension possible,
and not only the fourth, but dimensions of all degrees. The basis of
the non-Euclidean geometry may be found then in the notion of space
which has been predominant in the minds of the investigators.
[10] Vide _Nature_, Vol. XLV, 1892.
Finally, it should be pointed out that the non-Euclidean geometry,
though a consistent system of postulates, has been constructed upon a
misconception based upon the identification of real, perceptual space
with systems of space-measurements. Hyperspaces which are not spaces
at all should not be confounded with _real space_. But they constitute
the substance of non-Euclidean geometry; they are its blood and sinews.
Their study is interesting, because of the possibilities of speculation
which it offers. No mind that has thought deeply upon the intricacies
of the fourth dimension, or hyperspace, remains the same after the
process. It is bound to experience a certain sense of humility, and
yet some pride born of a knowledge that it has been in the presence of
a great mystery and has delved into the fearful deeps of kosmic mind.
To the mind that has thus been anointed by the sacred chrism of the
inner mysteries of creative mentality there always come that stillness
and calm such as characterize the aftermath of reflection upon the
incomprehensible and the transfinite.