The Mystery of Space

the Non-Eucl

uclidean Geometry-Space Curvature and Manifoldness-Some Elements of the Non-Euclidean Geometry-Certainty, Necessity and Universality as Bulwarks

esignate any system of geometry which

-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 i

ed by M. Dehn to denote all kinds of geometries which represented vari

forts 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 main

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 th

tween their notions and things as they actually are. The attitude of the metageomet

eries of orderly worlds, worlds that are possible and logically actual, and h

ations 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 prov

ptual 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 fr

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

ation 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 conce

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 the

ron 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

clidean 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 t

nd 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 quali

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 d

y 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 eas

es not proceed. Here again, we are led to the confession that however fantastic these two notions may seem and evidently are, there is nevert

s 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 o

e of the relative merits of the three formal bulwarks of geomet

f 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

d 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 s

hat 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

Euclidean geometry it is now thought permissible

as a consequence of efforts made both at proving

rmine a line it is

nsversal equal alternate angles t

straight is

y do not meet, then the sect joining their extremities makes equal angles wi

sis of the right (angle); is greater than a straight angle in the hypothesis of the

ypothesis of the acute is Bolyai-Lobachevski

el to a second the second

tinually appro

dle point of the sides of a triangle a

traight its extremity describes a curve calle

tial will slid

use a straight is of finite s

hts always

rd straight intersect at a point half

f a straight f

pole. Therefore, if the pole of one straight lies on another str

ights is the pole of t

s inclose a plane

ongruent if their

ircle with center at the p

n the Bolyai hypothesis of th

ight line AB. If D move off indefinitely on the ray CB, the

g.

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 give

than the angle of parallelism and less than its supplemen

s stated in the non-Eucl

the same side of it less than two right angles the two straight lines being produced inde

anning[9] in the

es on the same side of the cutting line is less than two right an

t 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

e, to the substitution of various other postulates more or less equivalent to it in

of by a restatement of the

two parallels make with a transversa

rallel straight li

intersects one of two paralle

r triangle can be constructed similar to

oints not lying on a straight l

bounding an angle a straight line can always

e angles of a triangle is

o 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

ory 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 determ

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 rema

n 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 f

ame 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 a

t 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 r

an 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.

idean 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

ological 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

on; 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 i

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