The Mystery of Space

Chapter X Is the Natural and Logical Sequence of the Inquiry Into the

question of spatiality. The conclusions reached therein and the obvious inferences which should be drawn from the arguments presented flow inevitably not only from the evidences of mathematical data but of the common observations of life. And while we disclaim any intention of demanding acceptance of them as final, authoritative declarations, we shall be satisfied if the readers of this volume be incited to solve for themselves the problems which these queries naturally suggest. Happy indeed shall be the outcome if there be any who, following the path sketched herein, shall find the solution of the Mystery of Space and apply its meanings to the enhancement of the values of the intuitive life.

In conclusion, the author esteems it a special obligation of gratitude that he should here acknowledge the debt which he owes to all of his friends who have in any way assisted or encouraged him in the completion of this work. Among those whom he is permitted to thank in this way is Mr. JAMES RINDFLEISCH who, having very kindly prepared the illustrations for the photo-engravers, is deserving of special mention.

ROBERT T. BROWNE.

New York City, 1919.

CONTENTS

PAGE

PREFACE vii

INTRODUCTION: Explanatory Notes x

PART ONE

CHAPTER I. The Prologue

THE PROLOGUE

On the Variability of Psychic Powers--The Discovery of the Fourth Dimension Marks a Distinct Stage in Psychogenesis--The non-Methodical Character of Discoveries--The Three Periods of Psychogenetic Development--The Scope and Permissibility of Mathetic License--Kosmic Unitariness Underlying Diversity 23

CHAPTER II. Historical Sketch of the Hyperspace Movement

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 44

CHAPTER III. Essentials of the Non-Euclidean Geometry

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 69

CHAPTER IV. Dimensionality

DIMENSIONALITY

Arbitrary Character of Dimensionality--Various Definitions of Dimension--Real Space and Geometric Space Differentiated--The Finity of Space--Difference Between the Purely Formal and the Actual--Space as Dynamic Appearance--The A Priori and the A Posteriori as Defined by Paul Carus 92

CHAPTER V. The Fourth Dimension

THE FOURTH DIMENSION

The Ideal and the Representative Nature of Objects in the Sensible World--The Fluxional, the Basis of Mental Differences--Natural Symbols and Artificial Symbols--Use of Analogies to Prove the Existence of a Fourth Dimension--The Generation of a Hypercube or Tesseract--Possibilities in the World of the Fourth Dimension--Some Logical Difficulties Inhering in the Four-Space Conception--The Fallacy of the Plane-Rotation Hypothesis--C. H. Hinton and Major Ellis on the Fourth Dimension 118

PART TWO

SPATIALITY: AN INQUIRY INTO THE ESSENTIAL NATURE OF SPACE AS DISTINGUISHED FROM THE MATHEMATICAL INTERPRETATION

CHAPTER VI. Consciousness the Norm of Space Determinations

CONSCIOUSNESS THE NORM OF SPACE DETERMINATIONS

Realism Is Determined by Awareness--Succession of Degrees of Realism--Sufficiency of Tridimensionality--The Insufficiency of Self-consistency as a Norm of Truth--General Forward Movement in the Evolution of Consciousness Implied in the Hyperspace Concept--The Hypothetical Nature of Our Knowledge--Hyperspace the Symbol of a More Extensive Realm of Awareness--Variations in the Methods of Interpreting Intellectual Notions--The Tuitional and the Intuitional Faculties--The Illusionary Character of the Phenomenal--Consciousness and the Degrees of Realism 161

CHAPTER VII. The Genesis and Nature of Space

THE GENESIS AND NATURE OF SPACE

Symbology of Mathematical Knowledge--Manifestation and Non-manifestation Defined--The Pyknon and Pyknosis--The Kosmic Engenderment of Space--On the Consubstantiality of Spatiality, Intellectuality, Materiality, Vitality and Kosmic Geometrism--Chaos-Theos-Kosmos--Chaogeny and Chaomorphogeny--N. Malebranche on God and the World--The Space-Mind--Space and Mind Are One--The Kosmic Pentoglyph 203

CHAPTER VIII. The Mystery of Space

THE MYSTERY OF SPACE

The Thinker and the Ego--Increscent Automatism of the Intellect--The Egopsyche and the Omnipsyche--Kosmic Order or Geometrism--Life as Engendering Element--The Mystery of Space Stated--Kathekos and Kathekotic Consciousness--Function of the Ideal--The Path of Search for an Understanding of the Nature and Extent of Space Must Proceed in an Inverse Direction 242

CHAPTER IX. Metageometrical Near-Truths

METAGEOMETRICAL NEAR-TRUTHS

Realism Is Psychological and Vital--The Impermanence of Facts--On the Tendency of the Intellect to Fragmentate--The Intellect and Logic--The Passage of Space--Kosmometer and Zoometer, Instruments for the Measurement of the Passage of Space and the Flow of Life--The Disposal of Life and the Power to Create--Space, a Dynamic, Creative Process--Numbers and Kosmogony--Kosmic Significance of the Circle and the Pi-proportion--Mechanical Tendence of the Intellect and Its Inaptitude for the Understanding of Life--The Criterion of Truth 284

CHAPTER X. Media of New Perceptive Faculties

MEDIA OF NEW PERCEPTIVE FACULTIES

The Spiritualization of Matter Is the End of Evolution--Sequence and Design in the Evolution of Human Faculties--The Upspringing Intuition--Evidences of Supernormal Powers of Perception and the Possibility of Attainment--The Influence and Place of the Pituitary Body and the Pineal Gland in the Evolution of Additional Faculties--The Skeptical Attitude of Empirical Science and the Need for a More Liberal Posture--The General Results of Pituitarial Awakening Upon Man and the Theory of Knowledge 327

BIBLIOGRAPHY 359

INDEX 367

THE MYSTERY OF SPACE

INTRODUCTION

EXPLANATORY NOTES

The following interpretation of words, phrases and notions occurring in the text, and also biographical sketches which the scope and purpose of the book itself make it impracticable to elaborate, are appended with the view to facilitating its perusal.

AT-ONE-MENT (state of unity, unitariness); denotes the ultimate state of oneness towards which all evolutionary movement tends; applied to consciousness, indicates the final expansion of consciousness wherein it coincides with the universal consciousness in extent and quality of comprehension. As applied to things, denotes the unification of all movements, tendencies, and evolutions as a singularity; the end of all evolutionary activity (vide p. 270).

BELTRAMI, EUGENIO, was born at Cremona, November 16, 1835; there he attended the elementary schools, the gymnasium and the lyceum, excepting the scholastic year 1848-49 when he was at the Gymnasium of Venice, now known as Marco Polo. He finished his lyceal studies in the summer of 1853, and in the following autumn (November) became a student in the Mathematical Faculty of the University of Pavia, after having obtained a scholarship there on the Castiglioni Foundation in the Collegio Ghisleri.

In 1854, the succeeding year, he was expelled from this college in company with five of his colleagues who were accused of promoting "disorders" against the Abbot LEONARDI, rector of the college. The expulsion brought him many hardships and disappointments, and for two years he drifted along merely existing as his family was too poor to have him matriculated at another university. But in 1856, he went to Verona where he succeeded in securing employment as secretary to the engineer, DIDAY, in the Government service of Lombardy-Venice. On January 10, 1857, he was dismissed from this position "for political reasons"; but as the annexation of Lombardy to Piedmont occurred soon thereafter, he became again attached to the office of DIDAY, his former employer, when it was transferred to Milan as a consequence of political changes.

At Milan BELTRAMI took up his mathematical education in real earnest as he now had access to Professor BRIOSCHI, his former tutor, and also LUIGI CREMONA. Through the influence of these two men he was designated (October 18, 1862) "Professore straordinario" in the University of Bologna. His work on Surfaces of Constant Negative Curvature, as the pseudosphere, and his application of the expression given by LOBACHEVSKI (q.v.) for the angle of parallelism, very definitely secure for him a place among the foremost workers in the field of the non-Euclidean geometry. He postulated a theorem, known as Beltrami's Theorem, which he stated as follows: "The center of a circle circumscribing a triangle is the center of gravity of the centers of its inscribed and escribed circles." He died in the year 1900. (Vide Amer. Math. Mo., Vol. IX, p. 59.)

BOLYAI, JANOS (1802-1860), was born at Kleansenburg, Hungary. He is said to have inherited his mathematical genius from his father, BOLYAI FARKAS (1775-1856), who was born at Bolya, Hungary. Being a very spirited youth, his progress in his studies was most remarkable. He completed the curriculum at the Latin school when only twelve years of age. Was graduated from the Philosophical Curriculum as a result of two years of study and then entered the Viennese Academy of Engineers. Was appointed lieutenant at Temesvárlin, 1823, whence on November 3, 1823, he wrote his father: "I have discovered such magnificent things that I am myself astonished at them. It would be damage eternal if they were lost. When you see them, my father, you yourself will acknowledge it. Now, I cannot say more, only so much: that from nothing I have created another wholly new world." This letter was written in the Magyar language and has been preserved at the Marcos Vásárhely, Hungary. The mathematical conceptions formulated by him became the appendix of the Tentamen, a book which his father had written on the Theory of Parallels.

His Science Absolute of Space was translated into the French in 1868 by the French mathematician, J. HOÜEL, to whom belongs the credit of popularizing the works both of BOLYAI and LOBACHEVSKI. (Vide Science, n. s., Vol. 35, No. 906, 1912.)

CAYLEY, ARTHUR, born at Richmond, Surrey, England, August 16, 1821; studied at King's College school; entered Trinity College, Cambridge, already a well equipped mathematician at the age of seventeen. When but twenty-one years of age he took two of the highest honors in the University of Cambridge. He was Senior Wrangler and First Smith's Prizeman. He published his first paper in 1841 and this was followed by eight hundred memoirs.

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