{"collection":"scripture-revelation","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_45691/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_45691","id":"gutenberg_45691","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/45691","summary":null,"tags":["Category: Mathematics","Category: Philosophy & Ethics","Category: Psychiatry/Psychology","Category: Religion/Spirituality","Hyperspace"],"text":"THE\n    MYSTERY OF SPACE\n\n\n\n    _The domain of the senses, in nature, is almost\n    infinitely small in comparison with the vast\n    region accessible to thought which lies beyond\n    them._--TYNDALL, \"On Radiant Heat.\"\n\n\n\n\n    THE\n    MYSTERY OF SPACE\n\n    _A Study of the Hyperspace Movement\n    in the Light of the Evolution of\n    New Psychic Faculties\n    and\n    An Inquiry into the Genesis and\n    Essential Nature of Space_\n\n    BY\n    ROBERT T. BROWNE\n\n\n    NEW YORK\n    E. P. DUTTON & COMPANY\n    681 FIFTH AVENUE\n\n\n\n\n    COPYRIGHT 1919, BY\n    E. P. DUTTON & COMPANY\n\n    _All Rights Reserved_\n\n\n    _Printed in the United States of America_\n\n\n\n\n\n    TO\n    THE CHERISHED MEMORY\n    OF\n\n    Mylie De Pr\u00e8\n\n    WHOSE WIFELY DEVOTION,\n    SYMPATHETIC ASSISTANCE AND ENCOURAGEMENT\n    DURING THE EARLY LABORS ON THE TEXT\n    WERE A CONSTANT SOURCE OF\n    INSPIRATION AND FORTITUDE TO THE AUTHOR,\n    THIS BOOK IS DEDICATED\n    AFFECTIONATELY\n\n\n\n\n  PREFACE\n\n\nMathematics is the biometer of intellectual evolution. Hence, the\ndetermination of the _status quo_ of the intellect at any time can be\naccomplished most satisfactorily by applying to it the rigorous measure\nof the mathematical method. The intellect has but one true divining\nrod and that is mathematics. By day and by night it points the way\nunerringly, so long as it leads through materiality; but, falteringly,\nblindly, fatally, when that way veers into the territory of vitality\nand spirituality.\n\nWherefore, when we have wished to ascertain the real status of the\nintellect, as well as its limitations, tendencies, possibilities, we\nhave turned to its conduct in the field of mathesis where it is least\ntrammeled in its ingressive and egressive motivations because of the\nnatural and easy accommodation which is offered for intellectual\nmovement. Whether there are signs of moribundity or symptomatic\nevidences of marked growth or of a termination of intellectual\nregnancy, or whatever may be the occasion for the examination, no surer\nindex than the mathematical may be found for the purpose. Full logical\njustification is, therefore, claimed for the choice of mathematical\nevidences to test the assumption that a new era of conscious mental\nconquest is opening for the vanguard of humanity and sequently for the\nhuman family.\n\nThe treatment of these evidences has fallen logically into two\ndivisions, namely, the first, a brief and elementary review of the\nprinciples of the non-Euclidean geometry and their bearings upon the\nquestion of space as the subject of mathematical study; and second,\nthe consideration of space as a psychological, vital and dynamic or\ncreative phenomenon.\n\nIn Part I an effort has been made to trace the growth of the notion\nof hyperspace and to show that it is a symbol of a new epoch of\nintellectual expansion, an actual seizure of a new domain of awareness\nby the mind. And for this purpose a critical examination of the\nfundamental question of dimensionality is entered upon from which it\ndevelops that the status of this primary norm of mathematical thought\nexhibits a relatively inchoate character because of its insufficiency\nas a definitive quantity, and further, because of its rather\nsuperfoetated aspect when utilized as a panacea for mathetic symptoms.\nAlso, it has been found necessary to survey the field of the four-space\nwhich has been accorded such a prominent place in the mathematical\nthought of the day. The reader should find in the chapter devoted to\nit adequate material for thought and sufficient comprehension of its\nmeanings as a mathematical contrivance.\n\nIn Part II an endeavor is made to interpret the evidences offered by\nhigh mathematic considerations in the light of the larger psychogenetic\nmovement. For this undertaking the quality of awareness has been\nstudied with the view to establishing its priority as a determining\nagency in the consideration of space in aspects broader and less\nrestricted than those embraced in the mathematical premises. Wherefore,\nit appears that there are massive implications arising out of the\nhyperspace propaganda which have hitherto been neglected in the\npreliminaries incident to the fabrication of its structure. A very\nbrief, and more or less symbolic, sketch of the genesis of space has\nserved to demonstrate its essential nature as a consubstantive of\nmateriality, vitality and intellectuality, the three major objective\nprocesses. Consequently it becomes imperatively necessary that any\nview of space which neglects its kosmic intent and purpose should be\nregarded as gravely fragmentary and insufficient. It is only by linking\nup the two aspects of space, the mathematical and the psychological, in\nsuch a manner that the one shall supplement the other, that we shall be\nable to arrive at a truly satisfactory understanding of its nature.\n\nIn Chapter IX attention is invited to some of the extremities of\nmathematical laws wherein it is shown that, because mathematical goods\nare strictly of intellectual texture and fabric, vain is the hope of\nreaching any reliable certainty with respect to many vital questions,\neven regarding space itself, by means of the mathematical method. The\nintellect, and, therefore, mathematics encounter the most formidable\nstricture when effort is made to maneuver in the field of vitality\nor realism. In addition, it is shown that, when pushed to the utmost\nlogical limits, metageometry proves not only futile, but emphasizes the\nneed for a sharp turning of the path of search from the intellectual\nor material to the spiritual or intuitional. Indeed, it becomes\npainfully certain that the Golden Fleece of profounder knowledge will\nbe discovered never by an expedition whose bark has its sails set for\nthe winds of mathematical seas. But, contrarily, a new bark, moored at\nthe furthermost shore of the sea of intellectuality with sails set for\nthe winds which come from the realm of intuitional perception, must be\nseized. Whereupon, by the straightest line, we shall, at the last, land\nupon the shore of realism, of truth all inclusive.\n\nMathematical evidences have been used in these discussions because\nthey, of all lines of knowledge, afford a more just exemplification\nof intellectual evolution. The science of mathematics is the measure\nof the quality of intellectual growth and, therefore, its data, its\npostulates, hypotheses and advances clearly mark the stages of the\nintellectual movement.\n\nChapter X is the natural and logical sequence of the inquiry into the\nquestion of spatiality. The conclusions reached therein and the obvious\ninferences which should be drawn from the arguments presented flow\ninevitably not only from the evidences of mathematical data but of the\ncommon observations of life. And while we disclaim any intention of\ndemanding acceptance of them as final, authoritative declarations, we\nshall be satisfied if the readers of this volume be incited to solve\nfor themselves the problems which these queries naturally suggest.\nHappy indeed shall be the outcome if there be any who, following the\npath sketched herein, shall find the solution of the _Mystery of\nSpace_ and apply its meanings to the enhancement of the values of the\nintuitive life.\n\nIn conclusion, the author esteems it a special obligation of gratitude\nthat he should here acknowledge the debt which he owes to all of\nhis friends who have in any way assisted or encouraged him in the\ncompletion of this work. Among those whom he is permitted to thank in\nthis way is Mr. JAMES RINDFLEISCH who, having very kindly prepared the\nillustrations for the photo-engravers, is deserving of special mention.\n\n ROBERT T. BROWNE.\n\n New York City, 1919.\n\n\n\n\n  CONTENTS\n\n\n                                                         PAGE\n\n  PREFACE                                                 vii\n\n  INTRODUCTION: Explanatory Notes                           x\n\n\n      _PART ONE_\n\n      CHAPTER I\n\n      THE PROLOGUE\n\n  On the Variability of Psychic Powers--The Discovery of\n  the Fourth Dimension Marks a Distinct Stage in\n  Psychogenesis--The non-Methodical Character of\n  Discoveries--The Three Periods of Psychogenetic\n  Development--The Scope and Permissibility of\n  Mathetic License--Kosmic Unitariness Underlying\n  Diversity                                                23\n\n      CHAPTER II\n\n      HISTORICAL SKETCH OF THE HYPERSPACE MOVEMENT\n\n  Egypt the Birthplace of Geometry--Precursors: Nasir-Eddin,\n  Christoph Clavius, Saccheri, Lambert, La Grange,\n  Kant--Influence of the _Mecanique Analytique_--The Parallel\n  Postulate the Root and Substance of the Non-Euclidean\n  Geometry--The Three Great Periods: The Formative,\n  Determinative and Elaborative--Riemann and the Properties\n  of Analytic Spaces                                       44\n\n      CHAPTER III\n\n      ESSENTIALS OF THE NON-EUCLIDEAN GEOMETRY\n\n  The Non-Euclidean Geometry Concerned with Conceptual\n  Space Entirely--Outcome of Failures at Solving\n  the Parallel-Postulate--The Basis of the Non-Euclidean\n  Geometry--Space Curvature and Manifoldness--Some\n  Elements of the Non-Euclidean Geometry--Certainty,\n  Necessity and Universality as Bulwarks of\n  Geometry--Some Consequences of Efforts at Solving\n  the Parallel-Postulate--The Final Issue of the\n  Non-Euclidean Geometry--Extended Consciousness           69\n\n      CHAPTER IV\n\n      DIMENSIONALITY\n\n  Arbitrary Character of Dimensionality--Various Definitions\n  of Dimension--Real Space and Geometric Space\n  Differentiated--The Finity of Space--Difference Between\n  the Purely Formal and the Actual--Space as Dynamic\n  Appearance--The _A Priori_ and the _A Posteriori_ as\n  Defined by Paul Carus                                    92\n\n      CHAPTER V\n\n      THE FOURTH DIMENSION\n\n  The Ideal and the Representative Nature of Objects in\n  the Sensible World--The Fluxional, the Basis of\n  Mental Differences--Natural Symbols and Artificial\n  Symbols--Use of Analogies to Prove the Existence of\n  a Fourth Dimension--The Generation of a Hypercube\n  or Tesseract--Possibilities in the World of the\n  Fourth Dimension--Some Logical Difficulties Inhering\n  in the Four-Space Conception--The Fallacy of the\n  Plane-Rotation Hypothesis--C. H. Hinton and\n  Major Ellis on the Fourth Dimension                     118\n\n\n      _PART TWO_\n\n      SPATIALITY: AN INQUIRY INTO THE ESSENTIAL NATURE\n      OF SPACE AS DISTINGUISHED FROM THE MATHEMATICAL\n      INTERPRETATION\n\n      CHAPTER VI\n\n      CONSCIOUSNESS THE NORM OF SPACE DETERMINATIONS\n\n  Realism Is Determined by Awareness--Succession of Degrees\n  of Realism--Sufficiency of Tridimensionality--The\n  Insufficiency of Self-consistency as a Norm of\n  Truth--General Forward Movement in the Evolution\n  of Consciousness Implied in the Hyperspace Concept--The\n  Hypothetical Nature of Our Knowledge--Hyperspace\n  the Symbol of a More Extensive Realm of\n  Awareness--Variations in the Methods of Interpreting\n  Intellectual Notions--The Tuitional and the\n  Intuitional Faculties--The Illusionary Character of\n  the Phenomenal--Consciousness and the Degrees of\n  Realism                                                 161\n\n      CHAPTER VII\n\n      THE GENESIS AND NATURE OF SPACE\n\n  Symbology of Mathematical Knowledge--Manifestation\n  and Non-manifestation Defined--The Pyknon and\n  Pyknosis--The Kosmic Engenderment of Space--On\n  the Consubstantiality of Spatiality, Intellectuality,\n  Materiality, Vitality and Kosmic\n  Geometrism--Chaos-Theos-Kosmos--Chaogeny and\n  Chaomorphogeny--N. Malebranche on God and the\n  World--The Space-Mind--Space and Mind Are\n  One--The Kosmic Pentoglyph                              203\n\n      CHAPTER VIII\n\n      THE MYSTERY OF SPACE\n\n  The Thinker and the Ego--Increscent Automatism of\n  the Intellect--The Egopsyche and the Omnipsyche--Kosmic\n  Order or Geometrism--Life as Engendering\n  Element--The Mystery of Space Stated--Kathekos\n  and Kathekotic Consciousness--Function of the Ideal--The\n  Path of Search for an Understanding of the Nature and\n  Extent of Space Must Proceed in an Inverse Direction    242\n\n      CHAPTER IX\n\n      METAGEOMETRICAL NEAR-TRUTHS\n\n  Realism Is Psychological and Vital--The Impermanence of\n  Facts--On the Tendency of the Intellect to Fragmentate--The\n  Intellect and Logic--The Passage of Space--Kosmometer\n  and Zoometer, Instruments for the Measurement of the\n  Passage of Space and the Flow of Life--The Disposal of\n  Life and the Power to Create--Space, a Dynamic, Creative\n  Process--Numbers and Kosmogony--Kosmic Significance of\n  the Circle and the Pi-proportion--Mechanical Tendence\n  of the Intellect and Its Inaptitude for the Understanding\n  of Life--The Criterion of Truth                         284\n\n      CHAPTER X\n\n      MEDIA OF NEW PERCEPTIVE FACULTIES\n\n  The Spiritualization of Matter Is the End of Evolution--Sequence\n  and Design in the Evolution of Human Faculties--The\n  Upspringing Intuition--Evidences of Supernormal Powers\n  of Perception and the Possibility of Attainment--The\n  Influence and Place of the Pituitary Body and the Pineal\n  Gland in the Evolution of Additional Faculties--The\n  Skeptical Attitude of Empirical Science and the Need\n  for a More Liberal Posture--The General Results of\n  Pituitarial Awakening Upon Man and the Theory of\n  Knowledge                                               327\n\n  BIBLIOGRAPHY                                            359\n\n  INDEX                                                   367\n\n\n\n\n  THE MYSTERY OF SPACE\n\n\n\n\n  INTRODUCTION\n\n  EXPLANATORY NOTES\n\n\nThe following interpretation of words, phrases and notions occurring in\nthe text, and also biographical sketches which the scope and purpose of\nthe book itself make it impracticable to elaborate, are appended with\nthe view to facilitating its perusal.\n\nAT-ONE-MENT (state of unity, unitariness); denotes the ultimate state\nof oneness towards which all evolutionary movement tends; applied to\nconsciousness, indicates the final expansion of consciousness wherein\nit coincides with the universal consciousness in extent and quality of\ncomprehension. As applied to things, denotes the unification of all\nmovements, tendencies, and evolutions as a singularity; the end of all\nevolutionary activity (vide p. 270).\n\nBELTRAMI, EUGENIO, was born at Cremona, November 16, 1835; there\nhe attended the elementary schools, the gymnasium and the lyceum,\nexcepting the scholastic year 1848-49 when he was at the Gymnasium of\nVenice, now known as Marco Polo. He finished his lyceal studies in the\nsummer of 1853, and in the following autumn (November) became a student\nin the _Mathematical Faculty of the University of Pavia_, after having\nobtained a scholarship there on the Castiglioni Foundation in the\n_Collegio Ghisleri_.\n\nIn 1854, the succeeding year, he was expelled from this college in\ncompany with five of his colleagues who were accused of promoting\n\"disorders\" against the Abbot LEONARDI, rector of the college. The\nexpulsion brought him many hardships and disappointments, and for two\nyears he drifted along merely existing as his family was too poor to\nhave him matriculated at another university. But in 1856, he went to\nVerona where he succeeded in securing employment as secretary to the\nengineer, DIDAY, in the Government service of Lombardy-Venice. On\nJanuary 10, 1857, he was dismissed from this position \"for political\nreasons\"; but as the annexation of Lombardy to Piedmont occurred soon\nthereafter, he became again attached to the office of DIDAY, his\nformer employer, when it was transferred to Milan as a consequence of\npolitical changes.\n\nAt Milan BELTRAMI took up his mathematical education in real earnest\nas he now had access to Professor BRIOSCHI, his former tutor, and also\nLUIGI CREMONA. Through the influence of these two men he was designated\n(October 18, 1862) \"Professore straordinario\" in the University of\nBologna. His work on _Surfaces of Constant Negative Curvature_, as\nthe pseudosphere, and his application of the expression given by\nLOBACHEVSKI (q.v.) for the angle of parallelism, very definitely\nsecure for him a place among the foremost workers in the field of the\nnon-Euclidean geometry. He postulated a theorem, known as _Beltrami's\nTheorem_, which he stated as follows: \"The center of a circle\ncircumscribing a triangle is the center of gravity of the centers of\nits inscribed and escribed circles.\" He died in the year 1900. (Vide\n_Amer. Math. Mo._, Vol. IX, p. 59.)\n\nBOLYAI, JANOS (1802-1860), was born at Kleansenburg, Hungary. He is\nsaid to have inherited his mathematical genius from his father, BOLYAI\nFARKAS (1775-1856), who was born at Bolya, Hungary. Being a very\nspirited youth, his progress in his studies was most remarkable. He\ncompleted the curriculum at the Latin school when only twelve years of\nage. Was graduated from the Philosophical Curriculum as a result of two\nyears of study and then entered the Viennese Academy of Engineers. Was\nappointed lieutenant at Temesv\u00e1rlin, 1823, whence on November 3, 1823,\nhe wrote his father: \"I have discovered such magnificent things that I\nam myself astonished at them. It would be damage eternal if they were\nlost. When you see them, my father, you yourself will acknowledge it.\nNow, I cannot say more, only so much: that from nothing I have created\nanother wholly new world.\" This letter was written in the Magyar\nlanguage and has been preserved at the Marcos V\u00e1s\u00e1rhely, Hungary. The\nmathematical conceptions formulated by him became the appendix of\nthe _Tentamen_, a book which his father had written on the Theory of\nParallels.\n\nHis _Science Absolute of Space_ was translated into the French in 1868\nby the French mathematician, J. HO\u00dcEL, to whom belongs the credit of\npopularizing the works both of BOLYAI and LOBACHEVSKI. (Vide _Science_,\nn. s., Vol. 35, No. 906, 1912.)\n\nCAYLEY, ARTHUR, born at Richmond, Surrey, England, August 16, 1821;\nstudied at King's College school; entered Trinity College, Cambridge,\nalready a well equipped mathematician at the age of seventeen. When\nbut twenty-one years of age he took two of the highest honors in the\nUniversity of Cambridge. He was Senior Wrangler and First Smith's\nPrizeman. He published his first paper in 1841 and this was followed by\neight hundred memoirs.\n\nFor fourteen years he practiced as Conveyancer. In 1863 Lady SADLER'S\nvarious trusts were consolidated, and a new Sadlerian professorship of\nPure Mathematics was created for the express purpose of affording a\nplace for Cayley. Meanwhile, as early as 1852 he was a fellow of the\nRoyal Society; in 1858 he joined SYLVESTER and STOKES in publishing the\n_Quarterly Journal of Pure and Applied Mathematics_.\n\nHe was for a considerable time principal adviser as to the merits of\nall mathematical papers which were presented for publication to the\nRoyal Society, the Astronomical Society, the Mathematical Society and\nthe Cambridge Philosophical Society. He is said to have been the \"most\nlearned and erudite of mathematicians,\" and much of the material,\ntherefore, which now constitutes the basis of the non-Euclidean\ngeometry is due to his laborious efforts and comprehensive knowledge\nof mathematics. (Vide _Review of Reviews_, Vol. II, 1895, Sketch,\nreprinted from _Monist_.)\n\nCHAOGENY (Gr. Chaos, disorder, geny--generating, evolution); the\nevolution of chaos into order. A kosmic process involving the\nelaboration of the original, formless world-plasm into the first faint\nsigns of orderliness; the beginnings of the movement of life or the\nCreative Logos in preparation of the field of evolution.\n\nCHAOMORPHOGENY (Gr. Chaos, disorder--Morphe, form--geny, becoming,\ngenerating); evolution of the space-form, the universe; the\nestablishment of the metes and bounds of the universe; also, the\norigination and characterization of all forms as to tendence, purpose\nand limitations.\n\nCONCEPTUALIZATION--The act of conceptualizing, the formulation of\nconcepts; the process by which the Thinker arrives at concepts; the\nlogical procedure by which the consistency of a scheme of thought is\nestablished.\n\nCONSTRUCTION, IDEAL--A purely formal conception; a theory, hypothesis;\na logical determination not necessarily based upon facts, but\npossessing virtue because of consistency; a self-consistent scheme of\nthought.\n\nCOSMOS--Whenever the term \"cosmos\" appears in the text spelled as here\nshown, it refers to phenomena pertaining to the earth or the solar\nsystem; when spelled \"kosmos\" reference is made to the universe as a\nwhole.\n\nCRITERION OF TRUTH--Defined in the text as a four-fold standard of\nreference, embracing the following elements, namely, the causal,\nthe sustentative, relational and developmental. Lacking any one of\nthese, no view of truth is more than fragmentary. Applied to space,\nit contemplates an inquiry into the genesis or causal aspect, an\naccounting for the duration aspect, a recognition of its relation\nto the totality of objects, and lastly, a prophecy of its telestic\nor perfective culmination. This test has been applied to the study\nof space as sketched in the text and the conclusions reached are an\noutcome of the inquiry directed along these lines.\n\nCURVATURE OF SPACE--A doctrine formulated by RIEMANN and which\nmaintains that space is curved, and consequently, all lines drawn\ntherein are curved lines. Professor PICKERING aptly describes the\nresults of movements in a curved space by pointing out that if we go\nfar enough east we arrive at the west; north, we arrive at the south;\ntowards the zenith, we arrive at the nadir, and _vice versa_.\n\nDEIFORM--The basic idea indicated is that the universe is the form or\nbody of the supreme deity, since He is not only immanent in the kosmos,\nbut sustains it by His life; that in order to create a manifested\nuniverse, it was necessary to limit, or sacrifice, in a measure, His\nown illimitability. Viewed in this light, the kosmos assumes an added\nsignificance.\n\nDIMENSION--(L. Dimensio, to measure), measurement; a system of space\nmeasurement. The Euclidean geometry recognizes three dimensions or\nco\u00f6rdinates as being necessary to establish a point position; witness,\nthe corners of a cube to form which three of the edges come together\nat a point. These edges represent co\u00f6rdinates. For the purposes of\nmetageometry, the term dimension has been variously defined, as,\ndirection, extent, a system of space measurement, or a system of\nco\u00f6rdinates. Regarded as a series of co\u00f6rdinates, it became possible\nto postulate a system which required four co\u00f6rdinates to establish\nthe position of a point, as in the hypercube. There may be five, six,\nseven, eight, or any number of such co\u00f6rdinate systems according to\nthe kind of space involved in the calculations. Determinations based\nupon the logical necessities of the various co\u00f6rdinate systems have\nbeen found to be self-consistent throughout and, therefore, valid for\nmetageometrical purposes. Much depends upon the definition; for, after\nthe definition has been once determined it remains then merely to make\ninferences and conclusions conform to the intent of the definition.\n\nDIVERSITY--Philosophically, the idea indicated has reference to all\ndissimilarities, differences, inequalities, divergent tendencies,\nmovements and characteristics to be noted in the universum of life;\nthe antithesis of kosmic unity; the natural outcome of life in seeking\nexpression; the result of the fragmentative tendency of life.\n\nDUODIM (duo, two; dim, abbreviation of dimension)--A hypothetical\nbeing supposed to be possessed with a consciousness adapted only to\ntwo dimensions; a dweller in \"Flatland\" or two-space whose scope of\nmotility is limited to two directions, as on the face of a plane; a\nterm invented by hyperspace advocates for the purpose of establishing\nby analogy some of the characteristics of the four space and also its\nrationality.\n\nDUOPYKNON (duo, two--pyknon, primary unit in the process of kosmic\ninvolution, a condensation)--Secondary phase in the elaboration of\nchaos into kosmic order. DUOPYKNOSIS (duo, two, secondary--pyknosis,\nprocess of condensation and origination)--The second period of\nthe involutionary movement of life during which the duadic plane\nof the kosmos is being established; the second, in the series of\nseven distinct phases, of space-genesis; dual differentiation of\nkosmic plasm. Duopyknosis contemplates that, in the passage of the\nkosmos-to-be from the plane of non-manifestation to the plane of\nmanifestation, there are seven distinct, though interdependent and\ninterrelational, stages through which life passes, and that, of these,\nit is the second. It relates to the plane of non-manifestation and is,\ntherefore, beyond the ken of the intellectuality, being a symbol.\n\nEGOPSYCHE (Ego, the self-conscious I--psyche, soul)--The mental,\nemotional and physical mechanisms of man, the Thinker. These include\nthe purely mental system, the emotional or affective mechanism, the\nnerve-systems (cerebro-spinal and sympathetic) and the brain; the\nobjective or sense-derived consciousness of the Thinker which is\nelaborated from the total mass of perceptions transmitted through\nthe senses; the medium of self-consciousness; the intellectual\nconsciousness as distinguished from the intuitional or omnipsychic\nconsciousness (q.v.).\n\nThe notion of the egopsychic consciousness is based upon data\nalready empirically determined from the mass of evidences everywhere\nobservable. It seems to be apparent that there is a consciousness,\na seat of knowledge, in man the content of which is unknown to the\nsense-consciousness. Dreams, premonitions, intuitions, impressions and\nthe totality of all such phenomena substantiate this view. Furthermore,\nit is agreed that the source of the intuition is not identical with\nthat of the intellect. The egopsychic consciousness, accordingly, is\npurely intellectual.\n\nFLUXION, PSYCHIC--The difference between a mental image and an\nobject; an image is the representation of certain salient or cardinal\ncharacteristics of an object, sufficient for identification; but an\nimage is not congruent, in every respect, with the object. Thus when we\nperceive an object, although as BERGSON contends we perceive it in the\nplace where it is and not in the brain, it is the image of the object\nwhich takes its place in memory and not the object itself. There is, of\ncourse, a marked disparity between this memory-image and the object.\nEven if the image possessed one of the properties of the object, as,\nsize, it could not take its place in memory, and neither could it do so\nif it possessed any of the properties of the real object. Consciousness\nis such that all due allowances are made for these conditions and the\nmind is able to retain more or less exact knowledge of these properties\nin the image; but there is a difference, small though it may be. This\ndifference is the psychic fluxional.\n\nFOHAT (Skt)--A term applied to the Creative Logos who is said to be\nthe generating element in the differentiation of chaos into kosmic\norderliness; the supreme deity in the r\u00f4le of Creator.\n\nFORM, PURE--An abstraction arrived at by subtracting the last vestige\nof materiality or substantiality from an idea and viewing the remains\nas a pure unsubstantial form or idealization; the shell or frame-work\nof a material object or condition; existing in idea or thought only;\na mental conception regarded as a type or norm; a purely hypothetical\nconstruction.\n\nFOUR SPACE--Often referred to as the fourth dimension (vide Chapter V);\na space in which four co\u00f6rdinates (four lines drawn perpendicular to\neach other) are necessary and sufficient to establish the position of a\npoint, as, a hypercube.\n\nGAUSS, CHARLES FREDERICK, born at Brunswick, April 30, 1777. His\nfather, being a bricklayer, had intended that he should follow the same\noccupation. So, in 1784, Charles was sent to the B\u00fctner Public School\nin Brunswick, in order that he might be taught the ordinary elements\nof education. But during his attendance at this school, his unusual\nintelligence and aptitude attracted the attention and friendship of\nProfessor BARTELS who later became the Professor of Mathematics at\nDorpat. In 1792, through the kindly representations of Professor\nBARTELS to the Duke of Brunswick, young Gauss was sent to the\nCollegium Carolinum. This greatly displeased his father as he saw in\nthis move the frustration of his plans for Charles. In 1794, however,\nGAUSS entered the University of G\u00f6ttingen still undecided whether he\nshould make mathematics or philology his life work. While residing at\nG\u00f6ttingen, he made his celebrated discoveries in analysis and these\nturned his attention definitely to the field of mathematics.\n\nHe completed his studies at G\u00f6ttingen and returned to Brunswick in\n1798, residing at Helmstadt where he had access to the Library in the\npreparation of his _Disquisitiones Arithmeticae_ which was published\nin 1801. He received his doctorate degree (Ph.D.) on July 16, 1799.\nHis next notable work was the invention of a method by which he\ncalculated the elements of the orbit of the planet _Ceres_ which had\nbeen discovered by PIAZZI, January 1, 1800, and who had left no record\nof his calculations by which other astronomers could locate the planet.\nGAUSS also calculated an ephemeris of _Ceres'_ motion by means of which\nDE ZACH rediscovered the planet December 31, 1800.\n\nHis _Theoria Motus Corporum Coelestium in Conicis Sectionibus Solem\nAmbientium_, in which the author gives a \"complete system of formul\u00e6\nand processes for computing the movements of a heavenly body revolving\nin a conic section\" is an outgrowth of his early researches and brought\nhim lasting fame.\n\nThrough the influence of his friend OLBERS, he was appointed, July 9,\n1807, first director of the new G\u00f6ttingen Observatory, and Professor\nof Astronomy in the University, a position which he held until the end\nof his life. He died February 23, 1855. (Vide _Astronomical Society\nNotices_, Vol. 16, p. 80, 1856; also _Nature_, Vol. XV, pp. 533-537,\n1877.)\n\nGEOMETRISM--Of geometrical quality; a notion derived from PLATO'S\ndeclaration: \"God geometrizes.\" It was his belief that the creative\nacts of the deity are executed in accordance with geometric design and\nlaws; that in the totality of such acts there necessarily inheres a\nlatent geometric quality. KANT closely adhered to this notion in his\ndiscussions of space as an aspect of divine intelligence. He believed\nthat the intellect merely rediscovers this latent geometrism when\nit turns to the study of materiality, and this belief is shared by\nBERGSON, the foremost metaphysician of the present time.\n\nHYPERSPACE (hyper, above, beyond, transcending--space)--That species of\nspace constructed by the intellect for convenience of measurement; an\nidealized construction; a purely arbitrary, conventional mathematical\ndetermination; the fourth dimension; any space that requires more than\nthree co\u00f6rdinates to fix a point position in it, as, a five space, an\n_n_-space.\n\nINTUITOGRAPH--The means by which the omnipsychic consciousness\ntransmits intuitional impressions to the egopsychic or intellectual\nconsciousness. An intuitogram is a direct cognition, an intuition;\na primary truth projected into the egopsychic consciousness by the\nThinker. It is recognized that, under the necessities of the present\nschematism of things, it is exceedingly difficult to propagate an\nintuition, especially with the same degree of ease as concepts are\npropagable; yet, this is believed to be a condition which will be\novercome as the evolution of the higher faculties proceeds.\n\nINVOLUTION--Process of enfolding, involving; antithesis of evolution;\nphilosophically, the doctrine of involution maintains that, during\nthe process of kosmic pyknosis (space-genesis), _all_ that is to be\nexpressed, developed and perfected as a result of the evolutionary\nmovement was first involved, enfolded or deposited as latent archetypal\ntendencies and radicles in the original world-plasm; that, as the\ninvolutionary movement proceeded through the various phases of\nspace-genesis, these became more and more phenomenal until at last\nthey terminated in the elaboration of a manifest universe: each\nstage, accordingly, of the involutionary procedure became the basic\nsubstructure of a plane of specialized substance or materiality\nand consciousness. Thus it appears that evolution really begins\nwhere involution ends (vide Fig. 18), and the two opposing processes\nconstitute the dualism of life as generating element. This notion has\nbeen symbolized in the _Lingam yoni_ of Hellenistic philosophies, also\nin _Yang_ and _Yin_ of Chinese philosophy, which represent the original\npair of opposites.\n\nKATHEKOS--A purely arbitrary term devised for the express purpose\nof providing a convenient symbol to convey the idea embodied in the\ntriglyph, _Chaos-Theos-Kosmos_, and is composed of the first three\nletters in each one of the terms of the triglyph; hence, symbolizes\nthe triunity and interaction involved in the resolution of chaos into\nan orderly kosmos by the will of the Creative Logos. Thus, \"kathekos\"\nembodies a quadruplicate notion, namely, chaos, Creative Logos,\nmanifested kosmos, and the creational activity of the Logos in the\ntransmutation of disorder into order. The justification for this term,\ntherefore, resides in its convenience, brevity and comprehensiveness.\n\nBy referring to figure 17, it will be seen that Kathekos divides into\ntwo kinds--involutionary, or that which pertains to involution, and\nevolutionary, or that that pertains to evolution. It thus comprises\nthe beginning and the end of the world age or cycle and pertains to\nnon-manifestation. The _raison d'\u00eatre_ of this differentiation is\nembodied in the notion that, on the involutionary arc of the cycle, the\nchaogenic period represents a phase of the world age when space-genesis\nis in an archetypal state wherein are involved all possibilities that\nare to become manifest in the kosmos, and on the evolutionary arc, the\nkathekotic period which is parallel to the chaogenic and represents a\nphase of the world age when the kosmos has reached ultimate perfection,\nembodying the perfected results of the possibilities which inhered in\nthe chaogenic period or in involutionary kathekos. Thus, kathekos is\ndual in nature, on the one hand representing kosmic potency, and on the\nother, kosmic perfection of these potencies. It is Alpha, as related\nto involution, and Omega, as related to evolution.\n\nKATHEKOSITY--A derivative, signifying creative activity and all that it\nimplies; the state of consciousness or cognition corresponding thereto.\n\nKLEIN, FELIX (1849--), born at Dusseldorf; studied at Bonn, and when\nonly seventeen years of age was made assistant to the noted PL\u00dcCKER\nin the Physical Institute. He took his doctorate degree in 1868, then\nwent to Berlin, and later to G\u00f6ttingen where he assisted in editing\nPL\u00dcCKER'S works. He entered the G\u00f6ttingen faculty in 1871; became\nProfessor of Mathematics at Erlangen in 1872; and subsequently held\nprofessorships at Munich, 1875; Leipzig, 1880, and G\u00f6ttingen, 1886.\nNo one else in Germany has exerted so great influence upon American\nmathematics as he.\n\nKOSMOS--_See Cosmos_.\n\nLA GRANGE, JOSEPH LOUIS, born at Turin, January 25, 1736; died at\nParis, April 10, 1813; regarded as the greatest mathematician since the\ntime of Newton. It may be interesting to note that LA GRANGE remarked\nthat mechanics is really a branch of pure mathematics analogous to a\ngeometry of four dimensions, namely, time, and the three co\u00f6rdinates\nof the point in space. (Vide _Ball's Account of the History of\nMathematics_.)\n\nLIE, SOPHUS, a noted mathematician, referred to as the \"great\ncomparative anatomist of geometric theories, creator of the doctrines\nof Contact Transformations and Infinite Continuous Groups, and\nrevolutionizer of the Theory of Differential Equations.\"\n\nLOGOS--The supreme deity of the phenomenal universe; Creator; Fohat; a\nplanetary god; the deity of a solar system.\n\nMANVANTARA (Skt.)--A world age; the periods of involution and evolution\ncombined; the stage during which the universe is in manifestation; a\nDay of Brahma.\n\nMATHESIS (Gr. mathein, to learn)--Erudition; profound learning; the\nrealm of metaphysical conceptions; the field of higher mathematics; the\nsphere of conceivability; the theoretical.\n\nMENTOGRAPH--A cognitive factor consisting of a complete perception\nfused or in coalescence with a memory-image. Pure memory, of itself,\nis without utility as an aid to cognition; but, when nourished or\nsupplemented by the substance of perception it becomes the basis of\nintellectual consciousness.\n\nMETAGEOMETRY (Gr. Meta, beyond, transcending--geometry)--Commonly, any\nkind of geometry that differs from the Euclidean, as the non-Euclidean;\na geometry based upon the assumption that the angular sum of a triangle\nis greater or less than two right angles; the highest form of geometry;\na system of idealized mathematical constructions. Sometimes called\n\"pangeometry\"; designated by GAUSS as \"Astral Geometry\"; the geometry\nof hyperspace. It consists of results arrived at by geometers in\nseeking a proof of the parallel-postulate.\n\nMETA-SELF--The higher self in man; the universal self; the one self\nof which all individual selves are but fragments or parts. In man, it\nis co\u00f6rdinate with the omnipsyche (q.v.) and as such is the medium of\nkosmic consciousness.\n\nMORPHOGENY (Gr. Morphe, form, vehicle, body--geny, evolution)--The\nevolution of forms, the production of individual bodies or vehicles\nfor life, including organs and faculties. Morphogenic--a derivative;\npertaining to morphogeny; a kosmic process (vide figs. 17 and 18).\n\n_N_-DIMENSIONALITY--Quality of conceptual space by virtue of which it\nmay be regarded as possessing an indefinite number of dimensions.\n\nNEAR-TRUTH--Any statement or view which is based upon partial\nknowledge; predicates concerning a class or genus derived from limited\nacquaintance with particulars of the class or genus; statements based\nupon logical determinations inhering in idealized constructions and\napplied to concrete or objective conditions; an abstraction viewed as a\nreality; the application of the qualities of abstractions to realities.\n\nNEUROGRAM--Psychologically, a movement received by the afferent\nnerves in the form of a stimulation and transmitted through the brain\nand efferent nerves as either a reflex or voluntary action; a nerve\nimpulse; a perception; a primary unit of intellectual consciousness;\ncf. _Intuitogram_.\n\nNEWCOMB, SIMON (1835-1909), born at Wallace, Nova Scotia; educated\nin his father's school and came to the United States in 1853. Began,\nin 1854, teaching in Maryland; was appointed computer on _Nautical\nAlmanac_ at Cambridge in 1857; was graduated at Lawrence Scientific\nSchool in 1858; appointed Professor of Mathematics in the U. S. Navy\nin 1861. He supervised the construction of the 26-inch equatorial\ntelescope at the Naval Observatory, and was secretary of the Transit\nof Venus Commission; was a member of nearly all of the Imperial and\nRoyal societies of Europe and of the various societies in the United\nStates, receiving the Copley Medal in 1874; the Huygens, 1878; the\nRoyal Society, 1890, and the Bruce Medal in 1898; held the presidency\nof the following learned societies, viz: American Association for\nthe Advancement of Science, 1877; Society for Psychical Research,\n1885-1886; American Mathematical Society, 1897-1898; the Astronomical\nand Astrophysical Society of America from its foundation in 1899. He\nrendered notable service in popularizing the doctrine of hyperspace.\n\nNORM--An authoritative standard; model or type; standard of reference.\nThe choice of a norm for spatial determinations cannot abide in\nany premise except that which naturally, and not artificially and\nconventionally, conforms to what is actually perceived; if so, there\nshould be justification for challenging the wisdom and utility of\nthe present schematism of things. There is an inherent conformity of\nspace with intellect and intellect with space, and because of this\nnatural complementarity of part with part and whole with whole, space\ncannot be otherwise than the intellectuality normally conceives it\nto be, provided, of course, that the cognitive movement is free and\nuntrammeled by arbitrary hindrances. Consciousness, therefore, is\nthe norm or standard of reference for all questions arising out of a\nconsideration of spatiality.\n\nOMNIPSYCHE--A term used to denote the Thinker's cognitive apparatus;\nthe universal soul manifesting in individuals; the consciousness\nof the Thinker in virtue of which he is at-one with the universal\nconsciousness; the medium of kosmic consciousness; the source of the\nintuition, cf. _Egopsyche_. The divinity in man (which is taken for\ngranted), or his highest self can in no way be said justly to take its\nrise from sense-experience or from any bodily process. If divine, then\neternal, and therefore, persistent. Broadly, the doctrine of evolution\nrecognizes the passage of life from form to form, adding a little to\neach successive form and inevitably pushing each to a higher degree\nof perfection. Now, what is it that passes from form to form? Is it\nundifferentiated life or is it a specialized form of life? From every\nevidence, it would be judged that the life that ensouls an individual\nform is a specialized principle, i.e., limited to the execution of a\ngiven purpose. If life as a specialized principle, limited to the\nexecution of a given purpose in each form, passes on, it must preserve,\nat least, the sublimated results obtained during its residence in each\nindividual form. It would thus become a sort of reservoir containing\nall these transmuted results. The omnipsyche, within the meaning of the\ntext, is precisely this specialized life principle.\n\nPARALLEL-POSTULATE--Variously referred to as the XIth, XIIth and XIIIth\naxiom of the _Elements of Euclid_; stated by MANNING, in _Non-Euclidean\nGeometry_, p. 91, in the following form: \"If two lines are cut by a\nthird, and the sum of the interior angles on the same side of the\ncutting line is less than two right angles, the lines will meet on that\nside when sufficiently produced.\" This celebrated postulate has proven\nto be the most fruitful ever devised; for it embodies in itself the\npossibility of three geometries based respectively upon the following\nassumptions, namely: I. That there exists a triangle, the sum of whose\nangles is congruent to a straight angle, the Euclidean; II. That there\nexists a triangle the sum of whose angles is less than a straight\nangle, the LOBACHEVSKIAN; III. That there exists a triangle the sum\nof whose angles is greater than a straight angle, the CAYLEY-KLEIN.\nSpeaking of the content of the last two named, EDWARD MOFFAT WEYER[1]\nsays: \"Hypothetical realms, wherein the dimensions of space are assumed\nto be greater in number than three, yield strange geometries, which\nare only card castles, products of a sort of intellectual play in\nthe construction of which the laws of logic supply the rules of the\ngame. The character of each is determined by whatsoever assumption its\nbuilder lays down at the start.\"\n\n  [1] Vide _Popular Science Monthly_, vol. 78, p. 554, 1911.\n\nPASSAGE OF SPACE--A phrase connoting the movement of space from chaos\nto perfect order, a process believed to be infinite. The genesis of\nspace necessarily implies an elaboration, a procedure, by which the\nmetamorphosis from disorder to kosmic order is made, and this movement\nis referred to as the \"passage of space,\" a phenomenon thought to be\nmeasurable by means of a suitable instrumentality.\n\nPERISOPHISM--See _Near-Truth_.\n\nPSEUDOSPHERE--A surface of constant negative curvature; basis of\nBELTRAMI'S metageometrical calculations; surface resembling a champagne\nglass or common spool. The assumption that space is pseudospherical\nhas given rise to the notion of space-curvature and various other\nconceptions.\n\nPSYCHOGENY (Gr. Psyche, Soul--geny)--History of the evolution of the\nsoul or the development of the senso-mechanism in organisms. ERNST\nHAECKEL has traced the psychogeny of man through twenty-two different\nstages from the moneron to the anthropoid apes, and man.\n\nPRALAYA (Skt.)--Kosmic quietude; the period during which the universe\nis not in manifestation; gestatory period; kosmic inactivity; opposed\nto manvantara (q.v.); figuratively, the kosmic womb; world egg.\n\nPYKNON (Gr. pyknon, hard)--The principle of kosmic condensation; the\nprimary basis of space-genesis; the initiation of the process by\nvirtue of which chaos is elaborated into kosmic order. PYKNOSIS--The\nprocess of spatial engenderment. There are seven of these processes,\neach indicating a phase of duration, namely: MONOPYKNOSIS, the primary\nphase; DUOPYKNOSIS, secondary; TRIPYKNOSIS, tertiary. These three\npertain to the plane of non-manifestation, the pralayic or gestatory\nduration-phase. The results arrived at during these duration-phases\nare concentrated in the Quartopyknotic which corresponds to the causal\nplane of manifestation or pure kosmic spirituality. QUINTOPYKNOSIS, a\nprocess concerned in the genesis of mentality; SEXTOPYKNOSIS, kosmic\nsensibility; SEPTOPYKNOSIS, kosmic materiality. These seven phases\nof duration constitute the scope of space genesis or kosmogenesis,\nand incidentally depose the substructure of kosmic materiality,\nsensibility, intellectuality and spirituality, as well as the higher\ntrinity of kosmic modes. The ramifications of these principles are\ninnumerable and omnipresent. (See Chapter VII.)\n\nQUARTODIM--A hypothetical being assumed to have a consciousness adapted\nto hyperspace or the fourth dimension, and whose scope of action is\nencompassed within a space which requires four co\u00f6rdinates, as the\nfour-space.\n\nREALITY (Realism)--Life; the harmony existing among the parts to\nmaintain their equilibrium in the whole; the principle of integrity\nsubsisting among parts; kosmic vitality.\n\nRIEMANN, GEORGE FREDERICH BERNHARD, was born September 17, 1826, in the\nvillage of Breselenz, near Dannenburg, in Hanover. Until he was eight\nyears of age his father was his sole tutor, but even at this age he\nexhibited great powers of arithmetical calculation. In the Spring of\n1840 young RIEMANN was sent to the Hanover Lyceum where he remained for\ntwo years, leaving in 1842 for the Gymnasium at Luneburg. Here, under\nthe direction of Professor SCHMALFUSS, he learned very rapidly, and is\nsaid to have required only one week thoroughly to familiarize himself\nwith LEGENDRE'S _Theory of Numbers_.\n\nOn April 12, 1846 (Easter), he entered the University of G\u00f6ttingen as a\nstudent of Theology in accordance with his father's wishes. His passion\nfor mathematics, however, was so aroused by the lectures of GAUSS that\nHe begged his father to be allowed to devote himself entirely to the\nstudies of his choice. For two years he studied under JACOBI at Berlin.\nHe then returned to G\u00f6ttingen, and was graduated, his thesis being a\ndissertation on the foundations of a general theory of functions of\na variable complex magnitude. In 1854 he qualified as a teacher by\ngiving a lecture on the \"Hypothesis on which Geometry is Founded.\" In\n1857 he became \"Professor Extraordinarius,\" and in 1859 was elected\nCorresponding Member of the Academy of Sciences of Berlin and in 1860 a\nmember of the Academy of Sciences of G\u00f6ttingen.\n\nAfter four years of failing health, during which he visited Messina,\nPalermo, Naples, Rome, Florence, Pisa and Milan, he died at Lago\nMaggiore, July 20, 1866, in full possession of his faculties and\nconscious of his approaching end.\n\nSCHWEIKART, FERDINAND KARL (1780-1857), studied from 1796 to 1798 in\nMarburg, attending the mathematical lectures of J. K. F. HAUFF. In\n1812 he became professor in Charkov, a position which he held for\nfour years. In 1816 he became a tutor in the City of Marburg where\nhe remained until 1820 when he transferred his labors to K\u00f6nigsberg.\nIt was during his tutorship at Charkov, Marburg and K\u00f6nigsberg that\nhe, entirely alone and without the slightest suggestion from any man,\ndeveloped and taught a non-Euclidean geometry to the students under\nhis care. For copy of his treatise on non-Euclidean geometry, see\n_Historical Sketch of the Hyperspace Movement_, Chapter II.\n\nSCOPOGRAPHIC IMPRESSIONS--Sight perceptions fused with an associated\nmemory-image, and forming the basis of action on external phenomena.\n\nSENSOGRAPHIC IMPRESSIONS--Perceptions or impulses transmitted through\nthe nerves of a sense-organ; any impression acting through the media of\nthe senses.\n\nSENSIBLE WORLD--The world of the senses; that which responds to the\nsenses; the domain of perception; the phenomenal world; world of\nperceptual space.\n\nSPACE-CURVATURE (see _Curvature of Space_).\n\nSPACE-GENESIS--The process of spatial engenderment; the movement of\nlife as engendering agent in bringing into manifestation the kosmos;\nthe story of the appearance of the organized kosmos. The genesis of\nspace can only be symbolized, as has been done in the text, for the\nlimitations of human consciousness do not otherwise admit of the\nempirical establishment of the notion of its detailed procedure.\n\nSPATIALITY--Space as a dynamic, creative movement; kosmic order, as\nopposed to disorder; the path of the engendering movement of life;\nthe place of life. Spatiality, materiality, intellectuality and\ngeometricity or the latent geometrism of the kosmos are thought of as\nbeing consubstantial and interdependent; but, of these, spatiality is\nregarded as the substance out of which the latter three are elaborated.\n\nSUPERCONCEPTUAL--The purely intuitional; an act of cognition performed\nwithout the detailed work of conception derived from sense-data;\nconception of intuitions and their inter-relations; the Thinker's\nconsciousness freed from intellectual characterization.\n\nSUPERPERCEPTION--Perception of conceptual relations; a state of\ncognition wherein, instead of receiving percepts or images from the\nexternal world, then elaborating them into concepts, the Thinker\napprehends composite images or concepts at first hand. It is a power\nwhich the liberated mind of the future will possess owing to the\ngrowing automatism of the intellect and the more facile expression of\nthe intuitional consciousness.\n\nTESSERACT (Gr. Tessera, four, cube, tessella)--A hypercube (see Chapter\nV.)\n\nTHINKER (Skt. Manu, thinker)--The real, spiritual man, as\ndifferentiated from his perceptive vehicles--mind, emotions and\nphysical body; the omnipsychic intelligence who receives, classifies,\ninterprets and preserves percepts; the manipulator of concepts; in\nfine, the higher, spiritual man.\n\nThe Thinker uses the various perceptive instrumentalities as so\nmany tentacles or antennae by which he contacts the sensible world\nand makes the necessary adaptations to environment. He is the pure\nintelligence which is the source of all cognitive motivation; opposed\nto ego, because the egopsychic instrumentality is essentially an\nindividualizing, separative agency; while the Thinker's omnipsychic\nintelligence is the basis of his unity with the universal intelligence.\nThis conception of the Thinker implies that, as a spiritual\nintelligence, he is within and without the body, filling it as the\nocean fills the sponge, encompassing, enveloping it and, at the same\ntime, originating the totality of activities which manifest in and\nthrough the body. He is limited, therefore, in his manifestations in\nthe sensible world only by the pliability of his vehicles.\n\nTRANSFINITY--A state or condition that is incomprehensible to finite\nintelligence; that which transcends the finite, yet is not infinite;\nless than infinity and greater than finity. Space is referred to as\nbeing transfinite rather than infinite in extent. But space transfinite\nshould be distinguished from space \"finite though unbounded.\" For,\nthere would seem to be little worthy of choice between a \"finite,\nunbounded space\" and an infinite one. The absence of boundary would\nnaturally suggest an infinite extent. And although RIEMANN who is the\nauthor of the \"unbounded\" space arbitrarily determined that such a\nspace should be a manifold possessing a measure of curvature which\ncould be determined either by counting or actual measurement, he\nundoubtedly knew, nevertheless, that while each manifold might be an\n\"unbounded\" space the totality of such manifolds, infinite in number,\nmust also be infinite in extent. It would seem to do violence to common\nsense, if not to logical necessity, to view space both as \"unbounded\"\nand finite in extent, yet there would be no such difficulty in the\nrecognition of space as being both transfinite and finite; because it\nis conceivable that the extent and character of space finite should\ntranscend a finite intellectuality, and yet not be infinite.\n\nTRIDIM--A being whose scope of consciousness is limited to a space of\nthree dimensions, as ordinary human beings. TRIDIMENSIONALITY--That\nquality possessed by perceptual space by virtue of which it is\nnecessary and sufficient to have three co\u00f6rdinates, and only three, to\nestablish the position of a point.\n\nUNODIM--A hypothetical being assumed to have a consciousness limited to\nlinear or one-space.\n\nZONES OF AFFINITY--Regions in the domain of intellectuality wherein\nminds, possessing a common differential, rate of vibration or quality,\nadhere to certain tenets from choice. Schools of philosophy, religions,\nand all those major divisions of intellectual effort which divide and\nsubdivide intellectual allegiance are believed to take their rise in\nthis property of intellectuality in virtue of which all minds having a\nsimilar coefficient gravitate towards a common agreement, especially\nwhere the movement is voluntary and untrammeled.\n\n\n  _PART ONE_\n\n  THE ESSENTIALS OF THE GEOMETRY OF HYPERSPACE AND THEIR SIGNIFICATIONS\n\n\n\n\n  CHAPTER I\n\n  THE PROLOGUE\n\n\n  On the Variability of Psychic Powers--The Discovery of the Fourth\n  Dimension Marks a Distinct Stage in Psychogenesis--The Non-Methodical\n  Character of Discoveries--The Three Periods of Psychogenetic\n  Development--The Scope and Permissibility of Mathetic License--Kosmic\n  Unitariness Underlying Diversity.\n\nIn presenting this volume to the public profound apologies are made\nto the professional mathematician for the temerity which is shown\nthereby. All technical discussion of the problems pertinent to the\ngeometry of hyperspace, however, has been carefully avoided. The reader\nis, therefore, referred to the bibliography published at the end of\nthis volume for matter relating to this aspect of the subject. The aim\nrather has been to outline briefly the progress of mathematical thought\nwhich has led up to the idea of the multiple dimensionality of space;\nto state the cardinal principles of the Non-Euclidean geometry and to\noffer an interpretation of the metageometrical concept in the light of\nthe evolutionary nature of human faculties and material characteristics\nand properties.\n\nThe onus of this treatise is, therefore, to distinguish between what is\ncommonly known as sensible space and that other species of space known\nas geometric spaces. Also to show that the notion which has been styled\n_hyperspace_ is nothing more nor less than an evidence of the faint,\nearly outcroppings in the human mind of a faculty which, in the course\nof time, will become the normal possession of the entire human race.\nThus the weight of all presentations will be to give currency to the\nbelief, very strongly held, that humanity, now in its infancy, is yet\nto evolve faculties and capabilities, both mental and spiritual, to a\ndegree hitherto viewed as inconceivable.\n\nOn this view it must appear that the faculty of thought including the\npowers of imagination and conceptualization are not psychological\ninvariants, but, on the other hand, are true variants. They are,\nconsequently, answerable to the principle of evolution just as all\nvital phenomena are. Some have thought that no matter what idea may\ncome into the mind of the human race or at what time the idea may\nbe born the mind always has been able to conceive it. That is, many\nbelieve that the nature of mind is such that no matter how complex an\nidea may be there has always been in the mind the power of conceiving\nit. But this view cannot be said to have the support of any trustworthy\ntestimony. If so, then the mind must at once be recognized as fully\nmatured and capable during every epoch of human evolution, no less\nin the first than in the latest, which, of course, is absurd. It is\nundoubtedly more reasonable and correct to believe that the powers of\nconceptualization are matters of evolutionary concern. For instance,\nthe assertion that the mind was incapable of conceiving, in the realm\nof theology, a non-anthropomorphic god, or, in the field of biology,\nthe doctrine of evolution, or, in the domain of invention, the wireless\ntelegraph, or, in mathematics, the concept of hyperspace before the\nactual time of these conceptions, cannot be successfully controverted.\n\nIn fact, it may be laid down as one of the first principles of\npsychogenesis that the mind rarely, if ever, conceives an idea until it\nhas previously developed the power of conceptualizing it and giving it\nexpression in the terms of prior experience. As in the growth of the\nbody there are certain processes which require the full development\nof the organ of expression before they can be safely executed so\nin the phyletic development of faculties there are certain ideas,\nconceptions and scopes of mental vision which cannot be visualized\nor conceptualized until the basis for such mentation has been laid\nby the appearance of previously developed faculties of expression.\nAnd especially is this true of the intellect. Inasmuch as the entire\ncontent of the intellect is constituted of sense-derived knowledge,\nwith the exception of intuitions which are not of intellectual origin\nthough dependent upon the intellect for interpretation, there can\nbe no doubt as to the necessity of there being first deposed in the\nintellect a sense-derived basis for intellection before it can become\nmanifest. The Sensationalists, led by LEIBNITZ, propounded as their\nfundamental premise this dictum: \"_There is nothing in the intellect\nwhich has not first been in the senses except the intellect itself_,\"\nand this has never been gainsaid by any school that could disprove\nit. The intuitionalist does not deny it: he merely claims that we are\nthe recipients of another form of knowledge, the intuitional, which,\ninstead of being derived from sense-experience, is projected into the\nintellectual consciousness from another source which we designate the\nThinker. Thus, from the two forms of consciousness, come into the area\nof awareness truths that spring from entirely different sources. From\nthe one source a steady stream of impressions flow constituting the\nsubstance of intellectual consciousness; from the other only a drop,\nevery now and then, falls into the great inrushing mass so as to add\na dim phosphorescence to an otherwise unilluminated pool. Obviously,\nwhen there is a lack of sensuous data from which a certain concept may\nbe elaborated there can be no conception based upon them, and as the\nvariety and quality of concepts are in exact proportion to the variety\nand quality of sense-experience there can be no demand for a particular\nspecies of notions such as might be elaborated out of the absent\nor non-existent perception. Hence, the power of conceiving springs\nforth from sense-experience. Sense-experience is essentially a mass\nof perceptions: these, creating a demand for additional adaptations,\nconspire, as if, to evoke the power or faculty to meet the demand, and\nconsequently, an added conceptualization is made.\n\nProgress in human thought is made in a manner similar to that which\nprevails in the development of other natural processes, such as, the\npower of speech in the child. In the development of this faculty there\nare certain definite stages which appear in due sequence. The child is\nnot gifted with the power of speech at once. It comes, by gradual and\nsometimes painful growth, into a full use of this faculty. Now, much\nthe same principle holds true in the evolution of the mind in the human\nspecies. It is an established biologic principle that the ontogenetic\nprocesses manifested in the individual are but a recapitulation of the\nphylogenetic processes which are observable in the progress of the\nentire species. The view becomes even more cogent when note is taken\nof the fact that the foetus, during embryogenesis, passes successively\nthrough stages of growth which have been shown to be analogous, if\nnot identical, with those stages through which the human species has\ndeveloped, namely, the mineral, vegetal and animal.\n\nWherefore it may be said that the fourth dimensional concept marks a\ndistinct stage in psychogenesis or evolution of mind. It required,\nas will be shown in Chapter II, nearly two thousand years for it to\ngerminate, take root and come to full fruition. For it was not until\nthe early years of the nineteenth century that mathematicians, taking\ninspiration from RIEMANN (1826-1866) fully recognized the concept as\na metaphysical possibility, or even the idea was conceived at all.\nSerious doubt is entertained as to the possibility of its conception by\nany human mind before this date, that is, the time when it was actually\nborn. Prior to that time, mathematical thought was taking upon itself\nthat shape and tendence which would eventually lead to the discovery\nof hyperspace; but it could not have reached the zenith of its upward\nstrivings at one bound. That would have been unnatural.\n\nSuch is the constitution of the mind that although it is the quantity\nwhich bridges the chasm between the two stages of man's evolution when\nhe merely thinks and when he really knows it is entirely under the\ndomain of law and must observe the times and seasons, as it were, in\nthe performance of its functions. The scope of psychogenesis is very\nbroad, perhaps unlimited; but its various stages are very clearly\ndefined notwithstanding the breadth of its scope of motility. And while\nthe distance from _moneron_ to man, or from feeling to thinking is\nvast, the gulf which separates man, the Thinker, from man, the knower,\nis vaster still. Who, therefore, can say what are the delights yet in\nstore for the mind as it approaches, by slow paces, the goal whereat\nit will not need to struggle through the devious paths of perceiving,\nconceiving, analyzing, comparing, generalizing, inferring and judging;\nbut will be able to know definitely, absolutely and instantaneously?\nThat some such consummation as this shall crown the labors of mental\nevolution seems only natural and logical.\n\nIt may be thought by some that the character and content of\nrevelational impressions constitute a variation from the requirements\nof the law above referred to, but a little thought will expose the\nfallacy of this view. The nature of a revealed message is such as\nto make it thoroughly amenable to the restrictions imposed by the\nevolutionary aspects of mind in general. That this is true becomes\napparent upon an examination of the four cardinal characteristics of\nsuch impressions. First, we have to consider the indefinite character\nof an apocalyptic ideograph which is due to its symbolic nature. This\nis a feature which relieves the impression of any pragmatic value\nwhatsoever, especially for the period embracing its promulgation. Then,\nsuch cryptic messages may or may not be understood by the recipient\nin which latter case it is nonpropagable. Second, the necessity of\nprevious experience in the mind of the recipient in order that he may\nbe able to interpret to his own mind the psychic impingement. The basis\nwhich such experience affords must necessarily be present in order\nthat there may be an adequate medium of mental qualities and powers in\nwhich the ideogram may be preserved. A third characteristic is that\nrevelations quite invariably presuppose a contemplative attitude of\nmind which, in the very nature of the case, superinduces a state of\npreparedness in the mind for the proper entertainment of the concept\ninvolved. This fact proves quite conclusively that revelational\nimpressions are not exceptions to the general rule. Lastly, a\ndissatisfaction with the conditions with which the symbolism deals or\nto which it pertains is also a prerequisite. This condition is really\nthat which calls forth the cryptic annunciation, and yet, preceding\nit is a long series of causes which have produced both the conditions\nand the revolt which the revelator feels at their presence. In view of\nthe foregoing, it would appear that objections based upon the alleged\nnonconformity of the revealed or inspired cannot be entertained as it\nmust be manifest that it, too, falls within the scope of the laws of\nmental growth.\n\nDiscoveries, whether of philosophical or mechanical nature, or whether\nof ethical or purely mathematical tendence, are never the results of\na deliberate, methodical or purposive reflection. For instance, let\nus take LIE'S \"transformation groups,\" mathematic contrivances used\nin the solution of certain theorems. Now, it ought to be obvious that\nthese mathetic machinations were not discovered by SOPHUS LIE as a\nconsequence of any methodic or purposeful intention on his part.\nThat is, he did not set out deliberately to discover \"transformation\ngroups.\" For back of the \"groups\" lay the entire range of analytic\ninvestigations; the mathematical thought of more than a thousand years\nfurnished the substructure upon which Lie built the conception of his\n\"groups.\" Similarly, it may be said with equal assurance that no matter\nhow great the intensity of thought, nor how purposeful, nor of how long\nduration the series of concentrated abstractions which led up to the\ninvention of the printing press, the linotype or multiplex printing\npress of our day could not have been produced abruptly, nor by use of\nthe mental dynamics of the human mind of remoter days. Its production\nhad to follow the path outlaid by the laws of psychogenesis and await\nthe development of those powers which alone could give it birth.\nThe whole question resolves itself, therefore, into the idea of the\ncomplete subserviency of the mind, in all matters of special moment, to\nthe laws aforementioned. The supersession of the law of its own life by\nthe mind is well-nigh unthinkable, if not quite so.\n\nIf we now view the history of the mind as manifested in the human\nspecies, three great epochs which divide the scope of mental evolution\ninto more or less well-defined stages present themselves. These are:\nfirst, the _formative stage_; second, _the determinative stage_; third,\nthe stage of _freedom_, or the _elaborative_ stage.\n\nIn all of the early races of men, through every step which even\npreceded the _genus homo_, the generic mind was being formulated. It\nwas being given shape, outline and direction. All of the first stage,\nthe _formative_, was devoted to organization and direction. Those\nelementary sensations which constituted the basis of mind in the\nprimitive man were accordingly strongly determinative of what the mind\nshould be in these latter days. To this general result were contributed\nthe effects of the activity of cells, nerves, bones, fibers, muscles\nand the blood.\n\nThe _formative_ period naturally covered a very extensive area in the\nhistory of mind or psychogenetic development. It was followed closely,\nbut almost insensibly, by the _determinative_ period during which all\nthe latent powers, capacities and faculties which were the direct\nproducts of the _formative_ period were being utilized in meeting the\ndemands of the law of necessity. The making of provisions against\ndomestic want, against the attacks of external foes; the combating\nof diseases, physical inefficiency, the weather, wild beasts, the\nasperities of tribal enmities; as well as furthering the production of\nart, music, sculpture, the various branches of handiwork, literature,\nphilosophies, religions and the effectuation of all those things which\nnow appear as the result of the mental activity of the present-day man\nmake up the essence and purpose of the determinative period.\n\nSigns of the dawn of the _elaborative_ stage, also called the stage of\n_freedom_, have been manifest now for upwards of three centuries and\nit is, therefore, in its beginnings. It is not fully upon us. Not yet\ncan we fully realize what it may mean, nor can we unerringly forecast\nits ultimate outcome; but we feel that it is even now here in all the\nglories of its matutinal freshness. And the mind is beginning to be\nfree from the grinding necessities of the constructive period having\nalready freed itself from the restrictive handicaps of the primeval\nformulation period. Already the upgrowing rejuvenescences so common\nat the beginning of a new period are commencing to show themselves\nin every department of human activity in the almost universal desire\nfor greater freedom. And this is particularly noticeable in the many\npolitical upheavals which, from time to time, are coming to the\nsurface as well as in the countless other aspects of the wide-spread\nrenaissance. Perhaps the time may come, never quite fully, when there\nwill be no longer any necessity to provide against the external\nexigencies of life; perhaps, the time will never be when the mind\nshall no more be bound by the law of self-preservation, not even when\nit has attained unto the immortality of absolute knowledge; yet, it\nis intuitively felt that it must come to pass that the mind shall be\nvastly freer than it is to-day. And with this new freedom must come\nliberation from the necessities of the elementary problems of mere\nphysical existence.\n\nThe inference is, therefore, drawn that the fourth dimensional concept,\nand all that it connotes of hyperspace or spaces of _n_-dimensionality\nare some of the evidences that this stage of freedom is dawning. And\nthe mind, joyous at the prospect of unbounded liberty which these\nconcepts offer, cannot restrain itself but has already begun to revel\nin the sunlit glories of a newer day. What the end shall be; what\neffect this new liberty will have on man's spiritual and economic life;\nand what it may mean in the upward strivings of the Thinker for that\nsublime perpetuity which is always the property of immediate knowledge\nno one can hope, at the present time, to fathom. It is, however,\nbelieved with KEYSER that \"it is by the creation of hyperspaces that\nthe rational spirit secures release from limitation\"; for, as he says,\n\"in them it lives ever joyously, sustained by an unfailing sense of\ninfinite freedom.\"\n\nThe elevating influence of abstract thinking, such as excogitation upon\nproblems dealing with entities inhabiting the domain of _mathesis_ is,\nwithout doubt, incalculable in view of the fact that it is only through\nthis kind of thought that the spirit is enabled to reach its highest\npossibilities. This is undoubtedly the philosophy of those religious\nand occult exercises known as \"meditations,\" and this perhaps was\nthe main idea in the mind of the Hebrew poet when he exclaimed: \"Let\nthe words of my mouth and the meditation of my heart be acceptable\nin thy sight, O Lord, my strength and my Redeemer.\" The principal,\nif not the only, value possessed by the \"summitless hierarchies of\nhyperspaces\" which the mathematician constructs in the world of pure\nthought is the enrichening and ennobling influence which they exert\nupon the mind. But admittedly this unbounded domain of mathetic\nterritory which he explores and which he finds \"peopled with ideas,\nensembles, propositions, relations and implications in endless variety\nand multiplicity\" is quite real to him and subsists under a reign of\nlaw the penalties of which, while not as austere and unreasonable\nas some which we find in our tridimensional world, are nevertheless\nquite as palpable and as much to be feared. For the orthodoxy of\nmathematics is as cold and intolerant as ever the religious fanatic\ncan be. But the reality and even the actuality which may be imputed to\nthe domain of mathesis is of an entirely different quality from that\nwhich we experience in our world of triune dimensionality and it is a\nregrettable error of judgment to identify them. It ought, therefore,\nnever be expected, nor is it logically reasonable to assume that the\nentities which inhabit the mathetic realm of the analyst should be\nsubmissive to the laws of sensible space; nor that the conditions which\nmay be found therein can ever be made conformable to the conditions\nwhich exist in perceptual space.\n\nIt was PLATO'S belief that ideas alone possessed reality and what\nwe regard as actual and real is on account of its ephemerality and\nevanescence not real but illusionary. This view has been shared by a\nnumber of eminent thinkers who followed, with some ostentation, the\nlead established by PLATO. For a considerable period of time this\nschool of thinkers had many adherents; but the principles at length\nfell into disrepute owing to the absurdities indulged in by some of the\nless careful followers. The realism, or for that matter, the actuality\nof ideas cannot be denied; yet it is a realism which is neither to\nbe compared with the physical reality of sense-impressions nor its\nphenomena. The character and peculiarity of ideas are in a class apart\nfrom similar notions of perceptual space content. It is as if we were\nconsidering the potentialities of the spirit world and the entities\ntherein in connection with incarnate entities which in the very nature\nof the case is not allowable. Furthermore, it is unreasonable to\nsuppose that the conditions on a higher plane than the physical can be\nmade responsible to a similar set of conditions on the physical plane.\n\nThere are certain astronomers who base their speculations as to the\nhabitability of other planets upon the absurd hypothesis that the\nconditions of life upon all planets must be the same as those on the\nearth, forgetting that the extent of the universe and the scope of\nmotility of life itself are of such a nature as to admit of endless\nvariations and adaptations. There is a realism of ideas and a realism\nof perceptual space. Yet this is no reason why the two should be\nidentified. On the other hand, owing to the diversity in the universe,\nevery consideration would naturally lead to the assumption that they\nare dissimilar. To invest ideas, notions, implications and inferences\nwith a reality need not logically or otherwise affect the reality of a\nstone, a fig, or even of a sense-impression.\n\nTo a being on the spirit levels our grossest realities must appear\nas non-existent. They are neither palpable nor contactable in any\nmanner within the ordinary range of physical possibilities. For us his\ngravest experiences can have no reality whatsoever; for no matter how\nreal an experience may be to him it is altogether beyond our powers of\nperception, and therefore, to us non-existent also. It should, however,\nbe stated that the state of our knowledge about a given condition can\nin no way affect its existence. It merely establishes the fact that\ntwo or more realities may exist independent of one another and further\nthat the gamut of realism in the universe is infinite and approaches a\nfinal state when its occlusion into absolute being follows as a logical\nsequence.\n\nRecurring to the consideration of the reality of spirit-realms\nas compared with that of sensible space, it comes to view that\nour idealism, that is, the idealism which is a quality of\nconceptualization, may be regarded as identical with their realism, at\nleast as being on the same plane as it. Stated differently, the things\nthat are ideal to us and which constitute the data of our consciousness\nmay be as real to them as the commonest object of sense-knowledge is to\nus. What, therefore, appears to us as the most ethereal and idealistic\nmay have quite a realistic character for them.\n\nUltimately, however, and in the final deeps of analysis it will\nbe found undoubtedly that both our realism and our idealism as\nwell as similar qualities of the spirit world are in all essential\nconsiderations quite illusionary. All knowledge gained in a condition\nshort of divinity itself is sadly relative. Even mathematical knowledge\nfalls far short of the absolute, the fondest claims of the orthodox\nmathematician to the contrary notwithstanding. It has been said\nfrequently that a mathematical fact is an absolute fact and that its\nverity, necessity and certainty cannot be questioned anywhere in the\nuniverse whether on Jupiter, Neptune, Fomalhaut, Canopus or Spica.\nBut having so declared, the fact of the sheer relativity of our\nknowledge is not disturbed thereby nor controverted. Happily, neither\ndistance nor a lack of distance can in any way affect the quality of\nhuman knowledge, mathematical knowledge not excepted. That can only\nbe affected by conditions which cause it to approach perfection and\nnothing but evolution can do that.\n\nIn the light of results obtained in analytic investigations the\nquestion of the flexibility of mathematical applications becomes\nevident and one instead of being convinced of the vaunted invariability\nof the laws obtaining in the world of mathesis is, on the other\nhand, made aware of the remarkable and seemingly unrestrained\nfacility with which these laws may be made to apply to any conditions\nor set of assumptions within the range of the mind's powers of\nconception. Mathematicians have deified the _definition_ and endowed\nit with omnific powers imputing unto it all the attributes of\ndivinity--immutability, invariance, and sempiternity. In this they have\nerred grievously although, perhaps, necessarily. Mathetic conclusions\nare entirely conditional and depend for their certainty upon the\nimputed certitude of other propositions which in turn are dependent,\nin ever increasing and endlessly complex relations, upon previously\nassumed postulates. These facts make it exceedingly difficult to\nunderstand the attitude of mind which has obscured the utter mutability\nand consequent ultimate unreliability of the fine-spun theories of\nanalytic machinations.\n\nThe apriority of all mathematical knowledge is open to serious\nquestioning. And although there is no hesitancy in admitting the basic\nagreement of the most primary facts of mathematical knowledge with the\nessential character of the intellect the existence of well-defined\nlimits for such congruence cannot be gainsaid. The subjunctive\nquality of geometric and analytical propositions is made apparent\nby an examination of the possibilities falling within the scope of\npermissibility offered by mathetic license. For instance, privileged to\nproceed according to the analytic method it is allowable to reconstruct\nthe sequence of values in our ordinary system of enumeration so as to\nadmit of the specification of a new value for say, the entire series\nof odd numbers. This value might be assumed to be a plus-or-minus one,\ndependent upon its posture in the series. That is, all odd numbers in\nthe series beginning with the digit 3, and continuing, 5, 7, 9, 11, 13,\n15, 17, 19, ... _n_, could be assumed to have only a place value which\nmight be regarded as a constant-variable. The series of even numbers,\n2, 4, 6, 8, 10, 12, 14, 16, ... _n_, may be assumed to retain their\npresent sequence values. Under this system the digit 1 would have an\nabsolute value; all other odd numbers would have a constant-variable\nvalue; constant, because always no more nor less than 1 dependent\nupon their place in the operations and whether their values were to\nbe applied by addition or subtraction to or from one of the values\nin the even number series; variable, because their values would be\ndeterminable by their application and algebraic use.\n\nThere would, of course, be utilitarian objection to a system of this\nkind; but under the conditions of a suppositionary hypothesis, it\nwould be self-consistent throughout, and if given universal assent\nwould suit our purposes equally as well as our present system. But\nthe fact that this can be done under the mathematic method verily\nproves the violability of mathematical laws and completely negatives\nthe assumption that the sum of any two digits, as say 2 plus 2 equals\n4, is necessarily and unavoidably immutable. For it can be seen that\nthe sum-value of all numbers may be made dependent upon the assumed\nvalue which may be assigned to them or to any collection thereof.\nFurthermore, it is a matter of historical knowledge that it was the\ncustom of ancient races of men to account for values by an entirely\ndifferent method from what we use to-day. The latter is a result of\nevolution and while experience teaches that it is by far the most\nconvenient, it is nevertheless true that earlier men managed at least\nfairly well on a different basis. Then, too, the fact of the utility\nand universal applicability of our present system, based upon universal\nassent, does not obviate the conclusion that any other system,\nconsistent in itself, might be made to serve our purposes as well.\n\nIt ought to be said, however, in justice to the rather utilitarian\nresults obtained by LA GRANGE, HELMHOLTZ, FECHNER, and others who\nstrove to make use of their discoveries in analysis in solving\nmechanical, physiological and other problems of more or less pragmatic\nimport that, in so far as this is true, mathematical knowledge\nmust be recognized as being consistent with the necessities of _a\npriori_ requirements. But even these results may not be regarded\nas transcending the scope of the most fundamental principles of\nsense-experience. It will be discovered finally, perhaps, that the\nenergy spent in elaborating complicate series of analytic curiosities\nhas been misappropriated. It will then be necessary to turn the\nattention definitely to the study of that which lies not at the\nterminus of the intellect's _modus vivendi_, but which is both the\norigin of the intellect and its eternal sustainer--the intuition,\nor life itself. This can result in nothing less than the complete\nspiritualization of man's mental outlook and the consequent inevitable\nrecognition of the underlying and ever-sustaining _one-ness_ of all\nvital manifestations.\n\nOne of the curiosities of the tendency in man's mind to specialize in\nanalytics, whether in the field of pure mathematics or metaphysics,\nis the fact that it almost invariably leads to an attempt to account\nfor cosmic origins on the basis of paralogic theories. This in times\npast has given rise to the theory of the purely mechanical origin\nof the universe as well as many other fantastic fallacies the chief\nerror of which lay in the failure to distinguish between the realism\nof mental concepts and that of the sensible world. In spite of this,\nhowever, one is bound to appreciate the beneficial effects of analytic\noperations because they serve as invigorants to mental growth. It could\nnot, therefore, be wished that there were no such thing as analytics;\nfor the equilibria-restoring property of the mind may at all times be\nrelied upon to minimize the danger of excesses in either direction.\nJust as the tide flowing in flows out again, thereby restoring\nthe ocean's equilibrium, so the mind ascending in one generation\nbeyond the safety mark has its equilibrium restored in the next by a\nrelinquishment of the follies of the former.\n\nThe four-space is one of the curiosities of analytics; yet it need\nnot be a menace to the sane contemplation of the variegated products\nof analysis. Safety here abides in the restraint which should\ncharacterize all discussion and application of the concept. If\nenthusiasts would be content not to transport the so-called fourth\ndimensional space out of the sphere of hyperspace and cease trying\nto speculate upon the results of its interposal into three space\nconditions, which is in every way a constructual impossibility, there\ncould not be any possible objection to its due consideration. This\nwould obviate the danger of calling into question either the sincerity\nor perspicacity of those whose enthusiasm tempts them to transgress the\nlimits of propriety in their behavior towards the inquiry.\n\nThere is but one life, one mind, one extension, one quantity, one\nquality, one being, one state, one condition, one mood, one affection,\none desire, one feeling, one consciousness. There is also but one\nnumber and that is unity. All so-called integers are but fractional\nparts of this kosmic unity. The idea represented by the word _two_\nreally connotates two parts of unity and the same is true of a\ndecillion, or any number of parts. These are merely the infinitesimals\nof unity and they grow less in size and consequence as the divisions\nincrease in number. The analysis of unity into an infinity of parts is\npurely an _a posteriori_ procedure. That it is an inherent mind-process\nis a fallacy. All our common quantities, as the mile, kilometer,\nyard, foot, inch, gallon, quart, are conventional and arbitrary and\nsusceptible of wide variations. As the basis of all physical phenomena\nis unity; it is only in the ephemeral manifestations of sensuous\nobjects that they appear as separate and distinct quantities.\n\nWe see on a tree many leaves, many apples or cherries; on a cob many\ngrains of corn. We have learned to assign to each of these quantities\nin their summation a sequence value. But this is an empirical notion\nand cannot be said to inhere in the mind itself. Let us take, for\ninstance, the mustard seed. If it were true that in one of these seeds\nthere existed all the subsequent seeds which appear in the mustard\nplant as separate and identifiable quantities, and not in essence, then\nthere would perhaps be warrant for the notion that diversity, as the\ncalculable element, is an _a priori_ conception. But, as this is not\nthe case and since diversity is purely empirical and pertains only to\nthe efflorescence of the one life it is manifestly absurd to take that\nview.\n\nUnder the most charitable allowances, therefore, there can be but two\nquantities--unity and diversity; yet not two, for these are one. Unity\nis the _one_ quantity and diversity is the division of unity into a\ntransfinity of parts. Unity is infinite, absolute and all-inclusive.\nDiversity is finite although it may be admitted to be transfinite, or\ngreater than any assignable value. Unity alone is incomprehensible.\nIn order to understand something of its nature we divide it into a\ndiversity of parts; and because we fail to understand the transfinity\nof the multitude of parts we mistakenly call them infinite.\n\nWhen analysis shall have proceeded far enough into the abysmal\nmysteries of diversity; when the mathematical mind shall have been\novercome by the overwhelming perplexity of the maze of diverse parts,\nit shall then fall asleep and upon awaking shall find that wonderfully\nsimple thing--_unity_. It is the one quantity that is endowed with\na magnitude which is both inconceivable and irresolvable. The one\nineluctable fact in the universe is the incomprehensibility and\nall-inclusivity of _one-ness_. It is incomprehensible, inconceivable\nand infinite at the present stage of mind development. But the goal\nof mind is to understand the essential character of unity, of life.\nIts evolution will then stop, for it will have reached the prize of\ndivinity itself whereupon the intellect exalted by and united with the\nintuition shall also become one with the divine consciousness.\n\n\n\n\n  CHAPTER II\n\n  HISTORICAL SKETCH OF THE HYPERSPACE MOVEMENT\n\n\n  Egypt the Birthplace of Geometry--Precursors: NASIR-EDDIN, CHRISTOPH\n  CLAVIUS, SACCHERI, LAMBERT, LA GRANGE, KANT--Influence of the\n  _Mecanique Analytique_--The Parallel-Postulate the Root and Substance\n  of the Non-Euclidean Geometry--The Three Great Periods: The Formative,\n  Determinative and Elaborative--RIEMANN and the Properties of Analytic\n  Spaces.\n\nThe evolution of the idea of a fourth dimension of space covers a long\nperiod of years. The earliest known record of the beginnings of the\nstudy of space is found in a hieratic papyrus which forms a part of the\nRhind Collection in the British Museum and which has been deciphered by\nEISENLOHR. It is believed to be a copy of an older manuscript of date\n3400 B. C., and is entitled \"_Directions for Knowing All Dark Things_\"\nThe copy is said to have been made by AHMES, an Egyptian priest\nbetween 1700 and 1100 B. C. It begins by giving the dimensions of\nbarns; then follows the consideration of various rectilineal figures,\ncircles, pyramids, and the value of pi ([Greek: p]). Although many of\nthe solutions given in the manuscript have been found to be incorrect\nin minor particulars, the fact remains that Egypt is really the\nbirth-place of geometry. And this fact is buttressed by the knowledge\nthat THALES, long before he founded the Ionian School which was the\nbeginning of Greek influence in the study of mathematics, is found\nstudying geometry and astronomy in Egypt.\n\nThe concept of hyperspace began to germinate in the latter part of the\nfirst century, B. C. For it was at this date that GEMINOS of RHODES\n(B. C. 70) began to think seriously of the mathematical labyrinth\ninto which EUCLID'S parallel-postulate most certainly would lead if\nan attempt at demonstrating its certitude were made. He recognized\nthe difficulties which would engage the attention of those who might\nventure to delve into the mysterious possibilities of the problem.\nThere is no doubt, too, but that EUCLID himself was aware, in some\nmeasure at least, of these difficulties; for his own attitude towards\nthis postulate seems to have been one of noncommittance. It is,\ntherefore, not strange that the astronomer, PTOLEMY (A. D. 87-165),\nshould be found seeking to prove the postulate by a consideration of\nthe possibilities of interstellar triangles. His researches, however,\nbrought him no relief from the general dissatisfaction which he felt\nwith respect to the validity of the problem itself.\n\nFor nearly one thousand years after the attempts at solving the\npostulate by GEMINOS and PTOLEMY, the field of mathematics lay\nundisturbed. For it was at this time that there arose a strange\nphenomenon, more commonly known as the \"Dark Ages,\" which put an\neffectual check to further research or independent investigations.\nMathematicians throughout this long lapse of time were content to\naccept EUCLID as the one incontrovertible, unimpeachable authority,\nand even such investigations as were made did not have a rebellious\ntendence, but were mainly endeavors to substantiate his claims.\n\nAccordingly, it was not until about the first half of the thirteenth\ncentury that any real advance was made. At this time there appeared\nan Arab, NASIR-EDDIN (1201-1274) who attempted to make an improvement\non the problem of parallelism. His work on EUCLID was printed in\nRome in 1594 A. D., about three hundred and twenty years after his\ndemise and was communicated in 1651 by JOHN WALLIS (1616-1703) to the\nmathematicians of Oxford University. Although his calculations and\nconclusions were respectfully received by the Oxford authorities no\ndefinite results were regarded as accomplished by what he had done.\nIt is believed, however, that his work reopened speculation upon\nthe problem and served as a basis, however slight, for the greater\nwork that was to be done by those who followed him during the next\nsucceeding eight hundred years.\n\nAbout twenty years before the printing of the work of NASIR-EDDIN,\nCHRISTOPH CLAVIUS (1574) deduced the axiom of parallels from the\nassumption that a line whose points are all equidistant from a straight\nline is itself straight. In his consideration of the parallel-postulate\nhe is said to have regarded it as EUCLID'S XIIIth axiom. Later BOLYAI\nspoke of it as the XIth and later still, TODHUNTER treated it as the\nXIIth. Hence, there does not seem to have been any general unanimity\nof opinion as to the exact status of the parallel-postulate, and\nespecially is this true in view of the uncertainty now known to have\nexisted in EUCLID'S mind concerning it.\n\nGIROLAMO SACCHERI (1667-1733), a learned Jesuit, born at San Remo,\ncame next upon the stage. And so important was his work that it will\nperpetuate the memory of his name in the history of mathematics. He was\na teacher of grammar in the Jesuit _Collegio di Brera_ where TOMMASO\nCEVA, a brother of GIOVANNI, the well-known mathematician, was teacher\nof mathematics. His association with the CEVA brothers was especially\nbeneficial to him. He made use of CEVA'S very ingenious methods in his\nfirst published book, 1693, entitled _Solutions of Six Geometrical\nProblems Proposed by Count Roger Ventimiglia_.\n\n    A +--------------------------------+ B\n      |                                |\n      |                                |\n      |                                |\n      |                                |\n      |                                |\n      |                                |\n    C +--------------------------------+ D\n\n    FIG. 1.\n\nSACCHERI attacked the problem of parallels in quite a new way.\nExamining a quadrilateral, _ABCD_, in which the angles _A_ and _B_ are\nright angles and the sides _AC_ and _BD_ are equal, he determined to\nshow that the angles _C_ and _D_ are equal. He also sought to prove\nthat they are either right angles, obtus acute. He undertook to prove\nthe falsity of the latter two propositions (that they are either\nobtuse or acute), leaving as the only possibility that they must be\nright angles. In doing so, he found that his assumptions led him into\ncontradictions which he experienced difficulty in explaining.\n\nHis labors in connection with the solution of the problems proposed by\nCOUNT VENTIMIGLIA, including his work on the question of parallels,\nled directly into the field of metageometrical researches, and perhaps\nto him as to no other who had preceded him, or at least to him in a\nlarger degree, belongs the credit for a continued renewal of interest\nin that series of investigations which resulted in the formulation of\nthe non-Euclidean geometry.\n\nThe last published work of SACCHERI was a recital of his endeavors at\ndemonstrating the parallel-postulate. This received the \"Imprimatur\"\nof the Inquisition, July 13, 1733; the Provincial Company of Jesus\ntook possession of the book for perusal on August 16, 1733; but\nunfortunately within two months after it had been reviewed by these\nauthorities, SACCHERI passed away.\n\nAll efforts which had been made prior to the work of SACCHERI were\nbased upon the assumption that there must be an equivalent postulate\nwhich, if it could be demonstrated, would lead to a direct, positive\nproof of EUCLID'S proposition. Although these and all other attempts\nat reaching such a proof have signally failed and although it may\ncorrectly be said that the entire history of demonstrations aiming at\nthe solution of the famous postulate has been one long series of utter\nfailures, it can be asserted with equal certitude that it has proven\nto be one of the most fruitful problems in the history of mathematical\nthought. For out of these failures has been built a superstructure\nof analytical investigations which surpasses the most sanguine\nexpectations of those who had labored and failed.\n\nIn 1766 JOHN LAMBERT (1728-1777) wrote a paper upon the _Theory\nof Parallels_ dated Sept. 5, 1766, first published in 1786, from\nthe papers left by F. BERNOULLI, which contained the following\nassertions:[2]\n\n1. The parallel-axiom needs proof, since it does not hold for geometry\non the surface of the sphere.\n\n2. In order to make intuitive a geometry in which the triangle's sum\nis less than two right angles, we need an \"imaginary\" sphere (the\npseudosphere).\n\n3. In a space in which the triangle's sum is different from two right\nangles there is an absolute measure (a natural unit for length).\n\nAt this time IMMANUEL KANT (1724-1804), the noted German metaphysician,\nwas in the midst of his philosophical labors. And it is believed that\nit was he who first suggested the idea of different _spaces_. Below is\ngiven a statement taken from his _Prolegomena_[3] which corroborates\nthis view.\n\n  \"That complete space (which is itself no longer the boundary of\n  another space) has three dimensions, and that space in general\n  cannot have more, is based on the proposition that not more than\n  three lines can intersect at right angles in one point.... That we\n  can require a line to be drawn to infinity, a series of changes to\n  be continued (for example, _spaces_ passed through by motion) in\n  indefinitum, presupposes a representation of space and time which\n  can only attach to intuition.\"\n\n  [2] Vide _New York Mathematical Society Bulletin_, Vol. III,\n  1893-4, p. 79, G. B. HALSTEAD on _Lambert's Non-Euclidean Geometry_.\n\n  [3] _Prolegomena_, KANT, p. 37, Trans. by J. P. MAHAFFY and J.\n  H. BERNARD.\n\nHis differentiation between space in general and space which may be\nconsidered as the \"boundary of another space\" shows, in the light of\nthe subsequent developments of the mathematical idea of space that\nhe very fully appreciated the marvelous scope of analytic spaces. His\nconception of space, therefore, must have had a profound influence\nupon the mathematic thought of the day causing it to undergo a rapid\nreconstruction at the hands of geometers who came after him.\n\nUnder the masterly influence of LA GRANGE (1736-1813) the idea of\ndifferent spaces began to take definite shape and direction; the\ngeometry of hyperspace began to crystallize; and the field of mathesis\nprepared for the growth of a conception the comprehension of which\nwas destined to be the profoundest undertaking ever attempted by the\nhuman mind. Unlike most great men whom the world learns tardily to\nadmire, LA GRANGE lived to see his talents and genius fully recognized\nby his compeers; for he was the recipient of many honors both from his\ncountrymen and his admirers in foreign lands. He spent twenty years in\nPrussia where he went upon the invitation of FREDERICK the Great who in\nthe Royal summons referred to himself as the \"greatest king in Europe\"\nand to LA GRANGE as the \"greatest mathematician\" in Europe. In Prussia\nthe _Mecanique Analytique_ and a long series of memoirs which were\npublished in the Berlin and Turin Transactions were produced. LA GRANGE\ndid not exhibit any marked taste for mathematics until he was 17 years\nof age. Soon thereafter he came into possession of a memoir by HALLEY\nquite by accident and this so aroused his latent genius that within one\nyear after he had reviewed HALLEY'S memoir he became an accomplished\nmathematician.\n\nHe created the calculus of variations, solved most of the problems\nproposed by Fermat, adding a number of theorems of his own contrivance;\nraised the theory of differential equations to the position of a\nscience rather than a series of ingenious methods for the solution\nof special problems and furnished a solution for the famous\nisoperimetrical problem which had baffled the skill of the foremost\nmathematicians for nearly half a century. All these stupendous tasks he\nperformed by the time he reached the age of nineteen.\n\nThe _Mecanique Analytique_ is his greatest and most comprehensive\nwork. In this he established the law of virtual work from which,\nby the aid of his calculus of variations, he deduced the whole of\nmechanics, including both solids and liquids. It was his object in the\n_Analytique_ to show that the whole subject of mechanics is implicitly\nembraced in a single principle, and to lay down certain formulae from\nwhich any particular result can be obtained. He frequently made the\nassertion that he had, in the _Mecanique Analytique_, transformed\nmechanics which he persistently defined as a \"geometry of four\ndimensions\"[4] into a branch of analytics and had shown the so-called\nmechanical principles to be the simple results of the calculus.\nHence, there can be no doubt but that LA GRANGE not only completed\nthe foundation, but provided most of the material in his analyses and\nother \"abstract results of great generality\" which he obtained in his\nnumerous calculations, for the superstructure subsequently known as the\ngeometry of hyperspace, and in which the fourth dimensional concept\noccupies a very fundamental place.\n\n  [4] In 1754 D'ALEMBERT (1717-1783) published an article in the\n  famous old _Encyclopedia_ edited by DIDEROT and himself on _Dimension_.\n  In this article the idea of the fourth dimension is dwelt upon at\n  length. The view which he expressed in this article, of course, served\n  greatly to popularize the conception among the learned men of the day,\n  and owing to the close relationship existing between D'ALEMBERT and LA\n  GRANGE, it is not surprising that the latter should have been very much\n  enamored of the idea.\n\nIt is as if for nearly seventeen hundred years workmen, such as\nGEMINOS, of RHODES, PTOLEMY, SACCHERI, NASIR-EDDIN, LAMBERT, CLAVIUS,\nand hundred of others who struggled with the problem of parallels,\nhad made more or less sporadic attempts at the excavation of the land\nwhereon a marvelously intricate building was to be constructed. There\nis no historical evidence to show that any of them ever dreamed that\nthe results of their labors would be utilized in the manner in which\nthey have been used. Then came KANT with the wonderfully penetrating\nsearchlight of his masterful intellect who from the elevation which\nhe occupied saw that the site had great possibilities, but he had not\nthe mathematical talent to undertake the work of actual, methodical\nconstruction. Indeed his task was of a different sort. However, he\nsucceeded in opening the way for LA GRANGE and others who followed\nhim. LA GRANGE immediately seized upon the idea which for more than\na thousand years had been impinging upon the minds of mathematicians\nvainly seeking lodgment and began the elaboration of a plan in\naccordance with which minds better skilled in the pragmatic application\nof abstract principles than his could complete the work begun.\nUnfortunately, on account of his intense devotion and loyalty to the\nstudy of pure mathematics, and when he had reached the summit of\nhis greatness where he stood \"without a rival as the foremost living\nmathematician,\" his health became seriously affected, causing him to\nsuffer constant attacks of profound melancholia from which he died on\nApril 10, 1813.\n\nWe come now to one of the most remarkable periods in the history of\nmental development. During the six hundred years between the birth of\nNASIR-EDDIN and the death of LA GRANGE the entire world of mathesis\nwas being reconstituted. Since there had been gradually going on an\ninternal process which, when completed, forever would liberate the mind\nfrom the narrow confines of consciousness limited to the three-space,\nit is not surprising that we should find, in the mathematical thought\nof the time, an absolutely epoch-making departure. The innumerable\nattempts at the solution of the parallel-postulate, all failures in the\nsense that they did not prove, have intensified greatly the esteem in\nwhich the never-dying elements of EUCLID are held to-day. And despite\nthe fact that there may come a time when his axioms and conclusions\nmay be found to be incongruent with the facts of sensuous reality; and\nthough all of his fundamental conceptions of space in general, his\ntheorems, propositions and postulates may have to give way before the\nsearching glare of a deeper knowledge because of some revealed fault,\nthe perfection of his work in the realm of pure mathematics will remain\nforever a master piece demanding the undiminished admiration of mankind.\n\nThe parallel-postulate, as stated by EUCLID in his _Elements of\nGeometry_, reads as follows:\n\n  \"If a straight line meet two straight lines so as to make the two\n  interior angles on the same side of it taken together less than\n  two right angles, these straight lines being continually produced,\n  shall at length meet upon that side on which are the angles which\n  are less than two right angles.\"\n\nOn this postulate hang all the \"law and the prophets\" of the\nnon-Euclidean Geometry. In it are the virtual elements of three\npossible geometries. Furthermore, it is both the warp and the woof\nof the loom of present-day metageometrical researches. It is the\ngolden egg laid by the god SEB at the beginning of a new life cycle in\npsychogenesis. Its progeny are numerous--hyperspaces, sects, straights,\ndigons, equidistantials, polars, planars, coplanars, invariants,\nquaternions, complex variables, groups and many others. A wonderfully\ninteresting breed, full of meaning and pregnant with the power of final\nemancipations for the human intellect!\n\nWhen the conclusions which were systematically formulated as a result\nof the investigations along the lines of hypotheses which controverted\nthe parallel-postulate were examined it was found that they fell into\nthree main divisions, namely: the synthetic or hyperbolic; the analytic\nor RIEMANNIAN and the elliptic or CAYLEY-KLEIN. These divisions or\ngroups are based upon the three possibilities which inhere in the\nconception taken of the sum of the angles referred to in the above\npostulate as to whether it is equal to, greater or less than two right\nangles.\n\nThe assumption that the angular sum is congruent to a straight angle is\ncalled the Euclidean or parabolic hypothesis and is to be distinguished\nfrom the synthetic or hyperbolic hypothesis established by GAUSS,\nLOBACHEVSKI and BOLYAI and which assumes that the angular sum is less\nthan a straight angle. The elliptic or CAYLEY-KLEIN hypothesis assumes\nthat the angular sum is greater than a straight angle. LOBACHEVSKI,\nhowever, not satisfied with the statement of the parallel-postulate\nas given by EUCLID and which had caused the age-long controversy,\nsubstituted for it the following:\n\n  \"All straight lines which, in a plane, radiate from a given\n  point, can, with respect to any other straight line, in the same\n  plane, be divided into two classes--the intersecting and the\n  non-intersecting. The boundary line of the one and the other class\n  is called parallel to the given line.\"\n\nThis is but another way of saying about the same thing that EUCLID had\ndeclared before, and yet, curiously enough it afforded just the liberty\nthat LOBACHEVSKI needed to enable him to elaborate his theory.\n\nFor the purposes of this sketch the field of the development of\nnon-Euclidean geometry is divided into three periods to be known as:\n(1) the _formative_ period in which mathematical thought was being\nformulated for the new departure; (2) the _determinative_ period during\nwhich the mathematical ideas were given direction, purpose and a\ngeneral tendence; (3) the _elaborative_ period during which the results\nof the former periods were elaborated into definite kinds of geometries\nand attempts made at popularizing the hypotheses.\n\n\n  The Formative Period\n\nCHARLES FREDERICH GAUSS (1777-1855) by some has been regarded as the\nmost influential mathematician that figured in the formulation of\nthe non-Euclidean geometry; but closer examination into his efforts\nat investigating the properties of a triangle shows that while his\nresearches led to the establishment of the theorem that a regular\npolygon of seventeen sides (or of any number which is prime, and also\none more than a power of two) can be inscribed, under the Euclidean\nrestrictions as to means, in a circle, and also that the common\nspherical angle on the surface of a sphere is closely connected with\nthe constitution of the area inclosed thereby, he cannot justly be\ndesignated as the leader of those who formulated the synthetic school.\nAnd this, too, for the simple reason that, as he himself admits in\none of his letters to TAURINUS, he had not \"published anything on the\nsubject.\" In this same letter he informs TAURINUS that he had pondered\nthe subject for more than thirty years and expressed the belief that\nthere could not be any one who had \"concerned himself more exhaustively\nwith this second part (that the sum of the angles of a triangle cannot\nbe more than 180 degrees)\" than he had.\n\nWriting from G\u00f6ttingen to TAURINUS, November 8, 1824, and commenting\nupon the geometric value of the sum of the angles of a triangle, he\nsays:\n\n  \"Your presentation of the demonstration that the sum of the angles\n  of a plane triangle cannot be greater than 180 degrees does,\n  indeed, leave something to be desired in point of geometrical\n  precision. But this could be supplied, and there is no doubt\n  that the impossibility in question admits of the most rigorous\n  demonstration. But the case is quite different with the second\n  part, namely, that the sum of the angles cannot be smaller than\n  180 degrees; this is the real difficulty, the rock upon which\n  all endeavors are wrecked.... The assumption that the sum of the\n  three angles is smaller than 180 degrees leads to a new geometry\n  entirely different from our Euclidean--a geometry which is\n  throughout consistent with itself, and which I have elaborated\n  in a manner entirely satisfactory to myself, so that I can solve\n  every problem in it with the exception of the determining of a\n  constant which is not _a priori_ obtainable.\"\n\nIt appears from this correspondence that GAUSS had in the privacy\nof his own study elaborated a complete non-Euclidean geometry, and\nhad so thoroughly familiarized himself with its characteristics and\npossibilities that the solution of every problem embraced within it was\nvery clear to him except that of the determination of a constant. He\nconcluded the above letter by saying:\n\n   \"All my endeavors to discover contradiction or inconsistencies in\n   this non-Euclidean geometry have been in vain, and the only thing\n   in it that conflicts with our reason is the fact that if it were\n   true there would necessarily exist in space a linear magnitude\n   quite determinate in itself; yet unknown to us.\"\n\nJudging from the correspondence between GAUSS and GERLING (1788-1857),\nBESSEL (1784-1846), SCHUMACHER and TAURINUS, the nephew of SCHWEIKART,\nand that between SCHWEIKART and GERLING, there had grown up a general\ndissatisfaction in the minds of mathematicians of this period with\nEuclidean geometry and especially the parallel-postulate and its\nconnotations. BESSEL expresses this general discontent in one of his\nletters to GAUSS, dated February 10, 1829, in which he says:\n\n  \"Through that which LAMBERT said and what SCHWEIKART disclosed\n  orally, it has become clear to me that our geometry is incomplete,\n  and should receive a correction, which is hypothetical, and if the\n  sum of the angles of the plane triangle is equal to 180 degrees,\n  vanishes.\"\n\nThe opinion of leading mathematicians at this time seems to have been\ncrystallizing very rapidly. Unconsciously the men of this formative\nperiod were adducing evidence which would give form and tendence to the\ndevelopments in the field of mathesis at a later date. They appear to\nhave been reaching out for that which, ignis fatuus-like, was always\nwithin easy reach, but not quite apprehensible.\n\nA bolder student than GAUSS was FERDINAND CARL SCHWEIKART (1780-1857)\nwho also has been credited with the founding of the non-Euclidean\ngeometry. In fact, if judged by the same standards as GAUSS, he would\nbe called the \"father of the geometry of hyperspace\"; for he really\npublished the first treatise on the subject. This was in the nature of\nan inclosure which he inserted between the leaves of a book he loaned\nto GERLING. He also asked that it be shown to GAUSS that he might give\nhis judgment as to its merits.\n\nSCHWEIKART'S treatise, dated Marburg, December, 1818, is here quoted in\nfull:\n\n  \"There is a two-fold geometry--a geometry in the narrower sense,\n  the Euclidean, and an astral science of magnitude.\n\n  \"The triangles of the latter have the peculiarity that the sum of\n  the three angles is not equal to two right angles.\n\n  \"This presumed, it can be most rigorously proven: (_a_) That the\n  sum of the three angles in the triangle is less than two right\n  angles.\n\n  \"(_b_) That this sum becomes ever smaller, the more content\n  the angle incloses. (_c_) That the altitude of an isosceles\n  right-angled triangle indeed ever increases, the more one\n  lengthens the side; that it, however, cannot surpass a certain\n  line which I call the constant.\"\n\n  Squares have consequently the following form:\n\n  [Illustration: FIG. 2.]\n\n  \"If this constant were for us the radius of the earth (so that\n  every line drawn in the universe, from one fixed star to another,\n  distant 90\u00b0 from the first, would be a tangent to the surface of\n  the earth) it would be infinitely great in comparison with the\n  spaces which occur in daily life.\"\n\nThe above, being the first published, not printed, treatise on the new\ngeometry occupies a unique place in the history of higher mathematics.\nIt gave additional strength to the formative tendencies which\ncharacterized this period and marked SCHWEIKART as a constructive and\noriginal thinker.\n\nThe nascent aspects of this stage received a fruitful contribution\nwhen NICOLAI LOBACHEVSKI (1793-1847) created his _Imaginary Geometry_\nand JANOS BOLYAI (1802-1860) published as an appendix to his father's\n_Tentamen_, his _Science Absolute of Space_. LOBACHEVSKI and BOLYAI\nhave been called the \"Creators of the Non-Euclidean Geometry.\" And this\nappellation seems richly to be deserved by these pioneers. Their work\ngave just the impetus most needed to fix the status of the new line\nof researches which led to such remarkable discoveries in the more\nrecent years. The _Imaginary Geometry_ and the _Science Absolute of\nSpace_ were translated by the French mathematician, J. HO\u00dcEL in 1868\nand by him elevated out of their forty-five years of obscurity and\nnon-effectiveness to a position where they became available for the\nmathematical public. To BOLYAI and LOBACHEVSKI, consequently, belong\nthe honor of starting the movement which resulted in the development of\nmetageometry and hence that which has proved to be the gateway of a new\nmathematical freedom.\n\nGAUSS, SCHWEIKART, LOBACHEVSKI, WOLFGANG and JANOS BOLYAI were the\nprincipal figures of the formative period and the value of their\nwork with respect to the formulation of principles upon which was\nconstructed the Temple of Metageometry cannot be overestimated.\n\n\n  The Determinative Period\n\nThis period is characterized chiefly by its close relationship to the\ntheory of surfaces. RIEMANN'S Habilitation Lecture on _The Hypotheses\nWhich Constitute the Bases of Geometry_ marks the beginning of this\nepoch. In this dissertation, RIEMANN not only promulgated the system\nupon which GAUSS had spent more than thirty years of his life in\nelaborating, for he was a disciple of GAUSS; but he disclosed his own\nviews with respect to space which he regarded as a particular case\nof manifold. His work contains two fundamental concepts, namely, the\n_manifold_ and the _measure of curvature_ of a continuous manifold,\npossessed of what he called _flatness_ in the smallest parts. The\nconception of the measure of curvature is extended by RIEMANN from\nsurfaces to spaces and a new kind of space, finite, but unbounded, is\nshown to be possible. He showed that the dimensions of any space are\ndetermined by the number of measurements necessary to establish the\nposition of a point in that space. Conceiving, therefore, that space\nis a manifold of finite, but unbounded, extension, he established the\nfact that the passage from one element of a manifold to another may\nbe either discrete or continuous and that the manifold is discrete or\ncontinuous according to the manner of passage. Where the manifold\nis regarded as discrete two portions of it can be compared, as to\nmagnitude, by counting; where continuous, by measurement. If the whole\nmanifold be caused to pass over into another manifold each of its\nelements passing through a one-dimensional manifold, a two-dimensional\nmanifold is thus generated. In this way, a manifold of _n_-dimensions\ncan be generated. On the other hand, a manifold of _n_-dimensions can\nbe analyzed into one of one dimension and one of (_n_-1) dimensions.\n\nTo RIEMANN, then, is due the credit for first promulgating the\nidea that space being a special case of manifold is generable, and\ntherefore, _finite_. He laid the foundation for the establishment of\na special kind of geometry known as the \"elliptic.\" Space, as viewed\nby him, possessed the following properties, viz.: generability,\ndivisibility, measurability, ponderability, finity and flexity.\n\nThese are the six pillars upon which rests the structure of hyperspace\nanalyses.[5]\n\n  [5] Vide _Nature_, Vol. VIII, pp. 14-17; 36, 37 (1873); also\n  _Mathematical Papers_, pp. 65-71.\n\n_Generability_ is that property of geometric space by virtue of which\nit may be generated, or constructed, by the movement of a line, plane,\nsurface or solid in a direction without itself. _Divisibility_ is that\nproperty of geometric space by virtue of which it may be segmented\nor divided into separate parts and superposed, or inserted, upon\nor between each other. _Measurability_ is that property by virtue\nof which geometric space is determined to be a manifold of either\na positive or negative curvature, also by which its extent may be\nmeasured. _Ponderability_ is that property of geometric space by virtue\nof which it may be regarded as a quantity which can be manipulated,\nassorted, shelved or otherwise disposed of. _Finity_ is that property\nby virtue of which geometric space is limited to the scope of the\nindividual consciousness of a unodim, a duodim or a tridim and by\nvirtue of which it is finite in extent. _Flexity_ is that property by\nvirtue of which geometric space is regarded as possessing curvature,\nand in consequence of which progress through it is made in a curved,\nrather than a geodetic line, also by virtue of which it may be flexed\nwithout disruption or dilatation.\n\nRIEMANN who thus prepared the way for entrance into a veritable\nlabyrinth of hyperspaces is, therefore, correctly styled \"The father\nof metageometry,\" and the fourth dimension is his eldest born. He\ndied while but forty years of age and never lived long enough fully\nto elaborate his theory with respect to its application to the\nmeasure of curvature of space. This was left for his very energetic\ndisciple, EUGENIO BELTRAMI (1835-1900) who was born nine years after\nRIEMANN and lived thirty-four years longer than he. His labors mark\nthe characteristic standpoint of the determinative period. BELTRAMI'S\nmathematical investigations were devoted mainly to the non-Euclidean\ngeometry. These led him to the rather remarkable conclusion that the\npropositions embodied therein relate to figures lying upon surfaces of\nconstant negative curvature.\n\nBELTRAMI sought to show that such surfaces partake of the nature of the\npseudosphere, and in doing so, made use of the following illustration:\n\n[Illustration: FIG. 3.]\n\n[Illustration: FIG. 4.]\n\nIf the plane figure _aabb_ is made to revolve upon its axis of symmetry\n_AB_ the two arcs, _ab_ and _ab_ will describe a pseudospherical\nconcave-convex surface like that of a solid anchor ring. Above and\nbelow, toward _aa_ and _bb_, the surface will turn outward with\never-increasing flexure till it becomes perpendicular to the axis\nand ends at the edge with one curvature infinite. Or, the half of a\npseudospherical surface may be rolled up into the shape of a champagne\nglass, as in Fig. 4. In this way, the two straightest lines of the\npseudospherical surface may be indefinitely produced, giving a kind of\nspace (pseudospherical) in which the axiom of parallels does not hold\ntrue.\n\nThe determinative period marks the most important stage in the\ndevelopment of non-Euclidean geometry and certainly the most\nsignificant in the evolution of the idea of hyperspaces and multiple\ndimensionality. RIEMANN and BELTRAMI are chief among those whose labors\ncharacterize the scope of this period. Their work gave direction and\ngeneral outline for later developments and all subsequent researches\nalong these lines have been conducted in strict conformity with the\nprinciples laid down by these pioneer constructionists. They laid out\nthe field and designated its confines beyond which no adventurer has\nsince dared to pass.\n\nThe great importance of the work of RIEMANN at this time may be seen\nfurther in the fact that it not only marked the beginning of a new\nepoch in geometry; but his pronouncement of the hypothesis that space\nis unbounded, though finite, is really the first time in the history\nof human thought that expression was ever given to the idea that space\nmay yet be only of limited extent. Before that time the minds of all\nmen seemed to have been unanimous in the consideration of space as an\nillimitable and infinite quantity.\n\n\n  The Elaborative Period\n\nThe elaborative stage includes the work of all those who, working upon\nthe bases laid down by LOBACHEVSKI, BOLYAI, SCHWEIKART and RIEMANN,\nhave sought to amplify the conclusions reached by them. Among those\nwhose investigations have greatly multiplied the applications of\nhyperspace conceptions are HO\u00dcEL (1866) and FLYE ST. MARIE (1871)\nof France; HELMHOLTZ (1868), FRISCHAUF (1872), KLEIN (1849), and\nBALTZER (1877) of Germany; BELTRAMI (1872) of Italy; DE TILLY (1879)\nof Belgium; CLIFFORD and CAYLEY (1821) of England; NEWCOMB (1835) and\nHALSTEAD of America.\n\nThese have been most active in popularizing the subject of\nnon-Euclidean geometry and incidentally the idea of the fourth\ndimension. The great mass of non-professional mathematical readers,\ntherefore, owe these men an immeasurable debt of gratitude for the work\nthat they have done in the matter of rendering the conceptions which\nconstitute the fabric of metageometry understandable and thinkable. A\nglance at the bibliography appended at the end of this volume will give\nsome idea of the enormous amount of labor that has been expended in an\neffort to translate the most abstract mathematical principles into a\nlanguage that could easily be comprehended by the average intelligent\nperson.\n\nThe characteristic standpoint of this period is the popular\ncomprehension of the hyperspace concept and the consequent mental\nliberation which follows. For there is no doubt but that unheard of\npossibilities of thought have been revealed by investigations into\nthe nature of space. An entirely new world has been opened to view and\nonly a beginning has been made at the exploration of its extent and\nresources.\n\nOne of the notable incidents of the early years of this period is the\nposition taken by FELIX KLEIN who stands in about the same relation\nto CAYLEY as BELTRAMI does to RIEMANN, in that he assumed the task of\ncompleting the work of his predecessor. KLEIN held that there are only\ntwo kinds of RIEMANNIAN _space_--the elliptical and the spherical. Or\nin other words, that there are only two possible kinds of space in\nwhich the propositions announced by RIEMANN could apply. SOPHUS LIE,\ncalled the \"great comparative anatomist of geometric theories,\" carried\nhis classifications to a final conclusion in connection with spaces of\nall kinds and decided that there are possible only four kinds of three\ndimensional spaces.\n\nBut whether men with peering, microscopic, histological vision shall\nestablish the existence of one or many spaces, and regardless of the\nmathematic rigor with which they shall demonstrate the self-consistency\nof the doctrines which they hold, the fact remains that the hypotheses\nthus maintained, while they may be regarded as true descriptions of\nthe spaces concerned, are, nevertheless, incompatible. All of them\ncannot be valid. It will perhaps be found that none of them are\nvalid, especially objectively so. The only true view, therefore, of\nthese systems of hyperspaces is that which assigns them to their\nrightful place in the infinitely vast world of pure mathesis where\ntheir validity may go unchallenged and their existence unquestioned;\nfor in that domain of unconfined mentation, in that realm of divine\nintuitability, the marvelous wonderland of ideas and notions, one is\nnot only disinclined to doubt their logical actuality, but is quite\nwilling to accede their claims.\n\n\n\n\n  CHAPTER III\n\n  ESSENTIALS OF THE NON-EUCLIDEAN GEOMETRY\n\n  The Non-Euclidean Geometry Concerned with Conceptual\n  Space Entirely--Outcome of Failures at Solving the\n  Parallel-Postulate--The Basis of the Non-Euclidean Geometry--Space\n  Curvature and Manifoldness--Some Elements of the Non-Euclidean\n  Geometry--Certainty, Necessity and Universality as Bulwarks\n  of Geometry--Some Consequences of Efforts at Solving the\n  Parallel-Postulate--The Final Issue of the Non-Euclidean\n  Geometry--Extended Consciousness.\n\n\nThe term \"non-Euclidean\" is used to designate any system of geometry\nwhich is not strictly Euclidean in content.\n\nIt is interesting to note how the term came to be used. It appears to\nhave been employed first by GAUSS. He did not strike upon it suddenly,\nhowever, as in the correspondence between him and WACHTER in 1816\nhe used the designation \"anti-Euclidean\" and then, later, following\nSCHWEIKART, he adopted the latter's terminology and called it \"Astral\nGeometry.\" This he found in SCHWEIKART'S first published _treatise_\nknown by that name and which made its appearance at Marburg in\nDecember, 1818. Finally, in his correspondence with TAURINUS in 1824,\nGAUSS first used the expression \"non-Euclidean\" to designate the system\nwhich he had elaborated and continued to use it in his correspondence\nwith SCHUMACHER in 1831.\n\n\"Non-Legendrean,\" \"semi-Euclidean\" and \"non-Archimedean\" are titles\nused by M. DEHN to denote all kinds of geometries which represented\nvariations from the hypotheses laid down by LEGENDRE, EUCLID and\nARCHIMEDES.\n\nThe semi-Euclidean is a system of geometry in which the sum of the\nangles of a triangle is said to be equal to two right angles, but\nin which one may draw an infinity of parallels to a straight line\nthrough a given point. The non-Euclidean geometry embraces all the\nresults obtained as a consequence of efforts made at finding a\nsatisfactory proof of the parallel-postulate and is, therefore, based\nupon a conception of space which is at variance with that held by\nEUCLID. According to the Ionian school space is an infinite continuum\npossessing uniformity throughout its entire extent. The non-Euclideans\nmaintain that space is not an infinite extension; but a finite though\nunbounded manifold capable of being generated by the movement of\na point, line or plane in a direction without itself. It is also\nheld that space is curved and exists in the shape of a sphere or\npseudosphere and is consequently elliptical.\n\nThe inapplicability of EUCLID's parallel-postulate to lines drawn\nupon the surface of a sphere suggested the possibility of a space\nin which the postulate could apply to all possible surfaces or that\nspace itself may be spherical in which case the postulate would be\ninvalidated altogether. Hence, it is quite natural that mathematicians\nfinding themselves unable to prove the postulate with due mathetic\nprecision should turn their attention to the conceptually possible.\nIn this virtual abandonment of the perceptual for the conceptual lies\nthe fundamental difference between the Euclidean and the non-Euclidean\ngeometries. It may be said to the credit of the Euclideans that they\nhave sought to make their geometric conceptions conform as closely as\npossible to the actual nature of things in the sensuous world while\nat the same time they must have perceived that at best their spatial\nnotions were only approximations to the sensuous actuality of objects\nin space.\n\nOn the other hand, non-Euclideans make no pretense at discovering any\ncongruency between their notions and things as they actually are.\nThe attitude of the metageometricians in this respect is very aptly\ndescribed by CASSIUS JACKSON KEYSER who says:\n\n  \"He constructs in thought a summitless hierarchy of hyperspaces,\n  an endless series of orderly worlds, worlds that are possible and\n  logically actual, and he is content not to know if any of them be\n  otherwise actual or actualized.\"[6]\n\n  [6] _Mathematics_, by C. J. KEYSER, Adrian Professor of\n  Mathematics, Columbia University.\n\nThe non-Euclidean is, therefore, not concerned about the applicability\nof ensembles, notions and propositions to real, perceptual space\nconditions. It is sufficient for him to know that his creations\nare thinkable. As soon as he can resolve the nebulosity of his\nconsciousness into the conceptual \"star-forms\" of definite ideas\nand notions, he sits down to the feast which he finds provided by\nsuperfoetated hypotheses fabricated in the deeps of mind and logical\nactualities imperturbed and unmindful of the weal of perceptual space\nin its homogeneity of form and dimensionality.\n\nFundamentally, the non-Euclidean geometry is constructed upon the\nbasis of conceptual space almost entirely. Knowledge of its content is\naccordingly derived from a superperceptual representation of relations\nand interrelations subsisting between and among notions, ideas,\npropositions and magnitudes arising out of a conceptual consideration\nthereof. In other words, representations of the non-Euclidean\nmagnitudes, cannot be said to be strictly perceptual in the same sense\nthat three-space magnitudes are perceived; for three-space magnitudes\nare really sense objects while hyperspace magnitudes are not sense\nobjects. They are far removed from the sensuous world and in order to\nconceive them one must raise his consciousness from the sensuous plane\nto the conceptual plane and become aware of a class of perceptions\nwhich are not perceptions in the strict sense of the word, but\nsuperperceptions; because they are representations of concepts rather\nthan precepts.\n\nNotions of perceptual space are constituted of the triple presentations\narising out of the visual, tactual and motor sensations which are fused\ntogether in their final delivery to the consciousness. The synthesis\nof these three sense-deliveries is accomplished by equilibrating\ntheir respective differences and by correcting the perceptions of one\nsense by those of another in such a way as to obtain a completely\nreliable perception of the object. This is the manner in which the\ncharacteristics of Euclidean space are established.\n\nThe characteristics of non-Euclidean space are not arrived at exactly\nin this way. Being beyond the scope of the visual, tactile and motor\nsense apprehensions, it cannot be said to represent judgments derived\nfrom any consideration or elaboration of the deliveries presented\nthrough these media. Yet, the substance of metageometry, or the science\nof the measurement of hyperspaces, may not be regarded as an _a priori_\nsubstructure upon which the system is founded. That is, the conceptual\nspace of non-Euclidean geometry is not presented to the consciousness\nas an _a priori_ notion. On the other hand, the _a posterioristic_\nquality of metageometric spaces marks the entire scope of motility of\nthe notions appertaining thereto.\n\nThe notions, therefore, of conceptual space are derivable only from the\nperception of concepts, or, otherwise consist of judgments concerning\ninterconceptual relations. The process of apperception involved in the\nrecognition of relations which may be methodically determined is much\nremoved from the primary procedure of perceiving sense-impressions\nand fusing them into final deliveries to the consciousness for\nconceptualization or the elaboration into concepts or general\nnotions. It is a procedure which is in every way superconceptual and\nextra-sensuous. The metageometrician or analyst in no way relies upon\nsense-deliveries for the data of his constructions; for, if he did, he\nshould, then, be reduced to the necessity of confining his conclusions\nto the sphere of motility imposed by the sensible world with the result\nthat we should be able to verify empirically all his postulations. But,\ncontrarily, he goes to the extra-sensuous, and there in the realm of\npure conceptuality, he finds the requisite freedom for his theories;\nthus, environed by a sort of intellectual anarchism, he pursues\nanalytical pleasures quite unrestrainedly. The difference between\nthe two mental processes--that which leads from the sensible world to\nconception and that which veers into the fields beyond--is so great\nthat it is hardly permissible to view the results arrived at in the\noutcome of the separate processes as being identical.\n\nTo illustrate this difference, let us draw an analogy. The miner\ndigs the iron ore out of the ground. The iron is separated from the\nextraneous material and delivered to the furnaces where the metal is\nmelted and turned out as pig iron. It is further treated, and steel,\nof various grades, cast iron and other kinds of iron are produced. The\ntreatment of the iron ore up to this stage is similar to the treatment\nof sense-impressions by the Thinker. Steel, cast iron, et cetera, are\nsimilar to mental concepts. Later, the steel and other products are\nconverted into instruments and numerous articles. This represents\nthe superperceptual process. Trafficking in iron ore products, such\nas instruments of precision, watch springs, and the like, represents\na stage still farther removed from the primary treatment of the\nore and is similar to that to which concepts are treated when the\nmetageometrician manipulates them in the construction of conceptual\nspace-forms. Perception is the dealing with raw iron ore while\nconception is analogous to the production of the finished product.\n\nSuperperception would be analogous to the trafficking in the\nfinished product as such and without any reference to the source\nor the preceding processes. Thus the notions and judgments of the\nnon-Euclidean geometry are arrived at as a result of a triple process\nof perception, conception and superperception the latter being merely\nsuperconceived as formal space-notions. But it is obvious that the\nmore complex the processes by which judgments purporting to relate to\nperceptual things are derived the more likely are those judgments to be\nat variance with the nature of the things themselves.\n\nIn view of the foregoing, the dangers resulting from identifying\nthe products of the two processes are very obvious indeed. But the\ndifference between the two procedures is the difference between\nEuclidean and non-Euclidean geometries or the difference between\nperceptual space notions and conceptual space notions. Hence, it is\nnot understood just how or why it has occurred to anyone that the two\nnotions could be made congruent. Magnitudes in perceptual, sensible\nspace are things apart from those that may be said to exist in\nmathematical space or that space whose qualities and properties have no\nexistence outside of the mind which has conceived them. It is believed\nto be quite impossible to approach the study of metageometrical\npropositions with a clear, open mind without previously understanding\nthe fundamental distinctions which exist between them.\n\nIt follows, therefore, as a logical conclusion that geometric space of\nwhatsoever nature is a purely formal construction of the intellect, and\nfor this reason is completely under the sovereignty of the intellect\nhowever whimsical its demands may be. Being thus the creature of the\nintellect, its possibilities are limited only by the limitations of the\nintellect itself. Perceptual space, being neither the creature of the\nintellect nor necessarily an _a priori_ notion resident in the mental\nsubstructure, but existing entirely independent of the intellect or\nits apprehension thereof, cannot be expected to conform to the purely\nformal restrictions imposed by the mind except in so far as those\nrestrictions may be determined by the nature of perceptual space. And\nfor that matter, it should not be forgotten that, as yet, we have no\nmeans of determining whether or not the testimony of the intellect is\nthoroughly credible simply because there is no other standard by which\nwe may prove its testimony. It is possible to justify the deliveries\nof the eye by the sense of touch, or vice versa. It is also possible\nto prove all our sense-deliveries by one or the other of the senses.\nBut we have no such good fortune with the deliveries of the intellect.\nWe have simply to accept its testimony as final; because we cannot\ndo any better. But if it were possible to correct the testimony of\nthe intellect by some other faculty or power which by nature might be\nmore accurate than the intellect it should be found that the intellect\nitself is sadly limited.\n\nThe possible curvature of space is a notion which also characterizes\nthe content of the non-Euclidean geometry. It is upon this notion\nthat the question of the finity and unboundedness of space, in the\nmathematical sense, rests. In the curved space, the straightest line\nis a curved line which returns upon itself. Progression eastward\nbrings one to the west; progression northward brings one to the south,\net cetera. On this view space is finite, but may not be regarded as\npossessing boundaries.\n\nSpace-curvature, reinforced by the idea that space is also a manifold\nis the enabling clause of metageometry and without them the analyst\ndares not proceed. Here again, we are led to the confession that\nhowever fantastic these two notions may seem and evidently are, there\nis nevertheless to be recognized in them a \"dim glimpse\" of a veritable\nreality--a slight foreshadowing of the revelation of some great kosmic\nmystery.\n\nThe manifoldness of space is the fiat of analysis. It is the inevitable\noutcome of the analyst's method of procedure. His education, training\nand view of things in general inhibit his arriving at any other\nresult and he may be pardoned with good grace for his manufacture of\nthe space-manifold. For by it perhaps a better appreciation of that\nwonderful extension of consciousness in the nature of which is involved\nthe explanation of the perplexing problems which the manifold and other\nmetageometrical expedients faintly adumbrate may be gained.\n\nIt is pertinent, in the light of the above, to examine into some of the\nrelative merits of the three formal bulwarks of geometrical knowledge.\nThese are _certainty_, _necessity_ and _universality_.\n\nGeometric certainty is derived solely from the nature of the premises\nupon which it is based. If the premises be contradictory, it is,\nof course, defective. But if the premises are non-contradictory\nor self-evident, then the certainty of geometric notions and\nconclusions is valid. Another consideration of prime importance in\nthis connection is the _definition_. From it all premises proceed.\nHence, the definition is even more important than the premise; for\nit is the persisting determinant of all geometric conclusions while\nthe premise is dependent upon the limitations of the definition. The\ndeterminative character of the definition has led to its apotheosis;\nbut this, admittedly, has been necessary in order to give stability\nand permanency to the conclusions which followed. But in spite of this\nit would appear that the certainty of geometric conclusions is not a\nquality to be reckoned as absolute or final.\n\nWith the same certainty that it can be said the sum of the angles of\nthe triangle is equal to two right angles it may be asserted that\nthat sum is also greater or less than two right angles. Certainty\nwhich is based upon the inherent congruity of definitions, premises\nand propositions is an entirely different matter from that certainty\nwhich arises out of the real, abiding validity of a scheme of thought.\nBut this difference is not lessened by the fact that the latter is\ndependent, in a measure, upon the correct systematization of our\nspatial experiences by means of methodical processes. Euclidean\ngeometry, accordingly, is not so certain in its applications as it is\nutilitarian; but non-Euclidean geometry is even less certain than the\nformer and consequently more lacking in its utilitarian possibilities.\n\nThe necessity of geometrical determinations is merely the necessity\nwhich inheres in logical inferences or deductions. These may or may\nnot be valid. Inasmuch as the necessariness of deductions is primarily\nbased upon the conditional certainty of premises and definitions it\nappears that this quality is in no way peculiar to geometry whether\nEuclidean or non-Euclidean. In like manner, the universality of\ngeometric judgments may not properly be regarded as a peculiarity of\ngeometry; but is explicable upon the basis of the formal character\nof the assumptions which underlie it. The chief value, then, of\nnon-Euclidean geometry seems to abide in the fact that it clarifies our\nunderstanding as to the complex processes by which it is possible to\norganize and systematize our spatial experiences for assimilation and\nuse in other branches of knowledge.\n\nWith the above statement of the case of the non-Euclidean geometry\nit is now thought permissible to state briefly some of the elements\nthereof.[7]\n\n  [7] The science of pure mathematics is perhaps indebted to no one in so\n  great a degree as to GEORGE BRUCE HALSTEAD, formerly of the University\n  of Texas, whose labors in connection with the popular exposition of\n  the non-Euclidean geometry have been most untiring and effectual. Vide\n  _Popular Astronomy_, Vol. VII and VIII, 1900, Dr. G. B. HALSTEAD.\n\nBelow will be found some of the elements obtained as a consequence of\nefforts made both at proving and disproving the parallel-postulate of\nEuclid:\n\n\"If two points determine a line it is called a straight.\"\n\n\"If two straights make with a transversal equal alternate angles they\nhave a common perpendicular.\"\n\n\"A piece of a straight is called a sect.\"\n\n\"If two equal coplanar sects are erected perpendicular to a straight,\nif they do not meet, then the sect joining their extremities makes\nequal angles with them and is bisected by a perpendicular erected\nmidway between their feet.\"\n\n\"The sum of the angles of a rectilineal triangle is a straight angle,\nin the hypothesis of the right (angle); is greater than a straight\nangle in the hypothesis of the obtuse (angle); is less than a straight\nangle in the hypothesis of the acute (angle).\"\n\n\"The hypothesis of right is Euclidean; the hypothesis of the acute is\nBOLYAI-LOBACHEVSKIAN; the hypothesis of obtuse is RIEMANNIAN.\"\n\n\"If one straight is parallel to a second the second is parallel to the\nfirst.\"\n\n\"Parallels continually approach each other.\"\n\n\"The perpendiculars erected at the middle point of the sides of a\ntriangle are all parallel, if two are parallel.\"\n\n\"If the foot of a perpendicular slides on a straight its extremity\ndescribes a curve called an equidistant curve, or an equidistantial.\"\n\n\"An equidistantial will slide on its trace.\"\n\n\"In the hypothesis of the obtuse a straight is of finite size and\nreturns into itself.\"\n\n\"Two straights always intersect.\"\n\n\"Two straights perpendicular to a third straight intersect at a point\nhalf a straight from the third either way.\"\n\n\"A pole is half a straight from its polar.\"\n\n\"A polar is the locus of coplanar points half a straight from its pole.\nTherefore, if the pole of one straight lies on another straight the\npole of this second straight is on the first straight.\"\n\n\"The cross of two straights is the pole of the join of their poles.\"\n\n\"Any two straights inclose a plane figure, a digon.\"\n\n\"Two digons are congruent if their angles are equal.\"\n\n\"The equidistantial is a circle with center at the poles of its basal\nstraight.\"\n\nA typical postulate based upon the BOLYAI hypothesis of the acute angle\nis the following:\n\n\"From any point _P_ drop _PC_, a perpendicular to any given straight\nline _AB_. If _D_ move off indefinitely on the ray _CB_, the sect will\napproach as limit _PF_ copunctal with _AB_ at infinity.\n\n[Illustration: FIG. 5.]\n\n_PD_ is said to be at _P_ the parallel to _AB_ toward _B_. _PF_ makes\nwith _PC_ an angle _CPF_ which is called the angle of parallelism for\nthe perpendicular _PC_. It is less than a right angle by an amount\nwhich is the limit of the deficiency of the triangle _PCD_. On the\nother side of _PC_, an equal angle of parallelism gives the parallel\n_P_ to _BA_ towards _AM_.[8] Thus at any point there are two parallels\nto a straight. A straight has, therefore, two separate points at\ninfinity.\"\n\n\"Straights through _P_ which make with _PC_ an angle greater than the\nangle of parallelism and less than its supplement do not meet the\nstraight _AB_ at all not even at infinity.\"\n\n  [8] NOTE.--_M_ may be any point on the line _BA_ indefinitely\n  produced.\n\nThe parallel-postulate is stated in the non-Euclidean geometry as\nfollows:\n\n\"If a straight line meeting two straight lines make those angles which\nare inward and upon the same side of it less than two right angles the\ntwo straight lines being produced indefinitely will meet each other on\nthis side where the angles are less than two right angles.\"\n\nIt is stated by MANNING[9] in the following language:\n\n\"If two lines are cut by a third and the sum of the interior angles on\nthe same side of the cutting line is less than two right angles the\nline will meet on that side when sufficiently produced.\"\n\nIt is rather significant that in this postulate which is really\na definition of space should be found grounds for such diverse\ninterpretations as to its nature. Of course, the moment the mind\nseeks to understand the infinite by interpreting it in the unmodified\nterms of the apparently unchangeable finite it entangles itself into\ninsurmountable difficulties. As a drowning man grasps after straws so\nthe mind, immersed in endless abysses of infinity, fails to conduct\nitself in a seemly manner; but gasps, struggles and flounders and\nis happy if it can, in the depths of its perplexity, discover a way\nof logical escape. The pure mathematician has a hankering after the\nlogically consistent in all his pursuits; to him it is the \"Holy\nGrail\" of his highest aspirations. He seeks it as the devotee seeks\nimmortality. It is to him a philosopher's stone, the elixir of\nperpetual youth, the eternal criterion of all knowledge.\n\n  [9] Vide _Non-Euclidean Geometry_, p. 91.\n\nFailures to demonstrate the celebrated postulate of EUCLID led, as a\nmatter of course, to the substitution of various other postulates more\nor less equivalent to it in that each of them may be deduced from the\nother without the aid of any new hypothesis.\n\nAmong those who sought proof by a restatement of the problem are the\nfollowing:\n\n1. PTOLEMY: The internal angles which two parallels make with a\ntransversal on the same side are supplementary.\n\n2. CLAVIUS: Two parallel straight lines are equidistant.\n\n3. PROCLUS: If a straight line intersects one of two parallels it also\nintersects the other.\n\n4. WALLIS: A triangle being given another triangle can be constructed\nsimilar to the given one and of any size whatever.\n\n5. BOLYAI (W.): Through three points not lying on a straight line a\nsphere can always be drawn.\n\n6. LORENZ: Through a point between the lines bounding an angle a\nstraight line can always be drawn which will intersect these two lines.\n\n7. SACCHERI: The sum of the angles of a triangle is equal to two right\nangles.\n\nThere were, of course, many other statements and substitutions used by\nmathematicians in their endeavors satisfactorily to establish the truth\nof the parallel-postulate. That their labors should have terminated,\nfirst, by doubting it, then by denying, and finally, by building up a\nsystem of geometries which altogether ignores the postulate is just\nwhat might naturally be expected of these men who have given to the\nworld the non-Euclidean geometry. In doing what they did many, if\nnot all of them, were not aware in any measure of the proportions\nof the imposing superstructure that would be built upon their\napparent failures. All of them undoubtedly must have sensed the vague\nadumbrations forecast by the unfolding mysteries which they sought to\nlay bare; all of them must have felt as they executed the early tasks\nof those crepuscular days of pure mathematics that the way which they\nwere traveling would lead to the inner shrine of a higher knowledge\nand a wider freedom; they may have been led by divine intuition to\nstrike out on this new path and yet they could not have known how fully\ntheir dreams would be realized by the mathematicians of the twentieth\ncentury. If so, they were truly gods and mathesis is their kingdom.\n\nThe analyst proceeds upon a basis entirely at variance with that which\nguides the ordinary investigator in the formulation of his conclusions.\nThe empirical scientist in arriving at his theories or hypotheses is\ngoverned at all times by the degree of conformity which his postulates\nexhibit to the actual phenomena of nature. He endeavors to ascertain\njust how far or in what degree his hypothesis is congruent with\nthings found in nature. If the dissidence is found to predominate he\nabandons his theory and makes another statement and again sets out to\ndetermine the degree of conformity. If he then finds that the natural\nphenomena agree with his theory he accepts it as for the time being\nfinally settling the question. In all things he is limited by the\nanswer which nature gives to his queries. Not so with the exponent of\npure mathematics. For him the truth of hypotheses and postulates is\nnot dependent upon the fact that physical nature contains phenomena\nwhich answer to them. The sole determining factor for him is whether\nor not he is able to state with _rational consistency_ the assumed\nfirst principles and then logically develop their consequences. If he\ncan do this, that is, if he can state his hypotheses with consistency\nand develop their consequences into a logical system of thought, he is\nquite satisfied and well pleased with his performances. But the fact\nthat this is true is of vital significance for all who seek clearly to\nunderstand the essential character of hyperspatiality.\n\nIt appears, therefore, that the science of consequences is the radical\nessence of pure geometry. The metageometrician enjoys unlimited freedom\nin the choice of his postulates and suffers curtailment only when it\ncomes to the question of consistency. He is at liberty to formulate as\nmany systems of geometry as the barriers of consistency will permit\nand these are practically innumerable. So long then as the laws of\ncompatibility remain inviolate his multiplication of postulate-systems\nmay proceed indefinitely. Is it strange then that under conditions\nwhere an investigator has such unbridled liberty he should be found\nindulging in mathetic excesses?\n\nKANT held that the axioms of geometry are synthetic judgments _a\npriori_; but it appears that in the strictest sense this is not the\ncase. It depends upon the type of mind which is taken as a standard of\nreference. If it be the uncultivated mind, it is certain that to it\nthe relations expressed by an axiom would never appear spontaneously.\nIf on the other hand, the standard be that of a cultivated mind it is\nalso equally certain that to it these relations would be discovered\nonly after methodical operations. All judgments arrived at as a result\nof logical processes should, it seems, be regarded as judgments _a\nposteriori_, i.e., the results of empirical operations. Confessedly,\nthe facts adduced in course of experimentation serve as guides in\nchoosing among all of the many possible logical conventions; but our\nchoice remains untrammeled except by the compulsion arising out of\na fear of inconsistency. The real criterion then of all geometries\nis neither truth, conformability nor necessity, but consistency and\nconvenience.\n\nThe difficulty with the non-Euclideans resolved itself into the\nquestion as to whether it is more consistent, as well as convenient, to\nestablish a proof of the postulate by taking advantage of the support\nto be found in other postulates or whether, by seeking a demonstration\nbased upon the deliveries of sense-experience as to the nature of\nspace and its properties, a still more consistent conclusion might\nbe reached. They had further perplexity, however, when it came to a\ndecision as to whether the organic world is produced and maintained\nin Euclidean space or in a purely conceptual space which alone can\nbe apprehended by the mind's powers of representation. Unwilling to\nadmit the existence of the world in Euclidean space, they turned their\nattention to the examination of the properties of another kind of\nspace so-called which unlike the space of the Ionian school could be\nmade to answer not only all the purposes of plane and solid figures,\nbut of spherics as well. And so, the manifold space was invented by\nRIEMANN and later underwent some remarkable improvements at the hands\nof his disciple, BELTRAMI. But it may be said here, parenthetically,\nthat the truth of the whole matter is that our world is neither in\nEuclidean nor non-Euclidean space, both of which, in the last analysis,\nare conceptual abstractions. Although it may not be denied that the\nEuclidean space is the more compatible.\n\nThe problem of devising a space, if only a very limited portion, in\nwhich could be demonstrated the assumed alternative hypothesis and\nits consequences logically developed, occasioned no inconsiderable\nconcern for the non-Euclidean investigators; but neither LOBACHEVSKI,\nBOLYAI nor RIEMANN were to be baffled by the difficulties which they\nmet. These only cited them to more laborious toil. Having succeeded in\nmentally constructing the particular kind of space which was adaptable\nto their rigorous mathetic requirements it immediately occurred to\nthem that all the qualities of the limited space thus devised might\nlogically be amplified and extended to the entire world of space and\nthat what is true of figures constructed in the segmented portion\nof space which they used for experimental purposes is also true of\nfigures drawn anywhere in the universe of this space as all lines\ndrawn in the finite, bounded portion could be extended indefinitely\nand all magnitudes similarly treated. From these results, it was\nbut a single step to the conclusion which followed--that either an\nentirely new world of space had been discovered or that our notion of\nthe space in which the organic world was produced is wholly wrong and\nneeds revision. But notwithstanding the insurmountable obstacles which\nstood in the way of the investigators who made the attempt to discover\nthe homology which might exist between the characteristics of the\nnewly fabricated space and the phenomenal world, investigations were\ncarried forward with almost amazing recklessness and loyalty to the\nmathetic spirit until it was discovered that all efforts to trace out\nany definite lines of correspondence were futile. Then the policy of\nignoring the question of conformability was adopted and has since been\npursued with unchangeable regularity by the analytical investigator.\n\nAmong the results obtained by the non-Euclideans in their profound\nresearches into the nature of hyperspace are these: 1. It was found\nthat the angular sum of a triangle, being ordinarily assumed to be a\nvariable quantity, is either less or greater than two right angles\nso that a strictly Euclidean rectangle could not be constructed. 2.\nThe angle sums of two triangles of equal area are equal. 3. No two\ntriangles not equal can have the same angles so that similar triangles\nare impossible unless they are of the same size. 4. If two equal\nperpendiculars are erected to the same line, their distance apart\nincreases with their length. 5. A line every point of which is equally\ndistant from a given straight line is a curved line. 6. Any two lines\nwhich do not meet, even at infinity, have one common perpendicular\nwhich measures their minimum distance. 7. _Lines which meet at infinity\nare parallel._ But it is apparent that these results have not followed\nupon any mathematical consequence of other supporting postulates or\naxioms such as would place them on a co\u00f6rdinate basis with those used\nas a support for the parallel-postulate; for they are based upon the\nenvisagement of an entirely new principle of space-perception and\nbelong to a wholly different set of space qualities.\n\nThe final issue then of the non-Euclidean geometry is neither in\nthe utility of its processes and conclusions nor in the increscent\ninclination towards a new outlook upon the world of mathesis; but\nresides solely in the possibilities yet to be developed in that vast\ndomain of analytical thought which it has discovered and opened to\nview. To say that it sheds any light upon the nature of the universe\nis perhaps to take the radical view; yet it cannot be doubted that the\nresearches incident to the formulation of the non-Euclidean geometry\nhave greatly extended the scope of consciousness. Whether the extension\nis valid and normal or simply a hypertrophic excrescence of mental\nfeverishness; whether by virtue of it we shall more closely approach\nan understanding of the true nature of the mind of the Infinite, or\nshall all fall into insanity, are certainly debatable questions. It\nnevertheless appears evident that humanity has gained something of\nreal, abiding permanence by this new departure. If that something be\nmerely an extended consciousness or an awakening to the fact that\nthere are stages of awareness beyond the strictly sensuous, and every\nobservable evidence points to this, then there has only begun the\nprocess by which the faculty of conscious functioning in this new world\nshall become the normal possession of the human species. But this new\nworld cannot be said to be of mathematical import; for it is doubtful\nif mathematical laws such as have been devised up to the present time,\nwould obtain therein. So that if anything, it must be psychological and\nvital.\n\nOn this view the worlds of hyperspace inlaid with analytic\nmanifoldnesses and constant curvatures are but the primal excitants\nwhich will finally awaken in the mind the faculty of awareness in the\nnew domain of psychological content. Then will come the blooming of\nthe diurnal flower of the mind's immortality and the outputting of the\norgan of consciousness wherewith the infinite stretches of hyperspaces,\nthe low-lying valleys of reals and imaginaries and the uplifting hills\nof finites and infinites shall be divested of their mysteries and\nstand out in their unitariness no longer draped in the veil of the\ninscrutable and the incomprehensible.\n\nThe fourth dimension, regarded by some as a new scope of motion for\nobjects in space, by others as a new and strange direction of spatial\nextent and by others still as the doorway of the temple of exegesis\nwherein an explanation may be found for the entire congeries of\nmysteries and supermysteries which now perplex the human mind, may also\nbe said to be the key to the non-Euclidean geometry. But it really\ncomplicates the situation; for one has to be capable of prolonged\nabstract thought even to envisage is as a conceptual possibility.\nPOINCAR\u00c9[10] says: \"Any one who should dedicate his life to it could,\nperhaps, eventually imagine the fourth dimension,\" implying thereby\nthat a lifetime of prolonged abstract thought is necessary to bring the\nmind to that point of ecstasy where it could even so much as imagine\nthis additional dimension. Nevertheless by it (the fourth dimension)\nwas the non-Euclidean geometry made and without it was not any of the\nhyperspaces made that were made. It is the view which geometers have\ntaken of space in general that has made the fourth dimension possible,\nand not only the fourth, but dimensions of all degrees. The basis of\nthe non-Euclidean geometry may be found then in the notion of space\nwhich has been predominant in the minds of the investigators.\n\n  [10] Vide _Nature_, Vol. XLV, 1892.\n\nFinally, it should be pointed out that the non-Euclidean geometry,\nthough a consistent system of postulates, has been constructed upon a\nmisconception based upon the identification of real, perceptual space\nwith systems of space-measurements. Hyperspaces which are not spaces\nat all should not be confounded with _real space_. But they constitute\nthe substance of non-Euclidean geometry; they are its blood and sinews.\nTheir study is interesting, because of the possibilities of speculation\nwhich it offers. No mind that has thought deeply upon the intricacies\nof the fourth dimension, or hyperspace, remains the same after the\nprocess. It is bound to experience a certain sense of humility, and\nyet some pride born of a knowledge that it has been in the presence of\na great mystery and has delved into the fearful deeps of kosmic mind.\nTo the mind that has thus been anointed by the sacred chrism of the\ninner mysteries of creative mentality there always come that stillness\nand calm such as characterize the aftermath of reflection upon the\nincomprehensible and the transfinite.\n\n\n\n\n  CHAPTER IV\n\n  DIMENSIONALITY\n\n  Arbitrary Character of Dimensionality--Various Definitions of\n  Dimension--Real Space and Geometric Space Differentiated--The\n  Finity of Space--Difference Between the Purely Formal and the\n  Actual--Space as Dynamic Appearance--The _A Priori_ and the _A\n  Posteriori_ as Defined by PAUL CARUS.\n\n\nIn previous chapters we have traced the growth and development of the\nnon-Euclidean geometry showing that the so-called fourth dimension is\nan aspect thereof. It is now deemed fitting that we should enter into\na more detailed study of the question of dimensionality with a view to\nexamining some of the difficulties which encompass it.\n\nThe question of dimension is as old as geometry itself. Without it\ngeometric conclusions are void and meaningless. Yet the conception\nof dimensionality itself is purely conventional. In its application\nto space there is involved a great deal of confusion because of the\ninferential character of its definition. For instance, commonly we\nmeasure a body in space and arbitrarily assign three elements to\ndetermine its position. The simplest standard for this purpose is the\ncube having three of its edges terminating at one of its corners.\n\n\n           D\n           +\n     A     |\n      \\    |\n       \\   |\n        \\  |\n         \\ |\n          \\|\n           +-------------------------\n           B                        C\n\n     FIG. 6.\n\nThus because it is found that the entire volume of a cube is actually\ncomprehended within the directions indicated by the lines _ab_, _bc_\nand _db_ it is determined that the three co\u00f6rdinates of the point _b_\nare necessary and sufficient to establish the dimensions of the cube\nand consequently of the space in which it rests. The conception may be\nstated in this way: If a collection of elements, say points or lines,\nbe of such a nature or order that it is sufficient to know a certain\ndefinite number of facts about it in order to be able to distinguish\nevery one of the elements from all the others, then the assemblage or\ncollection of elements is said to be of the same number of dimensions\nas there are elements necessary to its determination. In the above\nfigure there are three elements, namely, the lines _ab_, _bc_, and\n_db_, which are necessary and sufficient for the determination of the\nposition of the point _b_. In this way geometers have determined that\nour space is tridimensional; but it is obvious that this conclusion is\nbased not upon any examination of space itself but upon the measurement\nof bodies in space. Upon this view it is seen that conclusions based\nupon such a procedure render our notion of the extension of bodies\nin space identical with the notion of spatial extensity. In other\nwords, we take bodies in space and by examining their characteristics\nand properties arrive at an alleged apodeictic judgment of space. It\nis by means of this conventional norm of geometric knowledge that\nvarious other spaces, notably the one-, two-, four-and _n_-space, have\nbeen devised. It would appear that if some more absolute standard of\nmeasurement or definition of space were adopted the confusion which\nnow clings to the conception of dimension could be obviated. For if it\nbe true that three and only three elements are necessary to determine\na point-position in our space and that in this determination we also\nfind the number of dimensions of space, then it may also be true that\n_n_-co\u00f6rdinates would just as truly determine the dimensionality of\nan _n_-space, which is granted. But then the _n_-space would be just\nas legitimate as the three-space; for it is determined by exactly\nthe same standards. It is both quantitatively and qualitatively the\nsame. If, however, on account of the exigencies that might arise,\nwe are forced to seek solace in the notion of an _n_-space whither\nshall we turn for it? It cannot be found; for it is imperceptible,\nuninhabitable, non-existent, and therefore, absolutely and purely an\nabstraction. Consequently, there must be something radically wrong with\nthe definition of space or with its determinants.\n\nThe purely arbitrary character of dimensionality is very aptly\ndescribed by CASSIUS JACKSON KEYSER, who says:\n\n  \"... The dimensionality of a given space is not unique, but\n  depends upon the choice of the geometric entity for primary or\n  generating element. A space being given, its dimensionality\n  is not therewith determined, but depends upon the will of the\n  investigator who by a proper choice of generating element endows\n  the space with any dimensionality he pleases. That fact is of\n  cardinal significance for science and philosophy.\"[11]\n\n  [11] Vide _Monist_, Vol. XVI, 1896, Mathematical\n  Emancipations.\n\nIt is a fact of \"cardinal significance\" for science; because it\nemphasizes the necessity for some more rational procedure than that\nof the geometrician in arriving at an absolutely unique method of\ndetermining the dimension and essential nature of real space. Its\nsignificance for philosophy lies in the need of a logical, rigidly\nexclusive and absolutely peculiar standard of space definition. The\ndefinition of perceptual space should be such as rigorously inhibits\nits inclusion as a particular in any general class. The necessity for\nthis is warranted by its universality and uniqueness.\n\nThe lines of demarkation between what is recognized as perceptual\nspace and what has been called geometric or conceptual space should\nbe very sharply drawn. So that when reference is made to either there\nwill be no doubt as to which is meant. And then, too, conceptual\nspace is no space at all, properly speaking. It is merely a system of\nspace-measurement. And as such has no logical right to be put in the\nsame category as perceptual space.\n\nReal space is unique. Geometric space belongs to a class whose members\nare capable of indefinite multiplication. It is certainly most\nillogical to identify them. Perceptual space, figuratively speaking,\nis a quantity; analytic space is the foot-rule, the yard-stick, the\nkilometer, by which it is measured and apportioned. It is logically\nimpossible to predicate the same conclusion for both of them. That\nis, to do so causes a profound fracture of the fundamental norms of\nlogic. Such conclusions being thus illegitimate it is rather surprising\nthat an error of this nature should have been made. It is perhaps\naccountable for on the grounds of the geometer's complete _insouciance_\nas to how his postulates shall stand in their relation to things in the\nphenomenal world.\n\nIt is agreed that as convenient as is Euclid's system of\nspace-measurement it is not by any means congruent with the extension\nof real space objects. It does, however, approximate congruity with\nthese objects as nearly as possible. How then could it be expected\nthat a system of space-measurement so far removed from this primary\ncongruence as the non-Euclidean system is should exhibit more obvious\nsigns of correspondence? But the advocates of the _n_-dimensionality\nof space have illatively asserted the identity of space and its\ndimensions. Accordingly, there is not recognized any distinction\nbetween their conception of space itself and its qualitative\npeculiarities. They use the terms interchangeably. So that dimension\nmeans space and _vice versa_. In this lack of discrimination may\nbe found the source of much of the confusion which attaches to the\nconception of space.\n\nIf it were arguable that the relation between space and its dimensions\nis the same as that between matter and its properties then the\nrestriction of this relation to three and only three directions of\nextent would be disallowed; for the reason that if, as is commonly\ndone, dimension be made to mean direction of extent, there would\nbe an unlimited number of directions of extent and they would\nall be perceptible. But this is really another fundamental fault.\nNon-Euclideans have stretched the meaning of the term dimension so that\nit not only includes the idea of direction but an entirely new class of\nqualities--the fourth dimension. And despite this reformation of the\noriginal conception, they demand that it shall be called space.\n\nWe have just shown that the generic concept of dimensionality is that\nthree and only three co\u00f6rdinates are necessary and sufficient for\nits determination. Granting that this is true, are we not compelled\nconsequently to see that we have, by adding a fourth or _n_-dimensions,\ninvolved ourselves into a more complex situation than before? For by\npostulating a fourth dimension either we have created a new world whose\ndimensions are four in number or we have explicitly admitted that the\nthree dimensions have a fourth. Aside from the logical difficulties\nwhich beset these conclusions there is also set up a condition which is\nat variance with the most elementary requirements of common sense.\n\nThus far mathematical thought has not served to clarify our notions\nof space nor to shed any new light upon the vital processes which are\nalleged to have their explanation in the new discovery. Simply stated,\nmetageometricians have brought us to the place where we must either\nrecognize that the fourth dimension is another sphere lying dangerously\nnear the earth in which space extends in four primary directions and\nin which four co\u00f6rdinates are necessary for its determination or we\nare driven to the other horn of the dilemma where we are brought face\nto face with the conclusion that the three perceptual space dimensions\nhave in common a hitherto unknown property or extension in virtue of\nwhich it may be viewed as having an unlimited number of dimensions.\nTo accept the latter view is equivalent to saying that, in the above\nfigure, the three lines _ab_, _bc_ and _db_ have formed a triple\n_entente_ by which they have mutually and severally acquired a new\ndomain, hyperspace, and in which, because of the vast resources of the\nregion, they are able to perform wondrous things.\n\nLet us examine briefly the various current definitions of dimension.\nIt is assumed by not a few that dimension is the same as _direction_.\nBut can we grant this wholly to be true? If so, then a mere child\nmay see that there are and must necessarily be as many dimensions as\nthere are directions. Primarily, there are six directions of space\nand an unlimited number of subsidiary directions. On this view it\nis not necessary to invent a new domain of space if the object be\nmerely to discover and utilize a greater number of dimensions than has\nheretofore been allowed. For the identification of the term dimension\nwith direction already makes available an almost infinite number of\ndimensions. But this view is objected to by the advocates, for it is\ncontrary to the hypothesis of _n_-dimensionality.\n\nDimension also means _extent_. This is partially true. It cannot be\nwholly true. For, if it were, then, space would have only one dimension\nwhich is also not allowable under the hypothesis. Then the definition\nleaves out of account the idea that space is at the same time a\ndirection or collection of directions. The term extension is generic\nand when applied to space means extension in all possible directions\nand not in any one direction. So that it is not permissible to say\nthat space extends in this direction or that because it extends in all\ndirections simultaneously and equally.\n\nGeometers claim that space is a system of co\u00f6rdinates necessary for\nthe establishment of a point-position in it. This view, however,\nidentifies space with a system of space-measurement and is therefore\nfaulty. According to this view there may be as many spaces as there are\nsystems of space-measurement and the latter may be limitless. But if\nthe totality of spaces are to be viewed as one space then we shall have\none space with an indefinite number of dimensions; also an indefinite\nnumber of space measurements which would be confusing. Much, if not\nall, of such a system's utility and convenience would be unavailable\nor useless. That, too, would be in violation of the avowed purpose of\nthese investigations which is to enhance the utility and convenience of\nmathematic operations.\n\nNow it is evident that space is neither direction, extension, a system\nof space-measurement nor a system of manifolds whose dimensions are\ngenerable. And this is so for the same reason that a piece of cloth\nis not the elements of measurement--inches, feet, yards--by which it\nis apportioned. And because we find that the fabric of space lends\nitself accommodatingly to our conventional norms of measurement is not\nsufficient reason for identifying it with these norms. Here we have the\nsource of all error in mathematical conclusions about the nature of\nspace; because all such conclusions are based not upon the intrinsic\nnature of space, but upon artificial forms which we choose to impose\nupon it for our own convenience. But it should be remembered that the\nirregularities which we note are not in space itself but inhere in\nthe forms which we use. For these purposes space is extremely elastic\nand accommodates itself to the shape and scope of any construction\nwe may decide to try upon it. In this respect it is like water which\nhas no regard for the shape, size or kind of vessel into which it may\nbe posited. There is one thing certain that judging from the above\nconsiderations there has been not yet any absolute, all-satisfying\ndefinition devised for space by mathematicians.\n\nThe best definitions hitherto constructed are purely artificial and\narbitrary determinations. It is rather anomalous that there should be\nso little unanimity about what is the most fundamental consideration\nof mathematical conclusions which are supposed to be so certain, so\nnecessary and universal as to be incontrovertible. Confessedly, it\nis a condition which raises again the question as to just what are\nthe limits of mathematical certainty and necessity and just how far\nwe shall depend upon the validity of mathematics to determine for us\nabsolutely certain conclusions about the nature of space. In view\nof the uncertainty noted, are we justified in following too closely\nthe mathematic lead even in matters of logic, to say nothing of our\nconception of space? It seems that we shall have necessarily, on\naccount of the recognized limitations of mathematics in this matter,\nto turn to some more tenable source for the norms of our knowledge\nconcerning space. For in the light of the rather indefensible position\nwhich metageometricians have involved themselves there appears to be no\nhope in this direction.\n\nIt is undoubtedly safer not to rely altogether upon the purely\nabstract, even in the world of mathesis, for any absolute criterion\nof knowledge. It is perhaps well that we should expunge the word\n_absolute_ from our vocabularies. It is really a misnomer and has no\nmeaning in the lexicon of nature. There is in reality no _absolute_ in\nthe sense of final absolution from all conditions or restrictions.\n\nIn the ultimate analysis there is unquestionably no hue, tone, quality,\ncondition nor any imaginable posture of life, being or manifestation\nthat is absolved from every other one of its class or from the\ntotality. All these are relational and interdependent. There is no room\nfor the absolute. In fact, it is a quality which cannot in any way\nbe ascribed to any aspect of kosmic manifestation. It has existence\nonly in the mind and has been devised for the purpose of marking the\nlimits of its scope. All being is relative; all life is relative and is\ndestined to change its qualities as it evolves. All knowledge is also\nrelative and what is true of one state may not be true of another; what\nis true of one life may not be true of another life; the limitations\nof one degree of knowledge may not have any bearings upon another\ndegree. The norms of one will not satisfy the conditions of another\nstage of manifestation. It is always within limits that the criterion\nof knowledge will be found to satisfy a given set of conditions. Hence\nwithin certain limits mathematical conclusions will maintain their\nvalidity. Error is committed by pushing the validity of these limits\nto a position without the sphere of limitations. This seems to be the\ncrux of the whole matter. Mathematicians, notably non-Euclideans,\nhave sought to extend the comparatively small sphere of limits of\ncongruence between mathematic and perceptual space to such an extent\nas to cause it to encroach upon forbidden territory. In doing this they\nhave erred grievously, causing serious offense to the more sensitive\nspirit of the high-caste mathematicians among whom are none more truly\nconservative than PAUL CARUS,[12] who says:\n\n  \"Metageometricians are a hot-headed race and display sometimes all\n  the characteristics of sectarian fanatics. To them it is quite\n  clear there may be two straight lines through one and the same\n  point which do not coincide and yet are both parallel to a third\n  line.\"\n\n  [12] Vide _Monist_, Vol. XIX, p. 402 (1909).\n\nTo the student who has carefully followed the development of the\nnon-Euclidean geometry and the notion of hyperspace the above\ncharacterization is none too severe nor ill-deserved. Nothing could\nmore vividly yet correctly portray the impious tactics of the\nmetageometrician and establish his perceptual obliquity more surely\nthan the mere fact, mentioned by CARUS, that he can with evident lack\nof mental perturbation proclaim that two straight lines, noncoincident\nwith each other, may pass through a point and yet be parallel to\na third line. But this is a mere trifle, a bagatelle, to the many\nother infractions of which he is guilty. The wonder is that he is\nable to secure such obsequious acceptance of his offerings as many\nof the most serious minded mathematicians are inclined to give. Is\nit to be wondered at that, despite the profuse protestations of the\nadvocates, many who take up the study of the question of hyperspace\nshould experience a deep revulsion from the posture assumed by\nmetageometricians with respect to these queries?\n\nLinked with the idea of dimensionality is the notion that space is\ninfinite. This is a conception which has its roots imbedded in the\ndepths of antiquity. Primitive man, looking up into the heavens at\nwhat appeared to him as a never ending extension, was awed by its\nvastness; but the minds of the most learned of the present-day men\nare not free from this innate dread of infinity. It permeates the\nthought life of all alike and none seems to be able to rise above it.\nMathematicians, philosophers, scientists all share in the general\nbelief that space is without limit, unending in extent and eternally\nexistent. RIEMANN, whose thought life found its most convenient mode of\nexpression by means of pure mathematics, was the first in the history\nof human thought to surmise that space is not infinite but limited\neven though unbounded. But his conception has been much vitiated on\naccount of its entanglement with an _idealized_ construction by which\nspace is regarded as a thing to be manipulated and generated by act\nof thought. Were it not for this his conception would indeed mark\nthe beginning of a new era in psychogenesis. As it is, when all the\nnonsensical effusions have been cleared away from our space conceptions\nand men come really to understand something of the essential nature\nof space this new era will find its true beginnings in the mind of\nRIEMANN. Although it must be said, as is the case with all progressive\nmovements, the later development of a rationale for this conclusion\nwill vary greatly from his original conception. For he had in mind a\nspace that is generable and therefore a logical construction while\nultimately the mind will swing back to a consideration of real space.\n\nAlready men are beginning to see a new light. Already they are\nbeginning to take a new view of space in general. The departure is\nespecially noticeable in the attitude assumed by HIRAM M. STANLEY.[13]\nHe says:\n\n  \"If we seek the most satisfactory understanding of space we shall\n  look neither to mathematics nor Psychology but to Physics. The\n  trend of Physics, say with such a representative as OSTWALD, is\n  to make things the expression of force; the constitution and\n  appearance of things are determined by dynamism; and we may best\n  interpret space as a mode of this dynamic appearance.\"\n\nSpace, as a mode of dynamic appearance is a slight improvement upon the\nold idea of a pure vacuity; for in the light of what we now know about\nspace content much of the dignity of that view has been lost. Men now\nknow that space is not an empty void. They know that the atmosphere\nfills a great deal of space. They also have extended their conception\nin this direction to include the ether and occultism goes further\nand postulates four kinds of ether--the chemical, life, light and\npsychographic ethers. But it does not stop here. It postulates a series\nof grades of finer matter than the physical which fills space and\npermeates its entire extent even to identification with its essential\nnature.\n\n  [13] _Philosophical Review_, Vol. VII (1898).\n\nSTANLEY continues:\n\n  \"Everything does not, as commonly conceived, fall into some\n  pre-existent space convenient for it; but everything makes its\n  own spaciousness by its own defensive and offensive force, and the\n  totality of all appearance is space in general.\"\n\nAccording to STANLEY, not only do physical, perceptual objects, by\ntheir \"offensive and defensive force\" make their own space but the\nappearance of that in which no physical object is makes room for itself\nby its own dynamic force. In other words, that which we call \"pure\nextensity\" is by virtue of its dynamism the cause of its own existence.\n\nAt first hand there appears to be little worthy of serious\nconsideration in this view of STANLEY; yet, if carried to its logical\nconclusion, the merit of the hypothesis becomes apparent. Accordingly,\ninterstellar distances which are commonly said to be even without air\nor life of any kind are really an appearance possessed of a dynamism\npeculiar to itself. And this very force-appearance, constituting\nspace, is that which makes it perceivable. For instance, let us say\nthe space that exists between the earth and the moon, is not really\nempty nor does it have an existence prior to itself, but is a mode of\ndynamic appearance which is the cause of its own existence. Its dynamic\ncharacter makes it to appear perceptible to our senses. Logically, if\nthe dynamism were removed there would remain neither space nor the\nappearance of space. If this were true, and it is worthy of serious\nthought, then space is certainly finite, as in its totality, according\nto STANLEY'S view, it would have to be regarded as a \"phenomenon of the\ninner and finite life of the infinite.\"\n\nIt is believed that we may go a step further and unqualifiedly assert\nthat _space is finite_, even denying its infinity as a \"general mode\nof the activity of the whole.\" Yet it is transfinite in the sense\nthat it transcends the comprehension of finite minds or processes. It\nis _finite_ because it is in _manifestation_. Everything that is in\nmanifestation is finite. The infinite is not in manifestation. Infinity\nhas to be limited always to become manifest. The Deity has limited His\nbeing in order that there may be a manifested universe. All things,\nall appearances are finite; because they are phenomena connected with\nmanifestation.\n\nThis question may be viewed from another standpoint. All things in\nmanifestation or existence are polar in their constitution. For\ninstance: there cannot be a \"here\" without a \"there.\" There cannot\nbe an \"upper\" without a \"lower.\" Right is copolar with wrong;\ngood is copolar with evil; night with day; manifestation with\nnon-manifestation; truth with falsity; infinity with finity and so\non, throughout the whole gamut of the pairs of opposites. What is the\nlogical inference? Space is paired with a lack of space. There cannot\nbe what we call _space_ without there being at the same time the\npossibility, at least, of the _lack of space or spacelessness_. This is\na conclusion that is rigorously logical and incontrovertible.\n\nBut it has been urged that it is impossible for the mind to imagine\na condition where there is no space. It even has been asserted that\nit is contrary to the constitution of the mind itself to imagine \"no\nspace.\" But whether imaginable or not has no effect whatever upon the\nvalidity of the conception. Neither, it is said, can we imagine a\nfourth dimension but the mind has come dangerously near to imagining\nit. The distance from excogitating upon, discussing and describing\nthe properties of four-space to imagining it is not so great after\nall. Truly it is difficult indeed, it seems, to be able to describe a\nthing yet not be able to imagine or make a mental image of it. There\nis an evident fallacy here. Either the description of four-space is\nno description at all or it is a true delineation of an idealized\nconstruction which is well within the mind's powers of imagination.\nIndeed the question of imaginability is not determinative in itself;\nfor what the mind may now be unable to imagine, because of its more or\nless nebulous character, and owing to its infancy may in the course of\ntime be easily accomplished.\n\nThe universe is a compacted _plenum_. It is chock-full of mind, of\nlife, of energy and matter. These four are basically one. They exist,\nof course, in varying degrees of tenuity and intensity and answer to a\nwide range of vibrations. Together, in their manifestation of action\nand interaction, in their _dynamic appearance_, if you please, they\nconstitute space. If these were removed with all that their existence\nimplies there would result a condition of spacelessness in which no\none of the appearances which we now perceive would be possible. Even\nsheer extensity would be non-existent. All scope of motility would be\nlacking. Dimension, co\u00f6rdinates, direction, space-relations--all would\nbe impossible.\n\nA straight line is an ideal construction of the mind. It does not exist\nin nature. It can never be actualized in the phenomenal universe.\nBetween the ideal and the real, or actual, there is a kosmic chasm. It\nbroadens or narrows according as the phenomenal appearance approaches\nor recedes from the ideal. What, therefore, can be postulated of\nthe one will not apply with equal force to the other. They are not\ncongruent and can never be in the actualized universe. The moment the\nactual becomes identified with the ideal it ceases to be the actual.\nThe universe does not exist as _pure form_, neither does space.\nAs purely formal constructions of the intellect these can have no\nperceptible existence. The phenomenal or sensible may not be judged by\nexactly the same standard as the formal. The phenomenal or sensible\nrepresents things as they appear to the senses, or, so far as the\nactualized universe is concerned, _as they really are_. The _formal_\nrepresents things as they are made to appear by the mind. It cannot\nbe actualized. It may be said that the purely formal is the limit of\nevolution. The phenomenal may approach the ideal as a limit, but can\nnever become fully congruent with it. _The difference between the ideal\nand the actual is a dynamic one_; it is by virtue of this difference\nthat the universe is held in manifestation. Evolution is the decrement\nof this difference between the purely formal and the actual. So long\nthen as a kosmic differential is maintained the phenomenal continues\nto be manifest: when it is finally reduced to nothing it goes out of\nmanifestation. The phenomenal is finite; the ideal infinite.\n\nWherefore, it is undoubtedly improper to refer to space as being\ninfinite. The term really is inapplicable. Transfinity is much better\nand more accurate. Space is transfinite because its scope is greater\nthan any finite scope of motility can encompass, because it exceeds\nfinite comprehensibility.\n\nRIEMANN'S notion that space is limited gains weight in the light of the\nforegoing considerations. But he could not conceive of the limitability\nand unboundedness of space as such in its pure essence; but was\ncompelled, by his own limitations, to make an idealized construction in\nwhich he could actualize his conception. And for real, dynamic space,\nhe substituted his ideal construction and proceeded upon that basis.\nAnd of course, his view while it had no reference to perceptual space\nnevertheless possessed an illative relation thereto and should be\nrecognized as construable in that light.\n\nThe process of squaring the circle recognized as a geometric\nimpossibility is significant of the fluxional nature of the universal\nresiduum perpetually maintained between the archetypal and the\nmanifested kosmos. It seems that there is a profound truth embodied in\nthis problem. There is a lesson that may be learned by mathematicians,\nphilosophers, scientists and thinkers in general. There is an element\nof eternal necessity and universality about it which is truly symbolic\nof the finity of the universe and the infinity of the archetypal. Just\nas a square or a series of polygonal figures inscribed in a circle\ncannot be made to coincide exactly with the circle so cannot the actual\nbe made to coincide with the ideal. The circumference of the circle\nis the unapproachable limit of inscribed squares. If it were possible\nso to multiply squares thus inscribed that a figure coincident with\nthe circumference of a circle might be constructed, such a figure\nwould not be a square but a circle. The manifested universe is like\nthat--the process of inscribing squares within a circle. It is ever\n_becoming_, _evolving_, _developing_, but never quite attains. Infinity\nis a process. But no single stage in that process is infinite. Each\nis finite and their totality makes the infinity of the process. The\nuniverse manifested to the senses or the intellect is finite.\n\n\"Space,\" says PAUL CARUS, \"is the possibility of motion in all\ndirections.\"[14] To be sure, it is admitted that space offers\nopportunity for motion in all directions. But is space this opportunity\nof motility? Or is possibility of motion space? The possibility of\nmotion must rest in the thing that moves. It implies a potency in the\nmoving entity, not in space. If it is meant that space is the potency\nthat resides in the moving element it is still more difficult to\nunderstand the connotation. But even granting this view, are we not\ncompelled to recognize the dynamism of space as a necessary inference?\nAnother definition which CARUS gives is that space is a \"_pure form of\nextension_.\" If it be granted that space is a pure form of extension\nwe should have to conclude that it has no actual existence; for _pure\nform_ does not exist except as an idealized construction. It cannot\nbe found in nature. Pure form is _ideal_. Impure or natural form is\nactual. Therefore the space in which we live and in which the universe\nexists cannot be a \"pure form\" because life cannot exist in the purely\nformal. It is useless to talk about space as mere form so long as\nit maintains life. The difficulty which this phase of the question\npresents is another evidence of the inadequacy of our definitions.\n\n  [14] Vide _Foundations of Mathematics_, p. 107.\n\nIt is also found to be impossible to concur in CARUS' conception of\nknowledge _a priori_. His notion of the _a priori_ varies somewhat from\nthe Kantian view. He defines it as an \"idealized construction,\" the\n\"mind made,\" \"abstract thought,\" and places it in the same category as\na concept. This is undoubtedly born of his desire to get rid of KANT'S\n\"innate ideas\" which seem to be distasteful to him. But in doing so it\nappears that the real _a priori_ has been overlooked. Let us examine\nfor a moment this important question. The _a posteriori_ connotates all\nknowledge gained through the senses, or sense experience. All knowledge\ntherefore whose origin can be traced to the senses is knowledge _a\nposteriori_. Now, knowledge _a priori_ should be just the opposite of\nthis. It should indicate such knowledge as that which does not have\nits origin in the senses, or which is not dependent upon the ordinary\navenues of sense-experience. Abstract thought is as truly experience as\nsmelling, seeing or hearing. It is by traversing its scope of motility\nthat the mind finds out what the norms of logic are. It could not\nremain quiescent and discover them. It has to be active, examining,\ncomparing and judging. Almost the entire range of thought, its entire\nscope, is characterized by the _a posterioristic method_. In fact,\nall thought is _a posterioristic_. Despite the fact that, in thinking\nin the abstract, it is necessary mentally to remove all elements of\nconcreteness, all materiality and all actuality, the conclusions\nreached have to be referred to the standards maintained by the actual,\nthe concrete and the material. Then we do not start with the abstract\nin our thinking. We begin with the concrete and by mentally removing\nall physical qualities arrive at the abstract.\n\nThe mind has a constitution. It acts in a given way because it is its\nnature so to act. Not because it has learned to act in that manner. It\nperforms certain functions intuitively without previous instruction\nor experience for the same reason that water dampens or heat warms.\nIt is natural for it to do so. This naturalness, this performance of\nfunction without being taught or without experience constitute the\nprinciple of _apriority_ in the mind. _Aprioriness_ is a principle\nof mind partaking of the very nature and essence of mind. It is the\nvery mainspring of mentality. Perception and conception are processes\nwhich the mind performs intuitively. The mind perceives and conceives\nbecause it is impossible for the normal mind to do otherwise. We take\na view upon a given question; we assume certain mental attitudes of\naffirmation, negation or indifference because we have learned to do so\nby virtue of the tuitional capability of mind. These describe the _a\nposteriori_. That is, all knowledge obtained as a result of voluntary\nmental processes constitutes the mass of knowledge _a posteriori_. The\n_a priori_ is what the mind is by nature: the _a posteriori_ is what\nthe mind becomes. It is the mind-content.\n\nThe _a priori_ is not a mental construction; it is an essential\nprinciple of mind. It should not be identified with the \"purely\nformal,\" as is done by PAUL CARUS:[15]\n\n  [15] Vide _Foundations of Mathematics_, p. 42.\n\nHe says:\n\n  \"The _a priori_ is identical with the purely formal which\n  originates in our mind by abstraction. When we limit our attention\n  to the purely relational, dropping all other features out of\n  sight, we produce a field of abstraction in which we can construct\n  purely formal combination, such as numbers, or the ideas of types\n  and species. Thus we create a world of pure thought which has the\n  advantage of being applicable to any purely formal consideration\n  and we work out systems of numbers which, when counting, we can\n  use as standards of reference for our experience in practical\n  life.\"\n\nThus CARUS definitely links up the _a priori_ to a factor which is\nnothing more nor less than a mental by-product. For such is the\ncategory in which would be placed both the process of abstraction and\nits results. It is therefore exceedingly difficult to understand why\nso cursory a consideration should have been given to the principle of\n_apriority_ than which no other element of mind is more essentially a\npart of the mind itself.\n\nThe formal is symbolic. It signifies an informing quantity. Pure form\nitself is but a negation of that which formerly filled it. Then, too,\nthe formal is purely artificial because it is a mental construction.\nEssentially there is as much difference between the purely formal and\nthe _a priori_ as between creator and creature, as between potter and\nclay. The one is the builder, the other is the material; the one the\nknower and the other the known. Thus, the only reason that the formal\nis found to be answerable to the _a priori_ at all is due to the fact\nthat it is construable only upon the basis of the _a priori_. But\nbeing so is not sufficient warrant for its identification with the _a\npriori_. The formal merely represents the totality of possibilities in\nthe universe as viewed by the mind; but as the number of possibilities\nopen to the mind is, on account of its nature and purpose limited, it\nis not to be supposed that it (the mind) shall measure up to all the\npossibilities offered by the formal. Moreover, it is certain that no\nsane mind cherishes the hope that there shall ever be found in the\nuniverse of life and form a congruence for all of the possibilities\nheld out by the purely formal.\n\nAs an eternal principle of mind, the _a priori_ is in agreement with\nthe divine mind of the kosmos. In its _aposteriority_ the mind is of\ndiverse tendences, qualities and characteristics. Apriorily, it acts in\nunison with the eternal purpose of life and the universal mind. In its\naposteriority, it often goes awry. In its _apriority_ it can never be\ninsane; insanity is a symptom of the morbid _a posteriori_.\n\nThe mind in man acts the same as mind in the vegetal and lower animal\nkingdoms. Metabolism and katabolism, indeed all cell-activity, are\n_a priori_ performances of the mind. Growth and all its phenomena,\nthe cyclicism of natural processes, and every activity connected\ntherewith belong to the category of the _a priori_. Cells multiply,\ndivide, build up and tear down tissues and they do it intuitively. Most\ncertainly these functions are performed without any assistance from the\nintellect. All the myriad activities in nature with which the intellect\nin man has not the slightest concern, truly acting in accord with some\nprimordial impetus, are activities _a priori_.\n\nNow what is the attitude of the intellect, in the light of the _a\npriori_, towards space and the question of dimensionality? It is\nevident that no matter what this attitude may be it is in agreement\nwith the constitution of things and of the universe. And if so, it is\nright and without illusion. It is also evident that whatever notion\n_a posteriori_ the intellect may entertain with respect to these\nquestions is unavoidably liable to the illusionary drawbacks common to\nconclusions based upon limited experience. The geometric view of space\nbelongs to the category of the _a posteriori_. Hence it is subject to\nthe usual imposition of error.\n\nTersely stated, KANT's view of space is that it is a form of intuition,\na form _a priori_, a transcendental form. As such he considered it\nto be a native form of perception not belonging to the category of\nsense-deliveries. Accordingly, space is a form of intuition arising\nout of and inhering in the constitution of mind. It is a notion\nwhich constitutes the universal and eternal prerequisite of mind and\nis, therefore, intrinsically necessary to all phases of mentation.\nNow, this being true just what may be said to be the relation of\ndimensionality to this _a priori_ form of space which is found to\nexist in the mind as an eternal aspect of its nature? Does the mind\nintuitively measure its contents or its operations by the empirical\nstandard of space-measurement known as dimension? Is the attitude of\nthe mind towards the objectively real one of discrimination _a priori_\nas to the direction or dimension in which a percept may originate? In\nother words, does the mind habitually and intuitively refer its data\nto a system of co\u00f6rdinates for final determination? There is no other\nanswer but that the mind makes no such reference and is dependent upon\nno kind of co\u00f6rdinate system in any of its operations _a priori_. As a\nform of intuition, the space notion is present in the mind as a scope\nof existence, of motility, of being and of sheer roominess. The notion\nof direction or dimension, being an artificial construction, does not\nenter into this form of intuition at all. It is only when the mind\ncomes to elaborate upon its perceptive performances and possibilities\nthat the questions of relations, positions and directions arise.\nBut this latter is a matter separate and distinct from the state of\nawareness which embodies the notion of space.\n\nDimension is an arbitrary norm constructed by the mind for the\ndetermination of various positions in space. It is an accident or\nby-product of the process of elaborative cognition, a convenient\nand appropriate means of measurement for objects in space and their\nspace-relations. But it is no more _a priori_ than a foot rule or a\nsquare. But being purely an empirical product it may be said to be\nan aspect of psychogenesis because it relates to the evolutionary\naspect of mind. The assumption may therefore be allowed that the\nmind may, in the course of its evolution, find it convenient and\nappropriate to devise an additional ordinate or dimension to satisfy\nthe necessities of its more complex ramifications into the nature of\nthings and to determine their greatly increased space-relations. It\nmay be even possible for the mind to function normally in a space of\nfour dimensions. But this would simply be a new adjustment, not a\nchange in the essential nature of mind. It would be like the series of\nadjustments to environments which man has made in the onward movement\nof civilization. There has been no serious change in the manhood\n_per se_ of man. That has remained the same; there has been merely a\ncomplication of environmental influences. Similarly, in the acquisition\nof four-dimensional powers, granting that such an acquisition is\npossible, there is nothing to be added to the _aprioriness_ of mind\n_itself_. Is it not, therefore, logical to assume that the discovery\nof a fourth co\u00f6rdinate and the consequent conceptualization of the\nsame, point to the development in the mind of a greatly extended\nfaculty, more keenly penetrative powers of cognition and a further\ndiversification of its environments than it has hitherto enjoyed?\nIndeed, it seems so.\n\n\n\n\n  CHAPTER V\n\n  THE FOURTH DIMENSION\n\n  The Ideal and the Representative Nature of Objects in the\n  Sensible World--The Psychic Fluxional the Basis of Mental\n  Differences--Natural and Artificial Symbols--Use of Analogies to\n  Prove the Existence of a Fourth Dimension--The Generation of a\n  Hypercube or Tesseract--Possibilities in the World of the Fourth\n  Dimension--Some Logical Difficulties Inhering in the Four-Space\n  Conception--The Fallacy of the Plane-Rotation Hypothesis--C. H.\n  HINTON and Major ELLIS on the Fourth Dimension.\n\n\nThe world of mathesis is truly a marvelous domain. Vast are its\npossibilities and vaster still its sweep of conceivability. It is the\nkingdom of the mind where, in regal freedom, it may perform feats\nwhich it is impossible to actualize in the phenomenal universe. In\nfact, there is no necessity to consider the limitations imposed by\nthe actualities of the sensuous world. Logic is the architect of\nthis region, and for it there is no limit to the admissibility of\nhypotheses. These may be multiplied at will, and legitimately so. The\nchief error lies in the attempt to make them appear as actual facts of\nthe physical world.\n\nMathematicians, speculating upon the possibilities of mathetic\nconstructions and forgetting the necessary distinctions which should\nbe recognized as differentiating the two worlds, in their enthusiasm\nhave been led into the error of postulating as qualities of the\nphenomenal world the characteristics of the conceptual. Accordingly,\na great deal of confusion as to the proper limits and restrictions of\nthese conceptions has arisen and there still may be found those who are\nenthusiastically endeavoring to push the actualities of the physical\nover into the conceptual. But in assuming any attitude towards mathetic\npropositions, especially with a view to demonstrating their actuality,\nvery careful discrimination as to the essential qualities and their\nconnotations should be made. Hence, before taking up a brief study of\nthe fourth dimension proper, it is deemed fitting to indicate some of\nthe fundamental distinctions which every student of these questions\nshould be able to make with reference to the data which he meets.\n\nAll objects of the sensible world have both an essential or ideal\nnature and a representative or sensuous nature. That is, they may be\nstudied from the standpoint of the ideal as well as the sensuous.\nThe representative nature is that which we recognize as the mode of\nappearance to our senses which, as KANT held, is not the essential\nor ideal character of the thing itself. For there is quite as much\ndifference between the sensuous percept and the real thing itself as\nbetween an object and its shadow. In fact, a concept viewed in this\nlight, may be seen to have all the characteristics of an ordinary\nshadow; for instance, the shadow of a tree. View it as the sun is\nrising; it will then be seen to appear very much elongated, becoming\nless in length and more distinct in outline as the sun rises to a\nposition directly overhead. The elongation may again be seen when the\nsun is setting. Throughout the day as the sun assumes different angles\nwith reference to the tree the proportions and definiteness of the\nshadow vary accordingly. Thus the angularity of the sun, the intensity\nand fullness of the light and the shape and size of the tree operate to\ndetermine the character of the shadow.\n\nMuch the same thing is true of a sensuous representation. If we examine\ncarefully our ideas of geometric quantities and magnitudes, it will be\nfound that the concepts themselves are not identical with the objects\nof the physical world, but mere mental shadows of them. The angularity\nof consciousness, or the distinctness of one's state of awareness,\nbeing analogous to similar attitudes in the solar influence are the\nmain determinants of the character of the mental shadow or concept.\nWherefore mathematical \"spaces\" or magnitudes are not sensuous things\nand have therefore no more real existence than a shadow, and strictly\nspeaking not as much; for a shadow may be seen, while such magnitudes\ncan only be conceived. It may be urged that since we can conceive of\nsuch things they must have existence of some kind. And so they have,\nbut it is an existence of a different kind from that which we recognize\nas belonging to things in the sensible world. They have a conceptual\nexistence, but not a sensuous one. Therein lies the great difference.\n\nTo be sure, a shadow is a more or less true representation of the thing\nto which it pertains. That this is true can be established empirically.\nSimilarly, the degree of congruity between objects and concepts\nlikewise may be determined. If this were not true we should be very\nmuch disappointed with what we find in the phenomenal world and could\nnever be quite sure that the mentograph existing in our minds was a\nfaithful representation of the thing which we might be examining. But\nreally the foundation for such a disappointment is present in every\nconcept, every percept with which the mind deals. This disappointment,\nalthough in actual experience is reduced to an almost negligible\nquantity, is due to the failure of sensuous objects to conform wholly\nto the specific details of the mental shadow or mentograph. This\nlack of congruence between the mental picture and the object itself\nis necessary for obvious reasons. It is markedly observable in the\nearly efforts of a child in learning distances, weights, resistances,\ntemperatures and the like. No inconsiderable time is required for the\nchild to be able correctly to harmonize his sense-deliveries with\nactual conditions. Otherwise, the child would never make any of the\nludicrous mistakes of judgment of which it is guilty when trying to\nget its bearings in the world of the senses. In the course of time\nthe child gradually learns by experience that certain things are\ntrue of objects, distances, temperatures, resistances, etc., and\nthat certain things are not true of them. He learns these things by\nactually contacting various objects. He is then competent to render\ncorrect judgments, within certain limits, as to the conditions which\nhe finds in the sensible world. And the allowances, equations and\ncorrections which his motor, sensory and psychic mechanisms learn to\nmake in childhood serve for all subsequent time. And this is important\nto remember; for the mature mind is apt to forget or overlook the\nadaptations which the child-mind has made in its growth.\n\nIf there were no such differences between the concept and the thing\nitself, actual physical contact would not be necessary. For one could\nrely wholly upon the sense-deliveries and each sense might operate\nentirely independently of all the others as there would be no necessity\nto correct the delivery of one by those of the others. This, of course,\nraises the question as to the necessity of sense-experience at all\nunder conditions where there would be no disparity between the thing\nitself and the ideal representation of it in the mind. The absence of\nthis variable quantity would open to the mind the possibility of really\nknowing the essential nature of objects in the phenomenal world, a\ncondition of affairs which is admittedly now without the range of the\npowers of the mind.\n\nAt any rate, the essential \"thingness\" of objects can never be\ncomprehended by the mind until the diminution of this disparity between\nthe object of sense and the mental picture of it which exists in the\nconsciousness has proceeded to such a limit as either completely to\nhave obliterated it or to such an extent that the psychic fluxion is so\nslight as not to matter.\n\nIt is believed that the results of mental evolution, as the mind\napproaches the transfinite as a limit, will operate to minimize the\nfluxional quantity which subsists between all objects of sense and\ntheir ideal representation as data of consciousness. The conclusion\nthat the mind of early men who lived hundreds of thousands and perhaps\nmillions of years ago on this planet consumed a much longer time in\nlearning the adjustments between the objects which it contacted in the\nsensuous world and the elementary representations which were registered\nin its youthful consciousness than is to-day required for similar\nprocesses seems to be demanded, and substantiated as well, by what is\nknown of the phyletic development of the mind in the human race.\n\nIn view of the above, it is thought that the duration of such simple\nmental processes served not only to prolong the physical life of\nthe man of those early days, but may also account for the puerility\nand incapacity of the mind at that stage. Not that the slow mental\nprocesses were active causative agencies in lengthening the life\nof man, but that they together with the crass physicality of man\nnecessitated a longer physical life. This, perhaps in a larger\nsense than any other consideration, accounts for the fundamental\ndiscrepancies in the mind of the primitive man in comparison with the\nefficiency of the mind of the present-day man. In view of the potential\ncharacter of mind and in the light of the well graduated scale of its\naccomplishments, it is undoubtedly safe to conclude that the quality of\nmental capacities is proportional to the psychic fluxional which may\nexist at any time between the ideal and the essential or real. Mental\ndifferences and potentialities in general may be due to the magnitude\nof the psychic fluxional or differential that exists between the\nconceptual and the perceptual universe. In some minds it may be greater\nthan in others. The chasm between things-in-themselves and the mental\nnotion pertaining thereto may vary in a direct ratio to the individual\nmind's place in psychogenesis, and therefore, be the key to all mental\ndifferences in this respect.\n\nMost certain it is that there may be marked fluctuations in the\njudicial approach of minds towards any psychic end. In other words,\nthere is not only a fluxional or differential between the object and\nits representation, but also a differential between the approach\nof one mind and another in the judicial determination of notions\nconcerning ideas. In this way, differences of opinions as to the\nright and wrong of judgments arise. Indeed, there seem to be zones\nof affinity for minds of similar characteristics, or minds that have\nthe same degree of differential; so that, in choosing among the many\npossible judgments predicable upon a species of data, all those minds\nhaving the same degree of psychic differential discover a special\naffinity or agreement among themselves. Hence, we have cults, schools\nof thought, and various other sectional bodies that find a basis\nof agreement for their operations in this way. The outcome of this\nremarkable intellectual phenomenon is that there are as many different\nkinds of judgments as there are zones of affinity among minds. Various\nsystems of philosophy owe their existence to these considerations, and\nthe considerations themselves flow from the fact that all intellectual\noperations are essentially superficial; because there is no means by\nwhich they may penetrate to the steady flowing stream of reality which\npervades and sustains objects in the sensible world.\n\nIn view, therefore, of the foregoing and with special reference to\ngeometric constructions, it is necessary in approaching a study of\nthe four-space that it be understood at the outset that the fourth\ndimension can neither be actualized nor made objectively possible even\nin the slightest degree in the perceptual world; because it belongs\nto the world of pure thought and exists there as an \"extra personal\naffair,\" separate and distinct from the world of the senses.\n\nAs says SIMON NEWCOMB:[16]\n\n  \"The experience of the race and all the refinements of modern\n  science may be regarded as showing quite conclusively that, within\n  the limits of our experience, there is no motion of material\n  masses, in the direction of a fourth dimension, no physical agency\n  which we can assume to have its origin in regions to which matter\n  cannot move, when it has three degrees of freedom.\"\n\nThere is, however, no logical objection to the study of the fourth\ndimension as a purely hypothetical question, if by pursuit of the same\nan improvement of methods of research and of the outlook upon the field\nof the actual may be gained. Hence, it is with this attitude of mind\nthat we approach the consideration of the fourth dimension.\n\n  [16] Vide _Science_, Vol. VII, p. 2, No. 158, 1898.\n\nVarious efforts have been made to render the conception of a fourth\ndimension of space thinkable. The student of space has reasoned: \"We\nsay that there are three dimensions of space. Why should we stop here?\nMay there not be spaces of four dimensions and more?\" Or he has said:\n\"If 'A' may represent the side of a square, A^2 its area, and A^3 the\nvolume of a cube with edge equal to A; what may A^4, A^5 or A^{_n_th}\nrepresent in our space? The conclusion, with respect to the quantity\nA^4, has been that it should represent a space of four dimensions.\"\n\nAlgebraic quantities, however, represent neither objects in space nor\nspace qualities except in a purely conventional manner. All efforts to\njustify the objective existence of a fourth dimension based upon such\nreasoning will, therefore, fail; because the basis of such arguments\nis itself faulty. In the sentence: \"The man loves his bottle,\" the\nthing meant is not the bottle, but what the bottle contains. For the\npurpose of the figure the bottle signifies its contents. There is no\nmore real connection between the bottle and what it contains than\nbetween any word and the object for which it stands. Words are said\nto be symbols of ideas. But they are not natural symbols; they are\nconventional symbols, made for the purpose of cataloguing, indexing\nand systematizing our knowledge. Words can be divorced from ideas\nand objects, or rather have never had any real connection with them.\nThere are two classes of natural symbols, namely; _objects_ and\n_ideas_. These, objects and ideas, symbolize realities. Realities\nare imperceptible and incomprehensible to the intellect which has\naptitude only for a slight comprehension of the symbols of realities.\nFor instance, a tree is a natural symbol. It represents an actuality\nwhich is imperceptible to the intellect. The intellect can deal only\nwith the sensible symbol. It is a natural symbol; because it is\npossible directly to trace a living connection between the tree and\nthe _tree-reality_. That is, it would be possible so to trace out the\nvital connection between the tree and its reality if the intellect\nhad aptitude for such tracery. But, in reality, since it has no such\naptitude, it remains for the work of that higher faculty than the\nintellect which recognizes both the connection and the intellect's\ninability to trace it. Further, an object is called a natural symbol\nbecause it is the bridge between sensuous representation and reality.\nIt is as if one could begin at the surface of an object and by a\nsubtle process of elimination and excortication arrive at the heart\nof the universum of reality. No such consummation may be reached by\ndealing with words which have merely an artificial relationship with\nthe objects which they signify. Again, ideas, that is, ideas that are\nuniversal in application and have their roots in the great ocean of\nreality, are natural symbols; because if it were possible to handle\nan idea with the physical hands it would be possible to arrive at the\nheart of that which it symbolized without ever losing our connection\nwith the idea itself. In other words, ideas and objects, unlike words,\ncan never be divorced from that which they symbolize. Both, being of\nthe same class, are the opposite poles of realities. This then is\nthe difference between natural symbols and artificial symbols--that\na natural symbol, such as objects and ideas, is copolar with reality\nwhereas an artificial symbol, such as words, geometric constructions\nand the like not only lacks this copolarity but is itself a symbol of\nnatural symbols.\n\nIt is, therefore, inconceivable that because the algebraic quantity\nA^3 has been arbitrarily decreed to be a representation of the volume\nof a cube, every such quantity in the algebraic series shall actually\nrepresent some object or set of objects in the physical world. Even\nif it be granted that such may be the case, is it not certain that\nthere is a limit to things in the objective universe? Yet there may\nnot be any limit to algebraic or mathematical determinations. The\nmaterial universe is limited and conditioned; the world of mathesis is\nunlimited and unconditioned save by its own limitations and conditions.\nIt is irrational to expect that physical phenomena shall justify all\nmathematical predicates.\n\n_There is perhaps no single mathematical desideratum or consideration\nwhich may be said to be the natural symbolism of realities; for the\nwhole of mathematical conclusions is a mass of artificial and arbitrary\nbut concordant symbols of the crasser or nether pole of the antipodes\nof realism._ It is exceedingly dangerous, therefore, to predicate\nupon such a far-fetched symbolism as mathematics furnishes anything\npurporting to deal with ultimate realities. And those who insist upon\ndoing so are either blind themselves to these limitations or are madly\nendeavoring to befog the minds of others who are dependent upon them\nfor leadership in questions of mathematical import.\n\nAnalogies have been unsparingly used in efforts to popularize the\nfour-space conception and much of the violence which has been done\nto the notion is due to this vagary. The mathematical publicist, in\ntrying to give a mental picture of the fourth dimension, examines the\nappearances of three dimensional beings as they might appear to a two\ndimensional being or _duodim_. He imagines a race of beings endowed\nwith all the human faculties except that they live in a land of but\ntwo dimensions--length and breadth. He thinks of them as shadows of\nthree dimensional beings to whom there are no such conceptions as \"up\"\nand \"down.\" They can see nothing nor sense anything in any way that\nis without their plane. They can move in any direction within the\nplane in which they live, but can have no idea of any movement that\nmight carry them without that plane. A house for such beings might be\nsimply a series of rectangles. One of them might be as safe behind\na line as a _tridim_ or three dimensional being would be behind a\nstone wall. A bank safe for the _unodim_ would be a mere circle. A\n_duodim_ in any two dimensional prison might be rescued by a tridim\nwithout the opening of doors or the breaking of walls. An action of\na _tridim_ performed so as to contact their plane would be to them a\nmiracle, absolutely unaccountable upon the basis of any known fact to\nthe _unodim_ or _duodim_. A _tridim_ might go into a house where lived\na family of _duodims_, appear and disappear without being detected\nor its ever being discovered how he accomplished such a marvelous\nfeat. Our miracles, after the same fashion, are said to be the antics\nof some four dimensional being who has similar access to our three\ndimensional world and whose actions are similarly inexplicable to us.\nSo the analogies have been multiplied. But the temptation to apply the\nconsequences of such reasoning to actual three-space conditions has\nbeen so great that many have yielded to it and have consequently sought\nactually to explain physical phenomena upon the basis of the fourth\ndimension.\n\nThe utilitarian side of the question of hyperspace has not been\nneglected either. And so, early in the development of the hypothesis\nand its various connotations, the attention of investigators was turned\nto this aspect of the inquiry. Strange possibilities were revealed as\na result. For instance, it was found that an expert fourth dimensional\noperator is possessed of extraordinary advantages over ordinary\ntridimensional beings. Operating from his mysterious hiding place in\nhyperspace, he could easily appear and disappear in so mysterious a\nmanner that even the most strongly sealed chests of treasures would\nbe easily and entirely at his disposal. No city police, Scotland Yard\ndetective nor gendarme could have any terrors for him. DRS. JEKYLL\nand MESSRS. HYDE might abound everywhere without fear of detection.\nObjects as well as persons might be made to pass into or out of\nclosed rooms \"without penetrating the walls,\" thus making escape easy\nfor the imprisoned. No tridimensional state, condition or system of\narrangements, accordingly, would be safe from the ravages of evilly\ninclined four dimensional entities. Objects that now are limited to a\npoint or line rotation could in the fourth dimension rotate about a\nplane and thus further increase the perplexities of our engineering\nand mechanical problems; four lines could be erected perpendicular\nto each other whereas in three space only three such lines can be\nerected; the right hand could be maneuvered into the fourth dimension\nand be recovered as a left hand; the mysteries of growth, decay and\ndeath would find a satisfactory explanation on the basis of the\nfourth dimensional hypothesis and many, if not all, of the perplexing\nproblems of physiology, chemistry, physics, astronomy, anthropology\nand psychology would yield up their mysteries to the skill of the\nfourth dimensional operator. Marvelous possibilities these and much\nto be desired! But the most remarkable thing about these so-called\npossibilities is their impossibility. It is this kind of erratic\nreasoning that has brought the conception of a fourth dimension into\ngeneral disrepute with the popular mind. It is to be regretted, too,\nfor the notion is a perfectly legitimate one in the domain of mathesis\nwhere it originated and rightly belongs.\n\nIt is not to be wondered at that metageometricians and others should\nat first surmise that, in the four-space, they had found the key to\nthe deep mysteries of nature in all branches of inquiry. For so vast\nwas the domain and so marvelous were the possibilities which the new\nmovement revealed that it was to be expected that those who were\nprivileged to get the first glimpses thereof would not be able to\nrealize fully their significance. But the stound of their minds and\nthe attendant magnification of the elements which they discovered\nwere but incidents in the larger and more comprehensive process of\nadjustment to the great outstanding facts of psychogenesis which is\nonly faintly foreshadowed in the so-called hyperdimensional. The whole\nscope of inquiry connected with hyperspace is not an end in itself.\nIt is merely a means to an end. And that is the preparation of the\nhuman mind for the inborning of a new faculty and consequently more\nlargely extended powers of cognition. Metageometrical discoveries are\ntherefore the excrescences of a deeper, more significant world process\nof mental unfoldment. They belong to the matutinal phenomena incident\nto this new stage of mental evolution. All such investigations are but\nthe preliminary exercises which give birth to new tendencies which are\ndestined to flower forth into additional faculties and capacities. So\nthat it is well that the evolutionary aspect of the question be not\noverlooked; for there is danger of this on account of the magnitude and\nkosmic importance of its scope of motility.\n\nA geometric line is said to be a space of one dimension. A plane is a\nspace of two dimensions and a cube, a space of three dimensions. In\nfigure 7 below, the line _ab_ is said to be one dimensional; because\nonly one co\u00f6rdinate is necessary to locate a point-position in it.\nThe plane, _abcd_, figure 8, is said to be two dimensional because\ntwo co\u00f6rdinates, _ab_ and _db_ are required to locate a point, as the\npoint _b_. The cube _abcdefgh_, figure 9, is said to be tridimensional,\nbecause, in order to locate the point _b_, for instance, it is\nnecessary to have three co\u00f6rdinates, _ab_, _bc_ and _gb_. The tesseract\nis said to be four dimensional, because, in order to locate the point\n_b_, in the tesseract, it is necessary to have four co\u00f6rdinates, _ab_,\n_bc_, _bb'_ and _h'b_, figure 10.\n\n    A -------------------------------- B\n\n    FIG. 7.\n\n\n    A +------------------------------+ B\n      |                              |\n      |                              |\n      |                              |\n      |                              |\n      |                              |\n      |                              |\n    C +------------------------------+ D\n\n    FIG. 8.\n\nIt will be noted that in figures 8, 9 and 10, the element of\nperpendicularity enters as a necessary determination. In figure 8, the\nlines _ab_ and _bd_ are perpendicular to each other. Similarly, in Fig.\n10, lines _ab_, _bc_, _bb'_ and _h'b_ are perpendicular to one another.\nThat is, at their intersections, they make right angles. Similarly,\nfigures representing any number of dimensions may be constructed.\n\n[Illustration: FIG. 9.]\n\n[Illustration: FIG. 10.--The Tesseract.]\n\nThe line _ab_ represents a one-space. An entity living in a one space\nis called a \"unodim.\" The plane, _abcd_, represents a two-space,\nand entities living in such a space are called _duodims_. The cube,\n_abcdefgh_, represents a three-space and entities inhabiting such a\nspace are called _tridims_. Figure 10 represents a four-space, and its\ninhabitants are called _quartodims_. Each of the above-mentioned spaces\nis said to have certain limitations peculiar to itself.\n\nThe fourth dimension is said to lie in a direction at right angles to\neach of our three-space directions. This, of course, gives rise to the\npossibility of generating a new kind of volume, the hypervolume. The\nhypercube or tesseract is described by moving the generating cube in\nthe direction in which the fourth dimension extends. For instance, if\nthe cube, Fig. 9, were moved in a direction at right angles to each\nof its sides a distance equal to one of its sides, a figure of four\ndimensions, the tesseract, would result.\n\nThe initial cube, _abcc'e'fhh'_, when moved in a direction at right\nangles to each of its faces, generates the hypercube, Fig. 10. The\nlines, _aa'_, _bb'_, _cc'_, _dd'_, _ee'_, _ff'_, _gg'_, _hh'_, are\nassumed to be perpendicular to the lines meeting at the points, _a_,\n_b_, _c_, _d_, _e_, _f_, _g_, _h_. Hence _a'b'_, _b'd_, _dd'_, _d'a'_,\n_ef_, _fg_, _gg'_, _g'e_, represent the final cube resulting from\nthe hyperspace movement. Counting the number of cubes that compose\nthe hypercube we find that there are eight. The generating cube,\n_abcc'e'f'hh'_, and the final cube, _a'b'_, _b'd_, _dd'_, _d'a'_, _ef_,\n_fg_, _gg'_, _g'e_, make two cubes; and each face generates a cube\nmaking eight in all. A tesseract, therefore, is a figure bounded by\neight cubes.\n\nTo find the different elements of a tesseract, the following rules will\napply:\n\n1. _To find the number of lines_: Multiply the number of lines in the\ngenerating cube by two, and add a line for each point or corner in it.\nE.g., 2 \u00d7 12 = 24 + 8 = 32.\n\n2. _To find the number of planes, faces or squares_: Multiply the\nnumber of planes in the generating cube by 2 and add a plane for each\nline in it. E.g., 2 \u00d7 6 + 12 = 24.\n\n3. _To find the number of cubes in a hypercube_: Multiply the number of\ncubes in the generating cube, one, by two and add a cube for each plane\nin it. E.g., 2 \u00d7 1 + 6 = 8.\n\n4. _To find the number of points or corners_: Multiply the number of\ncorners in the generating cube by 2. E.g., 2 \u00d7 8 = 16.\n\n       *       *       *       *       *\n\nIn a plane there may be three points each equally distant from one\nanother. These may be joined, forming an equilateral triangle in which\nthere are three vertices or points, three lines or sides and one\nsurface.\n\nIn three-space there may be four points each equidistant from the\nothers. At the vertices of a regular tetrahedron may be found such\npoints. The tetrahedron has four points, one at each vertex, 6 lines\nand 4 equilateral triangles, as in Fig. 11.\n\nIn four-space, we have 5 points each equidistant from all the rest,\ngiving the hypertetrahedron. This four dimensional figure may be\ngenerated by moving the tetrahedron in the direction of the fourth\ndimension, as in Fig. 12. If a plane be passed through each of the\nsix edges of the tetrahedron and the new vertex there will be six new\nplanes or faces, making 10 in all, counting the original four. From\nthe new vertex there is also a tetrahedron resting upon each base of\nthe original tetrahedron so that there are five tetrahedra in all.\n_A hypertetrahedron is a four-dimensional figure consisting of five\ntetrahedra, ten faces, 10 lines and 5 points._\n\n[Illustration: FIG. 11.--Tetrahedron.]\n\n[Illustration: FIG. 12.--Hypertetrahedron.]\n\nPAUL CARUS[17] suggests the use of mirrors so arranged that they\ngive eight representations of a cube when placed at their point of\nintersection. He says:\n\n  \"If we build up three mirrors at right angles and place any object\n  in the intersecting corner we shall see the object not once, but\n  eight times. The body is reflected below and the object thus\n  doubled is mirrored not only on both upright sides but in addition\n  in the corner beyond, appearing in either of the upright mirrors\n  coincidingly in the same place. Thus the total multiplication of\n  our tridimensional boundaries of a four dimensional complex is\n  rendered eight-fold.\n\n  \"We must now bear in mind that this representation of a fourth\n  dimension suffers from all the faults of the analogous figure\n  of a cube in two dimensional space. The several figures are not\n  eight independent bodies but are mere boundaries and the four\n  dimensional space is conditioned by their interrelation. It is\n  that unrepresentable something which they inclose, or in other\n  words, of which they are assumed to be boundaries. If we were\n  four dimensional beings we could naturally and easily enter into\n  the mirrored space and transfer tridimensional bodies or parts\n  of them into those other objects reflected here in the mirrors\n  representing the boundaries of the four dimensional object. While\n  thus on the one hand the mirrored pictures would be as real as the\n  original object, they would not take up the space of our three\n  dimensions, and in this respect, our method of representing the\n  fourth dimension by mirrors would be quite analogous to the cube\n  pictured on a plane surface, for the space to which we (being\n  limited to our tridimensional space-conception), would naturally\n  relegate the seven additional mirrored images is unoccupied and if\n  we should make the trial, we would find it empty.\"\n\n  [17] Vide _Foundations of Mathematics_, pp. 93-94.\n\nThe utility of such a representation as that which CARUS outlines in\nthe above is granted, i.e., so far as the purpose which it serves in\ngiving a general idea of what a four-space object might be imagined\nto be like, but the illustration does not demonstrate the existence\nof a fourth dimension. It only shows what might be if there were a\nfour-space in which objects could exist and be examined. We, of course,\nhave no right to assume that because it can be shown by analogous\nreasoning that certain characteristics of the fourth dimensional object\ncan be represented in three-space the possible existence of such an\nobject is thereby established. Not at all. For there is no imaginable\ncondition of tridimensional mechanics in which an object may be said to\nhave an objective existence similar to that represented by the mirrored\ncube.\n\nBut there are discrepancies in this representation which well might be\nconsidered. They have virtually the force of invalidating somewhat the\nconception which the analogy is designed to illustrate. For instance,\nin the case of the mirrored object placed at the point of intersection\nof the three mirrors built up at right angles to each other. Upon\nexamination of such a construction it is found that the reflection of\nthe object in the mirrors has not any perceptible connection with the\nobject itself. And this, too, despite the fact that they are regarded\nas boundaries of the hypercube; especially is this true when it is\nnoted that these reflections are called upon to play the part of real,\npalpable boundaries. If a fourth dimensional object were really like\nthe mirror-representation it would be open to serious objections from\nall viewpoints. The replacement of any of the boundaries required in\nthe analogy would necessarily mean the replacement of the hypercube\nitself. In other words, if the real cube be removed from its position\nat the intersection of the mirrors no reflection will be seen, and\nhence no boundaries and no hypercube. The analogy while admittedly\npossessing some slight value in the direction meant, is nevertheless\nvalueless so far as a detailed representation is concerned. So the\nanalogy falls down; but once again is the question raised as to whether\nthe so-called fourth dimension can be established or proven at all upon\npurely mathematical grounds. It also emphasizes the necessity for a\nclearer conception of the meaning of dimension and space.\n\nThe logical difficulties which beset the hyperspace conception are\ndwelt upon at length by JAMES H. HYSLOP. He says:[18]\n\n  \"The supposition that there are three dimensions instead of one,\n  or that there are only three dimensions is purely arbitrary,\n  though convenient for certain practical purposes. Here the\n  supposition expresses only differences of directions from an\n  assumed point. Thus what would be said to lie in a plane in one\n  relation would lie in the third dimension in another. There\n  is nothing to determine absolutely what is the first, second,\n  or third dimension. If the plane horizontal to the sensorium\n  be called plane dimension, the plane vertical to it will be\n  called solid, or the third dimension, but a change of position\n  will change the names of these dimensions without involving the\n  slightest qualitative change or difference in meaning.\n\n  \"Moreover, we usually select three lines or planes terminating\n  vertically at the same point, the lines connecting the three\n  surfaces of a cube with the same point, as the representative of\n  what is meant by three dimensions, and reduce all other lines\n  and planes to these. But interesting facts are observable here.\n  1. If the vertical relation between two lines be necessary for\n  defining a dimension, then all other lines than the specified ones\n  are either not in any dimension at all, or they are outside the\n  three given dimensions. This is denied by all parties, which only\n  shows that a vertical relation to other lines is not necessary to\n  the determination of a dimension. 2. If lines outside the three\n  vertically intersecting lines still lie in dimension or are\n  reducible to the other dimensions they may lie in more than one\n  dimension at the same time which after all is a fact. This only\n  shows that qualitatively all three dimensions are the same and\n  that any line outside of another can only represent a dimension in\n  the sense of _direction_ from a given point or line, and we are\n  entitled to assume as many dimensions as we please, all within\n  three dimensions.\n\n  \"This mode of treatment shows the source of the illusion about\n  the 'fourth dimension.' The term in its generic import denotes\n  commensurable quality and denotes only one such quality, so that\n  the property supposed to determine non-Euclidean geometry must\n  be qualitatively different from this, if its figures involve the\n  necessary qualitative differentiation from Euclidean mathematics.\n  But this would shut out the idea of 'dimension' as its basis\n  which is contrary to the supposition. On the other hand, the term\n  has a specific meaning which as different qualitatively from the\n  generic includes a right to use the generic term to describe them\n  differentially, but if used only quantitatively, that is, to\n  express direction as it, in fact, does in these cases, involves\n  the admission of the actual, not a supposititious, existence of\n  a fourth dimension which again is contrary to the supposition\n  of the non-Euclidean geometry. Stated briefly, dimension as\n  commensurable quality makes the existence of the fourth dimension\n  a transcendental problem, but as mere direction, an empirical\n  problem. And the last conception satisfies all the requirements\n  of the case because it conforms to the purely quantitative\n  differences which exist between Euclidean and non-Euclidean\n  geometry as the very language about 'surfaces,' 'triangles,' etc.,\n  in spite of the prefix 'pseudo,' necessarily implies.\"\n\n  [18] Vide _Philosophical Review_, Vol. V, 1896, p. 352, et.\n  seq.\n\nThus it would seem that those who have been most diligent in\nconstructing the hyperspace conception have been the least careful\nof the logical difficulties which beset the elaboration of their\nassumptions. Yet it sometimes requires the illogical, the absurd and\nthe aberrant to bring us to a right conception of the truth, and when\nwe come to a comparison of the two, truth and absurdity, we are the\nmore surprised that error could have gained so great foothold in face\nof so overwhelming evidences to the contrary.\n\nThe entire situation is, accordingly, aptly set forth by HYSLOP when he\nsays, continuing:\n\n  \"There are either a confusion of the abstract with the concrete\n  or of quantitative with qualitative logic, ... so that all\n  discussion about a fourth dimension is simply an extended mass\n  of equivocations turning upon the various meanings of the\n  term 'dimension.' This when once discovered, either makes the\n  controversy ridiculous or the claim for non-Euclidean properties a\n  mere truism, but effectually explodes the logical claims for a new\n  dimensional quality of space as a piece of mere jugglery in which\n  the juggler is as badly deceived as his spectators. It simply\n  forces mathematics to transcend its own functions as defined by\n  its own advocates and to assume the prerogatives of metaphysics.\"\n\nShall we, therefore, assent to the imperialistic policy of\nmathematicians who would fain usurp the preserves of the metaphysician\nin order that they may exploit a superfoetated hypothesis? It is not\nbelieved that the harshness of HYSLOP'S judgment in this respect\nis undeserved. It is, however, regretted that the notions of\nmathematicians have been so inchoate as to justify this rather caustic,\nthough appropriate criticism. For it does appear that the moment the\nmathematician deserts the province of his restricted sphere of motility\nand enters the realm of the transcendental, that moment he loses his\nway and becomes an inexperienced mariner on an uncharted sea.\n\nIt is interesting to note that CASSIUS JACKSON KEYSER,[19] while\nrecognizing the purely arbitrary character of the so-called\ndimensionality of space, nevertheless lends himself to the view that\n\"if we think of the line as generating element we shall find that our\nspace has four dimensions. That fact may be seen in various ways, as\nfollows:\n\n  \"A line is determined by any two of its points. Every line pierces\n  every plane. By joining the points of one plane to all the points\n  of another, all the lines of space are obtained. To determine\n  a line, it is, then, enough to determine two of its points,\n  one in the one plane and one in the other. For each of these\n  determinations two data, as before explained, are necessary and\n  sufficient. The position of the line is thus seen to depend upon\n  four independent variables, and the four dimensionality of our\n  space _in lines_ is obvious.\"\n\n  [19: Vide _Monist_, Vol. XVI, 1896, Mathematical\n  Emancipations.]\n\nSimilarly he argues for the four dimensionality of space in spheres:\n\n  \"We may view our space as an assemblage of its spheres. To\n  distinguish a sphere from all other spheres, we need to know four\n  and but four independent facts about it, as say, three that shall\n  determine its center and one its size. Hence our space is four\n  dimensional also in spheres. In circles, its dimensionality is\n  six; in surfaces of second order (those that are pierced by a\n  straight line in two points), nine; and so on ad infinitum.\"\n\nThe view taken by KEYSER is a typical one. It is the mathematical\nview and is characterized by a certain lack of restraint which is\nfound to be peculiar to the whole scheme of thought relating to\nhyperspace. It is clear that the kind of space that will permit\nof such radical changes in its nature as to be at one time three\ndimensional, at another time four dimensional, then six, nine and\neven _n_-dimensional is not the kind of space in which the objective\nworld is known to exist. Indeed, it is not the kind of space that\nreally exists at all. In the first place, a line cannot generate\nperceptual space. Neither can a circle, nor a sphere nor any other\ngeometrical construction. It is, therefore, not permissible, except\nmathematically, to view our space either as \"an assemblage of its\nspheres,\" its circles or its surfaces; for obviously perceptual space\nis not a geometrical construction even though the intellect naturally\nfinds inhering in it a sort of latent geometrism which is kosmical.\nFor there is a wide difference between that kosmic order which is\nspace and the finely elaborated abstraction which the geometer\ndeceives himself into identifying with space. There is absolutely\nneither perceptible nor imperceptible means by which perceptual space\nin anywise can be affected by an act of will, ideation or movement.\nJust why mathematicians persist in vagarizing upon the generability\nof space by movement of lines, circles, planes, etc., is confessedly\nnot easily understood especially when the natural outcome of such\nprocedure is self-stultification. It is far better to recognize, as\na guiding principle in all mathematical disquisitions respecting the\nnature of space that the possibilities found to inhere in an idealized\nconstruction cannot be objectified in kosmic, sensible space. The line\nof demarkation should be drawn once for all, and all metageometrical\ncalculations and theories should be prefaced by the remark that: \"if\nobjective space were amenable to the peculiarities of an idealized\nconstruction such and such a result would be possible,\" or words to\nthat effect. This mode of procedure would serve to clarify many if\nnot all of the hyperspace conceptions for the non-mathematician as\nwell as for the metageometricians themselves, especially those who are\nunwilling to recognize the utter impossibility of their constructions\nas applied to perceptual space. We should then cease to have the\nspectacle of otherwise well-demeanored men committing the error of\ntrying to realize abstractions or abstractionizing realities. Herein\nis the crux of the whole matter, that mathematicians, rather than be\ncontent with realities as they find them in the kosmos, should seek\nto reduce them to abstractions, or, on the other hand, make their\nabstractions appear to be realities.\n\nKEYSER proceeds to show how the concept of the generability of\nhyperspace may be conceived by beginning with the point, moving it in\na direction without itself and generating a line; beginning with the\nline, treating it similarly, and generating a plane; taking the plane,\nmoving it in a direction at right angles to itself and generating a\ncube; finally, using the cube as generating element and constructing\na four-space figure, the tesseract. Now, as a matter of fact, a\npoint being intangible cannot be moved in any direction neither can\na point-portion of sensible space be removed. Nevertheless, we quite\nagree with him when he asserts:\n\n  \"Certainly there is naught of absurdity in supposing that _under\n  suitable stimulation the human mind may, in the course of time,\n  speedily develop a spatial intuition of four or more dimensions_.\"\n  (The italics in the above quotation are ours.)\n\nHere we have a tacit implication that the notion which geometers\nhave heretofore designated as \"dimension\" really is a matter of\nconsciousness, of intuition, and therefore, determinable only by the\nlimitations of consciousness and the deliveries of our intuitive\ncognitions. As a more detailed discussion of this phase of the subject\nshall be entered into when we come to a consideration of Chapter VI on\n\"Consciousness as the Norm of Space Determinations\" further comment is\ndeferred until then.\n\nNow, as it appears certain that what geometers are accustomed to call\n\"dimension\" is both relative and interchangeable in meaning--the one\nbecoming the other according as it is viewed--the conclusion very\nnaturally follows that neither constructive nor symbolic geometry\nis based upon dimension as commensurable quality. The real basis of\nthe non-Euclidean geometry is dimension as direction. For whatever\nelse may be said of the fourth dimension so-called it is certainly\nunthinkable, even to the metageometricians, when it is absolved from\ndirection although no specific direction can be assigned to it. It\nis agreed perhaps among all non-Euclidean publicists that the fourth\ndimension must lie in a \"direction which is at right angles to all the\nthree dimensions.\" But if they are asked how this direction may be\nascertained or even imagined they are nonplused because they simply do\nnot know. The difficulty in this connection seems to hinge about the\nquestion of identifying the conditions of the world of phantasy with\nthose of the world of sense. There are distortions, ramifications,\nsubmersibles, duplex convolutions and other mathetic acrobatics which\ncan be performed in the realm of the conceptual the execution of which\ncould never be actualized in the objective world. Because these antics\nare possible in the premises of the mathematical imagination is scarce\njustification for the attempts at reproduction in an actualized and\nphenomenal universe.\n\nOne of the proudest boasts of the fourth dimensionist is that\nhyperspace offers the possibility of a new species of rotation, namely,\n_rotation about a plane_. He refers to the fact that in the so-called\none-space, rotation can take place only about a point. For instance\nin Figure 7, the line _ab_ represents a one-space in which rotation\ncan take place only about one of the two points _a_ and _b_. In\nFigure 8 which represents a two-space, rotation may take place about\nthe line _ab_ or the line _cd_, etc., or, in other words, the plane\n_abcd_ can be rotated on the axial line _ab_ in the direction of the\nthird dimension. In tridimensional space only two kinds of rotation\nare possible, namely, rotation about a point and about a line. In the\nfourth dimension it is claimed that rotation can take place about a\nplane. For example, the cube in Figure 9, by manipulation in the\ndirection of the fourth dimension, can be made to rotate about the side\n_abgf_.\n\nA very ingenious argument is used to show how rotation about a plane\nis thinkable and possible in hyperspace. But with this, as with the\nentire fabric of hyperspace speculations, dependence is placed almost\nentirely upon analogous and symbolic conceptions for evidence as to the\nconsistency and rationality of the conclusions arrived at.\n\n      D             C            D1\n      +-------------+.............\n      |             |            :\n      |             |            :\n      |             |            :\n      |             |            :\n      |             |            :\n      +-------------+.............\n      A             B            A1\n\n    FIG. 13.\n\nIt is urged that inasmuch as the rotation about the line _bc_ in Figure\n13 would be incomprehensible or unimaginable to a plane being for the\nreason that such a rotation involves a movement of the plane into the\nthird dimension, a dimension of which the plane being has no knowledge,\nin like manner rotation about a plane is also unimaginable or\nincomprehensible to a tridim or a three dimensional being. It is shown,\nhowever, that the plane being, by making use of the possibilities of an\n\"assumed\" tridimension, could arrive at a rational explanation of line\nrotation.\n\n[Illustration: FIG. 14.]\n\nFigure 14 offers an illustration by means of which a two dimensional\nmathematician could demonstrate the possibility of line rotation.\nHe is already acquainted with rotation about a point; for it is the\nonly possible rotation that is observable in his two dimensional\nworld. By conceiving of a line as an infinity or succession of points\nextending in the same direction; by imagining the movement of his\nplane in the direction of the third dimension thereby generating a\ncube and at the same time assuming that the lines thus generated were\nmerely successions of points extending in the same direction, he could\ndemonstrate that the entire cube Figure 14, could be rotated about\nthe line _BHX_ used as an axis. For upon this hypothesis it would be\narguable that a cube is a succession of planes piled one upon the other\nand limited only by the length of the cube which would be extending\nin the, to him, unknown direction of the third dimension. He could\nvery logically conclude that as a plane can rotate about a point, a\nsuccession of planes constituting a tridimensional cube, could also\nbe conceived as rotating about a line which would be a succession\nof points under the condition of the hypothesis. His demonstration,\ntherefore, that the cube, Figure 14, can be made to rotate around\nthe line _BHX_ would be thoroughly rational. He could thus prove\nline-rotation without even being able to actualize in his experience\nsuch a rotation.\n\nAnalogously, it is sought by metageometricians to prove in like manner\nthe possibility of rotation about a plane. Thus in Figure 16 is shown a\ncube which has been rotated about one of its faces and changed from its\ninitial position to the position it would occupy when the rotation had\nbeen completed or its final position attained.\n\n[Illustration: FIG. 15.]\n\n[Illustration: FIG. 16.--Plane Rotation]\n\nThe gist of the arguments put forward as a basis for plane-rotation\nis briefly stated thus: The face _cefg_ is conceived as consisting of\nan infinity of lines. A cube, as in Figure 15, is imagined or assumed\nto be sected into an infinity of such lines, each line being the\nterminus of one of the planes which make up the cube. Each one of the\nconstituting planes is thought of as rotating about its line-boundary\nwhich intersects the side of the cube. The process is continued\nindefinitely until the entire series of planes is rotated, one by\none, around the series of lines which constitute the axial plane.\nHence, in order that the cube, Figure 16, may change from its initial\nposition to its final position each one of the infinitesimal planes\nof which the cube is assumed to be composed must be made to rotate\nabout each one of the infinitesimal lines of which the plane used as\nan axis is composed. In this way, it is shown that the entire cube\nhas been made to rotate about its face, _cefg_. This concisely, is\nthe \"QUOD ERAT DEMONSTRANDUM\" of the metageometrician who sets out to\nprove rotation about a plane. Thus it is made to appear that in order\nthat tridimensional beings may be enabled to conceive of four-space\nrotation, as in Figures 15 and 16, in which the rotation must also be\nthought of as taking place in the direction of the fourth dimension,\nthey must adopt the same tactics that a two dimensional being would use\nto understand some of the possibilities of the tridimensional world.\n\nIt is, of course, unwise to assume that because a thing can be\nshown to be possible by analogical reasoning its actuality is\nthereby established. This consideration cannot be too emphatically\ninsisted upon; for many have been led into the error by relying too\nconfidentially upon results based upon this line of argumentation.\nThere is a vast difference between mentally doing what may be assumed\nto be possible, the hypothetical, and the doing of what is actually\npossible, the practical.\n\nIn the first place, plane-rotation in the actual universe is a\nstructural impossibility. The very nature and constitution of material\nbodies will not admit of such contortion as that required by the\nrotation of a body, say a cube, about one of its faces. Let us examine\nsome of the results of plane rotation. 1. The rotation must take\nplace in the direction of the fourth dimension. Now, as it is utterly\nimpossible for any one, whether layman or metageometrician, even to\nimagine or conceive, in any way that is practical, the direction of\nthe fourth dimension it is also impossible for one to move or rotate\na plane, surface, line or any other body in that direction. We are\nin the very beginning of the process of plane-rotation so-called\nconfronted with a physical impossibility. 2. Plane rotation necessarily\ninvolves the orbital diversion of every particle in the cube. This\nalone is sufficient to prohibit such a rotation; for it is obvious\nthat the moment a particle or any series of particles is diverted from\nits established orbital path disruption of that portion of the cube\nmust necessarily follow. This upon the assumption that the particles\nof matter are in motion and revolving in their corpuscular orbits. 3.\nPlane-rotation necessitates a radical change in the absolute motion\nof each individual particle, electron, atom or molecule of matter in\nthe cube and a consequent retardation or acceleration of this motion.\nThis upon the hypothesis that the particles of matter are vibrating\nat the rate of absolute motion. 4. It presupposes a reconstitution\nof each atom, molecule or particle in the cube, changing the path of\nintra-corpuscular rotation either from a right to left direction or\nfrom a left to right direction, as the case may be. The particles of\nmatter in the cube will be acted upon in much the same manner as the\nparticles in a glove when it is maneuvered in the fourth dimension. In\ndescribing this phenomenon, MANNING says:[20]\n\n  \"Every part by itself, in its own place is turned over with only a\n  slight possible stretching and slight changing of positions of the\n  different particles of matter which go to make up the glove.\"\n\n  [20] Vide _Fourth Dimension, Simply Explained_, edited by\n  H. P. MANNING, p. 28.\n\nThe slight stretching and slight changing of the positions of the\nparticles referred to would be of small consequence if applied to\nponderable bodies. But when used in connection with particles of matter\nwhich are themselves of very infinitesimal size means far more--enough,\nas we have said, to militate severely against the integrity of the\ncube. It is not deemed necessary to go further into the physical\naspects of plane-rotation as it is believed sufficient has been said to\nnegative the assumption from a purely structural viewpoint.\n\nAmong the vagaries of hyperspace publicists none is perhaps more\nnotable than the view taken by C. H. HINTON:[21]\n\n  \"If it could be shown that the electric current in the negative\n  direction were exactly alike the electric current in the positive\n  direction, except for a reversal of the components of the\n  motion in three dimensional space, then the dissimilarity of\n  the discharge from the positive and negative poles would be an\n  indication of the one-sidedness of our space. The only cause of\n  difference in the two discharges would be due to a component in\n  the fourth dimension, which directed in one direction transverse\n  to our space, met with a different resistance to that which it met\n  when directed in the opposite direction.\"\n\n  [21] Vide _Fourth Dimension_, p. 75, C. H. HINTON.\n\nTo be sure. And with equal certainty it might be said that if the\nmoon were made of green cheese it might well be the ambition of the\nworld's chefs to be able at some time to flavor macaroni with it,\nthus serving a rare dish. Even so, if there were an actual, objective\nfourth dimension to our space we might be able to shove into it all the\nperplexing problems of life and let it solve them for us. But the fact\nthat the fourth dimensional hypothesis is itself a mere supposition\nseems to have been overlooked or rather completely ignored by HINTON.\nOr else, ought it not be an obvious folly to hope to construct a\nrational explanation of perplexing physical conditions upon the basis\nof a purely suppositionary, and therefore unproven, hypothesis?\n\nThe recognized domain of the four-space, mathematically considered,\nis according to the most generous allowance very small, so small, in\nfact, that the disposition of some to crowd into it the essential\ncontent of the manifested universe is a matter of profound amazement.\nThen, too, it cannot be denied that there is no appreciable urgency or\nnecessity for having recourse to a purely hypothetical construction\nfor explicatory data regarding a phenomenon which has not been shown\nto be without the scope of ordinary scientific methods of procedure to\nunravel.\n\nThe claim of certain spiritualists, notably ZOLLNER of Leipsig, that\nthe phenomena of spiritism is accountable for on the grounds that the\nfourth dimension affords a residential area for discarnate beings\nwhence spiritistic forayers may impose their presence upon unprotected\nthree dimensional beings is no less fatuous than the original\nsupposition itself. For upon this latter is built the entire fabric of\nmeaningless speculations so gleefully indulged in by those who glibly\nproclaim the reality of the four-space. Indeed, clearer second thought\nwill reveal that, when the pendulum of erratic thinking and trafficking\nin mental constructions swings back, hyperspaces, after all, are but\nthe _ignes fatuii_ of mathetic obscurantism.\n\nThen, why should it be deemed necessary to discover some more\nmysterious realm of four dimensional proportions in which the spirits\nof the dead may find a habitation? Are the spiritualists, too, reduced\nto the necessity of further mystifying their already adequately\nmysterious phenomena? If there were not quite enough of physicality\nupon the basis of which all the antics of these entities can be\nexplained, and that satisfactorily, one would, as a matter of course,\nbe inclined to lend some credence to these claims; but as it is clear\nthat all organized beings have some power, if no more than that which\nmaintains their organization, and as it ought also be an acceptable\nfact that such a being is directed by mind; and further, that owing to\nthe nature of a spirit body it can penetrate solid matter or matter of\nany other degree of density below the coefficient of spirit matter, it\nought likewise be unnecessary to go without the province of strictly\ntridimensional mechanics for an explanation of spiritistic phenomena.\n\nEqually unnecessary and uncalled for is the attempt of certain others\nwho lean toward the view of speculative chemists to account for the\nnone too securely established hypothesis that eight different alcohols,\neach having the formula C_{5}H_{12}O may be produced without variation.\nThis is said to be due to the fact that certain of the component atoms,\nnotably the carbon atoms, take a fourth dimensional position in the\ncompound and thus produce the unusual spectacle of eight alcohols from\none formula. Have chemists actually exhausted all purely physical means\nof reaching an understanding of the carbon compounds and are therefore\ncompelled to resort to questionable means in order to make additional\nprogress in their field? It is incredible. Hence the more facetious\nappears the mathematical extravaganza in which originates the tendence\namong the more sanguine advocates to make of the fourth dimension a\nsort of \"jack of all trades,\" a veritable \"Aladdin's lamp\" wherewith\nall kosmic profundities may be illuminated and made plain. Not until\nthe perfection of instruments of precision has been reached, and not\nuntil human ingenuity has been exhausted in its efforts to produce more\nrefined methods of research should it be permissible even to venture\ninto untried and more or less debatable fields in search of a relief\nwhich after all is unobtainable.\n\nNotwithstanding the fact that all attempts at accounting for physical\nphenomena on the basis of _n_-dimensionality (which is itself by all\nthe standards of objective reference a non-existent quantity and\ntherefore irreconcilable with perceptual space requirements) are to be\ncharacterized simply as a senseless dalliance with otherwise deeply\nprofound questions, many have fallen into a complete forgetfulness of\nthe logical barriers inhering in and hedging about the query and have\ncommitted other and less excusable errors in the premises. Take, for\ninstance, the suggestion that the action of a tartrate upon a beam\nof polarized light is due to the assumption of a fourth dimensional\ndirection by some component in the acid. This for the reason that\nexperimentation has shown that tartaric acid, in one form, will turn\nthe plane of polarized light to the right while in another form will\nturn it to the left. It is not believed, however, that there is any\nwarrant for such an assumption. There is also another kind of tartrate\nwhich seems to be neutral in that it has no effect whatever upon the\nbeam of light, turning it neither to the right nor to the left nor\nhaving other visible or determinable effect upon it. Indeed, it is\nnot clear how it is hoped to prove such a case by constituting as a\nnorm a hypothesis which is essentially indemonstrable. A more logical\nprocedure would be first to establish the objective, discoverable\nposture of four-space; show the actual movement of matter and entities\ntherein; locate it by empirical methods of research, and then, basing\nour assertions upon apodeictic evidences, assume a new attitude toward\nthese phenomena because of the support found in established and\nverifiable facts. Some hope of gaining a respectful hearing might then\nbe entertained; but at least to do so now appears to be quite untimely.\n\nMAJOR WILMOT E. ELLIS, Coast Artillery Corps, United States Army, in\n_The Fourth Dimension Simply Explained_,[22] remarks:\n\n  \"... in the ether, if anywhere, we should expect to find some\n  fourth dimensional characteristics. Gravitation, electricity,\n  magnetism and light are known to be due to stresses in, or\n  motions of, the infinitesimal particles of the ether. The real\n  nature of these phenomena has never been fully explained by\n  three dimensional mathematical analysis. Indeed, the unexplained\n  residuum would seem to indicate that so far we have merely been\n  considering the three dimensional aspects of four dimensional\n  processes. As one illustration of many, it has been shown both\n  mathematically and experimentally that no more than five\n  corpuscles may have an independent grouping in an atom.\"\n\n  [22] Q. v., p. 242, edited by H. P. Manning.\n\nThe weakness of this view may be due to the fact that at that\ntime MAJOR ELLIS was emphasizing in his own mind the necessity of\nsimplifying the conception so as to make it of easy comprehension\nrather than the establishment of any fealty to truth or the spirit\nof mathesis in his examination of the problem. What therefore of\nreality the student fails to find in his view may be attributed to the\nsacrifice which the writer (MAJOR ELLIS) felt himself called upon to\nmake for the sake of simplicity. Hence a certain expressed connivance\nat his position is allowable. But, on the other hand, if such were not\nthe conscious intent of MAJOR ELLIS it is not understood how it should\nappear that \"the unexplained residuum would seem to indicate that so\nfar we have merely been considering the three dimensional aspects of\nfour dimensional processes.\" Contrarily, it has yet to be proved that\nthree dimensional space does not afford ample scope of motility for\nall observable or recognizable physical processes and that there is\nno necessity for reference to hyperspace phenomena for an explanation\nof the \"unexplained residuum.\" It is, of course, understood that many\nof the possibilities predicated for hyperspace are purely nonsensical\nso far as their actual realization is concerned. Our concern is,\ntherefore, not with that class of predicates, but with those wherein\nreside some slight show of probability of their response to the\nconditions of n-dimensionality either as a system of space-measurement\nor a so-called space or series of spaces.\n\nMAJOR ELLIS concludes his simple study of four-space by proposing the\nfollowing query:\n\n  \"May not birth be an unfolding through the ether into the\n  symmetrical life-cell, and death, the reverse process of a\n  folding-up into four dimensional unity?\"\n\nIt is confessed that there seems to be nothing to warrant the giving\nof an affirmative reply to this query. It is, perhaps, sentimentally\nspeaking a very beautiful thing to contemplate death as a painless,\nunconscious involvement into a glorious _one-ness_ with all life, and\nbirth, as the reverse of all this. But where is the utility of such a\ndream if it be merely a dream and impossible of realization?\n\nSIMON NEWCOMB,[23] at one time one of the outstanding figures in\nthe early development of the fourth dimensional hypothesis, openly\ndeclared that \"there is no proof that the molecule may not vibrate in a\nfourth dimension. There are facts which seem to indicate at least the\npossibility of molecular motion or change of some sort not expressible\nin terms of time and the three co\u00f6rdinates in space.\"\n\nOf course, there is no proof that a molecule may not at times be\nensconced in a four-space neither is there proof nor probability that\nit is so hidden. Indeed, there is no proof that there is such a thing\nas a molecule for that matter.\n\n  [23] Vide _Science_, Vol. VII, 158, 1898, p. 4.\n\nIn all of the foregoing proposals it is assumed that the fourth\ndimension really exists and that it lies just beneath the surface of\nthe visible, palpable limits of the material universe; that lying\nin close juxtaposition to all that we are able to see, to hear or\nsense in any way is this mysterious, eternally prolific, all-powerful\nsomething, hyperspace, ever-ready to nourish and sustain the forms\nwhich have the nether parts firmly encysted in one or the other of her\n_n_-dimensional berths. Thus it would seem that while yet functioning\nin a strictly tridimensional atmosphere, some one, more reckless\nthan the rest, should at last stumble upon some up-lying portion of\nit and be instantly transformed into a mathetic fay of etherealized\nfour-dimensional stuff.\n\n\n  _PART TWO_\n\n  SPATIALITY\n\n  AN INQUIRY INTO THE ESSENTIAL NATURE OF SPACE AS DISTINGUISHED\n  FROM THE MATHEMATICAL INTERPRETATION\n\n\n\n\n  CHAPTER VI\n\n  CONSCIOUSNESS THE NORM OF SPACE DETERMINATIONS\n\n  Realism Is Determined by Awareness--Succession of Degrees of\n  Realism--Sufficiency of Tridimensionality--The Insufficiency of\n  Self-Consistency as a Norm of Truth--General Forward Movement\n  in the Evolution of Consciousness Implied in the Hyperspace\n  Concept--The Hypothetical Nature of Our Knowledge--Hyperspace\n  the Symbol of a More Extensive Realm of Awareness--Variations in\n  the Method of Interpreting Intellectual Notions--The Tuitional\n  and the Intuitional Faculties--The Illusionary Character of the\n  Phenomenal--Consciousness and the Degrees of Realism.\n\n\nThings have value for us only to the extent to which we can become\naware of their being. The appraisement of all objects, conditions,\nstates or qualities is determined directly by the degree or quality\nof awareness with which we apprehend them. Those elements which are\nwithout the intellect's scope of awareness have no interest and hence\nno value so far as the individual intellect is concerned. And this is\ntrue of all degrees and states of consciousness from the lowest to the\nhighest, from the human to the divine.\n\nThere enter into all conscious determinations three factors, namely:\n(_a_) the scope, or totality, of adaptations which an organism can\nmake in the sensible world, (_b_) the power of consciousness to make\nadaptations and (_c_) environment. These three are interdependent.\nThe totality of adaptations depends primarily, of course, upon the\nquality of conscious powers or faculties, and also, in a lesser\ndegree, upon opportunities afforded by environment. Faculties of\nconsciousness are derived directly from the influences exerted upon\nthe organism by his environment and the results of the struggle to\novercome them. Environment is of two kinds, artificial and natural. The\nartificial environment is such as has been modified by our conscious\naction upon external phenomena. The residue is natural. And thus the\nscope of adaptability becomes an unvarying witness to the quality of\nconsciousness manifesting through a given organism.\n\nThe universe is so constructed that the essential character of its\nvarious states and qualities is a fixed quantity for a given scope of\nconsciousness and varies only as the sphere of consciousness varies.\nStates of existence or scopes of adaptation which are registering upon\na higher plane or in a more subtle sphere of existence than that in\nwhich we may at any time be functioning can only appear evidential\nto us when the mechanism of our consciousness becomes congruently\nadjusted therewith. So that the focus of consciousness must always be\na variable quantity adaptable, under proper conditions, to any plane\nin the kosmos. Consciousness, then, becomes the sphere of limits both\nof knowledge and adaptability. But lest we seem to admit implicitly\npart of the contentions which mathematical publicists have made in\npostulating the unodim and duodim consciousness, it is necessary\ncarefully to differentiate between the results arrived at as a result\nof the two procedures. In the first place, analysts _assume_ the\nexistence of a unodim and duodim plane of consciousness and proceed\nto construct thereon an analogy designed to show the feasibility\nof another assumption, the fourth dimension. While, in laying the\nfoundation of consciousness upon a tridimensional plane we do not start\nwith an _assumption_, but with a fact. Therein lies the difference.\nEnormous advantages inhere in a procedure based upon facts, but in\na series of planes built upon assumptions no such advantages are\ndiscovered. For however much the series of hypothetical planes may be\nextended or elaborated there must inhere necessarily throughout the\nseries an assumptional value which vitiates the conclusions no less\nthan the premises. The sanity and integrity of intellectual operations\ndepend almost entirely upon the differentiation which we make between\nthe necessities arising out of assumptions and those which spring\nup empirically from established facts. No procedure is necessary to\nestablish the value of such a differentiation, nevertheless it may\nbe suggested that it is allowable, under the rules of logic, to make\nany assumption whatsoever so long as care is taken to see that the\nconclusions embody in themselves the characteristics of the original\npremise. For instance, it is permissible to assume that space is\ncurved. Under such an assumption, it is only necessary that the\nconstructions which follow shall be self-consistent. But the case is\ndifferent when we come to deal with spatiality and vitality. These are\nquantities which cannot, in the last analysis, be made to conform to\nthe rules of the game of logic.\n\nThus, when it is intimated that realism lends itself to an apparent\ndivision into degrees, and that each degree has a corresponding state\nof consciousness, it is by no means to be inferred that such apparent\ndivisions are of mathematical import. For, in reality, i.e., when the\nconsciousness has expanded so as to become congruent with the limits\nof even the space mind (vide Fig. 20), there appear to be no divisions\nin realism. It is only because of the fragmentariness of our outlook\nupon the kosmos that realism appears to be divided into various planes;\nfor all of these planes are one. The divisions exist for relative\nknowledge, but not for complete knowledge; they exist for a finite\nintelligence, but not for a transfinite intelligence. That is why we\nview realism as a series of planes. It is because we discover that, as\nwe proceed, as our consciousness expands and we take in more and more\nof the vital activities of the kosmos and understand better the causes\nunderlying that which we contact, we have passed from a state of lesser\nknowledge to one of greater knowledge. And so we say we have passed\nfrom one degree of realism to another, whereas, really we have not\npassed from one degree of realism to another degree. Instead, it is\nour consciousness that has expanded.\n\nIf now, we conceive reality to be a scale extending from one\nextremity to another (that is, from supreme consciousness to entire\nunconsciousness, from final knowledge to total ignorance), and the\nintellectual consciousness as the indicator which traverses the scale\ndenoting at all times the precise degree of our comprehension of\nreality, and hence the degree of expansion of consciousness, we shall\nconstitute a similitude closely approximating the real _status quo_ of\nhumanity with respect to the sensible and supersensible worlds. The\nquantity or force which causes the indicator to move along the scale\nis the quality of awareness. And this varies directly as the scope of\nadaptability varies. Realism is homogeneous throughout its extent;\nbut the scale marked upon it registers from _naught_ to _unity_. And\nbetween these every conceivable degree of awareness may be registered.\nThe indicator moves only as the scope widens, and thus is shown a\nchange in the quality of awareness. For, however paradoxical it may\nseem, the wider the scope of knowledge the better its quality: the more\none knows, the more complete and of higher quality becomes that which\nhe knows.\n\nThe intellect is of scientific tendence, studiously rejecting all\nphenomena which do not yield to its senso-mechanisms. Even intuitions\nsuffer the humility of rejection and do not escape the limitations\nwhich the intellect imposes upon them. This is so, because, as yet,\nthere is no adequate perceptive and conceptive apparatus for the\npropagation and classification of intuitions, as apart from concepts.\nThe outcome of these proscriptions is that intuitions--free, mobile,\nand more or less formless in themselves, must first be rehabilitated\nand vestured in garments _a la intellect_ to conform to the prevailing\nmode. But intuitions thus treated are no longer intuitions, but\nempirical concepts. True intuitions are like aqueous vapor--amorphous,\npermeating, diffusive: axioms or empirical concepts are like cakes of\nice--formal, inflexible and conforming to the shape of the mold into\nwhich they are poured. Because of this--the scientific tendence of the\nintellect and the consequent necessity of reforming so much of the\ndata which constitute its substructure, of pressing, condensing and\nreshaping it to suit its own ready-made patterns--it can be perceived\nhow profound is the influence of the intellectual consciousness in\ndetermining the character of the totality of data which the sensible\nworld, and for that matter, the supersensible, offer us. The intellect\nis the only means at hand for the interpretation of the meaning and\nsignificance of the world of phenomena. Consequently, whatever meaning\nor significance we are led to attach to that part of the universe which\nwe contact, in any way, is dictated by the intellectual consciousness.\nThere is no escape from the decisions of the intellect so long as the\npresent scheme of things endures.\n\nThus, by whatever standard of reference the matter may be determined,\nit remains indisputably established that the intellectual consciousness\nis the sole determinant of the phenomenal value of everything within\nour scope of awareness or adaptability. And whatever the fault,\nincongruity or discrepancy that may be revealed by a more intimate\nknowledge of the genesis and character of the appearance of the\nsensible world, it will be found to be due to the peculiar cut and mode\nof the intellect and not to things themselves. The value, qualitative\nor existential, which the intellect irrevocably assigns to objects\nand conditions in the world of the senses is the exclusive _norm_ not\nonly by which these are judged, but also, by which our action upon\nthem and their action upon us are determined. Images or objects which\ndo not act upon us and upon which we cannot act have no interest for\nus. But as an integral part of the totality of images or objects in\nthe sensible world, we must inevitably act upon all that is outside of\nourselves, and these, in turn, must react upon us. On the other hand,\nthere must be objects and images in the universe of life and form upon\nwhich, because of their inherent nature and on account of the lack of\nour interest in them and their interest in us, we can neither act nor\nbecome the object of their action.\n\nBut herein is a mystery. For, either we act upon and are recipients of\nthe action of the totality of images or objects in both the sensible\nand supersensible worlds, or we are so placed in the grand scheme of\nthings that both ourselves and the sphere of our interests and possible\nactions are closed circuits, hermetically sealed and non-communicative\nwith the other like spheres, which do not and cannot act upon us. There\nis yet a third possibility--that we are so fashioned, in the entirety\nof our being, that some part of us is exactly congruent with some part\nof every sphere of possible actions and interests in the kosmos, and\ntherefore, each of us has being or consciousness of a kind that is\nkeyed to and registering in the totality of such spheres; and that,\nat present, because our interests and possible actions are limited to\nthe plane of sensibility, we are conscious only there. And further,\nthat although those spheres of our consciousness which are fixed to\nregister in other planes do not answer to the lowest on which we now\noperate, having a character of which we are unaware, they nevertheless\ncannot be said not to exist, because of the lack of communication\nbetween them. Among these three possible choices, we have no hesitancy\nin expressing a decided preference for the last mentioned--that the\nrange of our being is co-extensive with the range of reality, and\nlike a pendulum, we oscillate, at long intervals, between two kosmic\nextremities--nescience and omniscience.\n\nThe intellectual consciousness is the touch-stone of realism. It\nis like a spreading light which, as it expands, reveals previously\ndarkened corners and conditions, only it has power both to reveal and\nto bring into manifestation. In its present state, man's consciousness\nis like a searchlight. It illumines and takes cognizance of everything\nthat falls within its scope of motility and is consequently able to\nstudy in detail that which it reveals. But that which is beyond its\nscope is as if it never existed so far as the individual consciousness\nis concerned. It is not reasonable to predict that the same\ncharacteristics that are observable in any given state shall persist\nthroughout all the various scopes through which the consciousness must\nproceed in its evolutionary expansion. For the scale of kosmic realism\nis one grand panorama extending from the grossest to the most subtle\nand refined. While in general the thread of realism may pervade the\nentire scale it is nevertheless marked by many and diverse changes\nin its characteristics as it is followed from one stage to another.\nSo that the realistic character of one stage may vary greatly from\nthat which next preceded it or from that which will succeed it. It\nwould appear, therefore, that in passing from one stage of realism to\nanother there need not remain anything but the mere fact of reality in\nits connection with ultimate reality; for it is obvious that in every\ncondition of realism which may be encountered in the kosmos there\nmust be a basic thread of ultimate reality running through the whole.\nThe entire gamut of realism may accordingly be traversed without the\ndanger of being diverted from the golden thread of realism which thus\npermeates all. It is always the phenomena of realism with which we are\nconcerned and which we are trying to understand rather than realism\nitself. It is this that confounds us. If it were not for the phenomena,\nwhich is the way realism or life presents itself to our consciousness,\nwe should experience no trouble in discovering the reality, all other\nthings being equal. For the former ever obscures the latter. It is\nthe supreme task of mental evolution to break through the clouds of\nphenomena in the search for the eternal substratum of reality which\nruns through the sensible universe of things.\n\nThe first view of conditions that the mind takes upon awakening to\nconsciousness in any new sphere of cognition is necessarily hazy\nand inchoate. There is more or less of astonishment, wonder and\nbewilderment upon first becoming aware of a new scope of realism. In\nthis state it is natural that the mind should overlook or ignore much\nthat is essential and perhaps all that is so even escaping the true\nimport of the phenomena which it senses. It is reasonable, too, that in\nsuch a state the main outlines of what is really seen may be greatly\ndistorted and exaggerated so that it is well-nigh impossible to secure\na correct comprehension of the character of a new scope of realism\nfrom any early survey. It is not until later years, after much study\nand circumspection that the mind, becoming used to the new conditions,\nbegins to get correct impressions and to make valid judgments as to\nthat which it discerns. And even then, it not infrequently happens\nthat the resultant view of things in general is found to be in need of\nrevision and correction. Hence, after everything is sifted down to the\nultimate allowance for the illusion incident to too much enthusiasm\nand wonder we have only a very small residuum of truth upon which to\nbuild and this latter we often find to be the single thread of reality\nwhich runs through all the phenomena and which is, therefore, the only\nquantity that remains worthy of much consideration.\n\nThus it is with religion. The path of progress over which our religious\nconceptions have come need not be outlined here, but to any one at\nall acquainted with the history of religious thought and ideals it\nat once must be patent that it has been one continuous surrender of\nthe old for the new, of one degree of realism for another newer and\nhigher degree; that always it has been the phenomena, the flora of\nthe ideals which have had to give way, while nothing was left but the\nroots of realism from which they have sprung. It has been the same\nwith scientific knowledge. Facts have been collected and hypotheses\nproposed to synthesize them and yet these have had to give way for\nothers, and others still, until the data of scientific knowledge to-day\nare quite different from what they were in earlier days. And yet\npermeating the scientific knowledge of all times has been the golden\nthread of reality, and of all facts and systems of facts which man\nhas successively assumed and surrendered nothing has remained but the\nreality; indeed, nothing could so remain, but reality. So it is with\nair phenomena with which consciousness has to deal. This perhaps is due\nto the fact that the mind interprets phenomena in accordance with the\nquality of its awareness, and as consciousness is a variable quantity,\nits standards of interpretation will likewise vary. Each new scope of\nawareness, after this manner, yields higher and more exact standards of\ninterpretation. And then, progressing in awareness from the segment to\nthe whole a fuller view of the phenomena as well as of reality itself\nis gained and also a more comprehensive judgment of the relations which\nexist between the segment and the whole. In other words, as the scope\nof consciousness widens it becomes more and more apparent that what was\nfirst thought to be a separate segment is in reality identified with\nthe whole in an indissoluble manner. For the Thinker is then not only\naware of the segment as such, but he is also conscious of the fact that\nit has definite relations with the entirety and that what he needs is\nmerely a more extended consciousness.\n\nIn denying the existence of the four-space or spaces of\n_n_-dimensionality as described and defined by geometricians, we do not\nthereby deny the existence of a plane of consciousness which is as\nmuch unlike the conditions of the tridimensional world as it is said\nto be unlike the four-dimensional world; but what we do deny is that\nsuch a higher plane of existence has necessarily to be conditioned\nby such characteristics as the metageometricians have proposed. It\nis maintained that there is no basis in consciousness for a world of\nfour dimensions; that the consciousness has no tendency for action in\nfour-space. Neither has matter nor life any inclination or potency\nto behave in a four-dimensional manner. It is indeed more rational\nto suppose that there is a higher plane, in fact, a series of higher\nplanes, in which the thread of realism is continuous, not broken as it\nnecessarily would have to be in extending to hyperspace, nor curved\nas in a manifold; that this series of subtler and finer planes of\nconsciousness are merely an elongation of our three dimensional scope\nof realism. It, therefore, remains only to master the phenomena of\neach in just the same manner as we have, in a measure, mastered the\nphenomena of tridimensionality. For it is easily conceivable that the\nquality of consciousness is such that it may adapt itself to a far\nwider range of possibilities than may be discovered in hyperspace and\nstill be a tri-space quantity.\n\nIt is believed, however, that in all the new and higher planes of\nconsciousness tridimensionality is the norm both of the phenomena\nand of the reality peculiar to them. And that, being such, does not\nchange or vary, but is a fixed quantity regardless of the plane of\nconsciousness. Furthermore, it is believed that the highest state of\nconsciousness in the entire kosmos could easily exist, and does so\nexist, upon the basis of three-space as the norm of its extent.\n\nA sharp line of demarkation should be drawn between the reality which\nis life and consciousness and that which belongs to the realm of\nphantasy. For it is the prerogative of the intellect to create, out\nof the remains and deposits which it finds in the pathway of life,\nwhatsoever it wills. This it does continuously; but it scarcely can be\nexpected that such creations shall be endowed with the same dynamic\ncharacter as that which life bestows upon its creations. The creations\nof the one are merely dead carcasses while those of the other are vital\nand real. Between them the same marked difference exists as between the\ngrowing tree and the lumber which the builder converts into a house.\nThe organization which we witness when we look upon a building made of\nthe dead body of a tree is not the same kind of organization as that\nwhich we see when we view the living, growing, vital tree. The dead\ntree is a deposit of life cast off by it when it passed on. Whatever\nthe intellect can do in disposing of the remains of the tree-life is\nconventional and artificial. If it convert it into an edifice it will\nthen bestow upon it a sort of consistency which is quite sufficient for\nall purposes. But the consistency which holds the organization of an\nedifice together is not the kind of consistency which holds a living\ntree together. In fact, there is a consistency that is not consistent.\nSuch is the consistency of metageometry. It is self-consistent and yet\ninconsistent with the consistency of the kosmos and its norm of being\nwhich is consciousness.\n\nSelf-consistency is one thing and kosmic consistency is quite\nanother. It does not necessarily follow that because a given scheme\nof thought is consistent in all its parts that it is also consistent\nwith universal truth or with life. This very vital fact was overlooked\nby GAUSS and all those who followed in his wake when he discovered\nthat his _Astral Geometry_ was consistent throughout in all its parts.\nThere is only one norm of truth and that is kosmic consistency. It\nmatters little that a thing shall be self-consistent; it matters much\nwhether it is consistent with the universal standard. It has been shown\nto be logically possible to elaborate at least two different systems\nof geometry, namely, the geometry of the acute angle and that of the\nobtuse, which, while each of them is self-consistent throughout, are\nnevertheless inconsistent with each other and with the geometry of the\nright angle (Euclidean). This, it would seem, appears to be sufficient\nfor the invalidation of either one or both of the non-Euclidean systems\nof geometric thought. Indeed, if it can be shown that the Euclidean\ngeometry is more representative of the true approach to the norm of\nspace-genesis and of creation so far as its mode of manifestation is\nconcerned, and consequently true of the norm set up by consciousness,\nthe rejection of both systems of non-Euclidean geometry seems to be\nthoroughly warranted. But this is obvious and requires no demonstration\nnor comment to make it clear. We have only to ask ourselves whether\nit has ever occurred to us that consciousness has either a tendency\nto or adaptability for action in a curvilinear manner; or, if when\nwe contemplate ideas or idea-relations we have the impression of\nperceiving a curvilinear or manifold tendence in them either of a\npositive or negative nature, and also whether it has been observed that\nour thought processes naturally assume four-dimensional attitudes.\nIf we find that such a query must be answered negatively, and indeed\nwe must so find, then, we have no basis for the assumption that any\none of the systems of non-Euclidean geometry is valid either for the\npresent status of consciousness or for a future existence, since it is\ntrue that the future is but an elongation of the present. Evolution is\nto bring no radical changes in the norms of reality; it has merely to\ndeepen and widen and make more intense, efficient and comprehensive\nthe present scope of our consciousness and thereby, while the Thinker\nis passing from one degree of realism to another, to bring him into a\nclearer conception of what his own limited scope of motility means to\nthe whole.\n\nThe four-space is a mathetic divertisement. That is, it cannot be\nsaid to lie in the direction of a straight line which proceeds either\nin a forward or lateral direction. Neither does it lie in a plane\nwhich is vertical or horizontal to the sensorium. It is, therefore,\na fractural departure from any conceivable tridimensional direction,\na geometric anomaly. Evolution, despite the minor aspects of its\nmovement, undoubtedly proceeds in a straight line and not by a zigzag\nnor discontinuous line and hence it is irrational to assume that it\nwill, when it passes on to the next advanced stage, emerge into the\nrealm of the four-space. For the so-called hyperspace of geometry\ncannot by any standards of reference be said to lie in the plane of\nany straight line which can be described in three-space. If life is\nto evolve more efficient forms and if the forms are to evolve into\nmore perfect organizations and mind and consciousness to become more\nintense and comprehensive expressions of the divine mind of the kosmos\nit is certainly not in the domain of hyperspace that these shall find\nthe substructure of their higher development; but, if at all, it shall\nbe found, as in all times past, in the realm of perceptual space where\nbodies are said to have three and only three possibilities of motion.\n\nWhat then is the significance of the more than a thousand years of\nmathematical labors; of all that has been said and done in an endeavor\nto bring into the popular consciousness a conception of hyperspace? Is\nit a question of _\"Love's Labour's Lost?\"_ Or is it a mere prostitution\nof mathematical talent? To answer these queries is the burden of this\ntreatise and it is hoped that as the text continues the reader may be\nable to arrive at his own conclusions as to the relative value of the\nwork of the mathematicians in this respect and be able to judge for\nhimself the true significance of it all.\n\nThe specific value of consciousness as a determinative factor in\nspace-measurement has been recognized by all who have sought to arrive\nat a logical justification for the conception of four-dimensionality by\nanalogous reasoning. The existence of the _unodim_ with consciousness\nlimited to a line or point has been assumed and it has been shown\nhow greatly such a being would be handicapped by his limited area of\nconsciousness, it having been proposed to confine his consciousness to\none dimension. An _unodim_ would, of course, be entirely unaware of any\nother dimension than that in which he could consciously function. So\nthat with respect to his own consciousness no other dimension would be\nnecessary for the continuance of his life processes. He might live his\nlife without any knowledge even of any limitations or barriers to other\nthings higher than those of his plane. He would be content to exist in\nthe one-space and enjoy the benefits which it offered. He could have no\nnotion of the two-space, but it has been allowed that a _super-unodim_,\nan _unodim_ metageometrician, if you please, could reason out a mental\nconception of what the two-space might be. Passing on to a space of\ntwo dimensions, the domain of the _duodim_, a greater freedom of\nmovement is allowed and instead of being able to function in only one\ndimension the inhabitants of this plane would find themselves able to\nmove about in at least two directions. Consciousness would accordingly\nenjoy a more comprehensive scope. But in a manner similar to that used\nby the _unodim_ metageometrician it is held that the _duodim_ could\nget a conception of the three-space by analogous reasoning and so\ngradually become conscious of a higher degree of spatiality than his\nown. In the conscious reasoning of both, however, is the condition of\nperpendicularity. That is, it must be assumed by both the _unodim_ and\nthe _duodim_ that the new dimension must lie in a plane perpendicular\nto their space. So, the _unodim_ would postulate that the two-space\nmust lie in a direction at right angles to his space, and yet he would\nnot be able to indicate the direction owing to his ignorance of any\nexperience that would acquaint him with the new space as well as the\nwant of possibility of motion therein. Similarly, the _duodim_ would\narrive at a conception of three-space. Thus, it has been argued that\n_tridims_, or people living in our tridimensional world, could, by\nusing a like line of argument or reasoning, arrive at a conception or\nunderstanding of the four-space, which, of course, must also lie in a\ndirection at right angles to three-space.\n\nThe implications of this mode of thought show how thoroughly the\nmathematician recognizes the limitations which consciousness imposes\nupon our knowledge of the world and the subtler conditions about us.\nBut, moreover, it is even obvious to all who stop to think about\nit; for it can readily be seen how little those things which do not\nenter our scope of awareness affect us either physically, mentally\nor spiritually. But no one can be so bold as to deny utterly that\nanything exists but what is found in our consciousnesses. It is even\ntrue that in the great centers of population where people are compelled\nto live many families in the same house, it is the usual thing for\nthese individual families to live in complete forgetfulness of all\nthe others in the house and live their lives so completely that it\nwould be exceedingly difficult to measure the effect the one has upon\nthe other. The mathematician, as is shown by the hyperspace movement,\nrecognizes that there are planes of superconsciousness the nature and\ncharacter of which persons confined to limited areas of consciousness\ncan have no knowledge and may only arrive at that knowledge by serious\nthought and contemplation. In other words, they tacitly admit the\nexistence of higher planes of consciousness as well as the necessity\nof elevating the personal consciousness in order to comprehend them.\nAlthough it was not expressly allowable in the analogy of the _unodim_,\nit is nevertheless one of the strongest implications of the process\nof reasoning that the _unodim_ could have easily raised the plane of\nhis consciousness by continuing his researches until he, too, became\nconscious of the three-space, mathematically, as well as the two-space.\nFor it was not necessary for him to raise the plane of consciousness in\norder to contact the two-space. He had need only to widen it. But in\norder to comprehend the mathematical three-space it would be necessary\nfor him to elevate his consciousness.\n\nThe fundamental error in the foregoing line of thought rests in the\nfact that awareness in the human family has not developed in the manner\noutlined. The human species has not come into conscious relations\nwith the three-space by outgrowing the one-space and the two-space in\nsuccession. The fact of the matter is that when consciousness first\ndawned it must have encompassed all three dimensions simultaneously\nand equally and there is nothing to indicate that its rise was\notherwise. Then, specifically there is no evidence that the evolution\nof consciousness has proceeded in a rectangular manner. Indeed, there\nis undoubtedly no warrant for the assumption that it has progressed\nin ways that are mathematically determinable at all. The question\nvery naturally rises in view of the above as to the relative value of\nmathematical knowledge in the scheme of psychogenesis. Can mathematical\nknowledge or laws be said actually and finally to settle once for all\ntime any question in which consciousness or life enters as a factor?\nUpon the response to this question hinges unanswerably the decision as\nto the category which mathematical knowledge should by right occupy in\nthe entire schematism of life. If it can be successfully maintained\nthat final judicative power abides in mathematics in the determination\nof these questions, then it would be useless to struggle against the\nfiat of mathematics and mathematicians; verily, we should be compelled\nto accept _nolens volens_ all that mathematicians have devised about\nhyperspace and its connotations. If, on the other hand, it can be shown\nthat no such judicative power inheres in mathematical knowledge we\nshall then be able to establish for mathematics a true category and to\ndispose of the hyperspace movement in a manner that shall at once be\nlogical and necessary.\n\nThat the discovery of hyperspace by the mathematician is merely an\naspect of a general forward movement in the evolution of consciousness\ncan be shown by a brief correlative study. Hyperspace is the artificial\nsymbol of a higher and more extensive realm of awareness. For it cannot\nbe doubted that to be able to think in the terms of hyperspace, to\nstudy the various relations and interrelations upspringing from the\noriginal premises, actually to become conscious in the hyperspatial\nrealm thus constructed, requires a different species or quality of\nconsciousness than that required for ordinary thinking. The period\ncovering the rise of artificial spatiality is contemporaneous with the\nrise of the phenomena identified with the spiritual life of SWEDENBORG;\nfor during the same time he began a series of visions which revealed to\nhim great knowledge of the unseen and supersensuous realities of life\nand existence. His consciousness was being flooded with the light from\nso-called celestial spheres and he was gradually becoming conscious\nof a \"new dimension,\" a new space, a higher world that is altogether\nunlike the world of the senses. During this period, too, DANTE, the\ngreat kosmic seer began to jot down the results of his \"hyperspace\"\nexperiences, after which he wrote his _Divina Commedia_ in which he\ndescribes more or less minutely some of the characteristics of the\nhyperspace domain which was revealed to his consciousness as he saw\nand interpreted it. Both SWEDENBORG and DANTE being deeply religious\nand pious-minded had their reports of the new world colored by their\nown mental experiences and proclivities. PLATO had at an earlier day\nset down what he conceived to be the ethical and civic characteristics\nof the new age, the _Utopia_ of mankind living on a higher plane.\nIt was during these days that the doctrine of evolution was born,\nalthough it remained for DARWIN to formulate and buttress it with a\nstupendous congeries of facts. MARTIN LUTHER, the great religious\nreformer, likewise contacted the radiating light-glow of a higher\nconsciousness into which the race was coming but of which only the\nforemost were able to get advance glimpses. KANT, one of the peerless\nleaders of the vanguard of humanity, at this time also, conceived and\nwrote his _Critique of Pure Reason_ which is likewise an evidence of\nthe upliftment of his consciousness on the side of pure intellectuality\nand the commencement of a general period of illumination. And then,\nlater, but embraced within the same period, artists began to get\nglimpses of this higher consciousness which showed itself in a new and\nstrange departure in art. In rapid succession new schools sprang up and\ncame to be known as the \"cubist,\" \"post-impressionist,\" \"futurist,\"\n\"orphimist,\" the \"synchromist\" and the \"vorticist.\" Art really began\nthe search for the \"plastic essence of the world\" trying to portray\nits conception of the \"real image of the spirit\" of the world. Color\nacquired a new kind of splendor and painting gave birth to a new\nintrinsic beauty of material and sheer loveliness of texture. All of\nwhich were evidences of an intellectual up-reaching in response to the\nsharp appulsions from above. DARWIN's mind, being of scientific bent,\nsaw and interpreted everything in the terms of materialistic science;\nbut there is no doubt but that the expansion of the area of awareness\nwhich his mind experienced in his great conception of evolution as a\ncontinuous process and all that it implies thereby was a result of the\nuniversal appulsion of the human intellect against the new domain of\nconsciousness. And KANT's conception of space in general perhaps may be\nsaid to have been the seed-thought for the metageometrician.\n\nBut thus it will be noted that in all the cases mentioned in the\nforegoing there is always present the personal element of the\ninvestigator, and that the reports of each of these have been colored\nand characterized by their individual consciousness and experiences.\nThat all reports would agree with respect to details connected with\nthe new domain of consciousness could scarcely be expected owing to\nthe wonder and bewilderment which seize the intellect under such\ncircumstances. No implication that the mathematicians have been unduly\nexcited by what they have discovered after years of patient research\nin this direction is indicated by the foregoing observations; but it\ncannot be denied that the enthusiasm of the moment and the consequent\nminimization of all other phenomena but the special line being\ninvestigated serve very effectively to obscure the mental vision of\nthe more partisan. It perhaps is sufficient that the investigator\nshould set down in as orderly manner as possible the things which he\nconceives, and that he should interpret them according to the standards\nof his own intellect. More than this cannot be expected. Moreover, it\nusually suffices that the future investigator, far removed from the\nbeclouding influences of partisanship, who successfully raises his\nconsciousness to that higher plane shall be able to synthesize the\nfindings of all and thereby with the aid which comes to him from a\nmore advantageous position arrive at sounder views and a more reliable\njudgment.\n\nIt will thus be seen that the metageometrician's method of\ninterpretation is no more entitled to final credence and general\nacceptance than that of the spiritualist, SWEDENBORG, or the occult\nseer, DANTE. For in their best moods and at their highest points of\nmental efficiency these have only succeeded in vaguely symbolizing what\nthey have conceived of the realities of the supersensuous realm in\nterms of their own experiences. Is there any more cogent reason, then,\nfor accepting the analyst's conception of a world of hyperspace peopled\nwith ensembles, propositions, spaces of _n_-dimensionality and other\nmathetic contrivances than the _Inferno_ of DANTE, inhabited by hideous\nshapes and repellent denizens, the remains of ill-spent earth-lives\nor SWEDENBORG'S _Celestial Realm_, wherein dwell numerous beings of\ncelestial character performing various tasks in the work of the world?\nThese observations should not lead the reader to come to the conclusion\nthat the visions of DANTE and SWEDENBORG are deemed to be more worthy\nof credence than mathematical knowledge when that knowledge is limited\nto the sphere where it rightfully belongs; but the proper view is that\nwhich would make it appear that it is the way these widely differing\nworkers interpret what they have seen; that it is the adaptation of the\nunseen realms to the peculiarities of the mentalities which observe\nthem. The mathematicians have simply portrayed as well as they could\ntheir conception of the new stage of consciousness and its contents,\nand following the _modus vivendi_ of all intellects have interpreted\nthese things in the terms of mathematics, merely because mathematics\nconstituted the best available symbology at hand for the purpose.\nSimilarly, the painter sees a new world of color; the politician, a new\nera of political freedom; the religious enthusiast, a new religious\nconception; the scientist, a new condition of matter and energy, and\nso on, to the most ordinary mind, each sees something new while at the\nsame time is necessarily limited to the confines of his own mentality\nwhen he comes to interpret what he sees and conceives. Hence, there\nwould appear to be only one way to regard all these advances and\nthat is by synthesizing them, by correlating, and by tracing them to\na common source, and finally by seeing them as one general forward\nmovement of intellectual evolution.\n\nMan, the Thinker, who in essence is a pure intelligence, has two\nmental mechanisms or organs of consciousness. One of these is the\nbrain-consciousness or the egoic. It is so called because the brain is\nits organ of expression and impression. It manifests through the brain\nand uses it to further the various objective cognitive processes. The\nbrain-consciousness is a child of the physical body and its life is\nintimately identified with the life of the body. This consciousness may\nbe called the _a posterioristic_ mechanism or organ of the Thinker and\nis therefore his means of interpreting the phenomena of the objective\nworld. Cell-consciousness is a phase of the ergonic functions of the\n_a posterioristic_ mechanism. The other organ of consciousness is an\naspect of the intelligence of the Thinker himself and perhaps may be\nsaid to be the active, organized portion of that intelligence. It is\nseparate and distinct from the _a posterioristic_ consciousness yet\nsustaining a substructural relationship with it, being the source of\nthe egoic or brain-consciousness. It may be called the _a priori_\nconsciousness. Its roots are buried deep within the heart of the\nspace-mind and it is therefrom nourished and developed by what it\nreceives in the way of intelligence. It is the intuitional faculty;\nknows without being taught; conceives without reason; interprets\naccording to the norms of the space-mind or the divine mind of the\nkosmos. It always resides on a higher plane than that of the brain-mind\nor consciousness, only at rare intervals being able to contact it with\nflashes of its own intelligence as intuitions.\n\nThe _a priori_ consciousness being the intuitive faculty of the Thinker\nis, therefore, a phase of his mental life on a higher plane than the\nsensuous. All its conceptions constitute the _a priori_ knowledge\nof the brain mind so-called. The _a priori_ faculty of man's higher\nconsciousness gives the character possessed by that form of knowledge\nknown to philosophy as the _a priori_. So that the _a priori_ has a\nmore substantial basis than has hitherto been surmised. It is not\nonly that which may be said to transcend experience but that which is\nthe organ of contact with the supersensuous realities as well as of\nexpression through the brain-consciousness.\n\nThe mind's method of apprehending objective phenomena is not a\ndirect process but an indirect process by virtue of which neurograms\nor nerve-impacts registered in the brain are interpreted. External\nsense-impressions are, of course, conveyed to the cortical area by\nmeans of appropriate vibrations which traverse the lines of the neural\nmechanism. These are recorded in the brain areas just as a telegraphic\ncommunication is registered in the apparatus of the receiving end,\nand in being so, they make terminal registrations which man, the\nThinker, interprets after a psychic code which has been built up\nempirically. That is, he comes to know that certain rates of vibration\nand certain peculiarities therein mean certain things when referred\nto the sensorium. He then interprets according to this experience the\nsymbolism of all neurographical impressions. But it is obvious that\nunder such circumstances, where the interpreter is far removed from the\nthing itself and finds it necessary to interpret rates of vibration or\nsymbols in order to arrive at a knowledge of the intelligence which is\nconveyed to him, the chances of inadequate conception are very great\nindeed. In fact, it is not possible through the use of neurographical\nsymbols alone to get any complete notion of the phenomena considered.\nAnd thus there stands between the Thinker and absolute knowledge a\nbarrier which prevents his arriving at a state of certitude in his\nknowledge of the world of sensible objects. It is, however, a barrier\nwhich will always remain, checking ever his approach to finality in his\nunderstanding of the universum of appearances.\n\nA markedly different condition obtains in the realm of the _a priori_\nor intuitional for the reason that the barriers which inhere in\nthe neurographical or _a posterioristic_ method are absent and the\nThinker has a more direct approach to the objects of cognition.\nHence the chance for error is very small indeed. This will account,\ntherefore, for the superiority of the intuitional over the rational\nor the perceptual. Indeed, it is doubtful whether the purely rational\npossesses any value whatsoever when its _modus vivendi_ is unsanctioned\nby the intuitional.\n\nElse why can we not be certain that the results of our rational\nprocesses are correct at all times? Is it not because we lack the power\nto perceive whether our premises are correct in the first place? Quite\ntruly. For if the Thinker can intuit the necessity and certitude of\nany given premise it follows that the consequences of that premise\nare true. It would, therefore, appear that the more the intuitional\nfaculty is developed the clearer will be our perceptions not alone\nof abstract values but of objective things themselves. Further, it\nis doubtlessly true that the more the space-mind is developed in the\nhuman race the deeper will become our perceptions of the essential\n_be-ness_ of things so that whatever may be the presentations of the\nspace-mind to the brain-mind they will be by far more accurate than the\nimpressions we receive through the latter as a medium of apprehension.\nIt is but natural, however, that in the present more or less chaotic\ncondition in which the faculty of the intuitional is found it should\nbe difficult even to interpret its presentations accurately. It is\nperhaps due to the fact that we are unused to its deliveries and mode\nof registration as well as to the fact that it has been overshadowed\nby the intellectual or rational faculty. But the mere fact that it is\npresent and functioning, even if but rudimentarily, is evidence of its\npotentiality and the possibility of its future development to a still\nhigher degree of efficiency.\n\nThere is no doubt but that the original impulse which resulted in the\ncreation of the faculty of perceptibility in the Thinker also marked\nout the metes and bounds of our entire range of perceivability which\nincludes not only the intuitional but something higher still. There is\nno doubting either the obvious fact that these metes and bounds cannot\nhave been other than rudimentary or general lines of denotation, and\nthat the work of their further elaboration and refinement is a matter\nof evolutionary detail. For if we assume that the general principles\nof evolution are true we immediately recognize the cogency of this\nview. That which we now call the hand has not always been the perfect\ninstrument that it is nor has the ear always been so keenly adjusted\nas at present. It has required undoubtedly many million of years for\nthe eye to reach its present degree of complexity and adaptability.\nYet in all these cases the different organs existed in potentiality\nfrom the beginning; the metes and bounds of the hand, the ear and the\neye were laid out primordially. Evolution has specialized and adjusted\nthem to environments and needs. Thus it will be seen that while the\nintuitional faculty was designed for manifestation from the beginning\nit has nevertheless required ages for its appearance even in the most\nrudimentary fashion.\n\nAlmost the entire content of human knowledge is based upon assumptions\nor hypotheses; in fact, is but a mass of these, and especially is\nthis true of mathematics, science and philosophy. Of course, there\nare certain minor observable facts which by reason of the seeming\npermanence of their existence have been eliminated from the category of\nassumptions. But even this elimination when it is traced to its depths\nmay be found to be erroneous, and perhaps after all, when we have\nreally begun to know something of the reality of things, may have again\nto be placed in this category. And then, too, the hypothetical nature\nof our knowledge is due largely to the Thinker's method of contacting\nthe objective world which is the subject of his knowledge. It is\nbecause it is necessary for him to interpret the neurographical symbols\nwhich sense-impressions make in the brain matter according to a psychic\ncode that renders his knowledge of things in general hypothetical. His\ninterpretations are based upon an assumed value which experience has\ntaught him to give to each neurogram. But even when his interpretations\nleave nothing to be desired in respect to their accuracy of\napprehension of what the neurographical vibration implies there is that\nfurther barrier to his cognition of reality which is due to his remote\nremoval from the object itself and the consequent extreme difficulty,\nif not present impossibility, of identifying his consciousness with the\nessence of the objects which he contemplates.\n\nWhen the Thinker's consciousness is presented with a neurograph\nof say, a cube, it is not the cube itself which he contemplates\nor observes; it is the neurograph or psychic symbol which the\nsense-impressions make in the brain. His consciousness deals not with\nthe object but with the symbols. It is true that when he verifies one\nneurograph by another, as the _scopographic_ or sight impressions by\nthe _tactographic_ or touch impressions it is found that the delivery\nthus determined is a true enough representation. It is also true that\nthe Thinker, as a rule, does not accept a neurograph as valid until it\nhas been verified by at least one or more presentations through his\nouter sense organs. It occurs, therefore, that all such deliveries\nare verified and corrected by one or more sense witnesses before\nfinal acceptance by the Thinker; but even then it cannot be said that\nhis notions thus gained are in all respects correct and true to the\nstandards set up by the brain-consciousness not to mention higher forms\nof consciousness. And then, when we consider that in addition to the\nnumerous chances for error which naturally inhere in this method of\ncognition it must also be apparent that the Thinker's approach to the\nreality of things is much impeded by his separation therefrom, the\nunreliability of our ordinary methods of cognition is much emphasized.\n\nBut aside from the egoic or brain-consciousness there is the higher\nconsciousness of the Thinker himself. For the brain-consciousness is\nmerely his method of regarding and comprehending the neurographical\ndeliveries, the psychic code by which he systematizes and organizes\nhis cognitions or impressions of the sensible world. This higher\nconsciousness constitutes the faculty _a priori_ for the Thinker on\na higher plane of existence, and because it deals with elements\naltogether unlike those which make up the data of brain-consciousness\nis, accordingly, less liable to error in its judgments of the\nsupersensuous presentations than is the objective or brain-mind. In\nfact, it is difficult to conceive of a state or conditions wherein,\nsupported as this view contemplates, the intuition should err in\njudgment. Viewed from the standpoint of external impedimenta, this\ncondition may be said to be due to the absence of sensuous barriers\nwhich would otherwise prevent the near approach of the Thinker's\nconsciousness to the essence of things which is the object of\nhis consciousness on this higher plane. Directly, however, it is\nundoubtedly due to the fact that, following the lead of life itself,\nyea, as the veritable handmaid of life, it cannot err where life\nis concerned. When dealing with notions _a priori_ or intuitograms\nthe Thinker is relieved of the onerous necessities and limitations\nincident to the examination and determination of neurographic symbols\nregistered in the brain cortex and so is free to study, to examine\nand judge at first hand the impressions which are received from his\nown plane of intuition. The difference is about the same as that\nwhich should exist between the methods of communication between two\ntelegraphic operators when in one instance they would have to depend\nupon the deliveries conveyed over the wires, while in the other, when\nthey stood face to face with each other, they could communicate by\ndirect conversation. In the one case the method of communication is\ndirect and simple, while in the other it is indirect, circuitous and\ncomplex. It can, therefore, be readily seen that in all cases where\nthe approach is made in a direct, simple manner the probability of\nerror is much less than in cases where the intellectual approach is\nless direct and more complicated. Hence in drawing conclusions as to\nthe relative importance of the two mechanisms of consciousness, the _a\nposterioristic_ and the _a priori_, it is necessary to bear in mind\nthe comparative superiority of the one over the other as a means of\ncognition. It matters little that the intuitional faculty is not so\nwell developed as the tuitional because it is but natural that inasmuch\nas the Thinker's needs are subserved in the sensuous realm by the\ntuitional consciousness it should, from more active use, gain somewhat\nover the intuitional in facility of expression and general utility. And\nthe more so, because the two faculties serve different purposes; one is\nattuned to receive impressions from a subtler plane while the other is\nfitted for contact with the phenomenal universe; one is related to the\nconceptual while the other is related to the perceptual. They differ\nnot only in function, but in nature as well. There is, of course, a\nnatural barrier consisting of the inherent limitations of each faculty\nwhich prevents the full mergence or unification of the two states of\nconsciousness so that there exists a state of consciousness the data of\nwhich the brain-mind is unaware, it being able only at rare intervals\neven to receive so much as slight impressions from it in the nature\nof intuitional flashes or inspirations and the like. Viewed in this\nlight it would appear that the cognitions which are most truthworthy\nare those which are presented by the intuitional faculty because\nthey are nearer to the essential reality of things; they have to do\nmore specifically with the nature of that which _appears_ while the\ntuitional mind can only regard that which is the appearance. Herein\nlies the whole difference.\n\nThe natural outcome of this division of labor between the tuitional and\nthe intuitional is the establishment of the fact of man's relationship\nboth to the phenomenal and the real; that in his psychic nature must\nreside the faculty of apprehending the real and that he shall one day\nawaken into activity this now latent faculty whereby he may make a\ndirect and naked contact with reality.\n\nIf it be true that, as PLATO said, God does geometrize, and that\nthe divine geometry, as will appear, is based upon a system, an\nalphabet which taken together are the point . , the line ----, the\ntriangle [Illustration], the square [Illustration] and the circle\n[Illustration], then, we have in this geometric alphabet the very\nsecret of the divine geometry. With these, and in the kosmic laboratory\nof _chaogeny_, the Creative Logos has measured off the limits and\nconfines of space; with them He has traced out its dimensions the\narcheological evidences of which we may view in the space-mind itself;\nand with them he has established the manner of its appearance to the\nThinker. In dimensions, three, and yet not three, but one, Space,\nthe eternal progenitor of all forms and energies, having received\nthe divine fiat in the beginning that thus far it should extend\nand no further, persists in faithful obedience to the law of its\nbeing--tridimensionality. It must be so because it is thus sanctioned\nby the highest faculty in man that can render judgment thereon. If\ntridimensionality inhere in the space-mind, as the law of its being and\nin the intuitional consciousness as the norm of its essential nature\nand as the easiest and simplest expression of the tuitional mind, how\ncan it be gainsaid that these considerations obviate the necessity of\nthe mathetical hyperspace?\n\nIf the reality of things is hidden from us and if we are, therefore,\nunable to perceive their real essences it is because our mode of\nthought and our consciousness have obscured our vision and limited us\nto this state of paucity of perception. It is not because reality is\nitself a hidden, inscrutable quantity nor that its _modus vivendi_\nis \"unknowable\"; but because we being multiformly limited, \"cabined,\ncribbed and confined\" are resultantly lacking in the power to discern\nthat which otherwise would be most obvious to us. It may well be set\ndown as axiomatic that when, in the process of our thinking, we arrive\nat the inscrutable, the unknowable and the infinite, it is evident\nthat our thought processes are dealing with a form of realism which is\nhigher and beyond the possibilities of our loftiest thought-reaches.\nAnd in order to symbolize to itself this condition the intellect\nposes such terms as \"inscrutable,\" \"unknowable\" and the \"infinite\"\nsimply because that is the best it can do. Hence when it is said that\nspace is infinite it is apparent that the mind recognizes that when\nit contemplates space it is dealing with something whose degree of\nrealism transcends its powers of comprehension. Infinity is a relative\nterm, and in fact, decreases in extensity in the proportion that the\nconsciousness expands and comprehends. It is not unlikely that should\nthe intellect one day discover that it had awakened into union with\nthe space-mind it would immediately reject its preconceived notion of\nthe infinity of space. But we need not wait until the coming of this\nfar off event in the path of psychogenesis; for we can here and now\nperceive with what must be a higher faculty than the intellect the\nverity of this conclusion.\n\nBut certain it is that the intellect, in the pride and arrogance of\nits traditional heritage, will not without a great struggle yield the\nground and prestige it has held for an aeon of time; and in vain does\nthe intuition serve notice of dispossessal in these premises; but\nhowever stubbornly fought the battle, however tenaciously held the\nposition time will discover the weakening of the intellect's hand.\nDeath for the intellect may ensue as a result of the conflict but it\nwill be a death wherefrom it will arise, quickened, revivified and\nuplifted by its disposer, the intuition, upon the remains of its dead\nself to a higher and grander state than it has ever enjoyed before.\n\nSpace is not static. It is dynamic, potential and kinetic. It is a\nprocess, a becoming. Its duration as a process is never ending. Its\nextensity is limited and finite. The so-called infinity of space is one\nof the capital illusions of the intellect which can only be removed\nby an expansion of the consciousness, by a mergence of the individual\nconsciousness with the space-consciousness. In the ever-widening circle\nof the individual consciousness lesser realities give way to greater as\nthe darkness recedes from the light--the lesser appearing in comparison\nwith the greater, as the consciousness broadens, as matter to spirit,\nas night to day or as limitation to non-limitation. Thus the most solid\nfacts and conditions of our limited life are but the shadows of the\ndeeper realities which shall be revealed to the Thinker in the days of\nhis larger and more glorious life of freedom from limitations.\n\nAnd now it will appear that the whole fabric of our knowledge shall\nhave to be reduced to the bare warp and woof; for nothing is real but\nthese. It is as if the Thinker, using the tuitional mind, had been in\nall times past studying the design woven in the surface of a very thick\nplush carpet. There are the warp and woof, the long vertical threads\nwhich make the plush and the intricate design appearing on the surface.\nOur knowledge may be likened to the design. It represents the contents\nof our knowledge. We have not even so much as begun the study of the\nnature of the vertical threads as they appear beneath the surface to\nsay nothing of beginning the study of the warp and woof. The warp and\nthe woof are the realism of the kosmos; the vertical threads are the\nroots and stem of the phenomenal world; the design is our sensible\nworld as it appears to the intellect. The life of the intellect has\nbeen spent in contemplating this design; while of the hands which wove\nthe carpet, of the mind which directed the hands and of the spirit\nwhich vitalized all, it knows nothing nor indeed can it know anything.\nWhere shall we say are those hands, that mind and that spirit which\nmade the carpet possible and an actuality? In vain do we search among\nthe remains, among the soft, glistening threads of the carpet or among\nthe intricacies of the design. For they are not there. They have passed\non. The intellect looks at the design or at the vertical threads and\nbecause it is unable to follow them to their source, it decides that\nthey are infinite, inscrutable and unknowable. But not so. All that is\nrequired are eyes to see and a mind (or shall we say a mind vitalized\nby the intuition) trained to discern the threads as they point upward\nwith their termini firmly rooted in the warp and woof of the fabric.\nBut we must first master the design, and then turning to the threads,\nmaster them. Then shall the doors of kosmic reality swing wide and the\nThinker shall be ushered into the eternal palace of kosmic realism\nwherein he shall find the great secret, the heart, the purpose, the\nbeginning and the end, the very nature of things-in-themselves.\n\nThe nature of every degree or condition of realism is so constituted\nthat its qualities, characteristics and limitations are exactly\nadequate for the satisfaction and fulfillment of all the requirements\nand needs of every possible state of normal consciousness. So that each\ndegree of reality and each state of normal consciousness is sufficient\nand complete in itself and mutually satisfies the necessities of each\nother. The substratum of reality or life which extends from the heart\nof the kosmos to the extreme limits of the phenomenal universe exists\nin degrees, not discrete, but continuous. And these merge into one\nanother by insensible stages. Such is the imperceptible continuity of\nthe whole as each degree is gradually immerged into the other that\nonly the limitations of consciousness itself make it to appear as if\nit were discontinuous. For every stage of realism there is a state\nof consciousness which answers to it completely and sufficiently.\nSo both the state of consciousness and that of reality, manifesting\nat any given stage, seem to be complete and final for that stage.\nRealism or life and consciousness possess only a relative finality\nfashioned upon the necessities and requirements for any given state\nof being. Consciousness alone fixes the apparent limits of life; it\nalso determines the state of our knowledge of life. And thus when the\nThinker is confined to any stage of reality and congruent degree of\nconsciousness it appears that what he there finds is ample for all his\npurposes. Accordingly he is convinced that that stage is the final\nconsideration of his scope of motility. It is only when he is able\nto raise his consciousness to a point where he can contact higher\nrealities that he becomes aware that there are higher stages in which\nhis consciousness may manifest. This peculiarity of the Thinker's\nconsciousness is accentuated when he allows himself to become wholly\nengrossed with a study of the phenomena of that stage in which he can\nconsciously function. Hence it constantly occurs that men exhausting\nthe study of the phenomenal find themselves floundering upon the beach\nof the outlying shores of consciousness where in sheer desperation\nthey fall into the illusion that they have indeed reached the limits\nof manifested life and that beyond those limits there is no organized\nbeing. Unconscious are they that in ever widening circles the fertile\nlands extend and await the awakening of their consciousness when they\nmay till the fallow ground of this new domain and begin again the\nsearch for the ultimately real.\n\nWith respect to the present powers of consciousness, it cannot be\nsuccessfully controverted that the concept of tridimensionality of\nspace is sufficient for all purposes. It must be so for it is not only\nan aspect of the phenomena of space but of reality as well. This fact\nis attested by the nature of mind that answers to the nature of space.\nTridimensionality characterizes the entire extent of consciousness and\nlife, and therefore, of space itself. This characterization may be\ntraced to the very doors of the heart of space where the three become\none. Nor would this conception be in the least vitiated if it were\nallowed that the mass of the phenomena of the supersensuous world,\nlying in close proximity to the sensuous world, does present itself to\nthe consciousness in a four-dimensional manner and that the phenomena\nof a still higher plane present themselves in a five or _n_-dimensional\nmanner to that state of consciousness which may be congruent with\nthem; because then we should be making allowances for the changes in\nphenomena and their mode of presentation to the consciousness which\nby no means implies a corresponding change in reality or life. All\nphenomena are fashioned by the intellect. The phenomenal world is\njust what the intellect interprets it to be. It is that and nothing\nmore. Its qualities, attributes and characteristics are such as the\nconsciousness gives to it. It exists only for the purposes of the\nevolving consciousness. And, as an instrument of consciousness, its\nexistence is strictly subject to the evolutionary needs thereof. In\nthat moment that the immediate needs of the consciousness shall no\nlonger be able to find satisfaction in the phenomena of any plane of\nnature, in that moment the phenomena of that plane disappear, recede\nand are swallowed up in the maelstrom of eternal reality.\n\nIn the gradual expansion of consciousness as it passes through the\ninfinite series of grades of awareness meantime becoming deeper,\nbroader and more comprehensive as it proceeds, there may be observed\nrunning through all these planes and orders that which is neither the\nphenomena of the various planes nor the consciousness; but which must\nbe the substructural basis of both, remaining the same, unchanged and\nunchangeable. That is the thread of reality, the passage of life itself\nwhich is the eternal basis of all. Now it is to this reality, life,\nthat the space-mind is related and in which its roots, its heart and\nthe very center of its being are at one with the divine mind of the\nkosmos.\n\nThe question of dimensionality is solely a concern of the objective\nor brain-mind which is the intellect. It is one of the ways in which\nthe intellect endeavors to understand phenomena. It is an arbitrary\ncontrivance devised by the intellect for its convenience in studying\nthe world of things. Without it, as obviously appears, the intellect\nwould not be able to go very far in its consideration of the minor\nproblems which inhere in material things. The fourth dimension is\nbut another attitude, another contrivance, which the intellect has\ndevised in order that it may study from another angle the evanescent\nphenomena of the world of appearances. Having apparently exhausted\nthe possibilities of motion in three dimensions, and being driven on\nto the acquirement of more picturesque views by the very necessity\nof its continued growth, it has betaken itself to another higher\nmountain peak, called \"hyperspace\" where with larger lenses and higher\npowered instruments it is beginning to scan the landscapes of a new\nintellectual realm of consciousness. Yet the celestial wonders of this\nnew-found realm of consciousness remain in undisturbed forgetfulness\nor neglect. But it is not by a scrutiny of mathetic landscapes nor\nby a study of the celestial wonders that the Thinker shall one day\nrealize the object of his eagerly pushed quest after the real; for he\nshall find it, if at all, in the temple of the kosmic mind which is not\nmade by the intellect nor meted and bounded by geometric systems of\nspace-measurement.\n\nIn all the learned pother incident to the mastery of the phenomenal,\nthe furniture of the world of the senses, it is as if the self in man,\nthe Thinker, sat secluded in a six-walled tenement, and hence six times\nremoved from the subject of his study, and endeavored to interpret that\nwhich appeared to his vision. And thus, thinking that what he sees is\nthe only reality, he remains in inglorious nescience of the reality of\nthat upon which he himself stands, unconscious that the tunnel-shaped\naperture through which he peers leads not outward, but _backward_ and\nwithin to the habitation of the real of which he himself is a part.\nMen are deeply and well-nigh hopelessly concerned with appearances,\nwith static views of life, with instantaneous exposures. Life, reality\nand all the eternal verities pass on and assume countless postures,\nattitudes, moods, tenses and nuances. The intellect is content to\noccupy itself with a single tense or mood. Indeed, it has no aptitude\nor power to consider more than one at a single time. It thus misses\nthe continuity, the ceaselessness, of life. What is more, every\nsingularity, every attitude, mood or tense which the intellect grasps\nfor consideration is immediately remade so as to fit its own moods\nand tenses. And upon each and every nuance the intellect immediately\nimposes its own form--actually and literally rehabilitates them with\nits own habiliments. Unfortunately, this peculiarity occludes the\nintellect from any approach to the true nature of that phase which it\ncan grasp.\n\nHyperspace is one of the illusions of the phenomenal; it is the dress\nwhich the intellect has superimposed upon a single nuance; it is a mask\nwhich is an exact replica of the mood of the intellect. Yet through\nthis mask the intellect grandly hopes to approach reality. The period\nthrough which the mind is now passing is a repetition of the evil\ndays of scholasticism when men set out to determine the exact number\nof celestial beings that could be perched upon the extremity of a\nneedle point. It is a time when men's minds easily assume grotesque\nand hideous shapes and their thoughts become the embodiment of\nfantastic entities. The exclusive occupation of such minds becomes the\nfabrication of mathetic monstrosities which rapidly deliquesce upon\nthe first approach of the real or the appearance of the first ray of\nintuition which may escape through the dim and misty condition of the\nintellectual over-hangings. It will not be ever thus; for the Thinker\nwill one day pass from a study of the arrangement of phenomena in space\nand by well-ordered steps come once again to himself. And then through\nthe maze of it all set out upon the true path----the tridimensionality\nof space following which he will inevitably approach the citadel of the\nreal, the kosmic space-mind.\n\n\n\n\n  CHAPTER VII\n\n  THE GENESIS AND NATURE OF SPACE\n\n  Symbology of Mathematical Knowledge--Manifestation and\n  Non-manifestation Defined--The Pyknon and Pyknosis--The\n  Kosmic Engenderment of Space--On the Consubstantiality of\n  Spatiality, Intellectuality, Materiality, Vitality and Kosmic\n  Geometrism--Chaos-Theos-Kosmos--Chaogeny and Chaomorphogeny--N.\n  MALEBRANCHE On God and the World--The Space-Mind--Space and Mind\n  Are One--The Kosmic Pentoglyph.\n\n\nGeometry is concerned primarily with a study of the measurement of\nmagnitudes in space. Three co\u00f6rdinates are necessary and sufficient for\nall of its determinations. Metageometry comprehends the study of the\nmeasurement of magnitudes in conceptual space. For its purposes four\nor _n_-co\u00f6rdinates are necessary and sufficient. Perceptual space is\nthat form of extension in which the physical universe is recognized\nto have been created and in which it now exists. Conceptual space is\nan idealized conception belonging to the domain of mathesis and has\nno actual, physical existence outside of the mind. Mathematical space\nrepresents the idealism of perceptual space.\n\nGeometrical magnitudes may be defined as symbols of physical objects\nand geometry as a treatise on the symbology of forms in space. In\nfact, all cognitive processes are simply efforts at interpreting\nthe symbolism of sense-deliveries; and the difference between mere\nknowledge and wisdom, which is the essence of all knowledge, is the\ndifference between the understanding of a symbol and the comprehension\nof the essential nature of the thing symbolized. So long as knowledge\nof space is limited to the understanding of a symbol or symbols by\nwhich it is presented to the consciousness so long will it fall short\nof the comprehension of the essential nature of space. In vain have\nwe sought in times past to understand space by studying relations,\npositions and the characteristics of forms in space; in vain have\nwe based our conclusions as to its real nature upon the fragmentary\nevidences which our senses present to our consciousnesses. It is\nas if one had busied himself with one of the meshes in a great net\nand confined his entire attention to what he found there, meanwhile\nremaining in complete ignorance of the nature of the net, how it came\nto be there, of what it is made and how great its extent may be.\n\nThere is ever a marked difference between a symbol and the thing which\nit symbolizes. Words are the symbols of ideas; ideas, as they exist\nin the mind, are the symbols of eternal verities as they exist in\nthe consciousness of the Logos of the universe. There may be a wide\ndiversity of symbolic forms which represent one single idea; as, for\ninstance, the variety of word forms which represent the idea of deity\nin the various languages. Likewise there may be a multiplicity of ideas\nwhich represent a single verity. But neither is the idea nor the word\nthe real thing in itself. That quality of a life-aspect which we call\nits _thingness_ has an essential nature which cannot become the object\nof consciousness except by virtue of its representation through ideas\nand their symbolisms, and even then, the thing which we conceive is\nnot the nature of a quality of the life-aspect but an idea of it--a\nsymbol which stands for that idea. In order, therefore, for the mind\nto arrive at an understanding of an eternal verity, such as space,\nit must first be able to synthesize all of the representative ideas\nand then abstract from their compositeness a notion of its essential\nnature. But this can be done only by identifying the consciousness\nwith the essential being of the object considered. In other words, the\nconsciousness and the intrinsic being of forms, principles, forces\nand processes must embrace each other in the intimacies of direct\ncognition; the life which is consciousness and that life which is\nessential being, being coeval, co\u00f6rdinate and mutually responsive, must\nin so close a contact as here intimated reach an understanding of the\nrealism shared by both. That is, the human consciousness, following in\nthe wake of life and consisting of a specialized aspect of life itself,\nwill, by such an intimate approach to the life-principle of forms,\nreadily understand; for it has only to recognize a replica of itself\nin rendering its judgment. But it is not claimed that such a state of\nrecognition by the consciousness of life itself can be attained at all\nby ordinary means, neither is it believed that it is the next stage\nin conscious evolution. However, it is not doubted but that such an\nexaltation of the consciousness is possible, yea practical; but the\ndifficulties which beset the path of attainment in this direction are\nso great that it may as well be considered unattainable. The mere fact\nof these difficulties, however, only re-emphasizes the insufficiency\nof the intellectual method. The identification of consciousness with\nessential being is a procedure which cannot be accomplished by an act\nof will directly and immediately. Because it is a process, a series of\nunfoldments, an adjustment of the focus of consciousness to the kosmic\nessentialities which constitute the substructure of the manifested\nuniverse. In the very nature of things, a kosmic essentiality cannot\nbe viewed as being in manifestation especially in the same degree\nas ordinary physical objects are manifest. The former is a state, a\npotentiality, a dynamic force, an existence which should be thought\nof as an extra-kosmic affair dwelling on the plane of unity or kosmic\norigins; while the latter are the exact opposite of this. The one can\nbe seen, felt and sensed while the other is the roots which are not\nseen but lie buried deeply in the heart of the universal plasm of being\nand beyond the ken of sensuous apperception. The term _manifestation_\nis both relative and flexible in its use. It is relative because\nit will apply equally to all stages of cognition. A thing is in\nmanifestation when it is presentable to the ordinary means of cognition\nbelonging to any stage of conscious functioning; it is not in\nmanifestation when it is beyond the scope of the Thinker's schematism\nof cognitive powers. Its flexibility is seen in its ready yieldance to\nthe entire range of implications inhering in the process of cognition,\nfitting the simplest as well as the highest and most complex.\n\nGreat is the gulf which is interposed between manifestation and\nnon-manifestation; and yet the two, in essence, are one. They are\nlinked together as the stem of a flower is joined to its roots.\nLikewise one is visible, palpable while the other is invisible,\nimpalpable though no less real and abiding. As the thin crust of\nearth separates the stem, leaves and flowers from the roots so the\nlimitations of man's consciousness separate the manifest from the\nunmanifest. Similarly, as when the surrounding earth is removed from\nthe roots and they are laid bare revealing their continuity and unity\nwith the outputting stem and flowers, so, when the limitations of\nconsciousness are removed by the subtle process of expansion to which\nthe consciousness is amenable so that it can encompass the erstwhile\nunmanifest, it, too, will reveal the eternal unity of the kosmic\npolars--manifestation and unmanifestation.\n\nThere is but one barrier to ultimate knowledge and that is the human\nconsciousness however paradoxical this may seem. The unutterable\ndarkness which shuts out the so-called \"unknowable\" from our\ncognition is the limitation of man's upreaching consciousness. These\nlimitations constitute the difference between the human intellect and\nthe mind of the Logos. Nevertheless the outlying frontiers of man's\nconsciousness gradually are being pushed farther and farther without.\nEvery new idea gained, each new emotion indulged, each new conception\nconquered, and every mental foray which the Thinker makes into the\nrealm of the conceptual, every exploration into the abysmal labyrinth\nof man's inner nature are the self's expeditionary forces which are\ngradually annihilating the frontier barriers of consciousness and thus\napproaching more closely upon the _Ultima Thule_ of man's spiritual\npossibilities.\n\nSpace is in manifestation. It exists and has being whether it is\nviewed as an object, an entity or the mere possibility of motion. That\nit offers an opportunity of motion and renders it possible for objects\nto move freely from point to point cannot be denied and yet this fact\nhas no bearing whatever upon the essential nature of space. The very\nfact of its appearance, its manifestation, makes it obvious that it is\nthe nether pole of that eternal pair of opposites--manifestation and\nnon-manifestation, being and non-being, which are essentially one.\n\nIt will be seen from figure 17 that the period of involution embraces\nseven separate stages, the _monopyknon_, _duopyknon_, _tripyknon_,\netc., being the unit principle or engendering elements of the\nrespective stages. Involution comprises all creative activity from the\nfirst faint stirrings of the void and formless chaos until the universe\nhas actually become manifest and dense physical matter has appeared.\nIt is divided into two cardinal periods, namely, the chaogenic period\nduring which primordial chaos is given its character and directive\ntendencies for the world age. It is a phase of duration wherein the\nfiat of kosmic order is promulgated, and consist of three stages,\nmonopyknosis, duopyknosis, tripyknosis[24] gradually, insensibly,\ngradating into the fourth or _quartopyknotic_; the chaomorphogenic\nperiod is likewise divided into three stages--_quintopkynosis_,\n_sextopyknosis_ and _septopyknosis_, developing out of the fourth\ngradually. The quartopyknotic stage is the stage of metamorphosis\nor transmutation wherein the transition from non-manifestation to\nmanifestation is completed; it is also the stage of kosmic causation,\nbecause from it spring the matured causes or \"vital impetus\" which\nengender all that follow.\n\n  [24] See Fig. 18.\n\n[Illustration: FIG. 17.--Involution and Evolution]\n\nThe close of the involutionary phase of the world age is marked by the\nfinal deposition of dense physical matter and this is closely followed\nby the beginnings of the evolutionary movement which, like the\ninvolutionary movement, is divided into two cardinal periods, namely,\nthe morphogenic (during which are produced, in turn, insensible forms,\nsensible forms and spiritual forms) and the kathekotic period which\nmarks the perfection, the consummation of the evolutionary movement.\nThese two cardinal periods of the evolutionary phase of duration and\nthe two cardinal periods of the involutionary phase complete the kosmic\nage, the \"Great Day of Brahma.\" The concentric circles, beginning with\nthe dot and ending with the seven concentric circles, and designated\nas a, b, c, d, e, f, g, are representations of the constitution of the\nrespective units corresponding to each of the seven subdivisions. They\nsymbolize the seven degrees of condensation or pyknosis which comprise\nthe genesis of space, on the one hand, and on the other, the stages\nof unfoldment. Because, during involution all potencies, powers and\ncharacters were being infolded, involved; but during evolution, these\nare being unfolded, expressed, evolved.\n\nThe figure 18 is another view of these two major movements, involution\nand evolution. The genesis of space is here shown symbolized by the\nKosmic Egg. The seven stages of involution are referred to as, the\nmonopyknotic, duopyknotic, tripyknotic, quartopyknotic, quintopyknotic,\nsextopyknotic and the septopyknotic; while the corresponding stages\nof evolution are referred to as, the physical, the sentient, mental,\ncausative or spiritual, the triadic, duadic and monadic, indicating\nthat the principle of physicality is succeeded by the principles of\nsentience, mentality, spirituality, and the three forms of kathekotic\nbeing. This symbolism, it should be stated, is designed with respect\nto the universe and man and has no reference to other possible\nevolutions than the human and contemporaneous animal, plant and mineral\nevolutions.\n\n[Illustration: FIG. 18.--The Genesis of Space]\n\nTo follow the ramifications of the symbolism above would involve a\nsurvey of all branches of knowledge, and indeed, would be out of place\nin this book. Only the widest general outlines can be suggested here\nand it is believed that this is sufficient to enable the reader to\ngrasp the magnitude of the symbol and to understand its purpose and\nintent.\n\nIt would appear, therefore, that if it is possible for the intellect to\ntraverse by means of a study of kosmic symbolism, used as a standard of\nreference, the entire length of the bridge which engages the antipodes\ninto an eternal unity, something may be gained in the way of a more\ndefinite and clearer understanding of the essential nature of space in\nits relation to kosmogenesis.\n\nIn the diagram, figure 17, is shown a table which represents the\nstages of space-genesis. It will be noted that the whole scheme is\ndivided into seven stages. It is not an arbitrary division simply but\na symbolic one and represents fullness, completeness, entirety. The\nnames given, namely, \"monopyknotic, duopyknotic,\" etc., represent the\nsymbolic characteristics of each stage in its relation to the universe\nin the process of becoming. The terms \"monadic,\" \"duadic,\" \"triadic,\"\netc., are representative of the seven planes of matter in the universe.\n\nA _pyknon_ is a kosmic principle and represents the typal aspect\nof kosmogenesis. It is a generic term and may be identified in its\nrelation to the various stages by the prefix. The monopyknon belongs to\nthe ulterior pole of the antipodes on the side of non-manifestation.\nSo do the _duopyknon_ and the _tripyknon_. Pyknosis is a process of\nkosmic condensation, or limitation for purposes of manifestation. It\nis a stage in the descent of the kosmic Spirit-Life, a degradation of\nnon-manifestation into manifestation, and is, therefore, the cardinal\ncausative principle of creation. The term pyknon being generic\nis applicable alike to a particle of matter, a state of being, a\ncondition of existence, a process or a principle. The _monopyknon_ is,\naccordingly, the primary aspect of the process of kosmic pyknosis. It\nis the archetype and therefore all inclusive and omnipotential. But\nwhether regarded as a singularity or as a whole it should never be\ndivorced in thought from the primal act of creation. It represents the\nfirst act of material differentiation in the being of the creative\nLogos on the plane of non-manifestation. It is the beginning of every\ngreat planetary or kosmic _manvantara_ or period of manifestation.\nDuring either a planetary, solar or kosmic _pralaya_, or gestatory\nperiod, the kosmic plasm is in a quiescent, undifferentiated condition.\nThis undifferentiated plasm when acted upon by the will of the Creative\nLogos, _Fohat_, as He is sometimes called in Eastern Philosophies,\nbegins to become conditioned, begins to differentiate. The primal\nact or stage in such a process is the formation or appearance of a\nmonopyknon. It then becomes the characteristic aspect of that stage.\n\nMonopyknons are the quiescent, unawakened, though potential and\narchetypal principles peculiar to the monopyknotic period of\nspace-genesis which are ultimately to become, on the physical plane,\nsingularities of life of whatsoever kind. _Thus the lineage of every\nsingle life-form or principle in the universe runs unbroken back from\nthe present of the Now to the present of monopyknosis. So great is the\ndesign of the kosmos that the entity which is now man or the atom was\nstarted on its journey to this culmination at the break of the Great\nKosmic Day when the omni-pregnant wheels of monopyknosis first began to\nturn._ Duopyknons and tripyknons constitute the two remaining stages\non the plane of non-manifestation. And their correspondences in every\nstage of involution or the descent of spirit into matter are eternal\nand kosmic. Likewise the lineage of the dual and triple aspects of\nall life forms on the path of evolution may be traced rearward to the\nduopyknotic and tripyknotic stages of kosmogenesis.\n\nThe metamorphosis by which the monopyknon becomes a duopyknon contrives\nthe differentiation of the pristine plasm of kosmic being so that the\nfirst becomes the ensouling or vitalizing principle of the second;\nand, in turn, the second becomes the vitalizing or inner principle\nof the third; the third of the fourth and the fourth of the fifth\nand so on throughout the series until the last is reached which is\nthe _septopyknon_. The septopyknon is, therefore, a seven-principled\nform. It is both unitary and septenary--unitary in the sense that\nthe seven are really one and septenary for the reason that each of\nthe seven principles, in the course of evolution, becomes a separate\nprocess specially adapted to functioning upon its peculiar plane of\nmatter. Thus it is seen that the utmost significance attaches to this\nseptenary pyknosis of the kosmic plasm of life. The implications of\nthis conception are, of course, too vast and multifarious to be set\ndown here. We shall have to dismiss it with one observation only, and\nthat is: _every single appearance of life and form in the totality of\nsuch appearances is rooted in kosmic pyknosis where it has received its\ninner vitalizing force, its form and the law of its mode and manner of\nappearance together with the metes and bounds of its existence_.\n\nThese processes, monopyknosis and duopyknosis, are to be regarded as\ntaking place, each in its own period, everywhere throughout the Body\nof Being of the Logos but on the plane of non-manifestation. They are\nstates of preparation for manifestation analogous to the germinative\nperiod of the seed or the egg. They represent the first stirrings of\nthe kosmic plasm and contain the promise and potency of all that is\nto succeed them. There is one other stage, co\u00f6rdinate with these two,\nand that is the tripyknotic which completes the unmanifest trinity\nand constitutes the archetypal vehicle whence proceeds the manifested\nuniverse. The ensouling principle of the tripyknon is the duopyknotic\nprinciple. But when the descent has reached the tripyknotic stage the\nthree have been merged into one and the characteristics which were\npeculiar to each as a separate pyknon are then fused into a single\nquality having three aspects which are mutually interdependent and\nco\u00f6rdinate.\n\nThe unmanifest trinity, now complete as a result of the triple\npyknotic process, is the imperishable and ever sustaining radix, the\nall-mother of the manifested universe. It is the Golden Egg laid by\nthe god _Seb_ at the beginning of a great life-cycle. It is also\n_Chaos-Theos-Kosmos_, i.e., kosmic disorder, divine will or generating\nelement, and kosmic order or space. In it, as in an egg, resides\nin kosmic potency all that the universe is to become in any \"Great\nDay of Brahma\" or any Great Kosmic Life Cycle. Its eldest born is\nSPACE, physiological and perceptual, and the latter is the eternal\nfather-mother of the universe. _Space is, therefore, the male-female\nprinciple of manifestation; in its kosmic womb all forms are created,\ndeveloped, evolved and sustained._ Into it again, at the close of\nthe Great Day, all existences, forms and all principles matured and\nripened by the vicissitudes of kosmic evolution will be inhaled with\nthe return of the Great Breath of Life. The unmanifest trinity is\nthe archetype, and therefore, the pattern or model for the manifest,\nembodying in potency all that the manifest may ever become. Forth from\nthe unmanifest proceeds space as a dynamic process endued with the\npotentiality for generating all that the manifested universe contains.\n\nThe terms \"unmanifest,\" \"unconditioned\" and \"unlimited\" have a special\nmeaning here and are used in the same sense as the mathematical term\n\"transfinite,\" and therefore, imply a transcending of any finite or\nassignable degree or quality of manifestation. Hence they should\nbe distinguished from the term \"absolute\" which has a different\nimplication. So that although the triple process outlined above\nmay not be viewed except as a characterization of the plane of\nnon-manifestation, and hence of the primordial activity of the Creative\nLogos, there is nothing in the symbolism to warrant the identification\nof this process in any way with the Logos in absolution. For on this\nview Absolute Being is, in a large measure, sacrificed when the\nmonopyknotic process is begun and the monopyknon (kosmic principle)\nbegins to appear. Absolute Being, while it may not be defined,\ndelineated or described may be symbolized by the ideograph: \"action in\ninaction\"; \"being in non-being\"; \"manifestation in non-manifestation\";\nfor these are symbols merely and do not describe or delineate.\n\nWe have observed the subtle connection which exists between\nmanifestation and non-manifestation and have seen how that, as the\nroots of a plant sustain the outer growth of stem and flowers, the\nformer being the matrix out of which proceeds the latter, and that\nin like manner does the unmanifest sustain the manifest; it should,\ntherefore, be clear that the Body of Being of the Unmanifest Logos, in\na similar manner, is the basis and cause of all that is manifest or\nin existence; _and yet it is more, it is essentially all that is and\nall that the all is to be in manifestation at any time throughout the\nimmeasurable process of kosmogenesis_. _It is the Self of the Universe,\nthe meta-self of the great diversity of selves._ The Self, however,\nalthough it cannot be said to exist except as a simple, homogeneous\nquantity is nevertheless, in the very nature of things, a triunity,\nand in essence, not only the basal element of the All-Space and the\nconcrete forms which exist therein, but is also identical in essence\nand substance with space.\n\nA stage has now been reached in the description of our symbolism when\nit may be assumed, upon the basis of the foregoing, that the meta-self\nhas become manifest, i.e., in potential and dynamic appearance, as a\nresult of the triple process of pyknosis hereinbefore outlined. The\nmeta-self may then be identified with the Supreme Manifest Deity;\nfor there is ever a subtle identification of the manifest with the\nunmanifest. Out of their action and interaction the universe is made\nmanifest and phenomenal and is thereby sustained. The reciprocity\nof action between these two kosmic polars, is the metamorphotic key\nto creation; it is the symbol of the procedure of Creative Will in\nthe act of creating. It is the transitional process whereby the\npassage is made from non-being to being; from the unconditioned to\nthe conditioned; from undifferentiation to differentiation and is\nreflected and symbolized in every natural process wherein matter is\ntransformed from one state to another, or life and mind and spirit\ndiffused, centralized and organized into ever new and higher forms and\nexpressions.\n\nAnother important notion to be gained in this connection is the fact\nthat it appears as a logical sequence to the foregoing that the being\nof the manifested Logos must necessarily fill all space, yea more,\nis that space in every conceivable essentiality. His limitations are\nthe limitations of space. His qualities, properties and attributes\nare the qualities, properties and attributes of space and are only\ndifferent from the original spatial character when manifested through\na diversity of forms by whose very inner constitution and exterior\nform the modifications are accomplished. _All matter in the universe,\nall energy, and indeed, all manifestations or emanations of whatsoever\ncharacter, hue, tone or quality are, in reality, His being and nothing\nbut His being. There is, therefore, no form nor ensouling principle\nwhether of life, mind or sheer dynamism which can exist outside of\nHis being and be, even in the slightest degree, absolved from an\neternal identity therewith._ Once this idea is grasped and its varied\nimplications noted it then is no longer conceivable that any other\norder or schematism can be possible in our universe, and that, too,\ndespite the multiform conceptions peculiar to the varied systems of\nphilosophy.\n\nThe matutinal dawn of creation came at the close of the tripyknotic\nmovement which resulted in the elaboration of materials, the initiation\nof principles, processes and types, and the final preparation of the\nfield of evolution. The three processes or aspects of non-manifestation\nprojected in preparation for manifestation, namely, monopyknosis,\nduopyknosis and tripyknosis represent the earliest stages of germinal\ndevelopment. When these had closed, the Great Kosmic Egg began to\ngerminate; the first faint, indescribable signs of manifested life\nbegan to appear. Involution set in. The fourth or quartopyknotic stage,\nthough only slightly differentiated, or rather representing that period\nof kosmic involution when that which is to become the manifested\nuniverse first begins to fall under the sway of kosmic order, is\nnevertheless the basis of all great world processes. It is just midway\nbetween the poles--manifestation and non-manifestation. During this\nstage the life elements are receiving the imprints of character, being\nendowed with directive tendencies and stored with such dynamism as will\npersist throughout the Great Life Cycle in which they are to manifest.\nIt is here that begins the movement of involution, the storing away\nof those elements and factors, no more and no less, that are to show\nforth on the upward path of evolution; it is here that matter begins to\nassume form; electrons, ions and atoms created, or, that those minute\nprocesses which on the evolutionary side are to culminate in these\nare originated. This is the metamorphotic stage. It is the laboratory\nof the universe wherein _Fohat_, the Creative Logos, prepares the\nmaterials out of which and in which the vast diversity of _morphons_\nor forms is created. Quartopyknosis, accordingly, is the first active\nstep, on the plane of manifestation, which results in the appearance\nof perceptual space and consequently of physical matter itself as well\nas all the other grades of matter in the kosmos. Space, brought into\nexistence by the act of the Creative Logos in imposing limitation upon\nHis being, is in its primordial form composed of quartopyknons or\nquadruplicate principles and tendencies which act in unison and to the\naccomplishment of a single end or purpose. On this plane or during the\ncontinuance of this period of space-creation, the roots of universal\nlaw and order are produced. In it are planted the principles of good\nand evil and a sharp line of demarkation established between all the\nconceivable pairs of opposites which exist. It accounts for the duality\nof life and form. Male-female; father-mother; positive-negative;\nRajah-Tamas (action-inaction)--all these find in this process their\neternal origination. It is the stage of harmony, bliss, ideality,\nperfection, perfect equilibrium and balance. Here, innumerable ages\nbefore they actually appear, the glow-worm and the daisy, the amoeba\nand the dynosaur, man and the planetary gods alike abode their time\nawaiting the toppling of the scales of kosmic potency when all would be\nplunged headlong into the endless labyrinth of becoming.\n\nThe quartopyknotic process is similar in all details to the three\npreceding processes, these latter being prototypes of all succeeding\nstages of involution. The quintopyknon, accordingly, symbolizes the\nquintuplicative action of life in its descending movement toward the\ncreation of matter in its densest form. That is, it is a five-fold\nprinciple acting in unison and kosmic consistency, infolding in the\nuniversal wherewithal that which is to become mental matter on the\nside of evolution. Just, as may be seen in the diagram, Figure 17,\nthe quartopyknotic process symbolizes, on the involutionary side of\nthe life current that which is to become on the evolutionary side,\nspiritual essence, so the quintopyknotic deposits the seeds of that\nwhich is to become mental matter. During both the quartopyknotic and\nthe quintopyknotic processes all the potencies and promises, residing\non the plane of non-manifestation and destined to show forth as\nspirit and mind are brought into a fuller and more marked degree of\nmanifestation and become the seeds of spirituality and mentality which\nare to ripen and be ladened with the fruitage thereof many ages hence.\n\nThe reader should bear in mind that the processes here described are\nthought of as taking place at the beginning and as having their roots\nplanted in eternal duration; that they refer to a period long before\nthe universe even resembled anything like its present aspect; at a\ntime even before there were individual minds to perceive it, before\neven the gods--solar, planetary, super-solar and super-planetary were\nin existence to take part in the matutinal ceremonials of creation's\nvast hour of stillness. The mind must accustom itself to go back\nof appearances, back of time, back of space itself and discern the\nfoundations of time, space and appearances being laid and to perceive\nthat which is no less than the action of the Supreme Deity Himself in\nbrooding over the primordial formlessness which is Himself, and from\nwhich will gradually evolve the universe of qualities, conditions and\nappearances.\n\nThe quintopyknon is, therefore, the base of the mind-principle in the\nkosmos. All the qualities of mind whether in man or in the planetary\ngods, whether in the moneron or the tyrannosaur, in the mountain or\nthe oak, reside in kosmic potency in the quintopyknon.\n\nNICHOLAS MALEBRANCHE,[25] in one of his very lucid moments, beheld\nthe essential character of the symbology of space and was led to the\nconclusion that God is space itself. To him it was equally certain that\nall our ideas of space, geometrical or purely physiological, as well\nas our notions of the great suprasensual domain of ideas, exist in\nthe kosmic deiform, or body of the Logos of Being. He saw \"all things\nin God.\" God did not create ideas; they are a part of God Himself;\nGod did not create mind; it is a part of Himself; no kind of matter\ndid He create; it is a veritable part of Himself and indissoluble\nfrom Himself. The great outstanding implication of this philosophy is\nthat our consciousness of God is but a part of God's consciousness\nof Himself; our consciousness of self and the not-self is but God's\nconsciousness of these things. There is no existence of anything,\neither of the self or the not-self, except in this consciousness. It is\nrefreshing, therefore, to note that although the approach is made from\nanother and entirely different direction, almost the same conclusion as\nto the ultimate resolution of all chaogenetic elements into what is the\nvery systasis or consistence of the great kosmic deiform, is reached.\n\n  [25] Vide _Recherche_, Chap. VII, also Philosophical Review,\n  V. 15, p. 401.--MALEBRANCHE.\n\nBut a marvelous vision comes with the dawn of this truth upon the lower\nmind. It establishes clearly the truth of KANT'S notion when he said:\n\"Since everything we conceive is conceived as being in space, there is\nnothing which comes before our mind from which the idea of space can\nbe derived; it is equally present in the most rudimentary perception\nand the most complete.\" The mind cannot get away from the conception\nof space, because, out of the very essence of space, as a result of\nthe quintopyknotic process, it was produced, created and organized.\nThe idea of space is, therefore, not derived from things in space nor\nfrom their relations in space. _It sprang up with self-consciousness._\nAs soon as the Thinker became conscious of himself he became aware of\nspace. The very state of self-consciousness implies space. The self in\nman is a specialized aspect of space. Indeed, it is a projection of\nspace. The moment the self can say: \"I am,\" it also can complete the\ndeclaration by saying: \"_I am Space_.\" When the self looks out from\nhis six-walled cabin of imprisonment into the immensity of what we\ncall space he looks out into that which is himself and his immensity;\nhe perceives the source and the ever-present sustenance of his being\nand recognizes his identity therewith, provided he does not allow\nhimself to become entangled in the philosophical difficulties which the\nintellect is prone to throw around the simple, yet marvelously complex,\nnotion of self-consciousness.\n\nThis should settle, once for all, the question of _apriority_. The\n_a priori_ inheres in quintopyknosis or kosmic psychogenesis. It is\nthe essential nature of mind; it is the mind's lines of organization;\nit is the law of the mind's being and action. All mental perception\noriginates from things in space. No thought of any detail, of any state\nor condition, whether limited or unlimited, related or isolated can be\nconceived except it be of things in space. And this is so, because\n_mind and space are one_. It is not so with our conception of time.\nTime is merely an aspect of consciousness in its limitation and does\nnot inhere in the mind in the same manner that does space. In fact,\nit is not a part of the mind's nature as it has been shown that space\nis. It would, therefore, seem to be a grave mistake so to co\u00f6rdinate\nthe two notions. Space is the progenitor of mind and is continually\nidentified with mind. Time is the child of consciousness. That is, it\nis one of the illusions of consciousness which the ego will shed as his\nconsciousness expands. Duration alone is a co\u00f6rdinate of space.\n\nThe mind now recognizes space as something apart and separate from\nitself only because of its unconsciousness of the identity existing\nbetween it and space. Just so, it is not by mind alone that the\n_at-one_ state of consciousness shall be attained; for although in\none form or another it is able to gain some knowledge of the apparent\noneness of all life it cannot directly realize this oneness. In order\nto do this fully it must be able consciously to identify itself with\nthe life, feel what it feels and experience what it experiences and\notherwise come into a conscious relationship with the root and source\nof life. Space-consciousness is a simple, direct cognitive process;\nwhile time-consciousness is a complex, and therefore, indirect process.\nThe former cannot be analyzed. That is, no analysis is necessary to\nits sufficient comprehension; the latter must always be analyzed and\ncategorized for its sufficient apprehension. Every moment of time\nwhether past, present or future, when presented to the consciousness,\nis determined by its relationship to some other moment of time. Space\nis indivisible; time is divisible. Space is an intuitional concept;\ntime is an intellectual concept. Time belongs to phenomena. Space is\nthe root and source of phenomena. Time is the leaves of a tree while\nspace is the life of the tree.\n\nSpace-consciousness, in its relation to the present status of\nmind-development, is itself an illusion; for despite the fact\nthat the Thinker's apperception of it as a state is simple,\ndirect and fundamental, it is only so because of the inability to\nrealize to itself the unity of the seeming two. The attainment\nof space-consciousness or the space mind, which contrives the\nunderstanding of the identity of mind and space also annihilates the\nconsciousness of space as a separate notion from the mind. Once the\nThinker's consciousness has arisen to that state where it perceives\nits unity with space all sense of separateness is lost. Just as when\ntwo molecules of hydrogen uniting with one molecule of oxygen to form\na new compound lose their identity in the new realization of unity, so\ndoes the consciousness when by the alchemy of psychogenesis it becomes\nidentified with space, not only lose its identity as such, but also\nany consciousness whatever that space exists as something separate\nand distinct from itself. Imagine the whole of the duration aspect of\nkosmogenesis crowded into an infinitesimal instant and the bulk of all\nmatter, suns, stars, worlds and planets, condensed into a space less\nthan the magnitude of an hydrogen ion and in this way a symbol of what\nit may mean to attain unto absolute knowledge or unto the space-mind,\nmay be obtained.\n\nRecurring to the process of quintopyknosis, it may be noted that the\nquintopyknon or five-fold kosmic principle of life which we have seen\nto be identical to the seeds of mental matter brought into existence\nby the reaction of Fohatic energy or the Will of the Creative Logos\nupon the substance of the quartopyknotic stage, is, symbolically\nspeaking, more dense and compact than the pyknons of the preceding\nstages. It is ensouled by the quartopyknon. It is a rather complex\nstate of ensoulment consisting of four condensations or pyknoses.\n\nThe next stage in the process of kosmic involution which is also\nconcerned with the preparation of the evolutionary field is that of\nsextopyknosis and implies the senary condensation of the original\nworld-stuff with the view to the formation of emotional or \"astral\"\nmatter. Identically the same process of ensoulment or involution\nobtains upon the plane of sextopyknosis as have been observed to obtain\nupon the preceding planes of involution. Involution must necessarily\nprecede evolution. That which has not been involved, enfolded or\nensouled cannot be evolved or unfolded. Whatever potencies, powers and\ncapabilities or qualities and characteristics that may appear at any\ntime in the universe of life and form must have first been involved\nor enfolded or else they could not have been evolved. Space itself is\nan evolution. It is a process of becoming, of unfolding, of flowering\nforth. As it evolves more and more there will appear new and added\ncharacteristics and qualities of life and form. New possibilities will\narise and in the end a supernal vision of a glorified universe will\nburst into view.\n\nThe scheme of space-genesis is completed during the septopyknotic\nprocess wherein the basal elements of dense physical matter and\nits various gradations are produced and given character, form and\ndirection. But this completion means merely a temporary estopment in\nthe process of kosmogenesis which actually results in the formation\nof physical matter in its crassest state. It does not mean a final\narrest of the entire process which is conceived of as continuing only\nin a regressive manner back to a kathekotic[26] condition wherein\nit embodies the fruitage of the entire scheme. The septopyknon,\naccordingly, is a seven-ply pyknon in which are embodied, in varying\ndegrees of manifestation and phanerobiogenic (life-exhibiting)\nquality, the essentialities of all that has preceded on all planes and\nduring all stages of space-genesis. That is to say--in the physical\nlife of the universe is confined the essence of all the series of\ngrades of life in the kosmos. In man's physical body are wrapped up\nall the glories attainable in his long, almost unending pilgrimage\nof evolution; in it are stored all the possibilities of the spirit;\nall powers, all qualities, all characteristics, ever intended for\nman's attainment are in the physical. But they must be evolved, they\nmust be unfolded and expressed. The physical must be _glorified_,\n_spiritualized_, _deified_. For by the way of the glorification and\nspiritualization of the flesh man may attain unto oneness with the\ndivinity in himself and consequently with the divine life of the world.\n\n  [26] _Kathekosis_ (from Chaos-Theos-Kosmos) is to evolution\n  what \"chaogeny\" is to involution. It is the end of evolution, but also\n  the beginning of involution, and in the latter function is known as\n  \"chaogeny.\" See diagram No. 17.\n\nTo summarize: The genesis of space embraces seven stages, namely, the\nmonopyknotic, the duopyknotic and the tripyknotic which belong to the\nplane of non-manifestation and are the primordial world-stuff and\ntogether make up the unmanifest body of the Logos of Being. These\nbecome the seed-germ of the universe of spatiality. The quartopyknotic\nis the fourth stage in the process of space-genesis and is the\n_metamorphotic_ or crucial stage during which non-manifestation is\nmetamorphosed into manifestation. In it the unmanifest becomes the\nmanifest. It corresponds to the plane of pure spirit, and indeed,\nembodies within itself all the qualities which spirituality is to show\nforth during the life of the kosmos. The quintopyknotic is the fifth\nstage and corresponds to the mental plane, embodying in itself all\nqualities of mentality in the universe and furnishing the basis and\nessence of that which is to become the kosmic mind in manifestation.\nThe sextopyknotic is the sixth process and symbolizes the sixth stage\nwhich embodies all the characteristics and properties of emotional\nmatter in the universe and is the basal element of the plastic essence\nof sentient existence in the kosmos. The septopyknotic is the seventh\nand final stage corresponding to the physical plane of the kosmos and\ncontains in its seven-fold constitution the seeds and potencies of all\nthe preceding stages, as well as all the characteristics and properties\nwhich physical matter is destined to show forth during the _manvantara_\nor world age. These seven processes result in the dynamic appearance\nof space, the mother and container of all things, and complete the\ninvolutionary aspects of kosmogenesis. Evolution began where involution\nceased and will end for this _manvantara_ when the last vestige of\nthose powers, capabilities and potencies which were involved shall have\nbeen evolved unto kosmic perfection.\n\nThe measure of the Great Kosmic Space-form was sealed at the close of\nthe involutionary movement of the Great Life Wave. Then its metes and\nbounds were fixed by the fringe of kathekosity which circumscribed it.\n\nIf it be true that the reader found it extremely difficult to grant\nthe connotations of the symbolism when the mental or quintopyknotic\nstage was reached when illative cognizance was given to the fact that\nspace is also composed of mental matter, it may be still more difficult\nto grant the claim that physical matter is also essentially a part of\nspace. But this is the implication. _And, therefore, it follows that\nall matter, all energy, all life and all mind wherever it may be found\nin the Great Space-Form is space itself and nothing but space._ Hence,\nit appears that space is indeed the dynamism of the universe. In its\nkosmic womb the great world egg was formed of its own substance solely\nand in it still the universe of form persists and evolves withal.\n\nIf it be suitable for the physicist to talk of gravitation,\nelectricity, magnetism and force let him do so, for these terms serve\nthe present category of human knowledge; but the human mind will not\nlament the day when it comes to recognize that these things, these\nforces, these aspects are nothing more than space-activities and\nspace-phenomena. If a planet's place be preserved in space it is\nbecause space, vital, dynamic, creative space, sustains it and from\nits gentle, yet eternally firm grasp there is no escape. All that\nthe planets, suns and worlds are and all that they may ever become\nin this _manvantara_ or world age have been derived from space, yea,\nare of the very essence of spatiality. If the chemist choose to\ntalk of chemism, negative and positive, of combining properties and\ndissociative phenomena let him also become aware that these phenomena\nare but the external aspects of the inner and ephemeral life-processes\nof space-forms and that ultimately these, too, may be traced back into\nan eternal originality within the bosom of the all-mother, spatiality.\n\nDense physical matter, such as constitutes the physicality of\ncelestial bodies in its ultimate dissociation would, accordingly, be\nresolved into the original chaogenetic formlessness which marked the\nchaogeny of non-manifestation although it would naturally be orderly\nand progressive passing through the seven stages, septopyknosis,\nsextopyknosis, quintopyknosis, etc., until the end had been reached,\nmeanwhile exhibiting in each plane the phenomena peculiar to the\ndissociative processes thereof. On this view space is a _plenum_ of\nmatter of varying degrees of intensity, ranging from the densest\nphysical to the most tenuous and formless matter of the highest levels\nof the manifested universe. But as neither the dense material forms\nnor the other grades of matter have an eternally enduring quality,\nbeing alike subject to mutation, space likewise falls under the law of\nbecoming whereby it, too, must yield to the edict of kosmic disorder.\n\nSome may be inclined to argue that since space and mind are one and\nthe same thing it must necessarily follow that whatever possibilities\nof measurement may be found to exist in the mind would logically be\nfound to exist in space; and that since all the necessary conditions\nof hyperspatial operations are proved to be existent in the mind the\ncase of the hyperspatiality of perceptual space is proved thereby. In\nother words, if the fourth dimension can be proved to be mentally\nconstruable it is also possible in perceptual space. But these\nhypotheses are not granted, and neither will they be acceptable to\nthose minds who choose to take that view when it is known that there\nis a marked difference between the mind that is purely intellectual\nand mind that is purely intuitional or mind _a priori_. The intellect\nis fashioned for matter only; it is so constructed as to fit squarely\ninto every nook and cranny, every groove and interstice in matter;\nyet for the generating element, life, it has no aptitude nor suitable\ncongruence.\n\nThe attainment of the space-mind or kosmic consciousness would then\nimply a mastery of all fundamental possibilities pertaining to all\ndegrees of matter. Thus by becoming conscious in the matter of all\nthe planes one makes a certain definite approach to this ultimate\nstate of consciousness until all the barriers between ordinary\nself-consciousness and the consciousness of the space-mind have been\nentirely obliterated.\n\nPyknosis, in all of its septenary aspects, is concerned primarily\nwith involution or the preparation of the chaogenetic elements for\nthe work of kosmic evolution. It may be thought of as being divided\ninto two great divisions, namely: _chaogeny_ and _chaomorphogeny_.\nIt is concerned with the organization of chaos, the establishment\nof kosmic geometrism in the formless, void, arupic substance and\npreparation for evolution. Chaogeny, of course, is that kosmic process\nby virtue of which space itself becomes manifest and in which there\nis no established order. Chaomorphogeny (from _Chaos + Morphe +\nGeny_) signifies the activities of the creative Logos in laying the\nfoundations in primordial space-matter of the various star-forms,\nincluding nebul\u00e6, worlds, planets, suns, etc., of which Canopus,\nJupiter, Fomalhaut and Sirius and our own sun are examples, giving\ndirection and general tendence to their varied life-processes. Both\nthese processes are concerned with the preparation of the field and\nits consequent fertilization in anticipation of its cultivation and\nharvest. These two constitute kosmic involution or the great life\nwave's passage on the downward arc of the Great World Egg or Circle.\nIt is during the chaomorphogenic cycle that the constitution of the\nuniverse of manifestation is promulgated; when laws for its government\nduring that _manvantara_ are sketched out in the world of nascent\nspatiality; when the archetype of every imaginable or possible form\nis projected upon the impregnated screen of creation, then folded in,\npushed toward the center, involved, awaiting that time when the life\nwave begins its passage upon the upward arc and evolution ensues,\ncalling forth all that has been enfolded in the bosom of the pyknotic\ncenters of manifestation. It is easily conceivable that here during the\ntroublous times of the chaomorphogenic enfoldment the now known six\ndirections of space were among the eternal edicts of space-genesis and\nthat that law which now makes it appear that three co\u00f6rdinates, and\nonly three, are sufficient for the determination of a point position\nin space was imprinted in the very nature of that which was to become\nspace.\n\nThe kosmic field having been prepared as a result of the\nchaomorphogenic activities, lowly and scarcely organized forms begin\nto appear and the ascent upon the upward arc of the Great Cycle\ncommences. Evolution begins. Its scope is likewise divided into two\ngreat stages, namely: (_a_) Morphogeny, the purpose of which is the\ndevelopment of life-forms or pyknons which are to appear on the various\nplanets, stars, worlds and suns of the universe. It embraces the whole\nspan of the life-aspect on the evolutionary side of manifestation.\nIn this aspect is included also every conceivable adaptation of the\nuniversal principle of life from the beginning of its movement to\nthe end. The universe is now functioning in and progressing through\nthe vicissitudes of this stage. That is, all the present observable\nadaptations which the life-pyknon or principle is making, has made or\nwill make, are embraced within the scope of what is here designated\nas the morphogenetic aspect of evolution. (_b_) The second and\nlast division of the arc of evolution is called _Kathekos_, thus\nsymbolizing the syncretism of the trinitarian aspects of kosmogenesis,\n_Chaos-Theos-Kosmos_, perfected and united as a result of the labors\nof manifestation. In this final summation of the labors of the\nlife-wave as it has progressed from involution through all the devious\nmanifestations of evolution are embodied the perfection and ultimate\nelaboration and expression of all the pyknotic tendencies which were\nestablished during the entire scope of space-genesis.\n\nThus it will be seen that the first three stages of space-genesis,\ncalled chaogeny, encompass the first three pyknotic processes or\nare analogous thereto while the latter called chaomorphogeny, the\norganization and ensoulment of space-forms, embraces the latter four,\nquartopyknosis, quintopyknosis, sextopyknosis and septopyknosis. This\ndivision obtains on the involutionary side of the great life-cycle.\nThe upward arc of the Great Kosmic Egg or Cycle is also divided into\ntwo great stages, namely, Morphogeny (manifestation of life through\nthe various forms which it assumes) and _Kathekos_, or the kathekotic\nplane of perfected triunity which is represented by the evolutionary\nunion of _Chaos-Theos-Kosmos_. _Kathekos_ would, therefore, symbolize\nthe ultimate elaboration of Chaos into a well-ordered kosmos wherein\nare expressed all the possibilities which inhered in the archetypal\nplan of the Creative Logos or _Theos_ and in which all had reached\nthe ultimate perfection in the body of being of the Logos Himself.\nBut the kathekotic plane is to be distinguished from the original\n_Chaos-Theos-Kosmos_ represented as functionating upon the plane of\nnon-manifestation during chaogeny. _Kathekos_ symbolizes the perfected\nmanifestations of the triune aspects of the Creative Logos through the\nperfected forms resulting from the labors of kosmic evolution, while\n_Chaos-Theos-Kosmos_ symbolized, as a triune _glyph_, the Unmanifest\nTrinity in the primordial beginnings of space-genesis. One is the seed;\nthe other the fully matured plant; one the egg; while the other is the\nfull grown bird; one the root; the other the fruitage; one Alpha, the\nother Omega; one the beginning, the other the end. The end, however,\nis reached only that, in due time, the entire scheme may be commenced\nagain, once more utilizing the results of the preceding scheme of\nevolution as the basis of the ensuing one. Thus after every Kosmic Day,\ncommences the Kosmic Night. The succession of kosmic days and nights is\ninfinite. This infinity of _becomings_ in the life of the kosmos is a\nnecessary outcome of eternal duration.\n\nThe above, thus briefly set down, is the symbolism of space-genesis.\nIt is commended to the reader as a basis for the conception that the\nreal, essential, perceptual space is something far more wonderful,\nmore fundamental than either the geometrician or the metageometrician\nhas ever dreamed of, and yet the latter's consciousness is undoubtedly\nbeing appulsed by the fingers of a new species of conceptualizations\nwhich, one day in the not too distant future, will arouse in it the\nfaint hungerings after the realization of the real space-nature. These\nmathetic appetites thus brought into being will finally lead the\nhuman mind into the Elysian fields of kosmic consciousness where for\nanother million years, perhaps, it may feed upon the mysteries and\nhypermysteries to be found in the granaries of the Space-Mind.\n\nThe study of space in its wider and deeper meanings is necessary in\norder that a clearer understanding of its true significance, as the\nsubject of geometric researches, may be gained. It is confessed,\nhowever, that there is neither direct evidence nor implicative\nauthority for any assumption that the view herein outlined affords any\njustification for the notion of the n-dimensionality of space. For,\nalthough the line of reasoning indulged in must lead inevitably to the\nconclusion that the worlds of spatiality, materiality, intellectuality\nand spirituality, essentially and fundamentally one so far as origins\nand qualities are concerned, were alike engendered by the same\ngenerating element, life; and that spatiality being the primal basis\nof the others is, nevertheless, under the exigencies of this aspect of\nthe kosmos, highly susceptible to the mensurative requirements of the\ngrossest, there appears to be no necessity for calling upon extraneous\nconsiderations for assistance in our efforts to comprehend the various\nconnotations of the symbolism. Then, too, it is easily conceivable that\nunder conditions where these elements, spatiality, intellectuality and\nmateriality, are not only co-extensive but interpenetrative, there is\nno justification for the assumption that they must exist in layers or\nmanifoldnesses or in discrete degrees, separated from one another as if\nthey were constituted of different substances and occupied different\nspheres. For every single point in perceptual space is a focus for\nlines drawn through every conceivable grade of materiality, spatiality\nor intellectuality in the kosmos. And the same system of co\u00f6rdinates\nwhich is necessary and sufficient for the localization of a point in\nour space is also sufficient for the location of a point anywhere in\nthe entire world of spatiality, intellectuality or spirituality. In\nfact, the external, visible worlds of materiality and spatiality are\nnothing more than the _mass-termini_ of lines extending from divinity\nto physicality; from primordial originality to kosmic modernity and it\nis intellectually conceivable that progression back over the grooves\nmade by these mass-termini of lines would lead directly and unerringly\nto originality itself. In spite of the manifold pyknoses which we have\nshown to characterize the symbolism of space-genesis it is a very\nsimple matter; for the entire scheme could and must have proceeded\nalong strictly tridimensional lines. Tridimensionality must have\ninhered in the primeval archetype of space or else it could not appear\nas an outstanding fact of perceptual space now; for all that we can now\nobserve in space as characteristics must have first been included,\nenfolded, involved, before it could have been evolved. Hence, it is\nto be remembered that we are to-day dealing with the expressions of\ntendencies and principles which inhered in the manifested universe as\npotentialities in the very beginning.\n\nThe alphabet of space-genesis consists of five characters, namely,\nthe point, the line, the triangle, the square and the circle. These\nare the pentagrammaton of space, of intellectuality, materiality and\nof spirituality. They constitute the basis of kosmic geometrism.\nWith these all geometrical figures may be constructed; with them\nall magnitudes may be delineated and projected. They describe every\nconceivable activity of the Creative Logos and designate the bounds of\nthe entire scope of motility of kosmogenesis.\n\nIn figure 19, are shown the dot, the line, triangle, square and the\ncircle which together form the kosmic pentoglyph. The point symbolizes\nkosmic inertia, inactivity or the beginning of motion; the line is the\nfirst aspect of motion, the beginning of creation; the triple aspect of\nkosmogenesis is symbolized by the triangle, _Chaos-Theos-Kosmos_, the\nUnmanifest Trinity; the square emblematizes kosmic being in evolution;\nwhile the circle is the syncretism of all these and stands for the\nperfected kosmos, or the kosmos in process of perfection. Very truly\ndid PLATO remark: \"God geometrizes\"; for the _pentagrammaton_--the\npoint-line-triangle-square-circle--is the deity's way of manifesting\nHimself. But there is here no need for space-curvature nor for\ntriangles whose value is greater or less than 180 degrees; there is no\nneed even for the mathematical fourth dimension.\n\n[Illustration: FIG. 19.--Kosmic Pentoglyph]\n\nIt cannot be believed, however, that metageometricians are really in\nearnest in what they suggest of hyperspace and _n_-dimensionality;\nit cannot be believed that they are entirely satisfied with what\nthey have found of the so-called hyperspatial, and yet, some of\nthem are fanatically patriotic over the new-found domain; some are\neven intolerant. But there are others who look upon the fabric of\nmetageometry as a stepping stone to space-realities, a mile-post on the\npath to the realization of a higher consciousness, the consciousness of\nthe space-mind or kosmic consciousness. And may this not, after all,\nbe the goal of the human intellect, now slightly distraught by the\nexuberances of youth and the joys of a new mental freedom? The work\nof the future mathematicians will be the destruction of the tumorous\ninconsistences to be found in the various non-Euclidean systems\nof geometrical thought, the elimination of the novelties and the\nnonsensicals, the synthesizing of those elements which are sanctioned\nby the space-mind and the building thereon a sane interpretation of\nspace-phenomena in the light of illuminations received from greatly\nextended faculties and a participation in that larger consciousness\ninto which the human race seems slowly to be immerging.\n\nThe domain of hyperspace is but the fairy-land of mathesis, peopled\nwith goblins, gnomes, kobolds, elves and fays which are the\nspaces, dimensions, propositions, ensembles and theorems of the\nmetageometrician. But like the fairies and nature spirits of the unseen\nabout us, they have their bases in the real, objective world of facts\nhowever difficult it may be to establish their direct connection with\nit. As the gay, invisible sprites of phantom-land represent intelligent\nnatural forces at work in the furtherance of the evolution of forms, so\nthe impalpable things of mathesis are emblems of kosmic forces at work\nin the upbuilding of structures of higher consciousness which shall be\ntowers of vision for the human soul whence it may view the hill-crests\nof infinite knowledge and the low-lying plains of kosmic mysteries.\n\nFinally, it has been noted that space is the very consistence of the\nkosmos; it is the life, the form and both the outer and the inner\nmanifestation of the combined life and form; it is reality, also\nillusion; it is concrete, also ideal. We have noted also that mind is\nconsubstantial with space and that space gives it its inner life and\nnature as well as nourishes its outer growth and development. In fact,\nwe have seen that space and mind are _one_ essentially and that they\nexist as aspects of the same thing, life. In whatever way, then, that\nthe mind normally views space that is the natural way. All attempts to\ndeviate from the natural way are, therefore, unsanctioned by the nature\nof things. So long as geometry remains true to the nature of mind and\nspace so long will it be valid universally and possessed of kosmic\nnecessity and invariance. It behaves most unseemly when it departs from\nits fealty to the nature of things _per se_.\n\nBoth the outlook of the mind upon the objective world as well as its\ninlook upon its own states of consciousness or the subjective world\nare tridimensional. Its growth is tridimensional, its nature is\nlikewise tridimensional, and there is not even the slightest tendence\neither to perceive, conceive or perform in a four-dimensional manner,\nmathematically speaking. Trace out the biologic development of each\nmental faculty, from the mind of the moneron to the mind of the most\nhighly developed man and it will be found that everywhere and always,\nwithout variation or exception, the nature of each of these has\nbeen to express itself tridimensionally and naturally. There is not\neven the slightest sign of so much as a germinal appetence for the\nfour-space; it would, therefore, seem almost a prostitution of mental\nfaculty to divert mental energy into the seemingly useless channel of\npresent-day metageometrical researches; yet, it must be admitted that\neven though the end sought cannot be attained, the final results of the\nintellectual delvings into the dread homogeneity of kosmic origins and\nthe consequent realization of the awesome coevalism of mind and space\nwhence shall arise the recognition of the wondrous unitariness of all\nexistences, will be that we shall come upon that thrice mysterious\ncontrivance--the Heart of Divinity, the Kosmic Space-Center in which\nabide the roots of the _Great All_ in a marvelously indescribable unity\nand infinite originality.\n\nWe may conclude, then, that hyperspatiality and all its appurtenances\nare but the toys of the childhood of humanity. But, as the years\npass and the days of maturity come on apace, it, toy-like, will also\nbe discarded. And the mind will seize then upon the seriousness of\nreality just as the matured youth responds to the stern realities of\nlife and manhood responsibilities. But no one can say that the toys of\nchildhood are wholly useless; no one can say that the joys which they\nbring are entirely fatuous and unreal nor shall we attempt to intimate\nthat mathetic contrivances are without utility, without purpose and\nsignificance in the life of the growing mind of humanity. But they,\ntoo, will pass away.\n\n\n\n\n  CHAPTER VIII\n\n  THE MYSTERY OF SPACE\n\n  The Thinker and the Ego--Increscent Automatism of the\n  Intellect--The Egopsyche and the Omnipsyche--Kosmic Order or\n  Geometrism--Life as Engendering Element--The Mystery of Space\n  Stated--Kathekos and Kathekotic Consciousness--Function of the\n  Ideal--The Path of Search for an Understanding of the Nature and\n  Extent of Space Must Proceed in an Inverse Direction.\n\n\nThe fragmentariness of the Thinker's outlook upon the universe of\nspatiality is due to the inhibitive action set up by the constrictive\nbonds which his complicate mechanism of intellectuality interposes\nbetween himself and reality. The Thinker, who stands back of and uses\nthe various media of objective consciousness, such as the neural\nmechanisms, brain, emotions, his individualized life-force and the\nmind which together make up the instruments with which he contacts the\nsensuous domain, by adapting his consciousness to these means, as the\nartisan utilizes his tools, constitutes his own intellectuality. The\nintellectuality, then, is the totality of media by which consciousness\neffects its entrance into the sensuous world and by which it receives\nimpressions therefrom. In other words, it is the sum of all those\nqualities, operations, processes and mechanisms which are recognized as\nconstituting the _modus vivendi_ of man's intellectuality, and these\nare, in reality, nothing more than the ego himself.\n\nMany have been inclined to regard that which has been called the\nego as the highest sovereign power in the state of manhood. He has\nbeen looked upon as the final consideration in the constitution of\nthe human being. But the ego is an evolutionary product and the\nconcomitant of self-consciousness which is the I-making faculty in\nman's psychic life. It is that quality of consciousness which makes\nman conceive of himself as a separate, detached and independent being.\nIt is a purely intellectual or tuitional product, and, as such, is to\nbe differentiated from the intuitional or life-quality which is the\nessence of man's real selfhood. With respect to the Thinker, the ego\noccupies precisely the same status as the agent to his principal. As\nthe agent is the representative of the principal in all matters which\ncome within the scope of his prescribed jurisdiction so is the ego the\nagent of the Thinker who is a spiritual intelligence. Accordingly,\nfrom an ethical viewpoint the Thinker is responsible for the acts of\nthe agent and can in no wise escape the penalties accruing as a result\nof the agent's violations. Just as a commercial firm sends out a\nrepresentative for the collection of data concerning certain phases of\nits business or it may be of any business or the entire world market\nso the Thinker projects his own consciousness into the mechanisms\nwhich are in their totality the egoic life. That is, he sends out his\nagent, the ego, into life and into the objective world of facts and\ndemands that he shall convey to him, from all points of the territory\nwhich he is expected to cover, reports of his findings. Of course,\nthese reports which are transmitted by the ego (the intellectual\nmechanism of the Thinker) are more or less well prepared summations\nof his individual observations and deductions. These are the percepts\nwhich the ego presents to the Thinker's consciousness. Concepts are\nformed by the Thinker in his treatment of these sense-presentations.\nIt very frequently happens that the ego transmits reports which, for\none reason or another, give very imperfect knowledge of the matter\nwhich his reports are designed to cover. Often it is necessary that\nadditional and supplemental reports be made about the same thing, and\neven then, it is well-nigh impossible, if not quite so, for him fully\nto cover every detail of the matter under consideration and in no case\nis it possible for him to do more than report on the superficialities\nof the question under scrutiny. If the ego, in his operations, be\nimagined to be hampered by similar circumstances and difficulties as\nthose which would ordinarily beset a commercial attach\u00e9 it will then\nbe clear that his reports must ever be fragmentary because of the\ninaccessibility of much of the data which would be necessary for a full\nreport, and further, because of the inadequacy of his methods and means\nof gathering data due to the inherent limitations of his capabilities,\nendurance and perspicacity and innumerable other limitations and\ndifficulties which must be faced in all search for the real. So that,\nwhile the sufficiency of the means which the ego enjoys at this stage\nfor all practical purposes is granted no hesitancy is entertained when\nit comes to a discovery of the reals of knowledge in declaring their\ninsufficiency.\n\nThen, too, when it is remembered that these egoic reports are in\nthe nature of neurographical communications which are similar to\ntelegraphic despatches and must pass through several stations, as\nganglia, etc., often being relayed from one to another, it will be\nquite apparent that much, even of the original quality of the missives\nforwarded, will have been lost or radically changed in some way before\nit is finally delivered for the inspection of the Thinker himself. It\nnot infrequently happens, even in perfectly normal beings, that the\nego in filing, recording, transcribing, interpreting, translating and\notherwise preparing these data for the Thinker's use, lets a cog slip,\nmisplaces some of the data, loses or destroys fragments of it and so is\nunable to maintain a complete portfolio of his materials.\n\nAs the Thinker is entirely dependent upon his agent, the ego, for\nthe trustworthiness of his information covering the matter of the\nsensuous world it is obvious that at best his information is very\nfragmentary indeed, and necessarily so when it is considered that the\n_modus operandi_ of his agent and the difficulty of his operations are\nso complicate as to magnify the obstructions in the way to complete\nfreedom in this regard.\n\nTo continue the similitude of principal and agent it may be asserted\nthat it is also true that the commercial house that sends out its\nattaches frequently will send letters containing directions as to\nprocedure, sometimes censuring for past delinquencies and sometimes\ncommending for praiseworthy deeds; and this, too, in addition to the\noriginal instructions which were given at the outset. It even comes to\npass that the home office, because of some meretricious accomplishment,\nas the marked increase of efficiency shown by the agent's close\napplication to his duties and the consequent success of his operations,\nconfers certain favors upon the agent or removes some of the\nrestrictions which were originally imposed, gives an increase in salary\nor promotes the agent to a higher and more lucrative office with larger\npowers and greater authority. This is analogous to what the Thinker\ndoes for the ego. For he not only receives reports from the ego, but\noften, in the shape of intuitions, gives additional information as to\nthe proper manner of doing things, sheds more light upon some obscure\noperation, commends for duty well performed, condemns for failures or\nfor wrong-doing, rewards arduous toil with greater powers of vision,\nkeener insight, greater capabilities; in fact, promotes the ego in the\nsight of other egos by marking him out as an exceptional ego. But the\ncurious aspect of this procedure is that, in time and after the ego\nhas been repeatedly commended and promoted and otherwise favored by\nthe Thinker, he begins to think that he owns the firm, that he is the\nlife and main support of the whole corporation. He becomes arrogant,\nself-willed and finally falls into the illusion that he alone is\nresponsible for the phenomenal success of the firm. This is the source\nof that illusion of the intellect which makes itself think that it, the\nego, is all there is to man, that his instruments of operation in the\nobjective world are the only kind of instruments that may be used; that\nhis method of gathering data about things is the only safe and sure\nmethod; and so it develops that the intellectuality is the source of\nman's separateness, his individuality and his apparent aloofness from\nother men and things. It is, of course, needless to point out that in\nthis way the intellect comes to be the tyrant of man, ruling with a\nrigid monopoly and as an all-exclusive autocracy.\n\nFrom the above implications it would appear that the intellect and the\nintuitive faculty are two separate and distinct processes, and so they\nare. One is the inverse of the other. The tendence of the egoic life or\nthe intellect is for the external while the intuition is an internal\nprocess. The intellect acts from without towards the interior while the\nintuition acts from within outward. The intellect is the product of the\nintuition which is another term for the consciousness of the Thinker on\nhis own plane. Just as the child lives a separate and distinct, though\ndependent, life from the parents so the intellect has a _modus vivendi_\nwhich is distinct and separate from that of the Thinker, and yet it is\nin all points dependent upon the life of the Thinker. Here again, we\nfind an analogy in the relation of the child to the parent. As some\nchildren are more amenable to the will of the parent than others,\nso, in some persons, the intellect is more amenable to the action of\nthe intuition than in others. Yet it is a certain fact that the more\nthe outward life is governed by the intuition, i.e., the more the\nintellect responds to the intuitive faculty of the Thinker, the higher\nthe order of the life of the ego and the more accurate his decisions\nand judgments. In fact, it assuredly may be asserted that the place\nof every individual in the scale of evolution is determined in a very\nlarge measure by the degree of agreement between the intuition and the\nintellect or by the ease with which the intuition may operate through\nthe intellect as a medium. At least, the quality of one's life may be\ndetermined directly by these considerations.\n\nThe Thinker being himself a pure spiritual intelligence, living upon\nthe plane of spirit and therefore unhampered by the difficulties\nwhich the ego meets in his operations in the objective sensorium, and\npossessed of far greater knowledge, is correspondingly free from the\nlimitations of the ego and very naturally closer to kosmic realities.\nHence, he is better situated for the procurement of correct notions\nof relations, essentialities and the like. It is believed, therefore,\nthat in the proportion that these two processes, the intellectual and\nthe intuitional, are brought, in the course of evolution, to a closer\nand more rigid agreement, in the proportion that the Thinker is able\nto transmit the intuitograms in the shape of concepts or that the\nintuition is made more and more conceptual, in just that proportion is\nhumanity becoming perfect and its evolution complete. The difficulty\nfound to inhere in the conceptualization of intuitions so that they may\nbe propagated from man to man seems not to lie in the Thinker himself,\nbut more essentially in the ego, in the intellectuality and its\ncomplicate schematism or plan of action. It would appear, therefore,\nthat the only way of escaping or transcending this difficulty is for\nthe ego so to refine his vehicles or so facilitate his plan of action\nby eliminating the numerous relays or sub-stations intervening between\nthe consciousness of the Thinker and that which may be said to be his\nown that the transmission of intuitograms may be accomplished with\nthe greatest ease and clearness. While no attempt will be made to\nindicate the probable line of action which the ego or objective man\nwill adopt for this purpose, it is believed that it may be said without\npedanticism that the only true method of attaining unto this much\ndesired state of things is, first of all, by assuming a sympathetic\nattitude not only towards the question of the intuition itself but\nto all phenomena which are an outgrowth of, or incident to, the\nmanifestations of the intuitive faculty through the intellectuality,\nand second, by the practice of prolonged abstract thought, this latter\nprocedure effecting a suspension of the intellectuality temporarily\nat the same time allowing it to experience an undisturbed contact\nwith the intuitional consciousness, thereby laying the basis for\nfuture recognition of its nature and quality. It would seem that\nthese two conditions are absolutely necessary in order that a more\ncongruent relationship may be promoted between these two cognitive\nfaculties. Ordinarily, it would appear that the philosopher who is\nundoubtedly inured to the necessities of continuous abstraction or\nthe mathematician whose most common tasks naturally fall in this\ncategory would be among all men most apt to develop to the point of\nconceptualizing intuitograms readily, yet it seems that this is not\nthe case. And there is good reason for it. The mind of the philosopher\nand the mathematician is intellectual rather than intuitional and is,\ntherefore, wedded to matter, to the action and reaction of matter\nagainst matter and hence operating in a direction at variance with the\ntrend of an intuitional mind. And this condition is undoubtedly due to\na lack of a sympathetic attitude towards this species of consciousness.\nAt any rate, it is thought that a too great anxiety in this respect\nneed not be entertained by humanity at all, for the reason that in\nthe case of a faculty, the rudimentary outcroppings of which are so\nmarked and universally observable and existing in greater or lesser\ndegrees in various human beings, there is ample evidence for the\nbelief that it is being carefully and duly promoted by a well-directed\nevolution of psychic faculties and powers, so that at the proper time,\ndeterminable by the state of perfection reached by the intellectuality\nor the ego in the operation of his cognitive processes, the much\ndesired agreement of these two faculties will have been realized and\nthe conceptualization of intuitograms into propagable conceptions an\naccomplished fact. Until this goal shall have been reached and the\nintuition shall have overshadowed the intellect as the intellect now\novershadows the intuition; or the consciousness of the ego, derived\nfrom the interplay of the Thinker's consciousness among the various\nelements which constitute the ego himself, shall have been merged with\nthat of the Thinker, the outlook must remain fragmentary, only becoming\na well-ordered whole as the barriers of dissidence are broken down in\nsuccession.\n\nThe evolution of consciousness, from the simple, undifferentiated\n_moneron_ to the differentiated cell and from that to the cell-colony\nand from the cell-colony to the organism, traversing in successive\npaces through all the stages of lower life--mineral, vegetable and\nanimal--to the stages of the simple, communal consciousness of the\nhigher animals, to the self or individual consciousness of the human\nbeing, each requiring millions of years for its perfection before a\nmore advanced stage is entered, has been one continuous relinquishment\nof the lower and less complicate for the higher and more complex\nexpression of itself through the given media. When a newer and higher\nstage of consciousness is being entered by humanity its appearance\nor manifestation is first made in the most advanced of the race and\nthat only in a dim, vague way. This rudimentary condition persists\nfor some time, perhaps many thousands of years, then the faculty\nbecomes more general in appearance, the number of advanced individuals\nincreases, and consequently, as in the case of the intuitive faculty,\nit becomes universally prevalent in all humanity; becomes transmissible\nas so-called \"acquired characters,\" and then appears as the normal\nfaculty of the entire human family cropping out in each individual.\nThus, in passing from the few advanced ones in the beginning to that\nstage where it becomes the common possession of all, a faculty requires\nmany thousands of years for its perfection, and especially has this\nbeen true in the past history of the development of human faculties.\nBut it is believed that the sweep of the life current as it proceeds\nfrom form to form, from faculty to faculty, gains in momentum as it\nproceeds, so that in these latter years due to the already highly\ndeveloped vehicular mechanisms at its disposal not so great a period\nof time as formerly is required for the _out-bringing_ of a new\nfaculty. It might well be that while in the past hundreds of thousands\nof years were necessary in the perfection of organs and faculties,\nin these latter days only a few thousand, perhaps hundreds, may be\nnecessary and that in the days of the future not even so many years\nmay be required to universalize a faculty. And especially does this\nappear to be true in a state of affairs where so large a number of\npersons are beginning consciously to take their evolution in hand and\nby volitional activities are supplying greatly increased impetus to\ntheir psychic processes which under ordinary, natural methods would\nbe considerably slower in their development. It is quite obvious that\nall cultural efforts when applied to the betterment of a given plant,\nanimal or faculty result in a corresponding hastening of the process of\ngrowth far in excess of what that growth would be under normal, natural\nconditions. All the present faculties possessed by man are remarkably\nsusceptible to cultural influences; in fact, the standing edict of\nethical and social law is that the human faculties must be cultivated\nas highly as possible, thereby giving the spirit a more perfect medium\nof expression. These observations, therefore, lead irresistibly and\nunavoidably to the conclusion that the time for the upspringing of\nthe intuitional faculty in the human organism is even now upon us,\nthat undoubtedly in certain very advanced ones it has already reached\na notable degree of perfection and is rather more general than would\nappear in the absence of careful investigation.\n\nNow, just as the intellect has made for individuality, has emphasized\nthe separateness of the Thinker's existence from that of other\nthinkers, has developed self-consciousness to a very high degree, even\npushing it far over into the domain of the higher consciousness to the\ntemporary obscuration of the latter, so the intuitional will make for\nunion, for the brotherhood of man, for co-operation and for the common\nweal. Through it man will come gradually into the consciousness that\nfundamentally, in his inner nature, in every respect of vital concern,\nhe is at-one with his fellowmen and not only with the apparent units\nof life but with all life as expressed in whatsoever form throughout\nthe universe. Then, too, he will be closer to the reality of things,\nof actions and natural processes; in fine, he will have begun the\ndevelopment of the space-mind which will bring him to the knowledge\nthat he is one with space also and, therefore, with the divine life of\nthe world.\n\nOne of the peculiarities of the vital force which shows itself in\nthe consciousness as man's intellect, is its growing _automatism_,\nor that tendency which enables the consciousness to perform its\nfunctions automatically and thus allow opportunity for the development\nof newer and higher faculties. Actions, oft repeated, tend to become\nautomatic. This is also true of thought and consciousness. It is one\nof the beneficent results of abstract thought that it develops, or\ntends to develop, a kind of automatism whereby a marked saving in time\nand energy is effected. This affords opportunity for other things.\nIt is undoubtedly true that in the days of the truly primitive man\nhis consciousness was more completely engaged in the execution of\nthe ergonic functions of cells, organs and tissues; that all those\nprocesses which are now said to be involuntary and reflexive were\nat one time, in the distant past of man's evolution, the results of\nconscious volitions. This is a condition which must have preceded\neven the development of the intellect itself. Indeed, there could be\nno intellect in a state where the entire modicum of consciousness was\nbeing utilized in the performance of cellular and histologic functions.\n\nThe rise of the intellect must have been in direct ratio to the\ndevelopment of automatism among the cells, tissues and organs, so that\nas these came gradually to perform their special labors reflexively\nthe intellect began to be formulated and to grow, at first only\nincipiently, then more and more completely until it reached its present\nstate. At the present stage of its evolution, a great deal of the\nlabor of the intellect is beginning to fall into a kind of increscent\nautomatism, although only rudimentarily, in many instances. Yet, as a\nresult of this tendency, quite the whole of the phenomena of perception\nis characterized by a sort of automatic action. And the mind perceives\nwithout conscious volition. Many of the steps of conceptualization\nare automatic, in part, if not wholly. Certain it is that impulses\nonce set in operation whether consciously or unconsciously continue\nto act along the same line until exhausted or until the end has\nbeen attained. Consequently, it is a proven fact that often serious\nmathematical and philosophic problems have been solved by the mind long\nafter any conscious effort to solve them had ceased. Often solutions\nhave been arrived at during sleep. Many such cases might be cited, but\nthe phenomenon is now so common that almost every one can cite some\nexperience in his own life that will substantiate the claim.\n\nThere is no doubt but that these phenomena are evidences of a reflexive\ndevelopment in the intellect. The time will come undoubtedly, and\nnecessarily so if the intellect is to give way to a higher faculty,\nwhich shall be as much above the intellect in its grasp of things\nas the intellect is now above the simple consciousness of the lower\nanimal, when quite the entirety of our intellectual processes will\nbecome automatic or self-performing. What then remains of the egoic\nschematism, after its transmutation or elevation as the organ of\nthe intuitional consciousness will be utilized as the organ of the\nThinker's involuntary cognitive processes. This will mean that all\nof that laborious ideation which is now the abstract thought of the\nThinker will be performed automatically, leaving the higher aspect\nof the egoic consciousness free to conceptualize or intuitograph\nthe intuitions. Perceptualization then will be replaced by\nconceptualization. This latter will occupy about the same status as the\nformer does now. And necessarily, perception will become more complex.\nIn other words, while we now perceive simple percepts which are again\narranged into concepts making a composite picture of the object, we\nshall then be taking in the composite picture of the object at first\nhand, thereby dispensing with the rather slow process of perception as\nit now operates. We shall still be perceiving, but what we perceive\nwill be concepts rather than percepts, as at present.\n\nThe increased powers of intellection gained as a result of the\nincrescent _automatism_ in the intellect, the flowering forth of\nthe intuitive faculty and the general enhancement of the intellect\nthroughout all its processes will enable it to entertain concepts or\ncomposite picture of things just as readily and as perfectly as it can\nat present deal with a single percept. Concepts will be replaced by\nsuper-concepts or intuitographs. Increased perspicacity will enable the\nThinker to manipulate the concepts and intuitographs with the same ease\nand readiness and withal the mind will have attained unto an almost\nunrealizable freedom in its search after truth.\n\nThe outcome of this new adjustment which, of course, will not spring\nup at once, but by insensible degrees, will be the clarification and\nunification of our knowledge. It will mean also the simplification\nof it; the obviation of diversities of opinions, the springing up of\na new and winnowed system of philosophy which shall be the true one;\nfurther, it will imply the lessening of the probability of error in our\njudgments and conclusions; the removal of illusion to a much larger\ndegree than to-day is possible and the realization by every one of\nsomething of the essence of things, of causes and effects, of actions,\noperations, natural forces and laws; in fact, a condition of mind\nwhich will present to the consciousness the simple truth above every\nconceivable phase of kosmic life which may come within the scope of the\nThinker's observation.\n\nThe further implications of this view are that there is a difference\nbetween the Thinker and the intellectuality. The Thinker is eternal\nand partakes, therefore, of the very essence of primordial originality\nwhile the mentality is an artificial process, the resultant of the\nadaptation of the Thinker's consciousness to his vehicular contrivances\nof objective cognition and the interplay of his life among them.\n\nIf the appearance of a choppy sea disturbed by the passage of a brisk\nbreeze over its surface be imagined, a similitude of the great ocean\nof life may be envisaged. The wavelet crests symbolize the egos; the\nbase of the wavelet which is one with the great sea of water represents\nthe Thinker which is one with the divine life and consciousness of\nthe kosmos. Just as wavelet crests are continually springing up and\nfalling back into the sea, so are egos continually being cast forth\nand reabsorbed into the universality of life only to be recast, as a\nwavelet crest or ego, upon the surface of the moving ocean of life.\n_And so, in this respect, the universum of life and consciousness\nwhich are essentially one is in a constant state of ever-becoming,\nun-becoming and re-becoming._\n\nAnother implication is that, on account of the diversity and complexity\nof the means of contact with the external world, it is not possible for\nthe ego to arrive at more than a fragmentary understanding of even the\nlatent geometrism of life, mind and materiality. In our examination\nof the sensuous world, we are very much like the three blind men set\nto examining an elephant. One set to scrutinizing his trunk by means\nof his sense of feeling. When asked for his judgment as to what the\nelephant was he declared it was a snake; a second who began with the\nlegs found it to be like huge pillars; and a third who caught hold of\nthe elephant's tail and declared the elephant to be like a rope. Each\none of the blind men described what he was able to perceive. To each\nwhat he felt was all there was upon which he could render judgment. And\nso, artists, philosophers, mathematicians, musicians, mechanicians,\nreligious seers, metaphysicians and all other types of mind, are\njust so many blind men set to the examination of an elephant, or the\nsensuous world. Each one confidently believes his view to be correct;\neach one is satisfied with the deliveries of his senses. Yet no one\nof them is wholly correct, no one of them has seen every phase and\naspect of the problem. Does it not, therefore, appear to be the more\nreasonable and urgent that the view which synthesizes the judgments of\nall the possible examiners thereby constructing a composite idea of\nthe entire mass of judgments is the more reliable and the more correct?\n\nReferring again to the dual intelligence, the ego and the Thinker,\nwhich together constitute man, it is deemed necessary, in order to\npresent the concept of this duality to the mind of the reader in the\nway that shall enable him easily to recall it, to designate the egoic\nintelligence as the _egopsyche_, and the Thinker's intelligence as the\n_omnipsyche_.\n\nThe egopsyche is the I-making faculty, the faculty of\nself-consciousness and the synthesis of all those psychic states and\nfunctions known as the intellect or mind and includes the ethical\naspect of man's nature. The omnipsyche is the organism of kosmic\nconsciousness, the space-mind, or man's higher self and that which\nconnects with or allies him to all life; it is the basis of human\nunity and of unity with divinity, just as the egopsyche is the basis\nof separation and individuality; it is the organ of direct and\ninstantaneous cognition and the permanent essence which has persisted\nthrough every form which the being, man, has ever assumed and through\nevery stage of human evolution. In it are stored up the memories of\nthe Thinker's past, the secrets of life, mind, being, reality, and the\nhistory of life from the beginning; in it also the plan of action for\nthe future of the life-wave as it passes from plane to plane, from\nstage to stage, and from form to form. It is the spark from the flame\nthat is never quite free from its source; it is the continuous spark,\nthe prolonged ray which does not go out and cannot be extinguished. It\nis that in man which when full union therewith has been attained makes\nhim a god in full consciousness.\n\nThe omnipsyche is really a neglected and overlooked factor in the\ndoctrine of evolution. Evolutionists, while they claim life to be\ncontinuous and that man has come through all the kingdoms of nature in\nsuccession and has spent millions of years in the perfection of his\nvarious organs, faculties and stages of consciousness, make no ample\nallowance for what is in reality the basal element in evolution--a\ncontinuous, persisting, permanent life-force which does not lose its\nidentity from the beginning to the end of the process. This fact--that\nthat spark of life which set out upon the evolutionary journey as a\nmoneron has glowed steadily from that stage to manhood, maintaining\nmeantime its original purposiveness and intent--seems to be the most\nobvious consideration of the whole doctrine, yet it has been more or\nless completely ignored. The elementary requirements of evolution\nwould seem to establish clearly the necessity for some such eternally\npersisting principle as the omnipsyche which is capable of such subtle\nadaptations to every conceivable form of life and in which should be\ngathered up the evolutionary results of every life-cycle. For this\npurpose the omnipsyche or unifying principle in man was designed\nfrom the beginning and it is that which constitutes the basis of his\nintellectual nature while in a far larger sense it is the divinity in\nman himself. It is indeed strange that so important a factor as the\nomnipsyche should have been omitted by evolutionists. Yet it can be\naccounted for upon the grounds of the purely mechanistic character\nof all intellectual attempts at solving the problems of vital\nmanifestations. But so long as men rely upon mechanical explanations\nof such phenomena so long will they be prone to overlook the very\nessentialities of the problems which they devoutly wish to solve.\nThe continuity of the physical germ-plasm of the human species,[27]\nnow quite generally admitted, would suggest, it seems, an analogous\ncondition to the continuity of the psychic plasm called the omnipsyche,\nthe only difference being that the omnipsyche is an intelligent\nfactor while the physical plasm is a medium of transmission though\nnon-intelligent. The omnipsyche is, therefore, the psychic reservoir\nof evolution into which are stored the transmuted psychics of moneron,\namoeba, jellyfish and every other form which it has ensouled and acts\nas the storeroom of man's psychic operations as well as the source of\nhis intellectuality.\n\nWe turn now from the study of a sketch of the mechanism of man's\nconsciousness which gives at its best only a fragmentary view of the\nuniverse of spatiality to a consideration of space itself in the light\nof its interrelational bearings upon the question of intellectuality.\n\n  [27] See _The Germ Plasm; A Theory of Heredity_, by A. WEISSMAN.\n\nIn the chapter on the \"Genesis and Nature of Space\" we have, in\ntracing out the engenderment of space, proved it to be basically one\nwith matter (and indeed the progenitor of matter), also with life and\nconsciousness. Further, it has been shown that all the characteristics\nof materiality are due to the adaptation of consciousness to it\nand that out of this adaptation grew the intellectuality. A close\napproximation to this view was maintained by KANT when he discovered\nthat our faculty of thinking or the intellect only finds again in\nmatter the mathematical order or properties which our faculty of\nperceiving or consciousness has deposed there. It appears, therefore,\nthat when the intellect approaches matter or spatiality it finds always\na ready yieldance to its demands simply because intellectuality has\npreviously established therein the delineation or map of the path\nover which it necessarily must traverse in its examination of the\nobject of its pursuit. In other words, the kosmic mind in engendering\nmateriality and spatiality has set up therein a kosmic order or\ngeometrism. Both motor and intellectual progress, therefore, can be\nmade through the world of spatiality because of the immanence of this\nkosmic geometrism which lies latent in the very fabric of the world of\nsubstance fashioning both the character and the nature of the intellect\nas well as of space itself. So that there is a perfect congruity\nsubsisting between spatiality and intellectuality. Accordingly it is\nimpossible for either one or the other to transcend the grim grasp of\nthe mathematical order which binds them in such lasting and fundamental\nagreement. Extra-spatiality may degrade itself into spatiality,\nand indeed in the very nature of the case, does so degrade itself,\nyet spatiality can never raise itself beyond the limits set by its\nengendering parent. Materiality may become more and more spatialized\nand consciousness more and more intellectualized, but they must proceed\nhand-in-hand one not superseding the other.\n\nBeing the essence of the natural geometry which is everywhere immanent\nin the universum of matter, space becomes an organized and ordered\nextension, in fact is the totality of such organized and ordered\nextension, which conforms to the latent geometrism the engenderment of\nwhich it is the sole cause in the last analysis. Does it not appear\nthen that all that mass of artificial geometry which has sprung\nup as a result of departures made from this natural geometry is\nutterly baseless and most certainly lacking in the kosmic agreement\nwhich spatiality lends to our primary conceptions? Of course, it is\nadmittedly possible to devise certain conventional forms of logic and\nendow them with all the evidences of a rigid consistency but which,\nbecause of their purely artificial character, will fall far short\nof any real conformity to the potential geometrism which has been\nestablished in spatiality. And this fact is of utmost significance for\nall those who seek to find justification either logically or naturally\nfor the existence of a multi-dimensional quality in space; for, if a\nclear, discriminative conception as to the categorical relationship,\neach to each, of the two kinds of geometry be carried in mind, it will\nnot be easy to confound them neither will it be difficult to discern\nwhere the one ends and the other begins.\n\nNow, the fourth dimension and the entirety of those mathematical\nspeculations touching upon the question of hyperspace, dimensionality,\nspace-curvature and the manifoldness of space are purely conventional\nand arbitrary contrivances and do not meet with any agreement or\nauthority in the native geometrism which we find inhering in space\nand which the intellect recognizes there. This conclusion seems to be\nobvious for the reason that, in the first place, the non-Euclidean\ngeometries have been constructed upon the basis of a negation of\nthe latent geometrism of space and intellectuality; and if so, is it\nreasonable to expect that either they or any of their conclusions\nshould accord with the nature of that form of geometry so admirably\ndelineated by EUCLID? Obviously not. It is a matter of historical\nknowledge that the whole of the artificial non-Euclidean geometries\nconsists of those purely conventional results which investigators\narrived at when they denied or controverted the norms supplied by the\nnatural geometry. When metageometricians found that they could neither\nprove nor disprove the Euclidean parallel-postulate they then set\nupon the examination of idealized constructions which negatived the\npostulate. The results, thus obtained, although self-consistent enough,\nwere compiled into systems of geometry which naturally were at variance\nwith each other and with this inherent geometrism which is found in\nspatiality and answered to by the intellect both normally and logically.\n\nFurthermore, there is another consideration which to us seems to be\nequally if not more forbidding, in its objections to the co\u00f6rdination\nof the two systems of geometry, and that is the fact that the geometry\nof hyperspace is denied the corroborative testimony of experience\nand this is true of practically the whole of its data. Indeed, there\nis perhaps no single element in its entire constitution which claims\nthe authority of experience. This is undoubtedly the weakest point in\nthe structure of the hyperspatial geometries. Contrarily, such is not\nthe case with the natural geometry; for, in this, the intellect in\nretracing its steps over the path laid out by that movement which has\nat the same time created both the intellect and spatiality, finds an\norderly and commodious arrangement into which it naturally and easily\nfalls. So exact is this agreement of the intellect with the kosmic\norder that if it were possible to remove the whole of spatiality and\nmateriality there would still be left the frame work which is this\nlatent geometrism of kosmogenesis. But the fact that the intellect\nnaturally fills all the interstices of materiality and spatiality,\nfitting snugly into all of them as if molded for just that purpose, by\nno means warrants the assumption that it would or does also fit the\nengendering factor which has created these interstices. The frame work,\nthe order or the geometrism of the kosmos has been established by life\nacting consciously upon the universum of materiality. And in order to\nestablish this geometrism life had to be mobile, active, creative. It\ncould not remain static, immobile, and accomplish it. Being mobile,\ndynamic, creative, it passes on. It is like a fashioning tool which\nthe cabinet makers use in cutting out designs upon a piece of wood. It\nmoves, and keeps moving until the design is finished, and then it is\nready for more designing. Life is like that. It cuts out the designs\nin materiality, fashions the form, molds the material, and passes on\nto other forms. The intellect fits into these designs gracefully. But\nwhat it finds is not life itself, only the design which life has made.\nHence, as there is neither an empirical spatiality nor materiality in\nconformity with which the artificial geometry of the analyst may be\nsaid to exist, and as it may not be said to conform to the path which\nlife has made in passing through either of these, it is absurd to\npredicate it upon the same basis as the natural geometry. And so, we\nare forced, in the light of these considerations to deny the validity\nand hence the acceptability of the non-Euclidean geometries as either\nreasonable or warrantable substitutes for the Euclidean, and denying\nwhich we also formally ignore the claims of the fourth dimension, as\nmathematically designed, to any legitimate anchorage in either our\nvital or intellectual movements.\n\nIt has been shown that the flow of life, as it describes that movement\nwhich we call evolution, engenders simultaneously and consubstantially\nspatiality, materiality and intellectuality, and these, in turn, the\nnatural order or geometrism everywhere immanent in the universe; and\nthat automatically, one out of the other and each out of the all,\nthese constitute the totality of kosmic fundamentals. Also we have\nsketched the mechanism of man's consciousness and discovered how,\nin its evolutionary development it has divided into two aspects,\nthe egopsychic and the omnipsychic, and these two factors ally him\ndefinitely and adequately to the world of the senses and to the world\nof supersensuous cognitions. And thus we have cleared up some of the\nmisconceptions which had to be confronted and made more easy the\napproach to the central idea, thereby conserving the substantiating\ninfluence which a general and more comprehensive view of the whole\nwould naturally give.\n\n_The totality of kosmic order is space. It is circumscribed by an\norderless envelope of chaos just as the germ of an egg is surrounded\nby the egg-plasm. The organized kosmos is the germ, kernel or central,\nnucleated mass, enduring in a state of becoming. Involutionary\nkathekos or primordial chaos is the egg-plasm which nourishes the\ngerm or the kosmos and is that out of which the germ evolves.\nKathekos or chaos is the unmanifest, unorganized, unconditioned,\nunlimited and undifferentiated plasm. Space is the manifest,\nlimited, finite, organized germ that, feeding upon the enveloping\nchaos, exists in a perpetual state of alternate manifestation\nand non-manifestation--appearing, disappearing and reappearing\nindefinitely._\n\nThe appearance of the kosmos as an orderly elaboration of the\ninvolutionary phase of kosmogenesis, in so far as kosmic order may\nbe said to be an accomplished fact, marked the turning point in that\nprocedure whose function it was to make manifest a universe possessing\ncertain definite characteristics of orderliness; but the kosmos, as it\nnow stands, may not be thought of as having attained unto a state of\nultimate orderliness. The idea meant to be conveyed is that between\nthe point of becoming and the actually pyknosed, or solidified stage\nin the process of creation there is a more or less well defined line\nof demarkation cutting off that which is spatiality from that which\nis non-spatiality. Beyond the limits of spatiality is an absence of\ngeometric order. Here geometry breaks down, becomes impotent, because\nit is an intellectual construction; at least, it is not so apparent\nas in the manifested kosmos. It is a state about which it is utterly\nfutile to predicate anything; because no words can describe it. The\nmost that may be said is that it is absence of geometric order as it\ninheres in space. And if so, all those movements comprehended under\nthe general notions of spatiality, materiality, intellectuality and\ngeometricity have both their extensive and detensive or inverse\nmovements nullified in their approach to it. Involutionary _Kathekos_,\ntherefore, may be said to be the primordial wilderness of disorder\nwhich outskirts the well laid-out and carefully planned garden of\nthe spatial universe. We may excogitate upon some of the obvious\nfunctions of this kathekotic world-plasm; but in doing so we must leave\noff all attempts at a description of its appearance, its magnitude,\nextent or other qualities, and think only of its kosmic function.\nWe cannot say that there is back of it a spatiality nor can we say\nthat it is a spatiality; for whatever may be its extent or volume,\nit suffices that it may not be said to be space. It is chaos. Space\nis order, organization, geometricity. It cannot be said that there\nis a latent geometrism in chaos; because geometric order is found\nonly in spatiality and is that which distinguishes spatiality from\nkathekosity or non-spatiality. Chaos is the lack of spatiality. This,\nof course, implies that it is impenetrable to the intellectuality or to\nvitality. All inverse movement such as is discovered as taking place\nin spatiality and which results in the phenomenalization of space runs\naground when it strikes against the rock-bound coast of kathekosity.\nWe can only say that it is both the point of origin for the evolving\nuniverse of life and form and its terminus. It is the nebulosity out of\nwhich the whole came and into which all is ultimately occluded.\n\nA great and far-reaching error is made in all our thinking with respect\nto the kosmogonic processes when we postulate the complete absorption\nof chaos as an early act of kosmogony. Customarily, we think of kosmic\nchaos as a primordial condition whose existence was done away as soon\nas the universe came into active manifestation. This because it has\nbeen exceedingly difficult, if not quite impossible, for those whose\nprivilege it was to determine the trend of philosophic thought to free\nthemselves from the bondage of a dogma which owed its existence to\na traditional or legendary interpretation of facts that ought never\nhave been so interpreted. Chaos IS and EVER SHALL BE, so long as the\nuniverse itself lacks completion, fullness or perfection in purpose,\nextent and possibility. It is undoubtedly being diminished, however,\nin proportion as the kosmos is approaching absolute perfection. And\nwhen the last vestige of chaos disappears from the outerskirts of\nthe maturing kosmos there shall appear a _glorified universe_ of\nindescribable qualities.\n\nSpace being a perception _a priori_ cannot be determined wholly by\npurely objective methods. The yard-stick, the telescope and the\nlight-year are objects which belong exclusively to the phenomenal and\nwith them alone never can we arrive at a true conception of the nature\nof space. We can no more demonstrate the nature of space by the use\nof objective instruments and movements than we can measure the spirit\nin a balance. Certainly, then, it cannot be hoped that by taking the\nmeasurement of space-distances in light-years, or studying the nature\nof material bodies, we shall be able to fathom this most objectively\nincomprehensible and ineluctable thing which we call space. It is such\nthat every Thinker must, in his own inner consciousness, come into the\nrealization of that awfully mysterious something which is the nature\nof space both as to existence and extent by his own subjective efforts\nunaided, uncharted and alone. When we measure, weigh, apportion and\notherwise try to determine a thing we are dealing with the phenomenal\nwhich is no more the thing itself than a shadow is the object which\ncasts it.\n\nWhat does it matter that metageometricians shall be able to\ndemonstrate that space exhibits itself to the senses in a four-or\n_n_-dimensional manner? Granting that they may be able to do this,\nif merely for the sake of the discussion, when they have finished,\nit will not be space that they have determined, but the phenomena\nof space, its arborescence, while space itself remain indeterminate\nand unapproachable by phenomenal methods. If there are curvature,\nmanifoldness and _n_-dimensionality these are not properties of space,\nbut of intellectuality in its cultured state and when it is, therefore,\nremoved from the native state of conception. Scientists may be able to\nweigh the human body, count every cell, name and describe every nerve,\nmuscle and fiber; they may even be able to know it in every conceivable\npart and from every physical angle and relationship, and yet know\nnothing of the life which vitalizes that body and makes it appear the\nphenomenal thing that it is. So it is not by instruments which man may\ndevise that we shall be able to determine the true nature and purpose\nof space. We must adopt other methods and means and assume other\nangles of approach than the purely objective in order to comprehend\nspace which, being the sole inherent aspect of consciousness, can be\nunderstood best by applying the measures which the latter provides for\nits understanding. It would appear, therefore, that the best study of\nspace is the consciousness itself, knowing which we shall know space.\n\nThe universum of space, including the phenomenal universe, and its\nrelation to consciousness may be likened to a conical funnel whose\nbase represents the phenomenal world of the senses and whose apex or\nsmallest point represents ultimate reality.\n\nIn Figure 20, we have endeavored to symbolize graphically this\nconception of space. The base marked \"Sensorium\" represents the\nsensible world. That marked \"Realism\" symbolizes the ultimate\nplane of reality, the inner essence of the world, the plane of\n\"things-in-themselves.\"\n\nThe cone arising from the base \"sensorium\" symbolizes the objective\nworld as compared with consciousness; the subverted cone, with apex in\nthe sensorium, represents the evolving human consciousness.\n\nThe successive bases have the following symbology: Self-consciousness,\nCommunal Consciousness, Mikrocosmic Consciousness, Makrokosmic or\nUniversal Consciousness, the Plane of the Space-Mind Consciousness,\nDivine Consciousness, Kathekotic Consciousness, or the Plane of Final\nUnion with the Manifest Logos.\n\nSelf-consciousness is that form of consciousness which enables the\nego to become aware of himself as distinguished from other selves or\nthe Not-self; the Omnipsychic or Communal Consciousness is that form\nof consciousness from which arises the realization by the Thinker of\nhis oneness with all other thinkers and with other forms of life.\nMikrocosmic consciousness denotes a still higher form of consciousness,\nas that which enables the Thinker to become conscious of his living\nidentity with the life of the world or the planet on which he lives.\nIt represents a stage in the expansion of consciousness when it\nbecomes one with the consciousness of the planet upon which it may\nbe functioning. Makrokosmic consciousness accomplishes the awareness\nof the Thinker's unity with the life of the kosmos or universe. The\nspace-mind and the consciousness which constitutes it enable the\nThinker to comprehend the originality and the terminality of kosmic\nprocesses. It is archetypal so far as the life-cycle of the universe\nis concerned because the beginning, the intermediate portion and the\nending of the kosmos are encompassed within it. Divine consciousness\nis that form of consciousness which arises upon the unification of the\nThinker's consciousness with that of the manifest deity; it is, in\nfact, omniscience. The kathekotic consciousness belongs to the ultimate\nplane of reality; to kosmic origins and chaogeny, and therefore,\npertains to the plane of non-manifestation.\n\n[Illustration: FIG. 20.--Kosmos and Consciousness]\n\nThe implications are that in comparison with the sensorium, the\nThinker's consciousness is a mere point in space. It is, in reality,\nso small and insignificant that the extensity of the physical world\nor universe seems unlimited, unfathomable in meaning and infinite in\nextent. But as his consciousness expands, as it passes, in evolutionary\nsuccession from one plane of reality to another and higher one,\nthe illimitability, the incomprehensibility and infinity of the\nuniverse grow ever smaller and smaller, until the plane of divine\nconsciousness is reached. Then the previously incomprehensible dwindles\ninto insignificance, lost in the real illimitability, infinity and\nunfathomability of consciousness itself. Kosmic psychogenesis, as\nexhibited and specialized for the purposes of the evolution of the\nThinker, can have no other destiny than the flowering forth as the _ne\nplus ultra_ of manifestation which is nothing short of unification\nwith the highest form of consciousness existent in the kosmos.\n\nIt is not to be considered really that the scope of space is diminished\nbut that the growing, expanding consciousness of the Thinker will\nso reduce the relative extension of it that illimitability will be\nswallowed up in its extensity. Consciousness, in becoming infinite\nin comprehension, annihilates the imaginary infinity of space.\nAccordingly, that which now appears to be beyond mental encompassment\nundergoes a corresponding diminution in every respect as the\nconsciousness expands and becomes more comprehensive. _The mystery\nof space decreases as the scope of consciousness increases._ As the\nThinker's consciousness expands the extensity of the manifested\nuniverse decreases. Thus the mystery of every aspect of kosmic life\nlessens, and fades away, as the intimacy of our knowledge concerning it\nbecomes more and more complete. There is no mystery where knowledge is.\nMysteriousness is a symbol of ignorance or unconsciousness, and that\nwhich we do not understand acts as a Flaming Sword keeping the way of\nthe Temple of Reality lest ignorance break in and despoil the treasures\nthereof.\n\nFigure 21 is a graph showing a sectional view of consciousness on\nall planes represented as seven concentric circles. This describes\nthe analogous enveilment of the consciousness when it ensouls a\nphysical body or when bound to the purely objective world of the\nsenses. The overcoming of the barriers of reality, represented by\nthe circumscribing circles is the work of the Thinker who is forever\nseeking to expand and to know. For only at its center, as symbolized\nhere, is the consciousness at one with the highest aspect of kosmic\nconsciousness and there alone is the mystery of space despoiled of its\nhabiliments.\n\n[Illustration: FIG. 21.--Septenary Enveilment of Consciousness]\n\nAccordingly, as consciousness or the Thinker is more and more divested\nof carnal barriers and illusions there develops a gradual recognition\nof the unitariness of spatial extent and magnitude; there arises the\ncertain knowledge that space has but one dimension and that dimension\nis sheer _extension_. The Thinker's sphere of awareness is represented\nas if it begins as a point in space and develops into a line which\ndivides into two lines, the boundaries of the space cones. Thus it may\nbe perceived that the ancients had a similar conception in mind when\nthey symbolized kosmogenesis with the dot (.), the line, and the circle\nwith diameter inscribed, which together represent the universe in\nmanifestation.\n\nWe realize the impossibility of adequately depicting the full\nsignificance of the inverse ratio existing between the extensity of\nspace and the increscent inclusivity of consciousness by means of\ngraphs; for neither words nor diagrams can portray the scope and\nmeaning of the conception in its entirety. Yet they aid the intellect\nto grasp a ray of light, an intimation of what the Thinker sees and\nunderstands interiorly.\n\nIn this connection it is interesting to note the function of the\nideal in the evolution and expansion of consciousness. The ideal has\nno perceptual value; it has no status in the world of the senses. It\nis unapproachable either in thought or action, and therefore, lies\nbeyond the grasp of both the intellectuality and the vitality. It is\nindescribable, inconceptible and searchless; for the moment that we\ndescribe, define, or approach the ideal, either intellectually or\nvitally, in that moment it ceases to be ideal, but actual. It flees\nfrom even the slightest approach; it never remains the same; it cannot\nbe attained, at least its attainment causes it to lose its idealty. It\nis then no longer the ideal. It is like an _ignis fatuus_; the closer\nwe come to it the farther away it recedes. It hangs suspended before\nthe mind like the luscious grapes which hung before the mouth of the\nhungry Tantalus. As the grapes and the water receded from his reach\nat every effort he made to seize them so the ideal remains eternally\nunseizable and unattainable. Whatever, therefore, is in our thought\nprocesses, or in our knowledge, that may be said to be _ideal_, does\nnot really exist. The ideal is a phantom growing out of the nature and\nessence of the intellectuality. Its purpose is to lead merely the mind\non; to allure it, to tantalize, and compel it to grow by exertion, by\nthe struggle to attain, by the desire to overcome. In this respect, it\nserves well its function in the economy of intellectual evolution. It\nis a mysterious aspect of the original and eternal desire to live which\nis the kosmic _urge_ present in all organized being and has its roots\nhidden in the divine purpose of creation.\n\nIdealized constructions, then, are like Arabian feasts conjured up by\na famishing mentality. They are like the dreams of a starving man in\nwhich he actualizes in phantom-stuff the choicest viands in abundant\nsupply for his imaginary delectation. The mind that is satisfied never\nidealizes, never makes an idealized construction. It is only when an\n\"aching void\" is felt, when a longing for the realization of that which\nit has not arises within itself, when a feeling of distinct lack, a\nwant, a hankering after something not in its reach, takes possession of\nthe mind that it begins to idealize. That is why some minds are without\nideals. It is because they are satisfied with what they have and can\nunderstand. They feel no hungering for better and grander things; they\nhave no desire to understand the unknown and the mysterious; hence they\ndo not idealize; they make no attempt to represent unto themselves\na picture of that which is beyond them. Such minds are dormant,\nhibernant, asleep, unfeeling and unresponsive to the divine urge.\n\nBut the ideal is neither obtainable objectively nor subjectively,\nneither phenomenally nor really, so that when we come upon the ideal\nin our mode of thinking we have arrived not at a finality or end,\nbut at that which is designed to lead us on to something higher, to\nnobler accomplishments and more extensive conquests. When we have\ndevised idealized constructions, therefore, we should not therewith be\ncontent but should scrutinize them, examine and study them for their\nimplications; for thereby we may discover the path and the guide-posts\nto a new domain, a new ideal, following which we shall, in time, come\nto a point in our search for the real where the fluxional is at a\nminimum; we shall reach that something which will admit of no further\nstruggle--the last chasm between the phenomenal and the real--and\nstanding on the bridge, consciousness, which engages the twain, shall\nhave a complete view of that Sacred and Imperishable Land of Kosmic\nRealism where like a fleeting cloud of sheerest vapor shall be seen the\nphantom-ideal deliquescing and disappearing in the cold, thin air of\nthe real and the eternal.\n\nSince space is judged to be infinite by the intellect occluded in\nsuch clouds of illusion and hampered by such constrictive bonds of\nlimitations, as it now endures, we have no right to conclude that the\nconcept of infinity would still linger before the mind's eyes when\nthe illusionary veil is removed; in fact, there is ample reason to\nbelieve, nay for the assertion, that the recession of the veil will\nreveal just the opposite of this illusion, namely that space is finite,\nand even _bounded_ by the fringe of chaogenetic disorderliness. Either\nwe perceive the real or we do not; either the pure _thingness_ of all\nobjects can be perceived or it cannot be perceived. If not, granting\nthat there is such a thing as the real, it must be within the ultimate\nrange of conceivability. It also seems reasonable that realism exists\nsomewhere, and if so, must be sought in a direction inverse to that in\nwhich we find the phenomenal and the approach thereto must necessarily\nbe gradual, continuous and direct and not by abrupt breaks, by twists\nand turns. The phenomenal must lie at the terminus of the real, and\n_vice versa_. So that by retracing the path blazed out by the real in\ncoming to phenomenalization we shall perhaps find that which casts our\nshadowy world, just as by tracing a shadow in a direction inverse to\nthat in which it extends we may find the object which projects it.\n\n_It is not out and beyond that we shall find the end of space; it is\nnot by counting tens of thousands of light-years that the supposed\nlimits of space shall be attained. The path of search must project\nin an opposite direction--not star-ward but Thinker-ward, toward the\nsubtle habitation of the consciousness itself._ We err greatly when\nwe think that by measuring distances we shall encompass space; for\nthat which we measure and determine is but the clouds caused by the\nvapor of reality. It is, therefore, not without, but within, in an\ninverse direction that the search must proceed. Going back over the\nlife-stream, beginning where it strikes against the shores of solid\nobjectivity, deeper and deeper still, past the innermost mile-stone of\nthe self-consciousness, back into the very heart of the imperturbable\ninterior of being where the Thinker's castle opens its doors to the\nGreat Kosmic Self, from that open door-way we may step out into that\ngreat mystery of space--limited, yet not limited, multi-dimensioned,\nand yet having only one dimension, veritably real and fundamental, the\nFather-Mother of all phenomena. Here the great mystery of mysteries\nis revealed as the citadel of the universal and the ultimate real.\nIn this citadel, the plane of kosmic consciousness, space loses its\nspaciousness and time its timeliness, diversity its multiplicity and\noneness alone reigns supreme.\n\nBut the movement towards the center and circumference of space, after\nthis manner, requires aid neither from the notion of space-curvature\nnor that of the space-manifold, except, indeed, only in so far as a\nstate of consciousness or a degree of realism may be said to be a\ntridimensional manifold. The feeling that space is single-pointed,\nand yet ubiquitously centered, has been indulged by mathematicians\nand others in a more or less modified form; but they have imagined it\nin the terms of an indefinite proceeding outward until in some manner\nunaccountable alike to all we come back to the point of origin. It\nhas been expressed by PICKERING when he says that if we go far enough\neast we shall arrive at the west; far enough north we shall come to\nthe south; far enough into the zenith we shall come to the nadir.\nBut this conception is based upon a notion of space which is the\nexclusive result of mathematical determinations and subject to all the\nrestrictions of mathetic rigorousness. It requires that we shall allow\nspace to be curved. This we decline to do for the reason that it is\nboth unnecessary and contrary to the most fundamental affirmations of\nthe _a priori_ faculty of the Thinker's cognitive apparatus. It would\nseem to be necessary only that we should extend our consciousness\nbackward, revert it into the direction whence life came to find that\nwhich we seek. By extension of consciousness is meant the ability to\nfunction consciously upon the various superkosmic spheres or planes\njust as we do on the physical. Yet it should be quite as easy to\ndevise an idealized construction which would imagify the results of\nthis ingressive movement of the consciousness as to represent the\nresults of a progressive outward movement star-ward. Having done so the\nexamination of them could be conducted along lines similar to those\nfollowed in the scrutiny of objective results.\n\nWhat would it mean to the Thinker if he were able to identify his\nconsciousness with the ether in all its varying degrees; what would it\nmean if he were able to identify his consciousness with life and with\nthe pure mind-matter of the kosmos; and lastly, with the spiritual\nessence of the universe? What if his various vehicles of awareness were\navailable for his purposes of cognition? What, indeed, if he could\ntraverse consciously the entire gamut of realism and consciousness\nfrom man to the divine consciousness? Does it not appear reasonable\nthat as he assumed each of these various vestures of consciousness,\nin succession, he would gradually and finally, come to a full\nunderstanding of reality itself? It seems so. This view is even more\ncogent when it is considered that the limitations, and consequent\nobscuration of consciousness are proportional to the number of vehicles\nor barriers through which the Thinker is required to act in contacting\nthe phenomenal universe. Common sense suggests that freedom of motility\nis determined by the presence or absence (more particularly the latter)\nof bonds and barriers; that the less the number of such barriers the\ngreater the scope of motility and consequently greater the knowledge.\n\nPLATO evidently had this in mind when he imagined the life of men\nspent in cave-walled prisons in which their bodies were so fixed that\nthey were compelled to sit in one prescribed position, and therefore,\nbe unable to see anything except the shadows of persons or objects\nas they passed by. He conceived that men thus conditioned would, in\ntime, suffer the diminution of their scope of consciousness to such an\nextent as to reduce it to identification with the shadows on the walls.\nTheir consciousness would be mere shadow-consciousnesses the entire\ndata of which would be shadowgraphs. So that for them the only reality\nwould be the shadows which they constantly saw. A similar thing really\nhappens to man's consciousness limited to the plane of the objective\nworld. Things which are not objective do not appear as real to him,\nif they do appear at all. It is not that there are no other realities\nthan those which appear to the egopsychic consciousness or that fall\nwithin its scope; but that this form of consciousness is incompetent\nto judge of the nature and appearance of those realities which do not\nanswer to the limitations under which it exists. And so, with men whose\ndata of consciousness or whose outlook upon the world of facts, or\nrather life, are confined to the narrow bounds of mathematic rigor and\nexclusiveness, there may appear to exist no realities which may not\nbe defined in the terms of mathematics. Similarly, to the empiricist,\nused to measurements of magnitudes, weights, and rates of motion,\nthere may also appear to be no realities which are not amenable to the\nmold of his empirical contrivances--the balance, the chromatic and\nthe scalpel. All of these are shadow men constricted to the metes and\nbounds of shadows which they observe only because they are ignorant of\nthe realities which lie without their plane.\n\nLife has so many ways of exhibiting its remains to the intellect; and\nthese remains have so many facets or viewpoints from which they may\nbe studied, that nothing short of a panoramic view of all the modes\nof exhibition and of all the facets and angles of appearances will\nsuffice to present a trustworthy and comprehensive view of the whole.\nThen, life itself is so illusive, so unseizable by the intellect that\nthe testimony of all investigators are required to summarize its modes\nof appearance. And, therefore, eventual contentment shall be secured\nonly when the mass of diverse testimonies is reduced to the lowest\ncommon divisor, and for this purpose the operations of every class of\ninvestigators must be viewed as the work of specialists upon separate\nphases, facets and angles of life's remains.\n\nAnd so it is manifestly absurd for the empiricists, by taking note of\nthe dimension, extent, quality and character of the shadows, or one\nsingle class of angles, to hope to predicate any trustworthy judgments\nabout either the realities which cast the shadows or underlie the\nangles; because whatever notion or conception they may be able to\ngain must of necessity be merely fragmentary and entirely inadequate.\nDespite this fact, however, we still have the spectacle of men who,\nstudying the sensible universum of space-content, endeavor either to\nmake it appear as a finality in itself, or that the world of the real\nmust necessarily be conformable to the precise standards which they\narbitrarily set up in their examination of the objective world. It can\nbe said with assurance that we shall never be able even so much as to\napproach a true understanding of the unseen, real world until we shall\nhave changed our mental attitude towards it and ceased to expect that\nit shall necessarily be fashioned and ordered in exactly the same way\nas the world of our senses, or that it shall be understood by applying\nthe same methods of procedure as those which we use in our examination\nof the phenomenal, sensuous world. It is a matter of logical necessity\nthat, as there are no senses which can respond to the real, as there\nare no organs which vibrate in accord with the rates of vibration of\nthe real, there can be no reasonable hope of understanding it by means\nof sensuous contrivances and standards.\n\nLet the consciousness, therefore, be turned not outward, but _inward_\nwhere is situated the temple of divine life; let there be taken away\nthe outward sheaths which enshrine the pure intelligence of the\nThinker; let him grow and expand his sphere of awareness; let there be\nan exploration of the abysmal deeps of mind, of life and consciousness;\nfor buried deeply in man's own inner nature is the answer to all\nqueries which may vex his impuissant intellectuality.\n\n\n\n\n  CHAPTER IX\n\n  METAGEOMETRICAL NEAR-TRUTHS\n\n  Realism is Psychological and Vital--The Impermanence of Facts--On\n  the Tendency of the Intellect to Fragmentate--The Intellect\n  and Logic--The Passage of Space, the Kosmometer and Zoometer,\n  Instruments for the Measurement of the Passage of Space and the\n  Flow of Life--The Disposal of Life and the Power to Create--Space\n  a Dynamic, Creative Process--Numbers and Kosmogony--The Kosmic\n  Significance of the Circle and the Pi-Proportion--Mechanical\n  Tendence of the Intellect and its Inaptitude for the Understanding\n  of Life--The Criterion of Truth.\n\n\nKosmic truth has many facets. The rays of light which we see darting\nfrom its surface do not always come from the core. Often they are\nreflections of rays whose light stops short at the superfice; and\nthese, in turn, are reflections of deeper realities. Thus the reflected\nlight may be traced to its source by following the lead of external\nreflections. It is now known that moonlight, and perhaps, in many\ncases, starlight, are reflections of sunlight, if not of our sun,\nsome other in the universe. But it is only at certain times and under\ncertain conditions that we can see the sun which is the source of the\nother kinds of light. The stars which owe their light to suns are so\nmany facets of sunlight. The moon is a facet of sunlight also. Facts\nare facets of truth. They are so many faces of eternal truth. They\nrepresent the many ways reality exhibits itself, or rather its effects,\nto the consciousness. When we, therefore, become aware of facts we have\nnot in virtue thereof become aware of the reality which produces the\nfacts. We have come to know only something of the termini of realism\nwhile the complexities and internal ramifications which lie between\nrealism itself and these termini altogether elude our cognition.\n\nLet us examine briefly an icosahedron, for instance. An icosahedron is\na figure comprehended under twenty equal sides. These various sides\nare so many faces by means of which the figure presents itself to the\nconsciousness. These faces, however, are not the real object. The\nfigure may be examined by viewing it from any one of its sides; yet,\nby simply examining a single face, or any number of faces, less the\ntotal number, we arrive at no satisfactory knowledge of the magnitude\nor its substance. We must first become conscious of all the faces,\nholding them in mind as a composite picture, before we can even begin\nto have anything like a complete notion of the icosahedron. Then by\ncontinuing the examination we may find that the magnitude is composed\nof wood fiber or stone or metal, as the case may be. In this way we\nmight carry the examination to indefinite limits and finally arrive at\na very comprehensive knowledge of the icosahedron and yet be unaware\naltogether of the forces which have been at work in the production of\nthe magnitude or of the reality which lies back of it.\n\nRealism is psychological and vital. In essence it is mind, spirit,\nlife. Yet these three are one. Mind is the outward vehicle of life;\nspirit is the form or the interior vehicle which life assumes in order\nto express itself. Realism, then, is life. Is the logician dealing\nwith reality when he collects and co\u00f6rdinates the various modes of\ninterpretation by which we learn to understand the symbolism of life?\nObviously not. The data of logic are simply a collection of rules for\ninterpreting concepts. It is a compendium of indices for the Book of\nLife. It is no more the book itself than a table of contents is a\nbook. But logic occupies about the same category as does an index to a\nvolume. A book, however, is more than its conventional contents. It is\nthe thought that is symbolized therein. The book of life, accordingly,\nis the sum total of life's expressions; but it is not life itself. That\nis the subtle, evasive something which the contents of the book of life\nsymbolize. Nature, both in her palpable and impalpable aspects, may be\nsaid to be the book of life wherein are recorded the movements, the\nexpressions, and the diacritics of life. The whole is a magnitude of\nmany facets (little faces). We shall have to know all the faces before\nwe can say that we have a comprehensive knowledge of nature. For so\nlong as we have only a fragmentary knowledge of the whole, so long even\nas we have merely a superficial knowledge of any aspect of nature,\njust so long will our knowledge be in vain. Just as it frequently\nhappens that, on account of the partial view of things, we are led\nto make incorrect judgments concerning them, so when we come to make\nassertions about life or nature in general, we are apt to fall into the\nerror of rendering judgments upon insufficient data. And it is not at\nall likely that judgments thus arrived at can possess true validity\nbecause it may happen, and undoubtedly does always so happen under the\npresent limitations of human knowledge, that the very elements which\nare ignored or neglected in forming a judgment possess enough of virtue\nto alter the intrinsic value of determinations based upon otherwise\ninsufficient data. Hence it develops not infrequently that our\njudgments repeatedly have to be changed in proportion as our data are\nmade more and more comprehensive. Men searching eagerly for the truth\nsometimes allow themselves to be carried away by the enthusiasm of the\nmoment which arises upon the discovery of a new facet of truth; but\nif all searchers were to bear in mind the fact that reality presents\nitself to consciousness in myriad ways and that there are innumerable\nfacets all leading eventually back to the source of all they not so\neasily would be induced to jump to the conclusion that they had covered\nthe entire ground. For when we have discovered a million facts, or many\nmillions of them, about nature we may say that we have only merely\nbegun and that what we have found is not to be compared with the\ntotality even of the directly observable phases of nature.\n\nLogic, therefore, deals with the symbolism existing between and\namong facets of truth, and not directly with truth itself, although\nthe conclusions reached by the logicians may be true enough from\nan intrinsic standpoint. Logic is not truth, however; it is merely\nthe consistence of relations and inter-relations between facts and\namong groups of facts. Truth is not established by logic; it stands\nin no need of the light of logic for its revelation; indeed, more\napt than not is logic to obscure truth. Truth is its own proof; it\nis self-evident. Logic is a mere modeler of facts; it is static,\nimmobile, fixed. All truly logical processes need a starting point, a\nfoundation, a premise, a base. Truth, being eternal, mobile, dynamic,\nvital, needs no starting point; needs no foundation because it is\nitself fundamental; it requires no premise because no premise is\ncomprehensive enough to encompass it. There is only one way of arriving\nat truth and that is not to arrive at all--just to recognize it without\nprocedure. The fact that facts are, and the fact of their relations and\ninter-relations, their sequence and implications, can be arrived at\nonly by logical processes. Life, in its passage through the universum\nof spatiality, carefully diacriticizes between the realm of facts and\nthe domain of truth, marking each off from the other by unmistakable\nsigns and barriers. Truth is perceived as an axiomatic, self-evident\nprinciple and no amount of logic could prove or establish its verity.\nFacts are intellectual creatures; truth is intuitional, vital. The\nintellect conceives the consistence of facts while the intuition\nrecognizes truth--is truth, and therefore, follows in the wake of life\nas consciousness.\n\nThere is no permanence in facts and the intellectual recognition of\ntheir consistence. The discovery of a single new fact may destroy the\nconsistence of a whole mass of previously correlated facts. Thus is\nrevealed the miracle power of logic over facts. It can take a mass of\nfacts, related or unrelated, mold them into hypotheses, endow them\nwith a sort of interior consistency, and make these hypotheses take\nthe posture of truth. Hence logic is often an effective mask which\nthe intellect commonly imposes upon its material; but it does so\ninstinctively and can no more escape the rigorous compulsion of this\ninstinctive functioning than water can escape its liquidity. Wherefore,\nwe conclude that true permanence abides alone in truth because truth\nis duration itself. For the foundations of the whole structure of\nfacts in religion, science, art and philosophy which man has toilfully\nbuilt up in the last million years might easily be destroyed or\noverturned by the discovery of some great fact or by appreciating the\ntrue value of truth. Let us suppose it should suddenly be realized\nby men that they are really and truly gods capable of creating and\npossessing all the other virtues, powers and capabilities which we are\naccustomed to impute to supreme divinity; and suppose that the fact\nof their omnipotence and divine omniscience always had been obvious\nbut that men were so engaged upon details and the non-essentials of\nlife and matter that they had not noticed nor realized it before,\nwould not this realization make a vast difference in the character\nof our knowledge and the attitude which we would necessarily assume\nthereafter towards matter, life and the problems which they present?\nWould not it completely revolutionize our arts, our sciences and our\nphilosophies? How much, then, of the facts of these would be left\nwhen the light of omniscience had been turned on--when truth itself\ncould be perceived and interiorly realized? Not much, to be sure. We\nshould undoubtedly have to dispense with the entirety of our fact-mass,\nfor it should then be entirely useless and meaningless in the light\nof the resplendent omniscience of truth. As at present constituted\nconsciousness is focused upon the material plane for the purposes of\nsuperficial observances. But if the focus of consciousness should be\nchanged so as to reveal conditions upon what must be a higher and more\ninterior level, the aspect of things would be entirely changed and the\nwhole of our theory of knowledge would have to be reconstituted. It\nis conceivable, yea obvious, that the stern reality of being is far\nremoved into the Great Interior of that which is; and there is a point\nin the path to the interiority of being where there is no illusion, no\nappearance, indeed, nothing but the cold, illuminating body of reality\nitself. It must appear also that along the journey interior-ward there\nare many apparent levels or planes, each of which requires a new focus.\nIt is unreasonable, then, to suppose that the conclusions arrived at\nas a result of purely logical processes, confined to the lowest levels\nof reality, are pertinent and valid for the entirety of realism which\nis neither of mathematical nor logical import. For instance, if we\ntake the purely axiomatic assertion: _x_ equals _x_, the intellect\nis at once certain that this is so, and cannot be otherwise, and yet\na proposition of this kind is purely conceptual, conventional and\narbitrary. _x_ may also equal 1, 2, 3, 4, or any other quantity. Then,\nif each _x_ in the above equation be replaced, by say, a horse, there\nimmediately arises a difficulty. For it is not possible to find two\nhorses which are in all respects mutually equal. So that as soon as\nwe pass from the conceptual into the actual, whether on the side of\nobjective reality or that of absolute reality, the validity of the\naxiom is immediately exposed to serious questioning. The truth of the\nmatter is that on both sides of the conceptual it is always found that\nthere is a variance from the standards set up by the conceptual, this\nvariance being more marked on the side nearest to absolute reality than\non the side of objectivism. Objectively, the conformity of the sensible\nwith the conceptual is of such approximation as to lend trustworthy\nutility to the conceptual in its application to the sensuous. Thus by\nsimply eliminating the vital factors from our equations we are enabled\nto proceed in a reasonably safe manner with our judgments. Really,\nhowever, no such approximate congruence can be found; for on the side\nof reality we are dealing with an indivisible something--something\nthat is eternally and absolutely unitary in its constitution while\nwhen we transfer the scene of our observations to the objective world\nwe discover a contrary situation. Here we are everywhere beset by\ndiversities, multiplicities and dissimilarities. This is so because the\nintellect naturally tends toward the objective where it finds a most\ncomfortable atmosphere for its operations. The conceptual is related to\nthe objective as a train of cars is related to the railway. That is to\nsay, the constitution of the intellect is such that it finds its most\nfacile expression in the objective world and is about as comfortable in\nthe domain of realism as the same train of cars would be on the ocean.\n\nThe intellectuality is designed to deal with facets of truth; it is\nmade to manipulate segments, parts, fractions, and cannot chart its\nway through a continuum such as reality. Being constitutionally a\nconventionality of the Thinker's own contrivance, and arising out of\nthe subtle adaptation of his vehicles to the environment afforded by\nthe sensuous world, it can only find congruence in that conventionality\nwhich is the instrumentality of a higher intellectuality expressed\nin a diversity of forms, into which reality divides itself for\nmanifestation. The human intellect is, therefore, the bridge over\nwhich is made the passage from the individual consciousness to the\nAll-Consciousness; simultaneously, the medium whereby the physics\nof the brain are converted into the psychics of unconsciousness. It\nmay be likened to a pair of specially constructed tongs which are so\nformed as to fit exactly the objects which a higher intellectuality\nhas made. It is without the province of the intellect to take note of\nwhat intervenes between physics and psychics; it is always oblivious\nof interstices while taking cognizance of objects or things. In this\nrespect, the intellect is much like a steerage passenger on board an\nocean liner who sees only his port of departure and port of arrival,\nknowing nothing in the meantime of what happens during the voyage,\nnothing of what the other passengers on the upper decks may experience\nand taking no part in any of the passing show until he lands. So\nthat the passage of the intellect from fact to fact is an altogether\nuninteresting voyage; it may as well be made unconsciously, and to all\nintents and purposes, is so made.\n\nAccordingly, the advocates of _n_-dimensionality find it quite\nimpossible to predicate anything whatsoever of the passage, say, from\ntridimensionality to quartodimensionality. They find themselves at ease\nin tridimensionality and have even contrived to find pleasant environs\nin the four-space having made therein such idealized constructions as\nwill afford ample hospitality to the intellect. But the questions as to\nhow the passage from the three-space to the four-space is to be made\nand how the intellect shall demean itself during the passage have been\ncompletely ignored and, therefore, left unanswered. What, then, shall\nbe said of an explorer who says he has found a new land and yet can\ngive no intimation as to how one may proceed to arrive at the new land,\nwhat changes are to be made en route, nor the slightest suggestion\nas to the direction one should take in setting out for it? It is not\nlikely that the report of such an explorer, in practical life, would\nbe taken seriously; and yet, there are those who, relying utterly\nupon similar reports made by certain enthusiastic analysts, dare to\nplace credence in their asseverations. Not only have they given wide\ncredence to these reports, but have, indeed, sought to rehabilitate\ntheir own territory in accordance with the strange descriptions given\nby unhappy analytical explorers. Now the question of greatest concern,\ngranting for the nonce that there is such a domain as hyperspace,\nis the _passage_. How shall we make the passage? Or, is the passage\npossible? In vain do we interrogate the analyst; for he does not know,\nnor does he confess to know. Evidently it is impossible for him to know\nby means of the intellect alone; for the intellect not being fitted to\ntake cognizance of the \"passage,\" but only the starts and stops, has no\naptitude for such questions. Hence, what seems to be the most important\nphase of the entire question will have to remain utterly inscrutable\nuntil the intellect nourished by the intuition shall be aroused from\nits lethargy and brought to a certain high point of illumination where\nit, too, may take note of the passage.\n\nSpace is the path which life makes in its downward sweep through\nall the stages of pyknosis or kosmic condensation by virtue of\nwhich it accomplishes the engenderment of materiality as also the\npath marked out by it in its upward swing whereby it accomplishes\nthe spiritualization of matter. It is the kosmic order which life\nestablishes by means of its outgoings and incomings. When we look out\ninto space we perceive that which is a dynamic appearance of life\nitself, and not a pure form. Nothing that is a pure form can exist in\nnature and in as much as space is not only indissoluble from nature\nbut partakes of its very essence it cannot be said to be a pure form.\nThe intellect, however, prone to follow the grooves laid out by pure\nlogic, never fails to seek to make everything that it contacts conform\nto these logical necessities. But, if the analyst were to make careful\ndiscrimination as to the respective categories--that into which life\nfalls and that in which the intellect is forced by its nature to\nproceed--he not so easily would be led into the fault of attempting to\nshape realities upon models which being strictly conventional were not\nmeant for such uses. But neither the logician nor the mathematician can\nbe condemned for such generosity if such condemnation were justifiable.\nFor they everywhere and at all times insist upon _realizing_\nabstractions and _abstractionizing_ realities, and they do this with\nan _insouciance_ that is at times surprising. Yet it is in this very\nvagary that is discovered the true nature of the intellect. There is\na sort of dual tendence observed in the method of the intellect's\noperation. A polarity is maintained throughout: the abstractive and the\nconcretional. It vacillates continually between the abstract and the\nconcrete and no sooner has it found a concrete than it begins to set up\nan abstract for it; and _vice versa_--as soon as it is has constructed\nan abstract it immediately seeks either its concrete or sets out to hew\nsome other concrete into such shape as will fit it. And between these\ntwo extremes numerous excuses are found for exercising this peculiar\ncharacteristic, and that too, without regard to consequences. It would\nseem that the intellect, in thus functioning, was really engaged at\na sort of sensuous play out of which it derived an intense and not\naltogether unselfish pleasure.\n\nOf course, it must be granted that diversity has its specific and\nwithal necessary uses in that it affords a field for the operation\nof human intellectuality and represents the adaptation of the kosmic\nintellect to the human for the purposes of evolution. This adaptation\nwhile necessary for the intellectual development is, however, not an\nend in itself. It is merely a means to a higher purpose. In fact, if\nwe regard materiality as a deposit of life, carried by it as a kind of\nimpedimentum, and consciousness, which _is_ life, as being identical\nwith the intellectuality which makes these adaptations, there should be\nno grounds for the statement that the one is adaptable to the other at\nall. And as this is really the view which we assume it would perhaps\nbe more strict to regard the adaptation as subsisting between the\nhuman intellect and materiality both of which having been constructed\nby kosmic intellectuality. Pursuant to the diversity of uses to which\nmateriality lends itself there arises in the intellect a supreme\ntendency to segment, to break up into separate parts, to multiply and\ndiversify. It is not content unless it is at this favorite and natural\npastime. It delights in taking a whole and dividing it into innumerable\nparts. This it will do again and again; because all its muscles, sinews\nand nerves are molded in that mold and can no more cease in their\ntendency to fragmentation than can the muscles of a dancing mouse\ncease in their circular twirling of the mouse's body. Yet, in this\nit is but creating a well-nigh endless task for itself--which task\nmust be performed to the uttermost. But in its performance, that is,\nin the intellect's complete understanding of the diversity of parts,\nin the knowledge of their relations and inter-relations and in their\nsynthesis, it may arrive at that one ineluctable something which is\ncalled _unity_. And so doing, become ultimately free.\n\nIn view of the foregoing, it is not surprising that the intellect\nshould have, finally, fallen upon the notion of _n_-dimensionality. It\nhas come to that as naturally as it has performed its most common task.\nLeft alone and unhampered in its movements, it has simply followed\nthe lead of the Great Highway through the domain of materiality. And\nnow it has arrived at a stage where it thinks it has succeeded in\nfractionalizing space. Time has long ago yielded to fragmentation,\nbeen divided into minute parts and each part carefully measured.\nSpace, not having a visible indicator like time to denote its passage\nor parts, suffered a long and tedious delay before it could boast\nof a measurer. As the sun-dial measured time in the past and became\nthe forerunner of the modern clock so _n_-dimensionality measures\nspace for the mathematician. What more practical instrument for this\npurpose may yet be devised is not ours to prophesy; yet it is not\nto be despaired of that some one shall find a suitable means for\nthis purpose. Seriously, however, it is not without possibility that\nshould some subtle mind devise an instrument for marking the passage\nof space as we have for denoting the passage of time a great stride\nforward would be accomplished in the evolution of the human intellect.\nFor the general outcome of the intellect's attention being turned to\nthe _passage of space_ would undoubtedly be to recognize not only\nits dynamism but its _becoming-ness_, as a process of kosmogenesis.\nBecause such an instrument would have to be so constructed as to take\nnote of the movement of life, and for this reason, it would have to be\nextremely sensitive necessarily and keyed to the subtleties of vitality\nand not to materiality. Mathematics shall have failed utterly in the\nutilitarian aspects of this phase of its latest diversion if it do not\njustify its claims by crowning its work in the field of hyperspace\nwith a \"Kosmometer,\" an instrument devised for the measurement of\nthe movement of space or a \"Zoometer,\" an instrument devised for\nthe measurement of the passage of life. We should like to encourage\ninventive minds to turn their attention life-ward and space-ward with\nthe end in view of constructing such an instrument. When once we have\nlearned accurately to measure life we shall then be able to dispose of\nit--to _create_. It is not doubted that if ever humanity is to arrive\nat that point in its evolution where it can understand life; if ever it\nis to attain unto the supreme mastery both of vitality and materiality\nand to come to the ultimate attainment of divine consciousness (all\nof which we confidently believe to be in store for humanity) it must\nbe accomplished after this manner: first, by syncretizing materiality\nwith vitality, and then, by intuitionally recognizing the truth of the\nimplications of the syncretism.\n\nThe history of consciousness in the human family is identical with the\nhistory of man's conquest over matter and physical forces. And this\nis clearly indicated in the incidentals contingent upon the toilsome\nrise of the _genus homo_ from the earliest caveman whose status denoted\na comparatively negligible transcendence of material forces, to the\npresent-day man who has gained a markedly notable conquest over these\nforces. Always consciousness seeks the means of adequately expressing\nitself in the sensible world. And to this end it engenders faculties,\norgans and processes in the bodily mechanism, and, in matter, devises\ninstruments of application whereupon and wherewith it may test,\nanalyze, combine and recombine the forces and materials it finds.\nThe unlimited range of expressions lying open to the consciousness\nmakes it necessary continually to devise higher and higher grades of\nappliances to meet its needs as it expands. It will not be gainsaid\nthat the telescope has served actually to lay bare to the consciousness\nan immeasurable realm of knowledge nor that the microscope, turning\nits attention in an opposite direction, marvelously has enlarged and\nenrichened our knowledge of the world about us. And similar declaration\nmay be made anent almost every invention, discovery and conquest which\nman has made over natural phenomena. Thus, by externally applying\nmechanical implements to the subject of his consciousness, man has\nextended actually his consciousness, his sphere of knowledge; has\ngreatly enhanced its quality, and, in the process, has urged the\nintellect to endeavors that have wrought its present unequaled mastery\nof things. Nor have the spiritual aspects of our advance along these\nlines been the least notable. For these have enjoyed the essence of all\nthat has been gained in the process and have, therefore, kept pace with\nthe onward movement of the intellectual consciousness. But heretofore\nno advance has been made as a result of methodic or reflexive\ndeterminations. That is, men did not set out from the beginning,\nequipped with foreknowledge of what their efforts would bring, to\ndevelop the present quality of human consciousness. They simply worked\non, their attention being absorbed by the problems that lay nearest and\ndemanded earliest consideration. So the advance has come as a resultant\nof man's close application to his ever-present needs--shelter,\nclothing, food, protection and other preservative measures--and it\nhas come naturally and inevitably and without prepense. Nevertheless,\nif man, knowing what to expect from the syncretization of matter and\nmind, after this fashion, should set out deliberately to accelerate\nthe intensification, expansion and growth of his consciousness, there\nis no doubt but that the consequence would be most far-reaching and\nsatisfactory.\n\nBut the path that leads to this grand consummation does not lie in the\ndirection of diversity; it lies in the opposite direction. In vain,\nthen, does the intellect fractionalize in the hope that by doing so it\nshall come to the solid substructure of life; in vain does the analyst\nsegment space into any number of parts or orders; in vain does he ask\nhow many and how much; for by answering none of these queries will he\nfind the satisfaction which he vaguely seeks.\n\nIf it be true that it is not by analysis but by synthesis that the true\nnorm of life, and therefore, of reality shall be found it is futile to\nentertain serious hope of finding it in any other way. As a perisophism\nor near-truth, then, _n_-dimensionality takes foremost rank. And this\nis so for the reason that when we proceed in the direction of multiple\ndimensions, that is, one dimension piled upon another dimension or\ninserted between two others we are traveling in a direction which, the\nmore we multiply our dimensions, leads us farther and farther away from\nthe truth. This is a simple truism. If we take, for instance, a wooden\nball and cut it up into four quarters, and divide each one of these\nquarters into eighths, into sixteenths, thirty-seconds, sixty-fourths,\netc., indefinitely, we shall have a multiplicity of parts, each one\nunlike the original ball. But from no examination of the multipartite\nsegments can we derive anything like an adequate conception of the\noriginal ball. Something, of course, can be learned, but not enough\nto enable the rendering of a correct judgment as to the nature, size,\nshape and general appearance of the ball. But this is precisely what\nhappens when the analyst divides space into many dimensions. He cuts\nit up into _n_-dimensional parts and the more minutely he divides it\ninto parts the more remote will each part be in its similarity to the\noriginal shape and form of space, and the farther away from the true\nconception of the nature of space he is led thereby.\n\nNow, _n_-dimensionality or that phase of metageometry which regards\nspace as being divisible into any number of dimensions or systems of\nco\u00f6rdinates is a direct and inevitable product of that tendency of the\nintellect to individuate and to singularize phenomena. Biologically\nspeaking, it is a peculiarity which harks back to the time when life\nwas manifested through the cell-colony and when the individual cells\nbegan, because of increasing consciousness, to detach themselves\nfrom the colony and set out for themselves, and thus each intellect\nrecapitulates in its _modus vivendi_ the salient tendencies of\nphylogenesis. Let it suffice, then, to point out that this universal\ntendency to segment and fragmentate which rigorously characterizes\nintellectual operations upon every phenomenon with which it deals is a\nculmination of the primordial tendency among cells to divide, inasmuch\nas this phase of cell life must be the work of the kosmic intellect.\nThe natural inference is that from the extreme of individualization\nthere shall be a gradual turning, whether of the intellect _per se_\nor of the intellect joined to the intuition does not matter, towards\nthat other extreme of _communalization_. And from this latter shall\ngrow up, as one of the inevitable and ineluctable tendencies of the\nThinker's consciousness a torrentious movement in human society towards\nco\u00f6peration, brotherhood, mutuality and union in everything. So that\nwhereas in the past and at the present time the intellect has been\ndeveloping under the dominant note of individuality it will then be\ncoming gradually under another dominant note--_communality_. The result\nof this development will be the unification of all things, and instead\nof many dimensions of space, many measures of time, and a general\ndiversification of all phenomena, we shall come to the only true notion\nof these things and realize pragmatically the true value and extent of\nunity in the universe.\n\nIt is admitted that the intellect, in treating objects singly and\ndealing only with the starts and stops of a movement, is withal\nloyal to the kosmic order, design and purpose which have priorly\ncharacterized manifested phenomena by segmentation. And in this\nloyalty it has been following merely a natural lead which, while\nadmitting of the widest development and experience, nevertheless at\nthe same time beneficently obscures the underlying reality in order\nthat in its adaptation to the sensuous world the intellect might\nhave the greatest freedom for the development suited to the given\nstage of its evolution. But in thus admitting the natural congruence\nbetween the intellectuality and the phenomenal or sensuous we do\nnot thereby unite with those who already believe that this kosmic\nagreement is the _ne plus ultra_ of psychogenesis. On the other\nhand, it is maintained that this is merely a phase of psychogenesis\nwhich shall be outgrown in just the same measure as other phases\nhave been outgrown. And notwithstanding the fact that judgments of\nthe intellect with respect to inter-factual relations or the ens\nof facts themselves are as valid as its judicial determination of\nself-consciousness, no more and no less, we are, by the very rigor\nand exclusiveness of this logical necessity and inherent limitation,\nled to view the intellect's interpretation of phenomena as partial\nand fragmentary; for the reason that the necessitous confinement of\nits understanding and interpretative powers to fact-relations quite\neffectively inhibits the use of these powers for the contemplation\nof the deeper causative agencies which have operated to produce the\nphenomena. But it is apparent that just as the transmuted results of\nother phases of psychogenesis are now being utilized as a basis for the\nefficient operation of the intellect in the sensuous world, thereby\nenabling the attainment of a very high mastery over matter, so will the\nfunctional dynamism acquired by it in the pursuit and comprehension of\ndiversity serve well when, in later days, it has acquired the power to\ndeal directly with reality, to _create_ and dispose of life just as\nthe kosmic intellect has and is now using it in the execution of the\ninfinite process of _becoming_ through which creation is proceeding. It\nwould seem that the necessary prerequisite to the development of any\nhigher functional capability is that the intellect should be capable of\ndisposing of innumerable details, indeed the totality of kosmic detail,\nbefore it can come wholly into the power and capacity to understand\nand manipulate life. Furthermore, it appears that the acquirement of\nthis power quite necessarily has been delayed awaiting that time when,\ndominated by the intuition, the intellect shall have attained the\nrequisite managerial ability for marshaling an exceedingly large number\nof details.\n\nThe supreme tendency of life is expression. And this expression,\nsingularly enough, reaches its most perfect phenomenalization by means\nof that movement which results in the multiplication of forms. Despite\nthe fact, therefore, that the comprehension of reality involves a\ngradual turning away from the exclusive occupation of organizing a\nmultitude of separate and apparently unrelated facts to a monistic\nview which at once recognizes the unitariness and co-originality of\nall things, of life, mind and form, the intellect will need the\ntraining and development which come from the mastery of diversity. It\nis, then, not difficult to perceive the wise utilitarianism of the\npresent schematism of things as shown in the universal tendency in the\nintellect to devote itself exclusively to parts or segments of truth.\n\nWhenever an individual intellectuality, on account of prolonged\nthought and the consequent inurement of the mind to higher and higher\nvibrations of the kosmic intellect, brings itself to such a high point\nof sensitiveness that it can receive so much as an intimation of some\ngreat truth, it begins to sense, in a more or less vague way, something\nof the substance and general tendence of the underlying reality of that\nwhich foreshadows its appearance. Then, confounded by the multiformal\ncharacteristics of kosmic truth because of the fact that it presents\nitself in such numerous ways and forms, men often are induced to\nattempt the reformation of all facts, or a great mass of kindred\nfacts, in accordance with the newly-found fact or principle. They\nforget evidently that no fact in the universe can be at variance with\nany other fact and still be a fact. So that in the totality of facts\nevery separate and distinct fact must be congruent with every other\nfact forming a beautiful, harmonious and symmetrical whole; but often\nthe whole is made to suffer in the attempt at making it conform to the\nsubstance of a mere intimation. Moreover, it is conceivable that even\nthe totality of facts may lack a rigid conformity with reality in all\nits parts and that having compassed the entire mass of facts one may\nfall short of the understanding of realism.\n\nThis is practically what has happened in the mind of the\nmetageometrician who having received an intimation as to the real\nnature of space as that whose center is everywhere and yet nowhere\nand whose nature is psychological and vital rather than mathematical\nand logical, misses the great outstanding facts and clings to the\nintimations which he experiences as to the nature of space. He,\ntherefore, concludes that the form of space is that of a flexure or\ncurve. There is a valid element in the notion of the curvature of\nspace but not enough of truth wholly to validate the notion. Since\nthe very reality of space is a matter which can be determined only\nby the conformance of the consciousness with it in such a manner\nas to render the conception of it entirely unintelligible to the\nintellect except in so far as it may be able to identify itself with\nthe space-process, there is much room for the serious questioning of\nthe mathematic conclusion upon the grounds of its fragmentariness if\nnot entirely upon the basis of its invalidity. Wherefore it may be\nseen that any search for either the center or the extreme outer limits\nwhich proceeds in a manner conformable to the external indications of\nthe intellectual order is vain, indeed. Although it is undoubtedly\ntrue that the attainment of a central or frontier position in space\ndoes not involve any lineal progression whatsoever, the same being\nattainable, not by progression nor by overcoming distances, but by a\nsubtle adjustment, yea, a sort of attachment of the consciousness to\nthe order of becoming which binds the appearance of space, wherever\none may be, it is nevertheless difficult and painful for the intellect\nto grasp the totality of this truth at one sweep. Indeed, it is not\npossible for it, alone and unaided by the intuition, to grasp it at\nall. Hence, the mathematician who depends entirely upon the deliveries\nof the intellect which conform, in their passage from the conceptual to\nthe written or spoken word, to all the rigors of mathetic requirements,\nfails utterly in perceiving the magnitude of this conception and all\nits connotations; he fails because his prejudices and the woof and warp\nof his intellectual habits prevent his assuming a sympathetic attitude\ntoward it and thereby precluding at the start any calm consideration\nof it. And not only is this true of the mathematician but of all those\nwhose endeavors are confined to the plane of purely sensuous and\nlogical data. It would, therefore, appear that our entire attitude\ntowards things spatial must be changed before we can even begin to\nperceive the reality which is really the object of all researches in\nthis domain. But, on the surface, there is after all little difference\nbetween the ultimate facts involved in these two totally different\nconceptions. Mathematically speaking, all progression eastward would\nterminate at the west, and _vice versa_; and the same would be true\nregardless of the point from which progression might originate. Always\nthe terminus would be the opposite of the starting point. Then, too,\nit might be said that if we sought the space-center we should arrive\nat the circumference. The difficulty with this view is that there is\na very remote, though important, connection between it and the truth\nof the matter. But the partiality of this view, and the absence of\neither experience or intuition to intimate a more reasonable view,\nserve effectively to buttress it as a hypothesis acceptable to many.\nThus it is ever more difficult to supplant a near-truth than it is to\ngain credence for the whole truth. On the other hand, according to the\nview which we maintain here, it is quite true that the seeking of the\nkosmic space-center will reveal the circumference; that the search for\nthe nadir will uncover the zenith; the east effloresces as the west,\nand a northward journey will wind up at the south, etc., but in quite a\ndifferent manner from that which the mathematician has in mind when he\npostulates the curvature of space. Our view involves no space curvature\nnor any other spatial distortion. _It deals with space as reality, as\na dynamic process, a flux which, like the sea, is continually casting\nitself upon the shores of chaos and falling back upon itself only to\nbe recast against the rock-bound coast of its chaotic limits._ Now,\nthat which falls back upon itself and rolls in a recurrent movement\nupon its own surface is _life_ which, in its recession is the natural\nand kosmic limitations of itself, generates matter in all its varied\nexpressions. Space, in its extensity, cannot transcend life; for it is\nthe path which life makes in its _out-coming_, its manifestation. Of\nthe chaotic fringe which circumscribes the manifested universe it is\nabsurd to say that it is vital or psychological in any sense of these\nterms. For notwithstanding the fact that out of its very substance are\nengendered life, intellectuality, spatiality and materiality, it is\nnevertheless none of these in its primary essence. It is Chaos-Kosmos;\nbecause from its content the kosmos is evolved, and it still remains;\nit is chaos-spatiality; chaos-materiality; chaos-intellectuality;\nchaos-geometricity; because these are engendered by the movement of\nlife in chaos while at the same time there remains a residuum of the\nchaogenetic substance which constitutes the limitations of all these\nsubsequent processes. In this sense, the chaogenetic fringe becomes\nthe limits of the manifested universe so that it would appear that all\nthose major processes outlined above are finite manifestations of the\neternal chaos. But none of those possibilities of motion which are\nfound in these major movements of the kosmos can be logically said to\nexist in chaos. It is the embodiment of everything that is the opposite\nof those qualities which may be found in them, that is, in materiality,\nvitality, spatiality, intellectuality and geometricity.\n\nApropos to this phase of the discussion let us examine briefly one\nof its most significant implications, both mathematical and kosmic,\nwhich arises out of the fact that space is an engendered product of\nlife that is bound by the fringe of chaos which sustains and limits\nit. The chaotic fringe plus manifesting kosmos constitute the absolute\nmagnitude of the kosmos. The manifestation factor is complemented by\nthe chaos factor and together the two define the _full_ universe.\nKosmogony is the universal movement of all kosmic elements or factors\nin diminishing the chaotic complement and reducing it to kosmic order\nor geometrism. It is undoubtedly impossible to determine mathematically\nthe exact volume of either complement or the ratio of the one to the\nother; yet it is conceivable that the chaotic fringe is greater in\nextent than the ordered portion of the kosmic uni-circle or universe.\nIt is even conceivable that the difference, upon the basis of the\nmeaning of the Pythagorean Tetragrammaton and the view outlined in the\nChapter on the \"Mystery of Space,\" is as seven to three wherefrom the\nconclusion might be drawn that the universe has yet seven complete\nstages more or less of evolution before the close of the Great Cycle of\nManifestation when the fringe of chaos shall have been totally used up\nin the work of creation. But for those who may experience impatience at\nthe infinitude of the process when viewed in this light the terms may\nbe reversed and the difference may be conceived as the ratio of three\nto seven wherefrom the conclusion would follow that the kosmogonic\nprocess is seven-tenths complete, as it will not vary the seeming\ninfinitude either way it may be determined. The notion, despite its\nspeculative character, offers an explanation of otherwise inexplicable\nconditions, and, on account of its profound connotations, may even\nbe found to be productive of the highest good in its equilibrating\ninfluence upon our mode of thinking.\n\nIn any event, there does appear to be a subtle relation subsisting\nbetween true numbers and kosmogony. Number is a phase of the kosmogonic\nmovement, a measurer of the intellect and the establisher of the\ngeometrism of space, answering tentatively to the numericity of pure\nbeing. In fact, being actually expresses number and number itself is an\nevolution and not a thing posited once for all as a pure, invariable\nform in the universe. It is, like the kosmos, in a state of becoming\nand there may yet appear to our cognitive powers a whole series of new\nnumbers pure in itself and altogether conformable to the conditions\nreigning at the time.\n\nThe symbology of the circle, in all times recognized as the true symbol\nof the kosmos in eternity, of eternity itself, of the archetypal, of\nspace, duration and Ultimate Perfection, is replete with profound\nsignifications. But it should be understood that the circle is a\nsymbol of the _perfected universe_ and not the universe in a state\nof evolution. It symbolizes perfection, completion and the ultimate\nunion of the manifesting with the archetypal which results in the\ncrowning deed of Perfection. The circle is, therefore, not a symbol\nof the universe as it now stands; it does not represent a snapshot\nview of the kosmos but the universe as a _full_. It cannot be a _full_\nuntil it has attained the _ne plus ultra_ of completion; for a kosmic\nfull is that state to be attained by the manifested kosmos upon the\ntermination of all the fundamental processes now in operation. But it\nis this state that the circle really represents, and by virtue of which\nit possesses its intrinsic qualities and also in virtue of which the\nintellect recognizes these qualities. The properties of understanding\nand recognizance in the intellect are veritably fixed by the _status\nquo_ of the universe during every stage. That is, the focus of the\nintellect, like the focus of a chromatic lens, is adjusted by the fiat\nof the nature and eternal fitness of things to correspond exactly with\nevery state through which the kosmos itself passes. This is one of the\nobvious implications of the phanerobiogenic behavior of the kosmos\nand is necessarily resident in the notion of the genesis of space and\nintellectuality as consubstantial and co\u00f6rdinate factors.\n\n_Wherefore the more cogent is the reason for the belief that the\ninherent qualities of the kosmogonic fundamentals; as, vitality,\nmateriality, spatiality, intellectuality and geometricity, are true\nvariants, and that their variability is proportional to the progress of\nthese major movements toward the ultimate satisfaction of the original\ncreative impulse._ May it not be, therefore, that the indeterminate\ncharacter of the ratio of the diameter to the circumference\n(3.1415926...), is due to causes far more profound than the crudity of\nour micrometers or the mere supposed fact of the circle's peculiarity?\nMay it not also be true that the _pi proportion_ shall become a whole\nnumber, and in its integration, keep apace with the perfecting process\nof the kosmos, diminishing, by retrogression to one or increasing, by\nprogression, to ten which, after all, is essentially unity, being the\nperfect numeral? It is not without the utmost assurance that these\nqueries will be categorically questioned by the orthodox, creed-loyal,\nstrictly intellectual type that we sketch these implications, but it is\nfelt to be an urgent duty to remind all such that the most effective\nbarrier to realization in the field of philosophy is an intolerant\nattitude towards all lines of thought which suggest the impermanence\nof conditions as we find them in the kosmos at the present time. The\nfact is that our lives are so distressingly short that we have neither\ntime nor opportunity to watch the changing moods of the kosmos nor\ndiscern the gradual reduction of mere appearance to the firm basis of\nreality, and accordingly, the intellect tenaciously clings to those\nnotions which it derives from the instant-exposure which the lens of\nintellectual conceivability allots to it. Once the view is taken it\nis immediately invested with everlastingness. This everlastingness is\nthen imputed to the kosmos in that particular pose, attitude or state.\nAlways the intellect beholds in that passing view, snatched from the\nfleeting panorama of eternal duration, a picture of itself which it\nmistakes for the reality of the not-self.\n\nThe inclination of the axis of the earth toward the plane of its orbit\nis approximately twenty-three and one-half degrees. No well-informed\nastronomer, however, doubts now the fact that this ecliptic angle\nis being gradually lessened; because, as a result of centuries of\nobservation, it has been found to be decreasing at the rate of about\n46.3 seconds per century. Yet no intellect is able to perceive in\nany given lifetime the actual decrement of this angle. It is only by\ncareful measurements after centuries of waiting that a difference can\nbe discovered at all. Thus it may even be so with the ratio of the\ndiameter to the circumference of a circle, the only difference being\nthat it has not yet been determined whether there is a _decrement_ or\nan _increase_ in the size of the ratio.\n\nThe _pi_ proportion is, then, a register or measurer of the slow,\nmeasured approach of the manifesting kosmos to the standard of ultimate\nperfection. Therefore, and in view of these considerations, we may not\nhesitate to confirm our belief in the validity of the notion that it\nactually and literally expresses the key to the evolutionary status\nof kosmogony. The mathematical determination which limits it as an\nunchangeable, inelastic quantity is, consequently, only partially true\nand leads to the inclusion of this quantity under the category of\nmathematical near-truths, for such it appears to be in spite of its\nrigorous establishment.\n\nThe formal topography into which the intellect spreads when seeking the\nideal and the abstract is not a condition which is derivable from the\nreal essence of life or matter, but, on the other hand, is a product\nof the intellect itself partaking of its nature rather than of the\nnature of reality. There is, therefore, a very important distinction to\nbe made between all deliveries of the intellect and the realism both\nof the objects and conditions to which the intellectual deliveries\npertain. One of the most marked peculiarities of the human intellect is\nthe fact that it always unavoidably stamps its own nature and features\nupon every datum which passes through it to the consciousness. The\nutmost importance attaches to this phenomenon, for the reason that\nit points to the necessity of carefully scrutinizing intellectual\ndeliveries and the making of allowances for those ever-present\ncharacteristics which the intellect superimposes upon its data. Perhaps\nthe inherent colorific quality which it imposes upon our knowledge\nwould be better understood if a similitude were indulged at this\njuncture. The intellect may be likened to a color-bearing instrument\nwhich, when it has once handled an object, leaves forever its own color\ntransfused into every cell and fiber of the object so that when the\nsame object is presented to the consciousness for purposes of cognition\nit bears always the same peculiar marks and colorations which the\nintellect, in its manipulation of it, places thereon. In this respect\nthe intellect may also be said to be like a potter who has but one mold\nand that of a peculiar formation. Hence, whatever wares it presents to\nthe consciousness will invariably be found to be molded in conformity\nwith that particular mold. If it were possible to view reality or the\nessential nature of things the difficulty which now the intellect\nlays in the path of direct and uncolored cognition would be obviated;\nfor then there no longer would be any necessity of viewing things as\nthey are colored or molded by the intellect. The intuition, being a\nprocess of pure consciousness, will, when it has arisen to a position\nwhere it may dominate the intellect as the intellect now dominates\nit, so modify this tendency which we see so ineradicably bound up in\nthe very nature of the intellect that the apparently insurmountable\ndifficulties which it has interposed between mere perception and a\ndirect cognitive operation will be quite completely overcome. Thus, in\nthe above, is discovered another obstacle which posits itself between\nthe notion of space as reality and the intellectual determination of\nit which the mathematician examines and to which his consciousness\nis necessarily limited. Furthermore, it may be perceived also how\neasily the mind may be deluded into thinking that the intellectual\nnotion which it entertains of space is necessarily correct, when\nobversely, it is simply examining a concept which has been remade by\nthe intellect into a form which is not at all unlike its own peculiar\nnature, and therefore, as much short of reality as the intellect\nitself is. Similarly, if the mathematical mind succeed in catching a\nglimpse of the reality of space in the form of an intimation, which,\nin itself though fragmentary, is nevertheless true, its consciousness\nis finally deprived of the true validity thereof simply because of the\nbehavior of the intellect in its manipulation of it. The importance\nof these intellectual difficulties cannot be over-estimated for they\nfurnish the grounds for the ineptitude of intellectual determinations\nmade in a sphere of motility to which the intellect is a stranger.\nAnd this fact will appear more evident when it is perceived that\nquite the entire content of human knowledge has been thoroughly\nvitiated by them. So that only in those very rare moments which (in a\nhighly sensitive mentality) enable the intuition to gain a momentary\nascendancy over the intellect is it possible for the Thinker to catch\nhold of realism itself, and project the truth of what he sees into the\nlower, intellectual consciousness. But so small is that portion of our\nknowledge which owes its origin to the intuition that when compared\nwith the totality of that which we seem to understand it is well-nigh\nnegligible. And then, when it is considered that at present there is\nno way of conceptualizing adequately the intuitograph so as to make it\npropagable the insignificance of this form of knowledge is even more\nnotable. It can now be seen in how large a measure the notion of the\ncurvature of space is merely an intellectual translation of a true\nintuition into the terms of the intellect which, in the very nature\nof the case, can only approximate the truth because of its colorific\nhabits.\n\nA similar declaration may be made of that other datum of\nmetageometrical knowledge which postulates the ultimate convergence\nof parallel lines. In fact, what has been said as to the perisophical\nnature of the notion of space-curvature will apply with equal force\nto the idea of parallel convergence since the latter is a derivative\nof the former. But there is yet another consideration, apart from the\ncolorific influence of the intellect, which, although it partakes of\nthe nature of this quality, is nevertheless a near-truth of quite\na different order. This may be better understood by referring to\nthe _graph_ showing the inverse ratio of objective space to the\nconsciousness.[28] Let us suppose that the _graph_ may also represent\nthe Thinker's outlook into the world of spatiality. It then appears\nthat, because of that movement of consciousness in its pursuit of\nlife which, as it expands, makes the objective world to appear to be\ndiminished in proportion to the extent of its expansion, it is quite\nnatural, under such circumstances, that parallel lines drawn anywhere\nin the limits of the objective world should seem to come to a point in\nthe ultimate extension of themselves. While this _graph_ is not meant\nto depict such a view, it may be found nevertheless, to be a true\ndelineation of the topography of that state of mind into which the\nmetageometrician brings himself when he visualizes space as _curved_;\nfor there is no doubt but that a state of intellectual ecstasy, such\nas that in which the mind of the metageometrician must be functioning\nin order to perceive space in that form, is quite different from the\nnormal and, therefore, in need of a different topographical survey.\nBut, if we grant that in the creational aspects of space there is\nconceivable an ever-present tendency to convolution, or a rolling back\nupon itself, it is imaginable that parallel lines inscribed either upon\nits surface or in its texture need not necessarily meet but maintain\ntheir parallelism regardless of the complexity of the convolutions. The\nconvergence of parallel lines is much like a tangent in the outgrowth\nof the idea from the notion of space-curvature. The more a tangential\nline is extended the farther away from the circumference it becomes\nand consequently less in agreement therewith. The more subsidiary\npropositions or corollaries are multiplied the more remote from the\ntruth the determinations become and especially is this true of the\nhypothesis of space curvature.\n\n  [28] Figure 20.\n\nIn the notion of the manifoldness of space, by virtue of which it is\nconceived as existing in a series of superimposable and generable\nmanifolds of varying degrees of complexity, are discernible traces\nof that intuitional intimation which underlies the assertion that\nbecause of the necessary phenomenalization of reality for the purpose\nof manifestation to the intellect it appears to exist in a series of\nseparate degrees, each one more refined and subtle than the preceding\none and requiring a more highly developed species of consciousness for\nits comprehension. In other words, that intuitional glimpse of the\nessential character of reality, as viewed by the human consciousness,\nwhich impinged upon the minds of RIEMANN and BELTRAMI leading them\nto postulate as a corollary proposition to space-curvature, its\nmanifoldness, is nothing more nor less than the intuition that the\nuniversum of spatiality cannot otherwise present itself to the\nintellect, owing to its peculiar adaptation to the sensuous, except by\na series of continuous degrees which are perceptible only in proportion\nas the understanding is magnified to conform with it. After all,\nhowever, it is not improbable that the very objectivism of the universe\nin manifestation subsists in just the manner in which this intuitive\nglimpse implies and that the wisdom and utilitariness of the kosmogonic\nprocess which engendered spatiality are clearly demonstrated in that\narrangement of the contents of the kosmos which presents the grossest\nelements of phenomena first to the intellect in its most impotent\nstate while reserving the less crass for that time when the Thinker\nshall have evolved a cognitive organ adaptable to its presentations.\nThose metageometricians who cling to the idea of the manifoldness of\nspace, based as we have shown upon the pseudo-interpretation of a\nrather vague hint arising out of an unquestionably true intuition, have\nallowed themselves to fall into the unconscious error of magnifying\nthe importance of the mere insinuation as to the space-nature to such\nan extent as wholly to obscure in their own minds and in the minds of\nthose who think after them whatever of the true vision that may have\nbeen grasped by them. Furthermore, it is indubitably true that that\nsame peculiarity of arrangement by which impalpable and invisible\nforces really subtend gross matter producing that subtle schematism in\nvirtue of which the visible is subjoined to the invisible, the sensuous\nto the non-sensuous, spirit to matter, etc., also characterizes the\nappearance of spatiality to the human understanding. While there is a\nsuperficial semblance of separate and discrete manifolds into which\nspace may be divided there are, in reality, no such sharp lines of\ndemarkation between the subtle and the gross, between the visible\nand the invisible or between spirit and matter, each of these being\ncapable of reduction, by insensible degrees, into the other regardless\nas to whether the reductional process originates on the side of the\nmost refined or on that of the grossest. Accordingly, there are no\nreasonable grounds upon which the notion of a space-manifold may be\njustified except as a metageometrical near-truth.\n\nIn addition to the foregoing, there are yet other very fundamental\nconsiderations which would seem to debar the totality of analytical\nconclusions as to the nature of space from any claim to ultimate\nreliability and trustworthiness. These are _first_: the fact that\nanalyses are absolutely incapable of dealing with life; that being\nthe direct product of a sort of mechanical consistency which marks\nthe intellectual operations it has adaptability only for dealing\nwith fragments or disconnected parts, and that without any reference\nwhatsoever to the current of life or the flow of reality which has\nproduced the parts. This fact is clearly shown in that attitude of the\nunderstanding which inevitably leads it to the declaration that a line\nis an infinite series of points, a plane an infinite series of lines,\nand a cube, an infinite series of planes, and so on, indefinitely.\nTo do this, to look upon all phenomena as a series of parts similar\nto each other and piled, one upon the other, or juxtaposed in the\nmanner which they are discovered in the sensible world, is the natural\ntendency of the intellect and this tendency finds its most facile\nexpression in analytics. Inadaptability of this sort is especially\nobservable in all problems of arithmetical analysis in which the vital\nelement is a factor. When these analyses are carried to their logical\nconclusion, as has been shown in the chapter on \"The Fourth Dimension,\"\ninvariably they end in an evident absurdity. But it is at their very\nconclusion where the life-element is encountered, where reality is\napproached, that they break down. The failure of analysis, then, to\nencompass life, to fit into its requirements and to satisfy its natural\noutcome seems clearly to establish the basis of the perisophical\nnature of the entirety of analytical claims, especially that species\nof analysis which seeks the remoter fields of the conceptual for its\ndeterminations. _Second_: the close connection which has been seen to\nsubsist between space and life as joint products of the same movement\nmakes it obvious that the same ultimate rule of interpretation must be\napplied to both in order to insure correct and dependable judgments\nregarding them. How different would be the intellectual attitude\ntowards space if it were considered in the same light as vitality,\nprovided one really understood anything about vitality! Moreover, as\nit appears certain that the path of the intellect does not run in\nthe same direction as the path which life makes, but in an inverse\ndirection, it is clear that the judgments of the former, as to the\naction and essence of the latter, must necessarily be ultimately\nunreliable. It can readily be seen, however, that should the intellect\nbe focused so as to follow the path of life, to attach itself to the\nvery stream of life, it would have necessarily to neglect materiality.\nAnd such an adjustment would, of course, obviate the need of a\nmaterial life at all for humanity. In fact, a physical life with an\nintellect would be impossible under such conditions. It is well to\nrecognize the suitability of the present schematism and not to become\nunwisely restive because of it; but it is also fitting that we should\ndiscriminate between that which is possible for the intellect chained\nto materiality and that which is impossible for it, in such a state,\nwhen foraying in a territory foreign to its nature, and beyond its\npowers to master.\n\nThe predominating tendency in the intellect to account for the\nuniverse of life, mind and matter upon a strictly mechanical basis\nis undoubtedly due to the constitution of the intellect which does\nnot admit of its direct consideration of the vital essence of things.\nWe are bound ineluctably to the surface of things. All our knowledge\nis therefore superficial. We are even bound to the surface of ideas,\nand cannot penetrate to the interior of these realities. Our art\nis the reproductions of superfices; our philosophies are the husks\nof eternalities; our religions, the habiliments of relations, and\nwhile it cannot be doubted that this arrangement is pre-eminently\nthe best possible one for the present stage of man's evolution, it\nis nevertheless worth while to note that it is this very restricted\nactivity of the intellect which shuts out from man's consciousness\nthose very elements about which he is most concerned when he goes into\nthe field of philosophy in search of a solution to his unanswerable\nqueries. But some progress most surely is made when the mind is\nenabled to see its plight and recognize what are the difficulties and\nlimitations that lie in its path of ultimate attainment.\n\nIt is believed that the mechanistic, or true, character of the\nintellect reached its zenith in the mind of LAGRANGE when he succeeded\nin reducing the entirety of physics to certain mechanical laws and\nformul\u00e6 which he embodied in his \"_Mecanique Analytique_\" This work\nis undoubtedly the capstone of intellectual endeavors and stands\nas a monument which marks the culmination of the present stage of\nintellectual development. In thus placing the _Mecanique_ at the apex\nof intellectual endeavors it is not thereby meant to be implied that\nthe intellect shall not make more progress nor that other formul\u00e6,\nequally as marvelous as those which LAGRANGE discovered, may be\ndevised, nor that other laws, heretofore undreamed of, may be found;\nbut what is maintained is the fact that while there will be growth and\ndevelopment these will run along other channels, perhaps in the realm\nof the intuitable, and not any longer, especially so notably as now,\nin an opposite direction against the current of life and reality; and\nfurther, that there will be a gradual turning away from mechanics to\nbiogenetics, from diversity to unity, from the purely intellectual\nto the intuitional, and withal a final getting rid of the bonds of\nillusion, of that thralldom of mechanics, whereupon will slowly arise\nthe obsolescence of all those disparities which may now be recognized\nin our knowledge and in the applications of the intellect to the data\nof the objective world.\n\nBecause the intellect is unsuited to deal with reality, and because of\nits peculiar adaptation for diversity, for multiplicity, due to its\nmechanistic _modus vivendi_, there has grown up a voluminous catalogue\nof systems of philosophy. These embody such a multitudinous array of\nbeliefs, ideas, conceptions, theories and conjectures and constitute\na movement in human thought which oscillates between the empiricist\non the one hand and the transcendentalist on the other; between the\nidealist and the realist, leaning sometimes towards the Platonic,\nthe Cartesian and the Kantian and at other times towards SPINOZA,\nARISTOTLE, SPENCER and SOCRATES, always terminating by multiplying the\nnumber of diverse beliefs rather than unifying them that the conclusion\nis unavoidable that so marked a lack of unanimity is indicative of\na profound mental prestriction. It was, therefore, inevitable that\nmathematics should fall under the same spell and brook no let nor\nhindrance until it had succeeded in devising several diverse systems\nof geometry which it has done for the mere joy of doing something, of\nfollowing its instinctive aptitudes. There is no other basis for the\nheterogeneity of our philosophies, our mathematics, indeed our beliefs\nthan this mechanical, and hence, radically illusionary character of the\nintellect in consequence of which we have had to be satisfied with mere\nglimpses, hints, intimations and faint glimmerings of reality, of life,\nand of those kosmic movements which, if we had the ability to trace\nthem from their source outward, would lead us unerringly to a truer and\ndeeper knowledge of those things that under the present schematism must\nremain for us a closed book.\n\nThe criterion of truth for us, constituted as we are and wedged\nin between the stream of life and its shore of materiality, must\nbe that which relates our knowledge both to the stream and to the\nshore. It must be so that all predicates which purport to approach\nit shall exhibit a dual reference--one that relates to materiality\nand another that relates to vitality, and yet a third that shall\ncombine these two relations into one. All assertions, therefore, which\npertain exclusively to either of these elements--to materiality or to\nlife--are necessarily partial, fragmentary and perisophical in nature.\nMathematics, because it relates to matter and the mechanical forces set\nup by matter acting against matter cannot be said to agree with such\na criterion; art, because it relates to snapshots or static views of\nmatter is even more remote in its agreement; philosophy, as it has been\nknown in the past and is known to-day, because it seeks to deal with\na vitality fashioned after the image of materiality has failed when\nposited alongside of this criterion; and thus, the intellectual toil of\nmillions of years has been in vain in so far as it has not succeeded\neven in raising a corner of the cover which hides reality from our view.\n\nA near-truth is any variation from this standard, this norm or\ncriterion. It may be either logical, cognitive, scientific or even\nmetaphysical. To define: a logical truth is a predicate based upon\nand involving the coherency and consistency of thoughts themselves; a\ncognitive truth is the conformity of knowledge with so much of reality\nas is known; scientific truth is the conformity of thoughts to things\nand conditions. All of these are obviously near-truths. Then, too, a\nnear-truth may be defined as an assertion based upon the criterion of\ntruth but falling within the category of cognitive truths owing to\ninsufficiency of data or vision. Such indeed are those metageometrical\npredicates--_n_-dimensionality, space-flexure, space-manifoldness and\nall other assertions based upon these in general and specifically. Any\nrecognition of truth must clearly embrace both the vital and material\naspects of its subject in order to be adequately inclusive, that is, it\nshould include the causative, the sustentative, the relational and the\ndevelopmental factors. These four factors are considered necessary and\nsufficient to determine the conformity of any view to the criterion of\ntruth for when we are cognizant of the cause of a subject, understand\nthe sustentative factors which keep it in existence, are conversant\nwith its relations to other subjects and can follow its developmental\nvariations until we come to its final status, why then, our knowledge\nis both sufficient and ultimate so far as that subject is concerned.\nIs it asking too much of mathematics or of philosophy or any system of\nthought that it conform to these standards or to this criterion before\nwe shall accept it as final? Or shall we be satisfied with less than\nthis? Let us hope not.\n\nIn the foregoing presentation stress has been placed upon the\nfragmentary, and therefore, illusionary character of the intellect in\norder to arrive at an understanding of the difficulties under which\nreal knowledge has to be acquired and to indicate the inanity of all\nattempts to resolve the riddle of space by means of mathematics though\nregarded as the most typical exemplification of the mechanistic nature\nof the intellect. And further, to show that, on account of the radical\nincongruity which estranges life, the producer of spatiality, from the\nintellect which returns again to scrutinize the passage of life in\nits outward expressions, no hope of ever gaining the true viewpoint\nby means of the intellect need be entertained. But in doing so, it is\ndeemed fitting that a note of warning should be sounded against any\nabortive attempts that may be made to obscure or distort the results of\nsuch a close discrimination lest the true import of the examination be\nlost for, if we emphasize the vanity of the intellect in the pursuit of\nthat which it is by nature unsuited to attain we also equally stress\nthe wise utilitarianism which limits it to the performance of the\ntasks assigned while at the same time reserving for the function of\nmore highly evolved powers, and indeed, for the intuition itself, the\nsolution of the riddle of spatiality. And if we declare the futility\nof the mathematical method in all endeavors aimed at unveiling the\nmysteries of life and mind, and of that movement which has its roots\nset in eternal duration from which it proceeds in an endless continuity\nof purpose and promise, we do also recognize that in the science of\nmathematics the intellect shall, as in no other method of cognition,\nmost fully fulfill the kosmic intent of its existence; and moreover, in\nthe pursuit thereof it shall push the frontiers of its possibilities\noutward until it can be said almost to be able to make disposition of\nlife itself--at least to that point where, when the intuition shall\nhave come into its own, the passage from the mechanics of matter to the\ndynamics of life, shall be comparatively easy and natural.\n\n\n\n\n  CHAPTER X\n\n  THE MEDIA OF NEW PERCEPTIVE FACULTIES\n\n  The Spiritualization of Matter the End of Evolution--Sequence\n  and Design in the Evolution of Human Faculties--The Upspringing\n  Intuition--Evidences of Supernormal Powers of Perception and\n  the Possibility of Attainment--The Influence and Place of the\n  Pituitary Body and the Pineal Gland in the Evolution of Additional\n  Faculties--The Skeptical Attitude of Empirical Science and\n  the Need for a More Liberal Posture--The General Results of\n  Pituitarial Awakening Upon Man and the Theory of Knowledge.\n\n\nEvolution is a continuous process and the primal impetus back of the\ngreat on-flowing ocean of life acts infinitively. It is not terminated\nwhen life has succeeded in perfecting a form for the perfection of\nforms in themselves is not the end of vital activity. _The end of\nevolution is the complete spiritualization of matter._ So that it does\nnot matter how perfect a form may be either subjectively or in its\nadaptation to environments; it does not matter how faultless a medium\nfor the ensouling life it may be, there is ever the eternal necessity\nthat life must drive it back over the path of its genesis until it\nshall be transmuted into pure spirit. Adaptation succeeds adaptation\nand with each there is a change in the form and this process continues\nuntil there is a more or less perfect congruence between form and\njuxtaposed environmental conditions. But no sooner than agreement\nhas been attained under one set of conditions new conditions arise\nand require a new setting, new adaptative movements. Thus there is a\ncontinuous proceeding from stage to stage, going from the grossest to\nthe subtlest and most refined, always the form is being pushed onward\nand upward by life. But adaptation is not undergone for the benefit\nof the form, but more truly for the informing principle. It is the\nprogression of the life-element which constitutes the adaptation of\nform to form and to their peculiar environs. The form is a tool or\ninstrument of life which it discards the moment it fails to respond\nto its requirements. Thus forms are constantly being assumed and as\nconstantly being relinquished. But no effort of life is lost regardless\nas to whether the action is performed in one or another form. The\ntotality of matter is perpetually being acted upon by the totality of\nlife. Every appulse of life against matter means an added push in the\ndirection of spiritualization. The totality of such appulses of life\nagainst matter may seem infinitely small in the visible results which\nthey produce in the process of spiritualization; but with each there\nis an eternal gain in that movement that shall end in the complete\ntransmutation of materiality into spirituality. This action of life in\nmetamorphosing matter, the nether pole of the great pair of opposites,\ninto spirituality, its copolar factor, in its outward, visible effects,\nis what we vaguely call evolution. And such it is; for life is merely\nunfolding that which it has enfolded. Matter, having been involved as a\nphase of kosmic involution, is now being evolved.\n\nIn the genesis of the kosmos there appear to be three great undulations\nin the universal current of life. The first of these prepares the\nfield by depositing that elemental essence which is to become the\nworld-plasm; the second precipitates the universum of materiality,\nspatiality and intellectuality, not as we now know them, of course,\nbut as potencies; the third great undulation in the current of life\neffects the endowment of the world-plasm with those tendencies that\nare to build around themselves forms appropriate to their fulfillment.\nThis ensoulment of the world-plasm with tendencies and the consequent\nsegmentation of it into separate forms by these tendencies constitute\nthe primary stages of that procedure of life which results ultimately\nin the up-raisement of matter and its final exaltation into pure\nspirit. Hence, the entire mass of materiality is besieged on all sides\nby the sum-total of life and the former is being raised slowly and\nirresistibly to heights that are immeasurably more sublime than its\npresent degree of grossness.\n\nIt appears paradoxical, therefore, that life, although in all respects\nvastly superior to matter, should become the apparent vassal of\nmateriality and give itself up to all the strict rules of imprisonment\nwhich are imposed upon it by the properties and qualities which we\nobserve in matter. It seems so subject to every whim and fancy of\nmatter that one is inclined to think that matter and not life is the\nchief designer of universal destiny. This is not a condition to be\nwondered at so much, for the reason that this apparent vassalage, this\nseeming enslavement of life by matter, is due to that superior and most\nmarvelous adaptability of life which it enjoys in contradistinction to\nthe relative unpliability of matter, and due also to the fact that life\nis kinetic and matter, being a mere deposit of life, is static. Life is\nmobility while matter is immobility and thus in possessing a greater\nrange of freedom is, of course, correspondingly superior; but in this\nadaptation of itself to the labyrinthine cavities and multiformed\ninterstices in matter it exhibits but a seeming serfdom which is really\nnot a serfdom but a mastery. It is as if a man had taken lumber,\nhardware and stone and built a house wherein he might dwell--life has\nmerely used matter, molded and fashioned it so as to make for itself\na medium, a dwelling-place wherein it operates, not as a slave but as\na master possessing unlimited freedom of motility. In the production\nof a form life stamps upon it, once for all time, the path of its\nengendering action. It leaves its finger-prints upon the mold which\nit makes for itself. So that if we would know where life _has been_\nor where it _is_ we should look for its finger-prints (organization);\nwe should observe the sinuosities which mark its pathway, remembering\nalways that it is life that has formed the intricacies and complexities\nof the form into which it pours itself so accommodatingly in order that\nit may raise that form, develop and transmute it into something higher\nand better.\n\nWhen we speak of _form_ it must not be understood thereby that\nreference is made only to the gross physical form, but to the entire\nrange of vital assumptions or vehicles which life ensouls for purposes\nof manifestation. This range we believe to cover the whole path of\nkosmogenesis seriating from the densest to the most subtle. Our chief\nconcern, however, is the immediate effect which the totality of life's\noperations will have upon humanity or the form which it ensouls as\nthe human organism. For it is impossible that humanity shall escape\neither the general or the specific results of the exalting power which\nlife exerts over materiality and its appurtenances. It is, of course,\nimpossible here to go into the various implications of this general\nforward movement of the universum of materiality or even to outline\nbriefly the divergent lines of operation into which a satisfactory\nexposition of this view would naturally lead. And then to do so would\nbe inappropriate in a volume of this kind. So we shall have to be\ncontent at this juncture to limit our study to a consideration of what\nwe believe to be some of the immediate indications of this vast and\nmost far-reaching phenomenon.\n\nIn the chapter on the \"Genesis and Nature of Space\" it is shown that\nthe material universe is engendered at the same time and by the same\nmovement or process as the universum of spatiality and intellectuality\nand that as the passage from chaos to kosmos proceeds the function of\nthis movement is changed gradually from engenderment to exaltation\nwherein materiality is transmuted into spirituality. It is, of\ncourse, obvious that as materiality is exalted so are spatiality and\nintellectuality; and that as the one becomes more and more refined,\ncapable of answering to higher and yet higher requirements so do\nall the others. For, at work in all and through all of these, is\nthe current of life which pervades them, engendering, sustaining\nand elevating as it proceeds. So that as matter has evolved added\ncharacteristics and properties, each answering to a given need\nand arising out of the necessities inhering in the stage at which\nit appeared, so has the intellect evolved faculties to correspond\ntherewith. In other words, the evolution of faculties for the\nexpression of the human intellect has proceeded synchronously with\nthe evolution of material qualities. And whenever a new faculty or an\nadditional scope of motility is achieved by humanity there is always\nfound a set of kosmic conditions which answers thereto. The cardinal\nprinciples of the doctrine of evolution are not, therefore, adverse to\nthe conclusion that the organs of sense-perception--hearing, touch,\nsight, taste and smell--have not been endowed upon the human race or\nattained by it at one time; but rather that each answering to a newly\nacquired need and opening a wider scope of motility for the intellect\nhas been evolved separately and in due order. It would also seem that\nthe quality of consciousness, as it has been manifested in the various\nstages of life through which it has passed, and especially the mineral,\nvegetal and lower animal, has not always been of the same degree of\nefficiency. Nor has it enjoyed the same kind of freedom which it now\nenjoys in the highly evolved _genus homo_. It is equally apparent that\nmatter itself has not always been in possession of the same qualities\nand characteristics which it now exhibits; but that it, too, has\ngone through various stages of evolution bringing forward into each\nnew stage the transmuted results of each preceding one as a basis\nfor further evolution and expansion. The innumerable arch\u00e6ological\nevidences which support this view make it unnecessary to do more than\nstate the facts, as they appear to be substantiated by indubitable\ntestimonies. Furthermore, it is believed that the outstanding\nimplications of these phenomena will not be successfully controverted\nby those who are disinclined to see such implications in the\nevolutionary process. In a previous chapter we have briefly sketched\nthe characteristics which mark the upspringing of a new faculty showing\nhow, at first, it appears as an abnormality which exhibits itself in\na very few individuals only, and that in a more or less indefinite\nmanner; and how later the number of individuals in which it appears\ngradually increases, the definiteness of the faculty, at the same time,\nappearing more marked; then, like a tidal wave, it recurs in a still\nlarger number of persons until, at last after a long period of time\nusually several thousands of years, it becomes universal exhibiting\nitself in every individual and appearing as a hereditary characteristic\nof the entire human race. It is, therefore, not without assurance as to\nthe ultimate soundness of this view that we make the assertions which\nfollow this brief introduction.\n\nIt has already been stated that for a very obvious reason, namely,\nthe satisfaction of the needs of our present humanity, the intuition\nis for the time dominated by the intellect and held in subjugation by\nit so that all of man's external operations are governed and dictated\nalmost entirely by the intellectuality, allowing the intuition only\nrare moments when it can come to the fore at all. This is the rule in\nthe evolution of faculties and characteristics. The higher faculty,\nalthough potentially present in every way, is ever held in abeyance\nwhile the lower is brought, under the rigors of its own evolution, to a\npoint where its joint operation with the higher may be executed with\nthe least possible friction and retardation as also with the greatest\npossible co\u00f6rdination and co\u00f6peration. Accordingly, notwithstanding the\nfact that materiality must possess in potentiality all the qualities\nwhich it will at any time reveal, it is nevertheless necessary that\nthese qualities shall come forth gradually and in due order. Similarly,\nhumanity has come into possession of its various faculties of mind, and\npowers of physiological functions, by insensible degrees, the higher\nalways being held in abeyance until the lower is fully developed.\nThose faculties which are to bestow added powers, additional freedom\nand a greater scope of motility are the ones which appear later than\nthose which are truly primitive in character. These facts have been\namply demonstrated by the science of embryology wherein it is shown\nthat _ontogeny_ is a recapitulation of _phylogeny_. That is, the\nhistory of the development of the individual is a recapitulation of\nthe development of the species. Thus the various stages of development\nthrough which the human embryo passes while _in utero_ are but a\nrepetition of similar stages through which the entire human species has\npassed in its phylogenetic development. Wherefore, it is certain that\nhumanity has not attained, at one and the same time, all the powers of\nmind and body which it now possesses; that the childhood of the human\nrace represented a time when it had but few faculties or organs of\nsense-perception--indeed a time when the higher sense-organs of smell,\ntaste and sight were entirely lacking although residing in potentiality\ntherein.\n\nIt is undoubtedly true that the earth has passed through a similar\nevolution with respect to its own material characteristics, that\nits childhood was, in all points, analogous to the childhood of\nhumanity; that the air, earth and water were wholly absent, except in\npotentiality, during the nebulous youth of its genesis. It is even\nprobable that there are at work to-day processes which in the future\nshall culminate in the evolution of newer, higher and more complicately\norganized species of plants, animals and minerals. Every year brings\nfresh evidences that crystallize the conviction that the earth has been\nthe scene for the appearance of many strange orders of animal life.\nFossiliferous strata are continually yielding incontestable testimonies\nof changing flora and fauna. We count the animal and vegetal life of\nto-day as being more highly developed than that of any other previous\nage, and it is well that this is so, for simplicity of organization and\nprimality of manifestation are always succeeded by complexity and a\ngreater scope of adaptability.\n\nWe have said that the whole of that movement of the intellect which\nhas brought forth the metageometrical creations of hyperspaces, the\ncurvature of space and its manifoldness together with the entire\nassemblage of mathetic contrivances are merely the early evidences of\nthe appearance in the human race of a new faculty, a new medium of\nperception whereby the Thinker shall acquire a still greater range of\nmotility than that now offered by the intellect. Attention has been\ncalled also to the fact that this phenomenon has been manifested not\nalone in the field of mathematics, but in art, religion, politics\nand also in science in which we have only to witness the marvelous\nstrides already made in the discovery of radio-active substances,\nthe ROENTGEN, BECQUEREL, LEONARD and other kinds of rays. It Is quite\nconfidently believed that these forward movements in every branch of\nintellectual pursuit, these combined efforts of the intellect, in\npeering into the occult side of material things, are in response to the\nevolutionary needs of the Thinker, and in addition, are the evidences,\nand shall in time be the cause, of the development of an additional set\nof faculties. Function, or the performance of acts, determines faculty\nor the power of action and ultimately the organ itself. Thus the mere\nwish to perform aroused by desire and vitalized by the will actually\nterminates, in the course of time, in the genesis of a faculty, or the\npower to perform. The constant upreaching yearnings of the Thinker\nthrough his intellect for greater freedom and a larger scope of action,\nthe desire to peer into the mysteries of life and mind, the infantile\nout-feelings of the mentality after some safer and surer basis for its\ntheory of knowledge cannot fail in producing not only the faculty or\npower to satisfy these cravings but the very organ or medium by virtue\nof which the satisfaction may be attained.\n\nIt is not strange that in mathematics the intellect should have found\nfirst the clue to the existence of a higher sphere of intellectual\nresearch wherein it might become the creator of the various entities\nwhich peopled the new found domain; it is not strange that the\nmathematician should, in this instance, have assumed the role of\nthe prophet proclaiming by various mathetic contrivances (although\nunconsciously) that the human race is nearing that time when it shall\nactually be able to function consciously in some higher sphere;\nneither is it to be wondered at that the voice of the prophet is heard\nand respected throughout the earth; for, indeed the mathematician is\na spokesman who, as a rule, is unmoved by sudden outbursts of passion\nand ecstatic frenzies of emotions but calmly and dispassionately\nverifies his conclusions, tests them for consistency and having found\nthem to satisfy the most rigorous mathetic requirements hesitates not\nto propound them. For this cause humanity respects the mathematician,\nand when he speaks listens to his voice. It is well, too, that this\nis so; for the history of mathematics is clearly the history of the\ndevelopment of the intellect. So exact a determinator of the quality\nof intellectual efficiency is it that its reign may be said to be an\nabsolute monarchism whose lines of dominance extend to the minutest\ndesire or appetency. It has always been the guide of the intellect,\ngoing before, as it were, blazing the trail, pushing back the frontiers\nof the intellect's domain and clearing away the _debris_ so that the\nintellect with its retinue of servitors might have an easy path of\nprogress.\n\nMathematics, however, has not the aptitude to serve the intuition as\nit serves the intellect. So the path into which the intuition would\nlead humanity the mathematician, because of his training and peculiar\nfunctions, is unprepared to enter. It is for this reason that when\nmathematics leads the intellect up to that point where it encounters\nlife it fails, it becomes confused and its dictatorship becomes a\nmockery, its decrees remain unexecuted and futile. In taking this view\nwe have certainly no desire to offend the mathematician or to detract\nfrom the glory of his monarchistic rulership over the intellectual\nprogress of the race; for, in truth, mathematics is the diadem of gold\nwherewith man has crowned his intellect. Yet it is well, yea imperative\nin the light of recent developments in the realm of hyperspace, that\na careful discrimination should be made as between the sphere of the\nintellect and that to which the intuition shall attain.\n\nThe intuition, long held in abeyance until the intellect should be\nfully crowned and reach the zenith of its powers, is now coming to the\nfront. It will be many centuries perhaps before it shall have grown to\nsuch proportions as those already attained by the intellect; perhaps\na few thousand years may pass before the intuition shall have evolved\nto that point where it may labor as coadjutor to the intellect; but\nundoubtedly the time will come when it, too, shall reward the Thinker's\nlabors with that which shall be more precious than the crown of gold\nwhich the intellect has won. Then, the intellect, grown old and\ndecrepit with years of reigning shall become dim and crystal-shaped and\nfinally pass into automatism or reflexive movements where without the\nurge of volitional impulses it will perform with exactness, precision\nand utter loyalty the tasks which it has learned so well to execute\nin the days of its forgotten glory. Mankind will then be free. A new\nfreedom, wherein the erstwhile lightning flashes of intuition will\nbecome fused into one glorious sheen of all-revealing light, shall come\nto men and thus the race resplendent will walk the earth enshrined in\nthe majesty of divine powers attained as a result of millions of years\nof aspiration.\n\nThat there are supersensuous realms so far above the range of our\nsenses as to be entirely beyond their ken needs now no proof or\nargument; for the scientist has demonstrated, by the invention of\ninstruments of extreme delicacy and precision, that such a world does\nreally exist. Already we know of stars so distant that, though light\ntraverses in the brief space of an hour six hundred million miles,\nthey might have ceased to shine before the pyramids were built and\nyet be visible to us in the skies. If the human eye were as sensitive\nas the spectroscope many thousand tints and shades might be added to\nthe world of color; if they possessed the magnifying powers of the\nmicroscope we should live in constant terror and awe of the monstrous\nentities that teem in the water which we drink and in the air which\nwe breathe; and if our ears could detect the microphonic vibrations\nwhich register in the delicate apparatus of some microphones the dead,\nvacuum-stillness of nature's great silences would appear as a babel of\nvoices by the seaside. The sense of touch, responding to the same range\nof vibrations as the micrometer, would reveal actually the interstices\nbetween particles of the densest elements; and gold, silver, platinum\nand mercury would seem but honeycombs of matter. But, to the\nforward-looking there is no element of absurdity in the expectation\nthat all these senses shall, one day, be able to dispense with the\nartificial aid of physical apparatus and perform, with even greater\nprecision and faithfulness, the task which they now perform so crudely\nand ineffectively. There are without doubt vibrations of taste and\nsmell which are so far above the range of these senses that they have\nno effect upon them whatsoever. Notwithstanding the fact, however, that\nthe galvanometer, microscope, the microphone, the spectroscope and the\ntelescope have extended thus the sphere of sense-knowledge there are\nyet subtler vibrations to which these delicate instruments do not and\nought not be expected to respond. But to say, as do many empiricists,\nthat since these phenomena cannot be detected by scientific instruments\nthey do not, therefore, exist seems to be expecting too much of\nmaterial means as well as exposing oneself unnecessarily to criticism\non the grounds of extreme materialistic appetences.\n\nThere is indeed need of a more liberal attitude among men of science\ntowards the world of the unseen. Intolerance of the data which it\noffers will for a time perhaps preserve the aloofness of scientific\ndogmatism inviolate but there will most surely come a reaction against\nthe dogmatism of science and men will seek freedom and attain it\ndespite their fetters. Sir OLIVER LODGE, in his book, the _Survival\nof Man_,[29] says: \"Man's outlook upon the universe is entering upon\na new phase. Simultaneously with the beginning of a revolutionary\nincrease in his powers of physical locomotion--which will soon be\nextended to a third dimension and no longer limited to a solid or\nliquid surface--his power of reciprocal mental intercourse is also in\nprocess of being enlarged; for there are signs that it will some day be\nno longer limited to contemporary denizens of earth, but will permit a\nutilization of knowledge and powers superior to his own, even to the\nextent of ultimately attaining trustworthy information concerning other\nconditions of existence.\"\n\n  [29] See pp. 338, 341.\n\nIt is the author's good fortune that he has for a period extending over\nseveral years been able to verify the conclusions which Sir OLIVER\nLODGE expresses in the above, and thus to satisfy his own mind that the\nprocess by which man's mental powers are \"being enlarged\" is indeed\ndemonstrable by actual observation and experimental methods.\n\nLODGE continues:\n\n  \"The boundary between the two states--the known and the unknown,\n  is still substantial, but it is wearing thin in places, and like\n  excavators engaged in boring a tunnel from opposite ends amid the\n  roar of water and other noises, we are beginning to hear now and\n  again the strokes of the pick-axes of our comrades on the other\n  side.\"\n\nCAMILLE FLAMMARION[30] cites 436 cases of psychic manifestations\nexamined by himself and which establish beyond any reasonable doubt\nthat there are certain perceptive faculties, namely, clairvoyance and\nclairaudience, that crop out in certain individuals, sometimes in\nabnormal conditions and sometimes normally, the very unusual character\nof which proves their rudimentary nature and the potency of their\nmaturescence in the humanity of the future. Among the cases cited by\nFLAMMARION are 186 instances of manifestations from the dying received\nby persons who were awake; 70 cases were manifestations received by\npersons asleep; 57 were observations of direct transmission of thought\nwithout the aid of sight, hearing or touch or other physical means; 49\nwere cases of sight at a distance or clairvoyance by persons awake,\nin dreams or in somnambulism and 74 cases of premonitory dreams or\npredictions of the future. Indeed, there are few persons now living\nwho have not had similar experiences, if not exactly like these, of\nthe same nature. These examples, of course, may be greatly multiplied\nin every country in the world, and it is unnecessary to enumerate\nthem further; for, when once the existence of such faculties has been\ndemonstrated in persons, either in a normal or an abnormal condition,\ntheir presence can no longer be questioned by the fair-minded. It is,\nthen, only a question of evolution before they will appear in the\nnormal way and their universalization, as transmissible characters, be\nan accomplished fact. When we are brought face to face with this sort\nof phenomenon which seems to be increasing rapidly the conclusion is\ninevitably forced upon us that since evolution must be a continuous\nprocess and matter destined to yield higher and more refined powers and\nhumanity to come into a far more extensive scope of motility because\nof the opening avenues of knowledge, it is not impossible that these\nacuter senses, these new faculties are now existing in the human\nrace in a rudimentary stage and are designed to become the universal\npossession of all. That this is to be the almost immediate outcome\nof the perpetual exalting power which life exercises not alone over\nmateriality but over human organs and faculties as well, seems to be\nthe one big, outstanding implication of the evolutionary process. The\npresence of such functions as the ability to sense the invisible and\nthe inaudible, to answer to vibrations far subtler than anything in the\nscope of our external sense-organs, certainly indicates the existence\nof rudimentary faculties which make these functions possible. _Back\nof these vague, indefinite functions, back of every supernormal or\nabnormal manifestation of man's mentality and back of all that class of\nphenomena which take their rise out of supersensuous areas must lie,\nin ever increasing potency, faculties and organs, however rudimentary,\nwhich are the source of these manifestations._ Life, that ineluctable\nagent of creation, which is incessantly pushing outward the confines\nof the intellect's scope of motility, never wearying, never tiring\nnor sleeping, has long ago, in the dim and distant past of man's\nevolution, laid the foundations; and in fact, with one stroke of its\ncreative hand, has molded the organs which are to become the active\nmedia of these new faculties. And now, these incipient demonstrations,\nthese infantile struggles which we see now and again outputting from\nthem, are but the specializing processes through which, in their later\ndevelopment, these organs are proceeding. These are the outward signs\nwhich should tell us that life is breaking up these organs into special\nparts, assigning to each a certain division of labor and making of\neach a perfect co\u00f6rdinate of all the others. It is, by these very\ndispread exhibitions, cutting up, specializing and by slow degrees\ndetermining the function, character and general tendence of the organs\nof expression wherewith these manifestations shall be centralized and\nput into effective operation. In doing this, it is but following its\naccustomed procedure, the procedure which it adopted when it produced\nthe eye, the ear, the heart and the spleen. We shall, therefore, gauge\nour understanding of the purport and end of evolution; in fact, we\nshall determine our exact intellectual comprehension of life itself, by\nthe attitude which we adopt towards it and the mode of its appearance.\nMuch depends, accordingly, upon the posture which we assume towards\nlife--whether we shall say the totality of life's creative powers has\nbeen dissipated in the bringing of the human body to its present degree\nof perfection; whether we shall say that it is neither necessary nor\npossible for life to produce other organs and faculties which shall\nrespond to the unseen world about us revealing its glories in a way\nfar more perfect than do our external sense-organs reveal the wonders\nof the world of sensation; or, whether we shall conclude from these\nmost palpable evidences that life has yet other powers and faculties\nwhich it designs to bestow upon the human mind and other organs and\ncapabilities with which it shall endow the human body so that man,\nin his evolution, shall be enabled to rise to still higher spheres\nwhile yet incarnate. There may be, and undoubtedly are, those who,\nfor various reasons prefer to take the former positions and there are\ncertainly those who like LODGE, FLAMMARION, HUDSON, CROOKES and a host\nof others, preferring the latter view, would rather believe in the\nstrength of the great mass of corroborative testimonies that we are\neven to-day in the midst of the matutinal hours of a newer, a better\nand a far more efficient era of human evolution than any through which\nwe have hitherto come.\n\n  [30] See _Unknown_, p. 485, et. seq.\n\nAlready, recent scientific investigations and the results obtained\ntherefrom have begun to turn the attention of medical authorities to\nthe activities of two very small organs situated in the mid-brain and\nknown as the _pineal gland_ and the _pituitary body_. These organs,\nand especially the _pineal gland_ hitherto supposed to be a vestige\nof the past, are now beginning to be recognized as rudimentary organs\nbelonging to the future evolution of humanity. Dr. CHARLES DE M.\nSAJOUS, who is an authority on the _pituitary body_, believes that it\nhas no active internal secretions but is an \"epithelio-nervous organ\"\nwhich controls, through nerves leading to the adrenals and thyroid\nbone, the processes of general oxygenation, metabolism and nutrition.\nLittle is known of the functions of the pineal gland except that it is\nan ovoid, reddish organ attached to the posterior cerebral commissure\nprojecting downward and backward between the anterior pair of the\n_corpora quadrigemina_. It is otherwise known as the \"_conarium_\" the\n\"_pinus_\" or \"_epiphysis_.\" Situated at the base of the brain, it\nis held in position by a fold of the _pia mater_ while its base is\nconnected with the cerebrum by two pedicles. It contains amylaceous\nand gritty, calcareous particles constituting the brain sand. There\nare, however, marked structural resemblances between the _pineal\ngland_ and the _pituitary body_ and their formation is perhaps the\nmost interesting feature of the development of the _thalamencephalon_\nor mid-brain. The _hypophysis cerebri_ or _pituitary body_ is a small,\novoid, pale, reddish mass varying in weight from five to ten grains\nand situated at the basal extremity of the brain in a depression of\nthe cranium known as the _sella turcica_, a configuration very much\nlike a Turkish saddle in shape. It is a composite, ductless gland and\nconsists of two divisions, an anterior and a posterior, connected\nby an intermedial portion--all of which are attached to the base of\nthe cerebrum by the _infundibulum_. The anterior lobe is larger than\nthe posterior and very vascular, springing in its development from\nthe buccal cavity of the embryo; the posterior lobe is situated in\na depression of the anterior and is a brain-process. The _pituitary\nbody_ itself is lodged in a cavity of the _sphenoid_ bone called the\n_pituitary fossa_. This is a most remarkable position, for the reason\nthat the _sphenoid, or wedge-shaped_, bone which lies at the base of\nthe skull articulates from behind with the occipital and in front with\nthe _frontal_ and _ethmoid_ bones and by lateral processes with the\n_frontal_, _parietal_ and _temporal_ bones. From this position it binds\ntogether all the bones of the cranium, and moreover, articulates with\nmany bones of the face. It is upon the upper surface of the _sphenoid_\nbone which occupies such a prominent and commanding position in the\ncranium, in a deep depression, that the _pituitary gland_ is located.\n\nEach nasal chamber is lined by a mucous membrane called the _pituitary_\nor _Schneiderian_. This membrane is prolonged into the meatuses and air\nsinuses which open into the nasal chambers. The _pituitary_ membrane is\nthick and soft and diminishes the size of the meatuses and air sinuses.\nIt is covered by a ciliated columnar epithelium and contains numerous\nracemose glands for the secretion of mucous or _pituita_. It is also\nvascular and the veins which ramify it have a plexiform or net-work\nlike arrangement. It divides into two membranes--a respiratory, which\nis concerned in breathing, and an olfactory region. The respiratory\nregion corresponds to the floor of the nose, to the inferior turbinated\nbone and to the lower third of the nasal septum. The olfactory region\nis the seat and distribution of the olfactory nerve and corresponds\nto the base of the nose, to the superior and middle turbinals and the\nupper two-thirds of the nasal septum.\n\nRecent developments prove that this gland has a profound influence\nover the animal economy. It is believed by some that the _pituitary\nbody_ actually destroys certain substances which have a toxic influence\non the nervous system; others believe that it secretes material\nmedia for the proper action of the trophic or nutritive apparatus;\nstill others believe that it influences blood-pressure. It is known,\nhowever, from experimentation, that its removal in dogs, cats, mice\nand guinea pigs causes a fall of temperature, lassitude, muscular\ntwitchings, dyspnoea or difficult breathing, and even speedy death.\nHypertrophy of the gland is directly associated with certain diseases,\nsuch as _giantism_ and _acromegaly_. The latter is a disease which\ncauses a general enlargement of the bones of the head, feet and hands,\nusually occurring between the ages of twenty and forty years, and most\nfrequently in females. The fact that these diseases are so closely\nassociated with a hypertrophic condition of the pituitary gland has led\nto the conclusion that perhaps the giants or Cyclops of ancient times\nwere cases of _giantism_ or _acromegaly_. This view, while interesting\nfrom the standpoint of the functions of the _pituitary_ gland, is not\nnecessarily a correct one; for the age of giants, when men attained\nto a much larger stature than at present, can be accounted for on\nother grounds, namely; that the early mesozoic man, on account of his\nhaving to live among animals, trees and other vegetation of such huge\nsize, had naturally to be fitted with a frame proportional to other\nanimals in order that he might successfully cope with his environing\nconditions. Nature thus wisely fitted him for the conditions which she\nhad prepared for the scenes of his life.\n\nThe facts adduced in the foregoing description are purely empirical\nand may be verified by any who seek to establish their correctness or\nincorrectness. But we are about to introduce a species of testimony\nwhich while it may also be verified will not be found so easy of\nverification as the above-mentioned physiological facts, and not by\nthe same means; yet they are nevertheless deserving of a place here.\nIt is the liberal attitude that we must adopt towards all phenomena,\nexcluding none that give promise of the widening and deepening of\nour knowledge and an explanation of much that has seemed heretofore\nunaccountable.\n\nWe have noted how subtle is the physical connection between these two\nbodies, the _pineal gland_ and the _pituitary body_; we have seen how\nprofound is the effect which the latter has been demonstrated, in a\nmeasure, to have over the entire bodily economy; but there is even\nother testimony to the effect that those gifted with the inner vision\ncan observe the \"pulsating aura\" in each body, a movement which is not\nunlike the pulsations of the heart and which never ceases throughout\nlife. In the development of clairvoyance it is known that this motion\nbecomes intensified, the auric vibrations becoming stronger and more\npronounced. The _pituitary body_ is the _energizer_ of the _pineal\ngland_ and, as its pulsating arc rises more and more until it contacts\nthe _pineal gland_, it awakens and arouses it into a renewed activity\nin much the same manner as current electricity excites nervous tissue.\nWhen the _pineal gland_ is thus aroused clairvoyant perception is\nsaid to become possible. These are facts which cannot be proved by\nthe materialistic man of science nor can they be demonstrated to the\nlayman who has to depend alone upon sense-deliveries for his knowledge.\nThis is true for the reason that, in the first place, it is necessary\nthat he shall either feel in his own mid-brain the energizing activity\nof these two organs and have his entire nerve-body shaken from crown\nto toe by the down rushing currents of that subtle energy with which\nthe _pituitary body_ floods it or be himself the perceiver of its\nactivities. Nevertheless attention is here called to these phenomena\nand the conclusions drawn therefrom are offered as a means of denoting\nthe probable line of investigations which will establish the directions\nwhich we should pursue and the source whence we shall find outcropping\nthe new faculties and their organs of expression.\n\nWe confess to a knowledge of the fact that men of empirical science\nhave long maintained a rather skeptical, if not contemptuous, attitude\ntowards all these phenomena but it is also felt that there is far more\nof discredit in their attitude than of credit; for, in so doing, they\nhave voluntarily adopted measures by means of which the knowledge\nthat they so eagerly seek is shut out from their attainment. In vain,\nthen, is appeal made to the intellect to remove the barriers which it\nunconsciously interposes between itself and the goal of its pursuit;\nin vain do we appeal to the materialist to give ear to testimony the\ndata of which cannot be made amenable to his knife and scalpel neither\nto the microscope nor microphone; in sheer vanity is he adjured to\nlook _within_--into the interior of life, of mind and the things which\nhe handles with his instruments--for the answers to his queries, for\nthe path which leads into the wake of life and consciousness. Because\nhis utter loyalty and devotion to the _modus vivendi_ of the intellect\nwill not permit this; but, after all, it is not wholly wise to allure\nhim away unbetimes from his search after truth through superficialities\nnor to inveigle him into giving up his tenacious prosecution of the\nphysically determinable. We would not have it so; for, perchance, he,\ntoo, one fine day, in the quiet of his laboratory shall come upon\nthe data which may substantiate in his own mind the long settled\nconclusions of the occultist who, frequently and not without cause,\ngrows impatient at the scientist's obstinate delay. These two workers,\nthe empiricist and the occultist, must ultimately come together as\ncollaborators--the one working upon the form, the vehicle, physical\nmatter and the other seeking to understand the life, the interior\nforces which produce, the creative element. They cannot remain always\naloof from one another; for they, too, are as men digging a tunnel from\nopposite ends. Finally, the partition will break and thus will dawn a\nnew day for the knowledge of humanity and men will see the rationale,\nthe truth and good sense of co\u00f6peration in this respect.\n\nIt can be said with confidence that whatever in the future may be\nlearned as to the physiologic functions of the _pituitary body_ and\nthe _pineal gland_, it suffices to know that it is life which they\nexpress and that, too, in a far superior manner than any of the other\nsense organs. The _modus_ of these two glands differs in a very\nmarked way from that of the organs of sight, hearing, taste, smell and\nfeeling. For these latter are designed for contact with the external,\nobjective world of sensations, their growth and evolution being\ndependent upon stimuli received from without while with the former the\ncase is far different, in fact, just the opposite. The mode of life\nof the _pituitary body_ and the _pineal gland_, instead of receiving\nsustenance and impetus from external stimuli, is rather dependent\nupon impacts received from the Thinker's own consciousness and made\nto impinge upon them by an exclusively interior process. Situated in\nthe mid-brain, safely secluded from all external interference, they\nare naturally limited to stimuli which come from within, or it may\nbe said, they are responsive to excitations that are more spiritual\nthan those which come through the external sense-organs. If, as has\nbeen said they control the internal processes of metabolism (anabolism\nand katabolism), oxygenation, nutrition, and other important internal\nmovements, none of which can be said to be under the control of the\nintellect, is it not, therefore, justly assumed that their response\nis directed towards stimuli which arise interiorly or upon a plane\nhigher than the intellectual? It is a matter of scientific knowledge\nthat those persons gifted with clairvoyance, and commonly known as\n\"sensitives\" are far more responsive to nervous excitation than those\nwho are not so gifted. This would seem to imply that, on account of\nthe superactivity of these two organs, the entire nerve-body has, in\nconsequence, become more delicately and subtly organized. They seem\nto act as a switchboard for the regulation of the flow of the current\nof life through the body. Not only do they come more nearly to an\nadequate expression of the physiologic function of life, but, as their\nenergization means an enlargement of the scope of perception by giving\nthe Thinker's active consciousness access to hitherto unapproachable\nrealities and by penetrating the outer mask which life ensouls and also\nlaying bare a domain of unlimited knowledge the manifestation of which\nis far more real than anything the senses can disclose, it is evident\nthat they constitute, in their collaborative functions, a more highly\nadaptable medium for the expression of the Thinker's consciousness. And\nif so, for the kosmic consciousness which is the source of all forms of\nconsciousness, they furnish a specializing and _adaptizing_ agency.\n\nNow, in all those cases of inspirations, revelations, telepathic\ncommunications, clairaudience, clairvoyance, dreams, visions, etc.,\nwherein the Thinker is enabled to perceive facts and verities which\nare then presented to his consciousness in a manner clearly without\nthe province of the common sense-organs, it must be apparent that\nthese manifestations are apprehended by a perceptual mechanism which\nis entirely independent of external sense presentations but which is\nan interior and subtler form of psychic activity. Sounds which are\nheard by so-called \"sensitives\" and objects which are perceived by eyes\nthat are keener than those organs said to have been evolved from the\n\"medusa\" cannot be heard by other persons nor perceived by them in any\nway. Thus it would seem that there are inner organs of perception which\nrespond to these finer vibrations and which enable the person so gifted\nto apprehend them.\n\nThere are those who, presumably basing their assertions upon actual\nobservation and knowledge, unqualifiedly assert that in order \"to\ngain contact with the inner worlds all that remains to be done is the\nawakening of the _pituitary body_ and the _pineal gland_. When this\nis accomplished man will again possess the faculty of perception in\nthe higher worlds, but on a grander scale than formerly (when humanity\nwas in its infancy and exercised a lower form of psychic power only);\nbecause it will be in connection with the voluntary nervous system, and\ntherefore, under the control of the will. Through this inner perceptive\nfaculty all avenues of knowledge will be opened to him and he will\nhave at his service a means of acquiring information compared with\nwhich all other methods of investigation are but child's play.\"[31]\nIt is the lack of this ability to see, with our physical eyes, as it\nwere, by the \"Roentgen rays,\" to penetrate the inwardness of things\nthat has baffled and confounded men for so long a time and which has\neventually led certain mathematicians and others to conjecture such\nstrange, and in many cases, illogical possibilities for the denizens\nof four-space. This inability together with the desire to fathom the\ninnermost complexities of solids and to handle, albeit with unholy\nhands, the supersensuous, the mysterious and the unapproachable\nidentity of \"things-in-themselves\" have induced the more zealous among\nthem to contrive some kind of hypothesis which would, at least, offer\nan explanation of these phenomena. It has driven them to wrestle\nwith metaphysical possibilities in a vain endeavor to grasp that\nwhich, _ignis-fatuus_ like, ever evades their slightest intellectual\napproach. But why this prolonged struggle, why this intellectual\nmaneuvering and sophistry? \"We can calculate, compute, excogitate,\"\nsays PAUL CARUS,[32] \"and describe all the characteristics of\nfour-dimensional space, so long as we remain in the realm of abstract\nthought and do not venture to make use of our motility and execute\nour plans in an actualized construction of motion; but as soon as we\nmake an _a priori_ construction of the scope of our motility, we find\nout the incompatibility of the whole scheme.\" Thus mathematicians are\nforced to relinquish all hopes of transforming the world of life into\na sort of four-space dwelling place where everything is done according\nto the laws of mathematics. But whether they shall accept it or not\nthere is a wider, truer and more rational view which recognizes all\nmetageometrical investigations, as well as all kindred phenomena, as\nuniversal evidences indeed, as the very causes which, in the future\nhumanity, will actually awaken and cause to be accelerated in their\ndevelopment these little inner sense-organs, the _pineal gland_ and the\n_pituitary body_, whose perfect development promises to provide for\nthe Thinker's consciousness an avenue of expression such as humanity\nhas possessed never before. And too, it is not without full knowledge\nof the fact that it has been customary, among certain scientists or\nperhaps all of them, to regard these bodies, at least the _pineal\ngland_, as vestigal organs belonging to the past of human evolution,\nthat we make these assertions. Yet, as man proceeds in the perfection\nof mechanical science, in the development of instruments of precision\nthat aid his external senses, responds more and more to the subtle\nvibrations teeming everywhere in the atmosphere about him, and comes,\nin the course of time, naturally to possess a more sensitively keyed\nnervous mechanism, a finer body and higher spiritual aspirations,\nthere will be a corresponding widening of his scope of vision and the\nattainment of larger powers of perception which must inevitably, in the\nvery nature of things, tend towards a deeper and truer knowledge.\n\n  [31] _Rosicrucian Cosmo-Conception_, p. 477, MAX HEINDEL.\n\n  [32] FOUNDATIONS OF MATHEMATICS, p. 90.\n\nIn view of the foregoing, it is believed that the general results of\nthis pituitarial awakening which may be expected as humanity continues\nto evolve should be seen in the marked effects which will be wrought\nin the entire metabolistic area of the human body whereby a gradual\nintensification and sensitization of the whole neural mechanism\nwill raise the peculiar efficiency of all the senses whether purely\nphysiologic or psychic. For there are undoubtedly notes so delicate in\ntheir intensity that they transcend the grasp of the audital nerves;\nscents and fragrances so subtle in their excelling purity that it is\nbeyond the powers of our present olfactorial contrivances to detect\nthem; colors and other external stimuli so sublimely supersensuous that\na nervous mechanism perhaps ten-fold more delicate and responsive than\nours is required to apprehend them. All these, and more than at present\nis conceivable, will come, with the aid of pituitarial stimulation,\nwithin the purview of a more highly developed humanity of the future.\nAnd because mathematics have led a movement into the very camp of\nthe intellectuals--logic-bound and tethered by the severest rigors\nof mathesis--whereby the intolerant intellect has been compelled,\nby rules of its own making, to recognize the existence of the\nsupersensuous, and by looking into the glaring light of the sun of the\nintuitable to gain strength of vision and boldness to press forward,\na great and far-reaching service has been wrought for humanity. And\nin the tower of hyperspace mathematics have erected a monument to the\nintellect which, as long as the human race remains, will mark the great\nturning point in man's path to the highest life.\n\nWhat if it were possible that the scientist, when he had carried\ninstruments to their utmost precision and penetration, should suddenly,\nor otherwise, be endowed with a clear-perceptivity of sight, hearing\nand smell, so that he could with his own powers of vision, feeling and\nhearing take up the task where the microscope, the microphone and the\nmicrometer left off and delve into depths far too unfathomable for his\nappliances, perceiving the innermost realities of things and processes?\nWhat if it were possible for him, with these added powers, to see and\nexamine without the aid of the magnifying lens the electron, the atom\nand the molecule? What if the cell, the bacterium, and other invisible\nforms of life would then deliver up their secrets to his knowing mind?\nWhat if he could sense with his own inner vision, the ultra-violet and\nthe infra-red rays; what indeed, if spirit itself, the innermost sheath\nof life, should be visible and palpable to him and he could note the\ninternal processes, the action and movements of the infinitesimals\nof life? Think you not that such direct contact, such immediate and\nincontrovertible knowledge would be far superior to any advantage which\nhis manufactured devices now bestow? It is even so.\n\nThus will react upon man's perceptive apparatus the flood of light\nwhich the awakened intuition will shed upon them and thus will man rise\nhigher, driven on by the current of life with the mass of materiality,\nto a point of complete spiritualization and take additional steps in\nthat direction which leads to Raja Yoga or the Royal Union with the\ndivine life of the universe.\n\nBefore this step is taken, however, and before the passage from\nmechanics to biogenetics is made, as made it must be, man must win a\ncomplete mastery over matter. But this he will do; for more and more\nhe is learning to put all those forms of labor which are so exacting\nas to leave him no time for the development of his higher powers into\nthe hands of machinery. He will not be free until he has done this\nwell-nigh completely. This is the task of the intellect and with it\nman must win his way to these higher faculties which are destined to\nsucceed the intellect whereupon he will be ushered out of a life bound\nand restricted by mechanics to a life of unimaginable freedom, the\n_intuitive life_.\n\nThe outcome of these new faculties of perception and the development of\nthe intuition will be the springing up of a new species of art that,\nturning away from appearances and sinking beneath or rising above,\nsuperficialities, will seek to portray in newly found colors, the\nplastic essence of things so that we shall have an art which pertains\nto the real, superseding that which pertains to the phenomenal.\nLanguage and the need of it will pass away; for man will have outgrown\nthe use of signs and symbols in his communion with his fellows and will\nuse the language of the intuition--direct and instantaneous cognition.\nPhilosophy will be regenerated, re-created. Speculation will give\nway to truth and there shall be but one philosophy and that shall be\nthe _knowledge of the real_. Mathematics, the royal insignia of the\nintellectual life, because it can deal only with immobilities, with\nsegments and parts and has no aptitude for the continuous flow, will\nyield its kingdom to a higher form of kinetics which will serve the\nintuitive faculty as mathematics now serve the intellect. Science will\nthen be no longer empirical in its method; but a system of direct and\nincontrovertible truths. Religion will rise to meet these changes which\nwill come in the path of human evolution; and faith will surrender its\nplace to knowledge. Ethics, recast in a new mold, will deal with the\nnew aspect of man's relation to his fellowmen. Man, for whose highest\ngood these ultimate changes will come, will be a new creature, a higher\nand better man; and humanity shall evolve a new race. There shall,\nindeed, be \"a new heaven and a new earth.\"\n\n\n  THE END.\n\n\n\n\n  BIBLIOGRAPHY\n\n\n  AMERICAN MATHEMATICAL MONTHLY:\n    Vol. VIII, p. 161, P. BARBARIN, for _Le Mathematiche_,\n     Trans, by GEO. B. HALSTED. See also pp. 31-35.\n    Vol. VIII, 1901, p. 217, Suppl. Report on Non-Euclidean Geometry,\n     GEO. BRUCE HALSTED.\n    Vol. IX, p. 59, GEO. B. HALSTED.\n    Idem, pp. 153-159.\n    Vol. XI, pp. 85-86, Simon's Claim for Gauss.\n\n  AMERICAN MATHEMATICAL SOCIETY BULLETIN, Lancaster, Pa.:\n    Vol. II, series 1895.\n    Vol. XX, 1914, pp. 409-412, R. C. ARCHIBALD.\n\n  ANNALS OF MATHEMATICS, 2d series, 1902:\n    Vol. XII, F. S. WOOD.\n\n  ASTRONOMICAL SOCIETY NOTICES:\n    Vol. XVI, p. 80, 1856.\n\n\n  BALL'S SHORT ACCOUNT OF THE HISTORY OF MATHEMATICS.\n\n  BOSTON COLLOQUIUM, for 1903, N. Y. 1905, _Forms of Non-Euclidean\n    Spaces_, Vols. I-III, pp. 31, et seq., 1891.\n\n  BULLETIN NEW YORK MATHEMATICAL SOCIETY:\n    Vol. II, pp. 1-144.\n    Vol. III, pp. 1, et seq., 1891, G. B. MCCLINTOCK.\n\n\n  CAMBRIDGE MATHEMATICAL JOURNAL:\n    Vol. IV, 1845, CAYLEY, on _Analytical Geometry of_ n-_Dimensions_.\n\n  CANADIAN INSTITUTE TRANSACTIONS:\n    Vol. VIII, pp. 315-339, 1909, AUGUST KIRSCHBAUM.\n\n  CAYLEY, A., vide _Science_, Vol. IX, pp. 59-63:\n    _Amer. Math. Mo._, Vol. II, pp. 102-106.\n    _Amer. Math. Mo._, Vol. VI, pp. 59-65.\n    _Review of Reviews_, June, 1895, pp. 693-694.\n\n  CLASS BOOK OF NON-EUCLIDEAN GEOMETRY:\n    Vide _Science_, XIII, pp. 462-465.\n\n  CORNHILL MAGAZINE (n.s.):\n     Vol. XXXVII, pp. 647-665, C. H. BRYAN, on the Fourth Dimension.\n\n\n  EASY NON-EUCLID:\n    _Monist_, XIX, pp. 399-402.\n\n  EDUCATIONAL REVIEW:\n    November, 1902, issue, pp. 346, et seq.\n\n  EUCLID'S PARALLEL-POSTULATE, by JOHN W. WITHERS.\n\n\n  FLATLAND, by A. SQUARE.\n\n  FOUNDATIONS OF GEOMETRY:\n    _Science_, Vol. VI, pp. 487-491.\n\n  FOUNDATIONS OF MATHEMATICS, by PAUL CARUS.\n\n  FOURTH DIMENSION:\n    By HERMAN SCHUBERT, C. H. HINTON, CLAUDE BRAGDON.\n\n  FOURTH DIMENSION, SIMPLY EXPLAINED:\n    Edited by H. P. MANNING.\n\n  FOURTH DIMENSION:\n    _Cornhill Magazine_, n.s., Vol. XXXVII, pp. 647-665, by C. H. BRYAN.\n\n  FOURTH DIMENSION, THE RIDDLE OF:\n    H. A. BRUCE, in _Scientific American Supplement_, Vol. LXVI, p. 146.\n\n\n  GEOMETRY, FOUNDATIONS OF, by O. VEBLEN:\n    _Popular Science Monthly_, Vol. LXVIII, pp. 21, et seq.\n\n  GEOMETRY, LOBACHEVSKI'S:\n    _Theory of Parallels_, trans. by GEO. BRUCE HALSTED.\n\n  GEOMETRY, NON-EUCLIDEAN:\n    _American Mathematical Monthly_, Vol. VII, pp. 123-133.\n\n  GEOMETRY, NON-EUCLIDEAN, REPORT ON PROGRESS IN:\n    Reprinted from _Science_, n. s., Vol. X, No. 251, pp. 545,\n     et seq., by GEO. BRUCE HALSTED.\n\n  GEOMETRY, NON-EUCLIDEAN, THESIS ON:\n    By M. L. GERARD.\n\n  GEOMETRY, NON-EUCLIDEAN:\n    By ROBERTO BONOLA, trans. by H. S. CARSLAW, University of\n     Sydney, N.S.W.\n\n  GEOMETRY, NON-EUCLIDEAN:\n    JULIAN LOWELL COOLIDGE, pp. 46, et seq.\n\n  GEOMETRY, NON-EUCLIDEAN:\n    By H. P. MANNING.\n\n  GEOMETRY, Space and, in Light of Physiologic, Psychologic and\n   Physical Inquiry, vide _Open Court_, 1906, by E. MACH.\n\n\n  HALSTED, GEORGE BRUCE:\n    Cayley, A., Biog. of, vide _Amer. Math. Mo._, Vol. II, p. 102.\n    Four-fold Space in Two-fold Time, vide _Science_, Vol. XIX, p. 319.\n    Ho\u00fcel, Biog. of, _American Math. Mo._, Vol. IX, pp. 99-101.\n    Lambert's Non-Euclidean Geometry, vide _N. Y. Math. Soc. Bull._,\n       Vol. III, p. 79.\n    Lobachevski, N., Biog. of, vide _Amer. Math. Mo._, Vol. II,\n       pp. 137-139; _Open Court_, Vol. XIII, pp. 411-415.\n    Message of Mathematics to Natural and Rational Theology, vide\n      _Hibbert's Journal_, Vol. VII, 1809.\n    Non-Euclidean Geometry, Inevitable, vide _Monist_, Vol. IV, p. 483.\n    Non-Euclidean Geometry, Historical and Expository, _American\n       Mathematical Monthly:_\n      Vol. I, pp. 70-72; 112-115; 149-152; 188-195; 222-223; 259-260;\n       301-303; 345-346; 378-379; 421-423 (1894).\n      Vol. II, pp. 10; 42-43; 67-69; 108-109; 144-145; 181-214;\n       256-257; 309-313; 346-348 (1895).\n      Vol. III, pp. 13-14; 35-36; 67-69; 109; 132-133 (1896).\n      Vol. IV, pp. 10; 77-79; 101-102; 170-171; 200; 247-249; 269-270;\n       307-308 (1897).\n      Vol. V, pp. 1-2; 67-69; 127; 218; 290-291 (1898).\n    _Review of Reviews_, Vol. II, 1895, pp. 693-694.\n    Space, New Ideas About, _Popular Science Monthly_, Vol. XI, p. 364.\n\n  HELMHOLTZ, H., _On Origin and Meaning of Geometric Axioms:_\n    _Vide Mind_, Part I, No. 3, July, 1876. For replies and discussions\n     of the above, vide _Nature_, JEVONS, Vol. IV, p. 481; J. L. TUPPER,\n    _Nature_, Vol. V, p. 202; HELMHOLTZ, in _Academy_, Vol. III, p. 52;\n    _Mind_, Part II, April, 1878.\n\n  HERMATHENA, Vol. XIII, pp. 491, et seq., 1906.\n\n  HIGHER SPACE, A PRIMER OF, by CLAUDE BRAGDON.\n\n\n  INGLEBY, C. M.:\n    A Non-Mathematical Criticism of a Part of Professor Clifford's\n    Lecture on \"Curved Space,\" _Macmillan's Magazine_, October, 1872;\n    also _Nature_, Vol. VII, 1873.\n\n\n  JOURNAL OF MATHEMATICS, AMERICAN:\n    Vol. I, pp. 262, 276, 384, 386; HALSTED (1878).\n\n  JOUR. PHILOS., PSYCHOL. AND SCIENTIFIC METHODS:\n    Vol. VI, pp. 617, et seq.\n\n\n  KANT, EMMANUEL, trans. by FRED. PAULSEN.\n\n  KANT, EMMANUEL, _Prolegomena_, trans. by J. P. MAHAFFY and\n    J. H. BERNARD, pp. 35-39.\n\n\n  LAMBERT, BIOGRAPHY OF:\n    _American Math. Mo._, Vol. II, pp. 209-211.\n\n  LIE, SOPHUS:\n    _Science_, Vol. IX, pp. 447-448.\n\n  LOBACHEVSKI, N.:\n    _Open Court_, Vol. XII, pp. 411-415,\n    _Amer. Math. Mo._, Vol. II, pp. 137-139.\n\n  LONDON, EDINBURGH AND DUBLIN PHILOSOPHICAL MAGAZINE:\n    Vol. V, series 6, p. 559.\n\n\n  MATHEMATICS, FOUNDATIONS OF: PAUL CARUS.\n\n  MATHEMATICAL PAPERS OF CHICAGO CONGRESS, p. 92.\n\n  MATHEMATICAL PAPERS, pp. 55-71.\n\n  MATHEMATICS (a brochure):\n    By C. J. KEYSER, Adrian Professor of Mathematics, Columbia Univ.\n\n  MATHEMATICS, HIGHER:\n    Edited by MANSFIELD MERRIAM and ROBERT S. WOOD, 1902, pp. 508,\n     et seq.\n\n  MATHEMATICS, SPIRITUAL SIGNIFICANCE OF:\n    C. J. KEYSER, reprinted from Columbia Univ. _Quar._, December, 1911.\n\n  MEMORIAL VOLUME OF SCIENTIFIC PAPERS:\n    Pub. by Univ. of St. Andrews, pp. 3-45, D. M. Y. SOMERVILLE.\n\n  MIND (Magazine):\n    Vol. I, 1876, pp. 303, et seq. J. N. P. LAND.\n    Vol. II, 1877, p. 40, J. N. P. LAND.\n    Vol. IX, pp. 522, et seq. 1900.\n\n  MONIST (Magazine):\n    Vol. II, pp. 50, 321, PAUL CARUS.\n    Vol. IV, 1894, Articles by G. B. HALSTED and J. DELBOEUF.\n    Vol. VI, 1895, HERMAN SCHUBERT.\n    Vol. VII, G. B. HALSTED, on Saccheri.\n    Vol. XIII, pp. 80-102; 218-234.\n    Vol. XIV, Article by E. MACH.\n    Vol. XV, p. 127, PAUL CARUS.\n    Vol. XVI, 1906, p. 433, DAVID P. ABBOTT.\n    Idem, 1906, p. 65, C. J. KEYSER, on _Mathematical Emancipations_.\n\n\n  NATURE (Magazine):\n    Vol. VIII, pp. 8, 14-17, 36-37 (1873).\n    Vol. XV, pp. 533-537 (1877).\n    Vol. XLV, P. 404 (1891).\n\n  NEWCOMB'S PHILOSOPHY OF HYPERSPACE:\n    _Science_, Vol. VII, p. 212.\n\n  NEW YORK MATH. SOC. BULLETIN:\n    Vol. III, p. 79 (1893), F. S. WOOD on _Forms of Non-Euclidean\n     Space_.\n\n\n  OPEN COURT (Magazine):\n    Vol. XVI, pp. 513-521.\n    Vide December, 1913 edition, p. 757.\n\n\n  PHILOSOPHY OF MATHEMATICS, by PAUL CARUS.\n\n  PHILOSOPHICAL REVIEW:\n    Vol. V, 1896, p. 175, F. C. S. SCHILLER.\n    Idem, p. 357, J. H. HYSLOP.\n    Vol. VII, p. 615.\n    Vol. X, pp. 113, 229, 314, 375, 488, 579, 583.\n    Vol. XII, p. 493.\n\n  POPULAR SCIENCE MONTHLY:\n    No. 66 (1905), pp. 639-646.\n    Vol. LXVII, pp. 639, et seq.\n    Vol. LXVIII, pp. 21, et seq.\n    Vol. LXXVIII. p. 554 (1911), SAMUEL MOFFAT WEYER.\n\n  POPULAR ASTRONOMY:\n    Vols. VII and VIII (1900), G. B. HALSTED.\n    Vol. XVIII, pp. 42, et seq. (1910), W. H. PICKERING.\n    Vol. LXXV, p. 179, W. W. PAYNE, on _Attraction and Figure of the\n     Earth_.\n    Vide No. 84 (1901), pp. 187-190.\n\n  POPULARIZATION, THE, OF NON-EUCLIDEAN GEOMETRY:\n    _Amer. Math. Mo._, Vol. VIII, pp. 31-35.\n\n  PSYCHOLOGICAL BULLETIN:\n    Vol. VIII, p. 22 (1911).\n\n\n  QUEEN'S QUARTERLY:\n    Vol. XX, pp. 431-446, A. L. CLARK (1913).\n\n\n  REVUE GENERALE DES SCIENCES:\n    No. 23, trans. _Nature_, Vol. XLV, 1892.\n\n  ROYAL ASTRONOMICAL SOCIETY PROCEEDINGS:\n    Vol. XVI, pp. 80-85.\n\n  ROYAL SOCIETY PROCEEDINGS, LONDON:\n    Obituary Notices of Deceased Fellows, Vol. XVI, p. 69 (1867-68).\n\n  ROYAL SOCIETY OF N. S. W. JOURNAL AND PROCEEDINGS:\n    Vol. XXXV, p. 243 (1902), G. H. KNIBBS.\n\n\n  SCIENCE (Magazine):\n    Vol. III (new series), No. 58, 1896.\n    Vol. VI, pp. 487-491 (1897).\n    Vol. VII, pp. 1-7, _Philosophy of Hyperspace_, S. NEWCOMB.\n    Idem, p. 861 (1898).\n    Vol. XIII, pp. 462-465.\n    Vol. XIV (new series), pp. 711-712, G. B. HALSTED.\n    Vol. XIX, pp. 401-413.\n    Vol. XX, pp. 353-456, JOSIAH ROYCE.\n    Vol. XXII, pp. 270-271.\n    Vol. XXXV, No. 906 (1912).\n    Vol. XXXVII, pp. 885, et seq., C. J. KEYSER, on _Concerning the\n     Figure of the Universe of Space_.\n\n  SCIENCE ET L'HYPOTHESE, R. POINCAR\u00c9.\n\n  SCIENTIFIC AMERICAN SUPPLEMENT:\n    Vol. LXVI, pp. 14-16 (1908).\n\n  SCIENTIFIC ROMANCES, C. H. HINTON.\n\n\n  THOUGHT, NEW ERA OF, C. H. HINTON.\n\n\n  WESTMINSTER REVIEW:\n    Vol. CLII, pp. 675, et seq.\n\n\n  ZOLLNER, J. C. F., _On Space of Four Dimensions_:\n    _Quar. Jour. of Science_, Vol. VIII (1878), pp. 227-237.\n\n\n\n\n  INDEX\n\n\n  A\n\n  Abeyance, higher faculties held in, until lower are fully developed, 334\n\n  -- intuition kept in, 338, _ff_\n\n  Ability, to sense the inaudible and the invisible, 342\n\n  Absolute, no room for the, 101\n\n  Abstract thought, invigorating power of, 33, 40\n\n  Abstractions, realizing, 144, 294\n\n  Absurdity, analysis inevitably ends in an, 319-20\n\n  Acrobatics, mathetic, 146\n\n  Acromegaly, 347\n\n  Action, automatic, of the intellect, 253-4\n\n  -- engendering, of life on the form, 330\n\n  Activities, space, as physical and chemical phenomena, 229-30\n\n  Actualities of the physical world pushed over into the conceptual, 119\n\n  Actuality imperceptible to the intellect, 126\n\n  Adaptability of life, 330\n\n  -- -- mathematical laws, 37-39\n\n  Adaptation, purpose of, 327-8\n\n  Adaptations made by an organism, 162\n\n  Additional freedom bestowed by higher faculties, 334\n\n  Affinity, zones of, 22, 124\n\n  Agency, interpretative, intellect as an, 166-7\n\n  Agent, ego as an, of the Thinker, 243, _ff_\n\n  -- engendering, life as an, 331\n\n  -- of creation, life as an, 343\n\n  Ahmes, an Egyptian priest, 44\n\n  Alchemy of psychogenesis, 225\n\n  Alcohols, eight different, from one formula, 155\n\n  Algebraic quantities and space, 125-7\n\n  Allowability of the rules of logic, 163\n\n  Aloofness, the, of scientific dogmatism, 340\n\n  Alphabet of space-genesis, 237\n\n  -- the geometric, 193\n\n  Amenability of mind to the laws of evolution, 28-31\n\n  Analogies, use of, to popularize the four-space, 128-9\n\n  Analogy, difference between two processes illustrated by, 74\n\n  Analysis incapable of dealing with life, 319\n\n  -- the manifold, the fiat of, 77\n\n  Analyst and the manufacture of the space-manifold, 77\n\n  -- disregards the conformity of the nature of things, 84\n\n  Analytics and the mechanical origin of the universe, 40-1\n\n  -- perisophical nature of, 320, _ff_\n\n  -- the four-space, a curiosity of, 40\n\n  Anchorage, fourth dimension denied, 265\n\n  Angularity of consciousness, 120\n\n  Answer of the senses to new needs, 332\n\n  -- the, of consciousness to realism, 97\n\n  Anti-Euclidean geometry, 69\n\n  Apotheosis of the definition, 78\n\n  Apparatus, man's perceptive, 357\n\n  Apparent vassalage of life, 329-30\n\n  Appearance, dynamic, space as, 104-7\n\n  Appetences, materialistic, of empiricists, 340\n\n  Appulses of life against matter, 328\n\n  Appurtenances of materiality, 331\n\n  _A priori_, 185, _ff_\n\n  -- -- knowledge, 110-5\n\n  Aprioriness, 112, 116\n\n  Apriority, the principle of, 113, 223\n\n  Arbitrariness of the common mensurative quantities, 41\n\n  Archeological evidences, 332\n\n  Archimedes, 70\n\n  Aristotle, 322\n\n  Arrangement of the contents of the kosmos, 317-8\n\n  Art and the criterion of truth, 323\n\n  -- -- -- higher consciousness, 181\n\n  -- evidences of new faculties in the field of, 335\n\n  Assemblage of mathetic contrivances, 335\n\n  Assumptions, category of, 189\n\n  -- procedure based upon, disadvantages of, 163\n\n  At-one-ment, 224, 252, 270\n\n  -- defined as the end of evolutionary activity, 1\n\n  Attachment of the consciousness to the order of becoming, 305\n\n  Attainment, the difficulty of, by the intellectual method, 206\n\n  -- of space-consciousness, 225\n\n  Attitude, the need of a more liberal, 340\n\n  -- towards things spatial, 306\n\n  Audital nerves, notes that transcend the grasp of, 355\n\n  Aura, pulsating, 348\n\n  Automatism of the intellect, 253, _ff_; 338\n\n  Avenues of knowledge opened to inner perceptive faculties, 353\n\n  Awakening of the faculty of awareness in a new domain, 90\n\n  -- -- -- pituitary body and the pineal gland, 353\n\n  -- pituitarial, general results of, 355-8\n\n  Awareness, a determinant of conception, 120\n\n  -- as a gauge of the existence of things, 161-202\n\n  -- degrees of, 165\n\n  -- hyperspace, a symbol of a more extensive, 180\n\n  -- progress in, 171\n\n  -- Thinker's sphere of, 274, 283\n\n\n  B\n\n  Baltzer, 66\n\n  Barrier, the, to the Thinker's certitude, 186-8\n\n  Barriers, freedom determined by absence of, 280\n\n  Becoming, endless labyrinth of, 220\n\n  -- the kosmos, in a state of, 265\n\n  -- the order of, 305\n\n  Becomings, infinity of, 234\n\n  Becquerel rays, 336\n\n  Being, identification of consciousness with, 205\n\n  -- kathekotic, 210\n\n  Being, the interiority of, 290\n\n  -- the numericity of, 309\n\n  -- the range of, co-extensive with reality, 168\n\n  Beings, discarnate, 154\n\n  Beltrami, Eugenio, 1-2, 63-5, 66-7\n\n  -- -- and the manifold, 317\n\n  -- -- and the pseudosphere, 64-5\n\n  Besiegement of matter by life, 329-30\n\n  Bessel, on geometry as incomplete, 58\n\n  Bewilderment, new realities cause, 169-70\n\n  -- of the mind, 182\n\n  Biogenetics, passage from mechanics to, 357\n\n  -- turning to, from mechanics, 322\n\n  Birth, as a fourth dimensional process, 159\n\n  Blind men and the elephant, 257\n\n  Bolyai, Janos, 2-3, 46, 60, 66, 83, 87\n\n  Boundaries of a hypercube, 136-8\n\n  Brain, the physics of, 292\n\n  Brain-consciousness, 184, 186\n\n  -- standards of, 190\n\n  Brotherhood, and the tendencies of the Thinker's consciousness, 301\n\n  -- the keynote of the intuition, 252\n\n  Bulwarks, the formal, of geometry, 77-9\n\n\n  C\n\n  Carpet, the sensible world as a, 196\n\n  Carus, Paul, on four-dimensional space, 254\n\n  -- -- on metageometricians, 102\n\n  -- -- on space, 110\n\n  -- -- on the representation of the tesseract, 136-7\n\n  Cayley, 3-4, 54-5, 66-7\n\n  Cell-activity as a performance _a priori_, 114\n\n  Cell-colony, 250, 301\n\n  Cell-consciousness as an aposterioristic phenomenon, 185\n\n  Centers, pyknotic, 232\n\n  Certainty, geometric, the basis of, 77-8\n\n  -- of mathetic conclusions, provisional, 37\n\n  Chaogenic period of involution, 208, _ff_\n\n  Chaogeny, 231, 233\n\n  -- the evolution of chaos into order, 4\n\n  -- the laboratory of, 193\n\n  Chaomorphogeny, defined, 4, 231-2\n\n  Chaos, as egg-plasm, 265\n\n  -- duration of, 268\n\n  -- movement of life in, 307\n\n  -- shores of, 307\n\n  Chaos-geometricity, 307\n\n  -- -intellectuality, 307\n\n  -- -materiality, 307\n\n  -- -spatiality, 307\n\n  -- -Theos-Kosmos, 215, 233-4, 237\n\n  -- -- -- a triglyph symbolizing kathekos, 11-2\n\n  Character of the universe, fixed by consciousness, 162\n\n  Characteristics, changes in the, of reality, 169\n\n  -- fourth dimensional, of the ether, 157\n\n  -- minds of similar, fall into zones of affinity, 124\n\n  -- of non-Euclidean space, 72-3\n\n  Chasm, kosmic, between the real and the ideal, 107\n\n  Chemists, speculative, and the ourth dimension, 155\n\n  Chrism, sacred, of creative mentality, 91\n\n  Circle, significance of process of squaring, 109\n\n  Circle, the symbology of, 309\n\n  Circuits, closed, our interests as, 167\n\n  Clairaudience, 352\n\n  Clairvoyance, 352\n\n  Clairvoyant perception, 349\n\n  Clause, enabling, of metageometry, 76-7\n\n  Clavius, Christoph, 46, 52, 83\n\n  Clifford, 66\n\n  Code, psychic, for systematizing cognitions, 190-1\n\n  Cognition, instantaneous, organ of, 258\n\n  -- intimacies of direct, 205\n\n  -- the method of, 186-8\n\n  Cognitions, intuitive, 145, 192-3\n\n  -- psychic code for systematizing, 190-1\n\n  Commensurable quality, dimension as, 140\n\n  Communal consciousness, 270, _ff_\n\n  Communalization and the intuition, 301\n\n  Conarium, the, 345\n\n  Conceivability and evolution of mind, 24\n\n  -- ultimate range of, 278\n\n  Conceiving, power of, derived from sense-experience, 26\n\n  Concept, as a shadow, 119\n\n  -- the hyperspace, gradual rise of, 27-8\n\n  Conception and awareness, 120\n\n  -- every, based upon prior experience, 25\n\n  -- freedom, the, of the mind from, 28\n\n  Conceptions, symbolic, dependence placed upon, 147\n\n  Concepts, and the Thinker, 244\n\n  -- intuitograms as, 248\n\n  -- quality of, dependent upon sense-experience, 26\n\n  Concepts, the perception of, 255\n\n  Conceptual, related to the objective, 291\n\n  -- the, and mathematicians, 71\n\n  Conceptualization, the act of, 4\n\n  -- the power of, dependent upon evolution of mind, 25\n\n  Conclusions, mathetic, complex dependence of, 37\n\n  Concrete hewn into shape by the intellect, 295\n\n  Congruency, the, between intellect and the universe, 319, _ff_\n\n  Congruity, the, between concepts and objects, 120\n\n  Connection, the, traced out between reality and object, 126\n\n  Co-originality of things, 303\n\n  Consciousness and kosmos, graph representing, 271\n\n  -- -- the character of the sensible world, 166\n\n  -- -- -- wake of life, 205\n\n  -- -- time, 224\n\n  -- angularity of, 120\n\n  -- _a priori_, 185\n\n  -- a replica of, and judgments, 205\n\n  -- as a scale, 165\n\n  -- -- barrier to ultimate knowledge, 207\n\n  -- -- determinant of dimension, 145\n\n  -- -- life, 295\n\n  -- -- variable quantity, 171\n\n  -- attachment of, to the order of becoming, 305\n\n  -- dawn of, and dimension, 179\n\n  -- degrees of, 162\n\n  -- deprived of the validity of notions, 314\n\n  -- evolution of, 250\n\n  -- expansion of, 164, 272\n\n  -- extended, 89-90\n\n  -- focus of, 290\n\n  Consciousness and the planes of the kosmos, 163\n\n  -- fragmentary view of, 260\n\n  -- freedom of, in the _genus homo_, 332\n\n  -- habitation of, 278\n\n  -- higher, and art, 181\n\n  -- -- planes of, 178\n\n  -- history of, 298-9\n\n  -- identification of, with objects, 189-90\n\n  -- intuitional, 255\n\n  -- keyed to the entire range of reality, 167-8\n\n  -- kinds of, 270, _ff_\n\n  -- kosmic, Elysian fields of, 235\n\n  -- of the primitive man, 253\n\n  -- organ of, 184, _ff_; 90\n\n  -- psychics of, 292\n\n  -- raised from the sensuous to the conceptual, 72\n\n  -- recipient of truths from dual sources, 26\n\n  -- sensible world as instrument of, 199\n\n  -- triple presentation of notions of space to, 72\n\n  -- turned inward, 283\n\n  -- unification of the states of, 192\n\n  -- youthful, of mankind, 122\n\n  Consequences, the science of, 85\n\n  Considerations, mathematical, and reality, 128\n\n  Consistency, kosmic, 174\n\n  -- self, of hyperspatial hypotheses, 67\n\n  -- the criterion of geometry, 86\n\n  -- -- inconsistence of, 173\n\n  -- -- metageometrician curtailed by, 85\n\n  Construction, idealized, cannot be objectified, 144, 276\n\n  -- ideal, the meaning of, 4\n\n  -- mental, trafficking in, 154\n\n  Consubstantiality of intellectuality and spatiality, 331, _ff_\n\n  Continuity of the psychic plasm, 260\n\n  Contrivances, mathetic, the passing of, 241\n\n  Convergence of parallel lines, 315\n\n  Convolutions, duplex, 146\n\n  Co\u00f6rdinates, as spatial determinants, 203\n\n  -- of hyperspace, 94, 97, 99, 115-6\n\n  -- systems of, 301, _ff_\n\n  -- three, 132, 232\n\n  Copolarity of ideas and objects, 127\n\n  _Corpora quadrigemina_, 345\n\n  Corpuscular orbits of particles, 152\n\n  Cosmos, significance of, 4\n\n  Creation, impregnated screen of, 232\n\n  Creations, the, of the intellect, 173\n\n  Criterion of geometry as consistency and convenience, 86\n\n  -- -- truth, 323-4\n\n  Crookes, 344\n\n  Cube, as succession of planes, 148\n\n  -- generation of the, 144\n\n  -- illustrated, 133\n\n  -- the generating, 134\n\n  Curiosities, analytic, energy spent in elaborating misappropriated, 39\n\n  Current, electric, as component in the fourth dimension, 153\n\n  Curvature of space, the doctrine of, formulated by Riemann, 5\n\n  -- -- -- valid element in the notion of, 305\n\n  -- the measure of, 61\n\n  Curved space and metageometricians, 316\n\n\n  D\n\n  D'Alembert, _note_, 51\n\n  Dante, 181\n\n  Darwin, 181\n\n  Day, the Great Kosmic, 213\n\n  De Tilly, 66\n\n  Death, as fourth dimensional unity, 159\n\n  -- of the intellect, 195\n\n  Decrement of the diametrical ratio to the circumference of circle, 312\n\n  Deeps, fearful, of kosmic mind, 71, 91\n\n  Definition, apotheosis of, 77-8\n\n  -- as an arbitrary determination, 100\n\n  -- the deification of, 37\n\n  Degrees of realism, 164-5\n\n  Deiform, basic idea of the, 5\n\n  Demarkation between reality and phantasy, the line of, 173\n\n  Deposits of life, 173\n\n  Descartes, 322\n\n  Designs, cut out in materiality by life, 264\n\n  Details, the power to dispose of innumerable, 303\n\n  Determinations, geometrical, the necessity of, inheres in logical\n   deductions, 78\n\n  -- the factors of conscious, 162\n\n  Determinative period of mental development, 31\n\n  Development of the intuitive faculty, future, 188\n\n  Diacritics of life, 286\n\n  Diameter, the ratio of, to the circumference, 311-2\n\n  Difference between concept and thing, 121-2\n\n  -- between mathematical and perceptual space, 120\n\n  -- between the ideal and the actual is dynamic, 108\n\n  Differential, among minds, 124\n\n  Difficulty of imagining the fourth dimension, 90\n\n  -- of propagating an intuition, 315\n\n  Difficulties in the acquisition of real knowledge, 325\n\n  -- of hyperspace, the logical, 139-41\n\n  Dilemma of metageometricians, 97-8\n\n  Dimension, a distinct stage in psychogenesis, 27\n\n  -- -- system of space measurement, 5\n\n  -- and analysis, 163\n\n  -- -- D'Alembert, _note_, 51\n\n  -- -- the action of a tartrate in, 156\n\n  -- as an arbitrary contrivance, 262\n\n  -- -- -- assemblage of elements, 93\n\n  -- -- direction, 140\n\n  -- -- extent, 98\n\n  -- -- lying near the surface of things, 159-60\n\n  -- -- space, 96\n\n  -- current definitions of, 98-100\n\n  -- denied legitimate anchorage, 265\n\n  -- determined by consciousness, 145\n\n  -- does not explain spiritualism, 155\n\n  -- evolution of the fourth, 44\n\n  -- fourth, direction of, 134\n\n  -- as \"jack of all trades,\" 156\n\n  -- impossible to actualize, 124\n\n  -- key, to non-Euclidean geometry, 90\n\n  -- movement of a plane into, 147-8\n\n  -- no motion of material masses into, 125\n\n  -- not _a priori_, 116\n\n  -- proof of the existence of the fourth, 136\n\n  Dimensionality, and the intellect, 200\n\n  -- as dependent upon the will of the investigator, 95\n\n  Dimensionality, conception of, purely conventional, 92\n\n  -- logical difficulties which beset, 97\n\n  -- of perceptual space, 72\n\n  -- -- space, the four, 142-3\n\n  Dimensions, four, no basis in consciousness for, 172\n\n  -- piled upon one another, 300\n\n  -- vanity, the, of segmenting space into many, 299\n\n  Direction as dimension, 99\n\n  Disappointment with the sensible world, 120-1\n\n  Discoveries, metageometrical, as excrescences of mental unfoldment, 131\n\n  -- never the result of methodic reflection, 29-30\n\n  Disorder, the edict of, and space, 230\n\n  Disorderliness, the fringe of, 277\n\n  Dispossessal of the intellect by the intuition, 195\n\n  Dissimilarities, besetment of, 291\n\n  Distinctions, between conceptual and perceptual space, fundamental, 73-5\n\n  Diversity, a result of the fragmentative tendency of life, 6\n\n  -- as transfinite quantity, 42\n\n  Divinity, prize of, won by the intellect, 43\n\n  Duadic phase of evolution, 210-1\n\n  Duodim, defined, 6; 128, 129, 134, 177\n\n  -- consciousness, 163\n\n  Duopyknon, 212, 213\n\n  -- the meaning of, 6\n\n  Duopyknosis, 208-10, 218\n\n  -- as a stage in the evolution of space, 6\n\n  Duration and space, 224\n\n  -- eternal, 234\n\n  Dynamic appearance, space as, 104, 107\n\n  Dynamism, of the intellect, functional, 303\n\n\n  E\n\n  Earth, the nebulous youth of, 335\n\n  -- a new, 358\n\n  East effloresces as West, 306\n\n  Ego, the, 270\n\n  -- and percepts, 244\n\n  -- as sovereign, 244\n\n  -- Thinker's dependence upon, 245\n\n  -- -- treatment of, 246\n\n  Egopsyche, 265\n\n  -- as an agency of self-consciousness, 258\n\n  -- defined, 7\n\n  Egos, compared to choppy sea, 256-7\n\n  Egypt, birth-place of geometry, 44\n\n  Eisenlohr, 44\n\n  Elaborative period of the non-Euclidean geometry, 66-70\n\n  -- -- of mental development, 31-32\n\n  Element of evolution, basal, 259\n\n  Elements of the non-Euclidean geometry, 79-80\n\n  Elephant and blind men, 257\n\n  Ellis, Wilmot E., on ether as four-dimensional, 157-9\n\n  Embrace of direct cognition, 205\n\n  Engenderment of space, 260\n\n  Ensoulment of the world-plasm, 329\n\n  Entities, hyperspace, and the phenomenal world, 128-130\n\n  Enumeration, reformation of the system of, 38\n\n  Enveilment of consciousness, 273-4\n\n  Environment, artificial and natural, 162\n\n  Epiphysis, the, 345\n\n  Equidistantial, described by hyperspatial perpendicular, 80\n\n  Essence, elemental, as world-plasm, 329\n\n  Ether as possessing fourth dimensional characteristics, 157\n\n  Ethics recast in a new mold, 358\n\n  Euclid, 46, 70, 83, 263\n\n  -- and the parallel-postulate, 45\n\n  -- never-dying elements of, 53\n\n  Evidences, mathematical, exemplifies intellectual evolution, x\n\n  -- the vanity of fragmentary, 204\n\n  Evolution, a continuous process, 327\n\n  -- and the norms of reality, 175\n\n  -- basal element of, 259\n\n  -- commencement of, 232-3\n\n  -- intellectual, forward movement of, 184\n\n  -- kosmic, vicissitudes of, 215\n\n  -- laws of, govern mind, 28\n\n  -- mental, results of, 122\n\n  -- of faculties, synchronous with evolution of matter, 332\n\n  -- -- material characteristics of the earth, 335\n\n  -- -- organs, time required for, 188\n\n  -- preparation of the field of, 218\n\n  Exaltation of matter into spirit, 329\n\n  Examples of new perceptive evidences, 341-3\n\n  Existence on a higher plane, states of, 162\n\n  Experience, corroborative testimony of, denied hyperspace, 263\n\n  -- prior, and conception, 25\n\n  Experiences, spatial, systematization of, 78\n\n  Extension, space as an unbounded, 61\n\n  Extra-spatiality, degraded into spatiality, 261\n\n  Extravaganza, mathematical, and the fourth dimension, 156\n\n  Eyes, as Roentgen rays, 353\n\n\n  F\n\n  Fact-mass, 289\n\n  Factors of conscious determinations, 162\n\n  -- four, of the criterion of truth, 324-5\n\n  Facts as facets of truth, 284\n\n  -- attempts to reform, 304\n\n  -- logic as symbolism of, 287\n\n  -- -- the modeler of, 288\n\n  Faculties, dual derivation of, 162\n\n  -- evolution of, synchronous with evolution of material\n   qualities, 332, _ff_\n\n  -- extended, 239\n\n  -- foreshadowed by the hyperdimensional, 131\n\n  -- higher, man must win his way to, 357\n\n  -- new faculties evidenced by four-space, 24-29\n\n  -- rudimentary nature of, 341, _ff_\n\n  -- the source of, 349\n\n  Faculty and the intellect, 247, _ff_\n\n  -- as transmissible character, 251\n\n  -- determined by function, 336\n\n  -- greatly extended, 117\n\n  -- higher than the intellect, 126\n\n  -- I-making, self-consciousness as, 243\n\n  -- intuitive, 185\n\n  -- of perception in higher worlds, 353\n\n  -- -- awareness, the awakening of, 89-91\n\n  -- overshadowed by the intellect, 188\n\n  -- of perception in higher worlds, 353\n\n  -- rudimentary condition of, 192, _ff_\n\n  Faculty, outcropping of, 249-51\n\n  Failure of efforts to justify the objective existence of\n   four-space, 125-6\n\n  Failures at solving the parallel-postulate, outcome of, 48, 83\n\n  Faith, dispossessed by knowledge, 358\n\n  Fay, mathetic, 160\n\n  Fechner, 39\n\n  Finity and unboundedness of space, 76\n\n  Flammarion, Camille, 341-2, 344\n\n  Flexity, as property of hyperspace, 63\n\n  Flexure, space as a, 305\n\n  Fluxional, between sense objects and ideal representations, 122-3\n\n  Fluxion, psychic, as difference between memory image and\n   object, 7-8; 122\n\n  Focus of consciousness, 163, 290\n\n  -- -- the intellect, 310\n\n  Fohat, and the creation of morphons, 219\n\n  -- as creative energy, 213\n\n  -- Creator, 8\n\n  Fohatic energy, 226\n\n  Form, as vehicle of life, 330\n\n  -- definition of, 8\n\n  -- driven back over the path of its genesis, 327\n\n  -- pure, cannot exist in nature, 294\n\n  -- the idealty of, 110\n\n  -- the universe not a pure, 108\n\n  Formative period of mental development, 31\n\n  -- -- of non-Euclidean geometry, 55\n\n  Formula, eight different alcohols from one, 155\n\n  Four-dimensionality, justification of, 176\n\n  Four-space, a curiosity of analytics, 40\n\n  -- and Riemann, 27\n\n  -- as a divertisement, 175\n\n  -- consciousness does not act in the, 172\n\n  -- existence of, denied, 171-2\n\n  -- movement of matter in, 157\n\n  -- reality of, glibly proclaimed, 154\n\n  -- the, 8, 240\n\n  -- -- denizens of, 353\n\n  -- -- domain of, 154\n\n  -- -- study of, 124\n\n  -- use of analogies to popularize, 128\n\n  Fourth dimension, analogical reasoning of, 177-9\n\n  -- as a transcendental problem, 140\n\n  -- -- an attitude of the intellect, 200\n\n  -- electric current as a component in, 153\n\n  -- imaginability of, 106-7\n\n  -- Simon Newcomb on, 125\n\n  Fragmentariness of the intellectual method, 164\n\n  Fragmentary view of the universe, 260\n\n  Fragmentation, tendency to, 296\n\n  -- harks back to cell-division, 301\n\n  Freedom, a new, 338\n\n  -- determined by absence of bonds, 280\n\n  -- mathematical, 60\n\n  -- mental, 66-7\n\n  -- now dawning for the mind, 32\n\n  -- of consciousness in _genus homo_, 332\n\n  -- regal freedom of the mind, 118\n\n  -- three degrees of, 125\n\n  -- unrealizable, for the Thinker, 255\n\n  Fringe, chaogenetic, 308, 309\n\n  -- of disorderliness, 257\n\n  -- -- kathekosity, 229\n\n  Frischauf, 66\n\n  Full, the universe as a, 310\n\n  Function determines faculty, 336\n\n  Functions back of latent faculties, indefinite, 343\n\n  -- cellular and histologic, 253\n\n  Functioning, instinctive, of the intellect, 289\n\n  Fundamentals, totality of, kosmic, 265\n\n\n  G\n\n  Gamut of realism, 169\n\n  Gauge, awareness as a, 161-202\n\n  Gauss, Charles Frederick, 8-9, 56-8, 59, 174\n\n  -- -- -- as formulator of the non-Euclidean geometry, 57\n\n  Geminos of Rhodes, 45, 52\n\n  Generability, as property of space, 62\n\n  -- of hyperspace, Keyser on, 144\n\n  -- -- space by lines, 143\n\n  Generation, the, of the hypercube, 134\n\n  -- of the hypertetrahedron, 135-6\n\n  Genesis, of space, 211, 227, _ff_\n\n  -- -- the earth, its nebulous youth, 335\n\n  -- -- -- form, 327\n\n  -- -- -- sensible world, 167\n\n  _Genus homo_, freedom of consciousness in, 332\n\n  Geometricity, 266\n\n  Geometries, non-Euclidean, based upon a negation of the latent\n   geometrism, 262-3\n\n  -- three possible, 54\n\n  Geometrism, engenderment of, 262\n\n  -- established by life, 264\n\n  -- kosmic geometrism, establishment of, 231\n\n  -- latent, 257\n\n  -- native geometrism of space, 262\n\n  -- rediscovered by the intellect, 9-10\n\n  Geometrism, the basis of, 237, 261\n\n  Geometry, anti-Euclidean, 69\n\n  -- and the study of magnitudes in space, 203\n\n  -- artificial, 262-4\n\n  -- a two-fold, 59\n\n  -- breakdown of, 266\n\n  -- determinative period of the non-Euclidean, 61-4\n\n  -- diverse systems of, 323\n\n  -- Euclid's Elements of, 53-4\n\n  -- formal bulwarks of, 77-9\n\n  -- _Imaginary Geometry, The_, 60\n\n  -- natural geometry, 261\n\n  -- Plato and the divine geometry, 193\n\n  -- possible systems of, 174\n\n  -- radical essence of pure, 85\n\n  Geometry, non-Euclidean, at variance with the parallel-postulate, 83-5\n\n  -- -- based upon a misconception, 91\n\n  -- -- determined by qualitative differences, 140\n\n  -- -- elaborative period of, 66-70\n\n  -- -- first published treatise on, 60\n\n  -- -- formative period of, 55\n\n  -- -- the final issue of, 88-9\n\n  -- -- growth and development of, 92\n\n  -- -- invalidation of, 174\n\n  -- -- key to, 90\n\n  -- Lambert's Non-Euclidean geometry, _note_, 51\n\n  -- -- popularization of, 66-7\n\n  -- -- Schweikart's treatise on, 59\n\n  -- -- self-consistence of, 57\n\n  -- -- some elements of, 79-80\n\n  -- -- superperceptual knowledge of, 72\n\n  Geometry, non-Legendrean, 70\n\n  -- symbolic, and commensurable quality, 145\n\n  Gerling, as correspondent of Gauss, 58\n\n  Germ-plasm, continuity of, 260\n\n  Giantism, 347\n\n  Glimpse, a, of the reality of space, 314\n\n  Glorification of the flesh, 227\n\n  Groups, transformation, discovery of, 30\n\n  Guide-posts to a new domain, 273, 277\n\n  Gulf, interposed between manifestation and non-manifestation, 206\n\n\n  H\n\n  Halley, memoir of, mastered by La Grange, 50\n\n  Halstead, G. B., _note_, 49, 66; _note_, 79\n\n  Heindel, Max, _note_, 353\n\n  Helmholtz, 39, 66\n\n  Hewer, the, of the concrete, the intellect as, 295\n\n  Hinton, C. H., 154\n\n  -- on the fourth dimension, 153\n\n  History, the, of mind, three great epochs of, 31-32\n\n  Homogeneity, the, of realism, 165\n\n  Ho\u00fcel, J., 60, 66\n\n  Hudson, 344\n\n  Humanity and the exalting power of life, 331, _ff_\n\n  Hypercube, boundaries of, and mirrors, 136-9\n\n  -- the generation of, 134, 145\n\n  Hyperdimensional, as a prophecy of new faculties, 131\n\n  Hyperspace, 175\n\n  -- a monument to the intellect, 356\n\n  -- and the involved procedure of arriving at a recognition of its\n   relations, 73\n\n  -- and the passage thither, 293, _ff_\n\n  Hyperspace as a figurative mountain-peak, 200\n\n  -- -- -- movement, 44-68\n\n  -- -- an all-powerful something, 160\n\n  -- -- -- idealized construction, 10\n\n  -- -- evidence of new faculties, 24\n\n  -- -- _ignes fatuii_, 154\n\n  -- -- illusion, 202\n\n  -- a symbol of higher consciousness, 180-4\n\n  -- concept of, as an evolutionary quantity, 44-5\n\n  -- confounded with real space, 91\n\n  -- creation of, 335\n\n  -- denied the corroborative testimony of experience, 263\n\n  -- direction of, baffling to mathematicians, 146\n\n  -- discovery of, a sign of mental evolution, 180\n\n  -- domain of, a fairy-land, 239\n\n  -- magnitudes of, the non-sensuous, 72\n\n  -- mysterious hiding place in, 129\n\n  -- the logical difficulties of, 139-141\n\n  -- the six pillars of, 62-3\n\n  Hyperspatiality as the toys of childhood, 241\n\n  Hypertetrahedron, the, 135\n\n  Hypertrophy of the pituitary body, 347\n\n  Hypervolume, the, 134\n\n  Hypothesis, a superfoetated, 141\n\n  -- four-space, utility of, 129; 154\n\n  Hypotheses, admissibility of, 118\n\n  -- incompatibility of the non-Euclidean, 67; 71\n\n  -- Riemannian, 80\n\n  -- solution of, Bolyai, 31\n\n  Hyslop, James H., on the logical difficulties of hyperspace, 139-141\n\n\n  I\n\n  Icosahedron, examination of the, 285\n\n  Ideal, and the real, kosmic chasm between, 107\n\n  -- perceptual value of, 275, _ff_\n\n  Ideas and words as symbols, 126-7\n\n  -- Malebranche on, 222\n\n  -- realism of, 24-35\n\n  -- the symbolism of, 205\n\n  Identification of consciousness with being, 205-6\n\n  -- with the objects of study, 189-90\n\n  Identity of things-in-themselves, 353\n\n  _Ignes fatuii_ and hyperspaces, 154\n\n  Illustration of plane-rotation, 148-150\n\n  -- -- the tesseract, 133\n\n  Images, totality of, recoils upon us, 167\n\n  Imaginability of the fourth dimension, 90, 106-7\n\n  Imagination, premises of, the mathematical, 146\n\n  Impossibility of plane-rotation, the structural, 151-3\n\n  Impressions, the symbolism of, neurographic, 186\n\n  Impulse, the satisfaction of the original creative, 310-1\n\n  Incomprehensibility of reality to the intellect, 126\n\n  Incongruity, life estranged by a radical, 325\n\n  Individual as space, 223\n\n  Ineptitude of intellectual determinations for vitality, 314\n\n  Infinite, interpreted in the terms of the finite, 82\n\n  Infinitesimals of unity, numbers as, 41\n\n  Infinity as a process, 109\n\n  -- of becomings, 234\n\n  Infinity of parallels through a given point, 70\n\n  -- -- space, a capital illusion, 195\n\n  -- the concept of, 277\n\n  -- -- innate dread of, 103\n\n  -- -- relativity of, 194\n\n  Influence of abstract thought, 33\n\n  -- -- Kant on the non-Euclidean geometry, 49-50\n\n  -- -- La Grange, 51-2\n\n  -- -- the intellect, 315\n\n  Infundibulum, the, 345\n\n  Inner organs of perception, 352\n\n  Innermost, the, realities of things, 356\n\n  _Insouciance_ of the geometer, 96, 294\n\n  Instant-exposure and intellect, 311\n\n  Instrument, intellect likened to a color-bearing, 313\n\n  -- for the measurement of the passage of space, 297\n\n  -- of consciousness, the sensible world as, 199\n\n  -- -- life, form as an, 328\n\n  Integers, as fractional parts of unity, 41\n\n  Intellect and its domination of the intuition, 333\n\n  -- -- -- final union with the space-mind, 194\n\n  -- -- -- topography, 312\n\n  -- -- spatiality, 263\n\n  -- -- the deposits of life, 173\n\n  -- -- -- designs cut by life in materiality, 264\n\n  -- -- -- dictum of Sensationalists, 26\n\n  -- -- -- instant-exposure, 311\n\n  -- -- -- intuitive faculty, 247, _ff_\n\n  -- -- -- prize of divinity, 43\n\n  -- as a color-bearing instrument, 313-4\n\n  -- -- -- searchlight, 168\n\n  Intellect as a fashioner of phenomena, 199, _ff_\n\n  -- -- hewer of the concrete, 295\n\n  -- -- sole interpretative agency, 166-7\n\n  -- automatism of, 253, 338\n\n  -- cannot seize life, 282\n\n  -- crowned by a diadem of gold, 338\n\n  -- dominated by the intuition, 250\n\n  -- fashioned for matter only, 231\n\n  -- follows in the grooves of logic, 294\n\n  -- hyperspace as a monument to, 356\n\n  -- in the field of vitality, ix\n\n  -- its aptitude for starts and stops, 292, 302\n\n  -- -- instinctive tendency to fragmentate, 296\n\n  -- quality, determined by mathematics, 337\n\n  -- makes for individuality, 252\n\n  -- misses the ceaselessness of life, 201\n\n  -- _modus vivendi_ of, its influence upon knowledge, 184\n\n  -- the constitution of, 291\n\n  -- -- cut and mode of, 167\n\n  -- -- focus of, 310\n\n  -- -- illusion of, 246\n\n  -- -- illusionary character of, 323\n\n  -- -- incomprehensibility of reality to, 126\n\n  -- -- instinctive functioning of, 289\n\n  -- -- judgments of, 302\n\n  -- -- moods of, 202\n\n  -- -- predominating tendency in, 320-1\n\n  -- -- scientific tendence of, 165-6\n\n  -- -- struggle of, against dispossessal, 195\n\n  -- unsuited to deal with reality, 322, _ff_\n\n  Intellectuality and reality, 304\n\n  -- -- spatiality, consubstantial, 331\n\n  Intellectuality as co-extensive with spatiality and materiality, 236\n\n  -- the source of, 260\n\n  -- -- Thinker makes his own, 242\n\n  Intelligence and automatism, 253, _ff_\n\n  -- the Thinker as a pure, 243\n\n  -- transfinite intelligence and the degrees of realism, 164\n\n  Intent, the kosmic, of the intellect, 326\n\n  Interests, the sphere of our, as closed circuits, 167\n\n  Interior, the great, 290\n\n  Interiority of being, 290\n\n  Interpretation, the standards of, vary as consciousness varies, 171\n\n  Interstices of materiality, 264\n\n  Intuition and brotherhood, 252\n\n  -- -- communalization, 301\n\n  -- -- the riddle of spatiality, 325\n\n  -- as dispossessor of the intellect, 195\n\n  -- cannot be served by mathematics, 337\n\n  -- dominated by the intellect, 333\n\n  -- held in abeyance, 338, _ff_\n\n  -- its domination of the intellect, 250\n\n  -- the development of a spatial, not absurd, 145\n\n  -- -- need of a sympathetic attitude towards, 249\n\n  -- -- results of the development of, 357\n\n  Intuitional consciousness, 255\n\n  -- the superiority of, over the rational, 187\n\n  Intuitions and the lead of life, 191\n\n  -- -- -- Thinker, 27\n\n  -- free, mobile and formless, 166\n\n  -- the conceptualization of, 248\n\n  -- -- humility of, 165\n\n  -- -- nature of, 185\n\n  Intuitograms, as concepts, 248\n\n  Intuitograph as means of contacting the egopsychic consciousness by\n   the Thinker, 10\n\n  -- as super-concepts, 255\n\n  -- the difficulty of transmitting, 315\n\n  Invariability, the vaunted, of the laws of mathesis, 37\n\n  Invariants, psychological, 24\n\n  Investigations, metageometrical, and the new sense-organs, 354\n\n  Involution, as antithesis of evolution, 10-11\n\n  -- kosmic involution, 226\n\n  -- of matter, 328\n\n  -- the movement of, 219\n\n  -- -- seven stages of, 208-12\n\n  Ions, creation of, 219\n\n  -- magnitude of a hydrogen ion, 225\n\n\n  J\n\n  Judgments _a priori_, Kant on, 85-6\n\n  -- and the faculty _a priori_, 190-1\n\n  -- -- zones of affinity, 124\n\n  -- based upon a replica of consciousness, 205\n\n  -- no trustworthy, can be predicated upon fragmentary knowledge, 282\n\n  -- of the intellect, 302\n\n  -- the lessening of error in, 256\n\n  -- -- more complex the more at variance with the nature of things, 75\n\n  -- -- synthesis of, 257-8\n\n  -- valid judgments long delayed, 170-1\n\n  Judicative power of mathematics, 180\n\n  Justification for a multi-dimensional quality in space, 262\n\n  -- of four-dimensionality, 176\n\n  Justification of sense-deliveries by one another, 76\n\n  -- -- the existence of the fourth dimension, 125\n\n\n  K\n\n  Kant, 85, 181, 182, 322\n\n  -- and the faculty of thinking, 261\n\n  -- -- -- idea of space, 322-3\n\n  -- influence of, on the hyperspace movement, 49-50\n\n  -- on the nature of things, 119\n\n  -- -- space as an intuition, 115\n\n  Kathekos, 233-4\n\n  -- as chaos, 266\n\n  -- -- symbol of Chaos-Theos-Kosmos, 11-12\n\n  -- symbology of, 234\n\n  Kathekosis, _note_, 227\n\n  Kathekosity, fringe of, 229\n\n  -- rock-bound coast of, 267\n\n  -- significance of, 12\n\n  Kathekotic consciousness, 272\n\n  -- period, 209\n\n  Key to the mysteries of nature, the fourth dimension as, 131\n\n  Keyser, Cassius Jackson, and freedom of the mind, 33\n\n  -- -- -- on attitude of metageometricians, 71\n\n  -- -- -- -- dimensionality, 94-5\n\n  -- -- -- -- four-dimensionality of space, 142-3\n\n  -- -- -- -- generability of hyperspace, 144\n\n  Klein, Felix, 12, 54-5, 66-7\n\n  Knowledge, all, relative, 101\n\n  -- barrier to the certitude of the Thinker's, 186-7\n\n  -- fabric of, 196\n\n  -- hypothetical nature of, 189, _ff_\n\n  -- immeasurable realm of, laid bare by the telescope, 298\n\n  Knowledge, mathematical, apriority of, questioned, 37\n\n  -- nature of the non-Euclidean, superperceptual, 72\n\n  -- real, difficulties of acquiring, 325\n\n  -- related to the stream of life and to the shore of materiality, 323\n\n  -- relative, degrees exist for, 164\n\n  -- sphere of, 184\n\n  -- systematization of, 126\n\n  -- ultimate, consciousness as a barrier to, 207\n\n  -- unification of, 256\n\n  Kosmogenesis, the latent geometrism of, 264\n\n  -- -- scope of, 237\n\n  Kosmometer, the, 297\n\n  Kosmos, and consciousness, graph representing, 271\n\n  -- arrangements of the contents of, 317-8\n\n  -- in a state of becoming, 265\n\n  -- magnitude of, 308\n\n  -- moods of, 311\n\n  -- space as the consistence of, 239\n\n  -- _see_ Cosmos, 12\n\n\n  L\n\n  Labor, division of, between the tuitional and the intuitional\n   faculties, 193-4\n\n  -- mathematical labors, significance of, 176\n\n  La Grange, Joseph Louis, 12, 39, 50-3, 321, 322\n\n  -- -- -- -- and the parallel-postulate, 51-2\n\n  Lambert, John, 58\n\n  -- -- and the theory of parallels, 48-9\n\n  Language, the passing of, 357\n\n  Legendre, 70\n\n  Leibnitz' dictum, 26\n\n  Leonard rays, 336\n\n  License, mathematic, permissibility of, 38\n\n  Lie, Sophus, 12, 67\n\n  -- -- and transformation groups, 30\n\n  Life, analysis incapable of dealing with, 319\n\n  -- and consciousness as one, 224\n\n  -- -- form rooted in pyknosis, 214, 328\n\n  -- -- the fourth dimension, 172\n\n  -- -- -- inaptitude of mathematics for, 179\n\n  -- -- -- power to create, 297\n\n  -- as agent of creation, 343\n\n  -- -- creative agent, 264\n\n  -- -- expression, 303\n\n  -- -- vassal of materiality, 329-30\n\n  -- causative agencies in prolonging, 123\n\n  -- current of, as engendering element, 331\n\n  -- deposits of, and the intellect, 173\n\n  -- estranged by a radical incongruity, 325\n\n  -- exalting power of, and humanity, 331, _ff_; 342\n\n  -- exhibition of its remains, 282\n\n  -- flow of, 265, 297\n\n  -- form as an instrument of, 328\n\n  -- indescribable signs of, 219\n\n  -- infinitive action of, 327\n\n  -- intellect has no aptitude for, 231\n\n  -- intuitive, values of, x\n\n  -- larger life of the Thinker, 196\n\n  -- lead of, followed by intuition, 191\n\n  -- limits of, fixed by consciousness, 198, _ff_\n\n  -- most solid facts of, as shadows, 195\n\n  -- motility of, admits of endless variations, 35\n\n  Life, movement of chaos in, 307\n\n  -- passage of, 201\n\n  -- passage through spatiality, 288\n\n  -- power to manipulate, 303\n\n  -- recurrent movement of, 307\n\n  -- totality of egoic, 243\n\n  -- undulations in the current of, 329\n\n  -- uniqueness of, 41\n\n  -- wake of, and consciousness, 205, 288\n\n  Life-cycle, evolutionary results of the, 259\n\n  Life-stream, the, 278\n\n  Light, polarized, and the fourth dimension, 156\n\n  -- consciousness as a spreading, 168\n\n  Light-years and space, 278\n\n  Limits, the sphere of, consciousness as, 163\n\n  Line, as generating element, 144\n\n  -- the straight, a curved, 76\n\n  Lineage of every principle runs back to monopyknosis, 213\n\n  Lines, perpendicularity of, in four space, 130\n\n  Lobachevski, 55, 60, 66, 87\n\n  Lodge, Sir Oliver, 340-1; 344\n\n  Logic as architect, 118\n\n  -- conventional forms of, 262\n\n  -- data of, 286\n\n  -- intellect follows in the grooves laid out by, 294\n\n  -- miracle power of, over facts, 288-9\n\n  -- rules of, allowability under, 163\n\n  -- -- -- the game of, 164\n\n  -- as symbolism of facts, 287\n\n  Logos and the limits of space, 193\n\n  -- being of, 218\n\n  -- body of being of, 214\n\n  -- consciousness of, 204\n\n  -- creative, 12, 213\n\n  -- the, 216-7\n\n  Lorenz, 83\n\n  Luther, Martin, 181\n\n\n  M\n\n  Magnitudes, geometry, a study of, 203\n\n  -- non-sensuous, of hyperspace, 72\n\n  Makrokosmic consciousness, 270\n\n  Malebranche, N., 222; see _note_\n\n  Manifestation and non-manifestation, 206\n\n  Manifold, finite, though unbounded, 70\n\n  -- manufactured by the analyst, 77\n\n  -- the, 61-2, 172\n\n  Manifoldness, analytical manifoldnesses, as mental excitants, 89-90\n\n  -- as a conventional construction, 262\n\n  -- -- an intuition, 318\n\n  -- of space, as a near-truth, 317\n\n  Manning, 82, 157\n\n  -- and maneuvers in the fourth dimension, 152\n\n  Manvantara, 213, 228-9, 232\n\n  -- as evolution and involution combined, 13\n\n  Mask, logic as a, 288-9\n\n  Mass-termini, of lines, 236\n\n  Mastery of life over matter, 330\n\n  -- of the sensible world, 196-7\n\n  Materiality as a deposit of life, 295\n\n  -- -- consubstantial with spatiality, 331, _ff_\n\n  -- becoming spatialized, 261\n\n  -- characteristics of, 260\n\n  -- engendered by kosmic mind, 261\n\n  -- engenderment of, 294\n\n  -- interpenetrative with spirituality, 236\n\n  -- interstices of, 264\n\n  -- neglect of, 320\n\n  Materiality, shore of, 323\n\n  -- transmuted into spirituality, 328\n\n  Mathematicians and the definition, 37\n\n  -- -- -- limitations of consciousness, 178\n\n  -- -- -- phenomenal world, 118-9\n\n  -- as prophets, 336\n\n  -- respected by humanity, 337\n\n  -- the gods of mathesis, 84\n\n  Mathematics, a determinant of the quality of the intellect, vii\n\n  -- and the criterion of truth, 323\n\n  -- -- -- kosmic intent of the intellect, 326\n\n  -- as symbology, 184\n\n  -- Euclidean, 140\n\n  -- fails when it encounters life, 337\n\n  -- its inaptitude for life, 179\n\n  -- -- kingdom yielded to kinetics, 358\n\n  -- possesses no judicative power over life, 180\n\n  -- the orthodoxy of, 34\n\n  Mathesis, and conceptional space, 203\n\n  -- conduct of the intellect in the field of, vii\n\n  -- definition of, 13\n\n  -- domain of, as origin of fourth dimension, 130\n\n  -- gods of, 84\n\n  -- marvelous domain of, 118\n\n  -- realm of, not submissive to laws of sensible space, 34\n\n  -- things of, as emblems of kosmic forces, 239\n\n  -- world of, 67, 127\n\n  Matter and mind, syncretization of, 299\n\n  -- appulses of life against, 328\n\n  -- as a deposit of life, 330\n\n  -- honeycombs of, 339\n\n  Matter metamorphosed by life, 328\n\n  -- mind as wedded to, 249\n\n  -- movement of, in four-space, 157\n\n  -- qualities of, and faculties, 332\n\n  -- seven planes of, 212\n\n  -- spiritualization of, as end of evolution, 327\n\n  -- totality of, acted upon by totality of life, 328\n\n  -- unity of, with space, 260\n\n  -- universum of, 261\n\n  Matutinal ceremonials of creation, 221\n\n  Measurability, as property of space, 62-3\n\n  Measurement of hyperspaces, the science of, 73\n\n  Measurements, space determined by the number of, 61\n\n  -- systems of space, 91\n\n  Measurer of space, _n_-dimensionality as a, 296-7\n\n  _Mecanique Analytique_, 50-1, 321\n\n  Mechanics of matter, the passage from, to the dynamics of life, 326\n\n  -- to biogenetics, the passage from 357\n\n  -- the turning from, to biogenetics, 322, _ff_\n\n  Mechanism, the doctrine of, due to analytics, 40\n\n  Memories, stored in the omnipsyche, 258\n\n  Men as gods, 289\n\n  Mental evolution, determinative period of, 31\n\n  -- -- elaborative period of, 31-32\n\n  -- -- formative period of, 31\n\n  Mentality, inner mysteries of creative, 91\n\n  -- infantile out-feelings of the, 336\n\n  -- the principle of, 210, _ff_\n\n  Mentalities, the adaptation of phenomena to, 184\n\n  Mentograph, the, 121\n\n  -- the basis of intellectual consciousness, 13\n\n  Mesozoic man and his environments, 347-8\n\n  Metageometrical investigations and the new sense-organs, 354\n\n  Metageometricians and hyperspace, 238\n\n  -- -- proof of rotation about a plane, 149-151\n\n  -- -- the curved space, 316\n\n  -- -- -- fourth dimension, 145\n\n  -- -- -- key to the mysteries of nature, 131\n\n  -- baffled by the direction of hyperspace, 146\n\n  -- dilemma of, 97-8\n\n  -- eschew sense-data, 73-4\n\n  -- Keyser on the attitude of, 71\n\n  -- perceptual obliquity of, 102\n\n  -- -- seed thought of, 182\n\n  -- turned to idealized constructions, 263\n\n  Metageometry, as a stepping stone, 238\n\n  -- defined, 13\n\n  -- Riemann, the father of, 63\n\n  -- the fabric of, 66\n\n  Metamorphosis of matter by life, 328\n\n  -- of the monopyknon, 214\n\n  Metamorphotic stage of space-genesis, 228\n\n  Metaphysician, the preserves of, usurped by mathematicians, 141\n\n  Meta-self as medium of kosmic consciousness, 13, 217\n\n  Methods of the ego, 344\n\n  -- scientific, and the ample explication of phenomena, 154\n\n  Mikrokosmic consciousness, 270\n\n  Mind, amenability of, to the evolutionary movement, 28-32\n\n  -- and freedom from conception, 28\n\n  -- -- -- -- problems of physical existence, 32\n\n  -- -- the new freedom, 33\n\n  -- as vehicle of life, 285-6\n\n  -- -- wedded to matter, 249\n\n  -- coevalism of, and space, 240\n\n  -- consubstantial with space, 239-40\n\n  -- divine, of the kosmos, 176\n\n  -- evolution of, and conceivability, 24\n\n  -- of the duodim, 128\n\n  -- powers of, not attained simultaneously, 334\n\n  -- profound influence of hyperspace on the, 91\n\n  -- space as progenitor of, 224\n\n  -- three great epochs of the history of, 30-31\n\n  -- tuitional, the limitations of, 192-3\n\n  -- the unity of, with space, 224\n\n  Mind-principle, the quintopyknon as the basis of, 221, _ff_\n\n  Mirrors, three, and the hypercube, 136-8\n\n  Mobility, life as, 264\n\n  Molecule, the four-dimensional, Newcomb on, 159\n\n  Monadic phase of evolution, 210-1\n\n  Moneron and man, the gulf between, 28\n\n  Monopyknon, the, 213, 216\n\n  Monopyknosis, defined, 17, 208-10, 218\n\n  Monstrosities, mathetic, 202\n\n  Moods of the kosmos, 311\n\n  Morphogeny, 13, 233-4\n\n  Morphons, the creation of, 219\n\n  Motility, the scope of, 131\n\n  Movement, forward, in knowledge, 336\n\n  -- hyperspace as an evolutionary, 44-68\n\n  -- -- influence of Kant on, 49-50\n\n  -- of life in chaos, 307\n\n  Multi-dimensional quality of space, justification for, 262\n\n  Multiplication of hypotheses, 118\n\n  Mutuality, intuition as promoter of, 301, _ff_\n\n  Mysteries, inner, of creative mentality, 91\n\n  -- kosmic, low-lying plains of, 239\n\n  --- of life and mind, 326\n\n  -- -- the sciences and the four-space operator, 131\n\n  Mystery, kosmic, the revelation of, foreshadowed in manifold, 77\n\n  -- of reaction, 167-8\n\n  -- -- space decreases as copes of consciousness increases, 273, _ff_\n\n\n  N\n\n  _n_-Co\u00f6rdinates, 203\n\n  _n_-Dimensionality, as a phase of geometry, 300\n\n  -- -- property of intellectuality, 269\n\n  -- -- quality of conceptual space, 14\n\n  -- irreconcilable with perceptual space, 156\n\n  -- no justification for, 235\n\n  -- not surprising that the intellect fell upon, 296, _ff_\n\n  -- predicates concerning, 158\n\n  _n_-Space, imperceptibility of, 94\n\n  Nasir-Eddin, 46, 53\n\n  Natural symbols, ideas as, 126\n\n  Nature, representative, of things, 119\n\n  -- the vacuum-stillness of, 339\n\n  _Naught_ to _unity_, consciousness registers from, 165\n\n  Near-truth, 300, _ff_\n\n  -- as variation from criterion of truth, 324\n\n  -- based upon partial knowledge, 14\n\n  -- space-curvature as a, 306\n\n  -- space-manifold as a, 318\n\n  Nebulosity, 267\n\n  Necessity, as a bulwark of geometry, 77-8\n\n  Nerve-body, the, 349\n\n  Neurogram, 14\n\n  Neurograph, the Thinker's scrutiny of, 190\n\n  Neurographical communications, 244\n\n  Newcomb, Simon, 14-5, 66\n\n  -- -- and the four-dimensional molecule, 159\n\n  -- -- on the fourth dimension, 125\n\n  Non-Euclideans and the meaning of dimension, 97\n\n  -- -- things as they actually are, 71\n\n  -- dilemma of, 86\n\n  -- meaning of the term, 69\n\n  -- results obtained by, 88\n\n  Non-manifestation, as antithesis of manifestation, 106\n\n  -- and manifestation, 206\n\n  Non-methodic advancement of human consciousness, 299\n\n  Non-spatiality, cut off from spatiality, 266\n\n  Norm, as a value assigned by the intellect, 167\n\n  -- consciousness as a, 15; 161-202\n\n  -- of life, found by synthesis, 300\n\n  -- -- the natural geometry, 263\n\n  Note, the dominant, of communality, 301\n\n  Notions of space, triple presentations to the consciousness, 72\n\n  Number, as a phase of kosmogenesis, 309\n\n  -- numbers as infinitesimals of unity, 41\n\n  Numericity of being, 309\n\n\n  O\n\n  Objects and ideas, 126-7\n\n  -- passage of, into the fourth dimension, 130\n\n  -- of study, identification of consciousness with, 189\n\n  Obliquity, perceptual, of metageometricians, 102\n\n  Obscurantism, mathetic, 154\n\n  Occultist and the scientist, 350\n\n  Omnipsyche, 265\n\n  -- as agency of kosmic consciousness, 258\n\n  -- -- neglected factor of evolution, 259\n\n  -- defined, 15\n\n  One-space represented by a line, 134\n\n  Ontogeny recapitulates phylogeny, 27, 334\n\n  Order, kosmic, the difference between, and hyperspace, 143\n\n  -- mathematical, discovered by the intellect, 261\n\n  -- the fiat of, 208\n\n  -- the totality of, 265\n\n  Orderliness, 266\n\n  Organ, determined by function to be performed, 336\n\n  Organs broken up by life into special parts, 343\n\n  -- the evolution of, 188\n\n  Orthodoxy of mathematics, 34\n\n  Outcome of new adjustments, 256\n\n  Out-feelings of the Thinker, infantile, 336\n\n  Outlook, the spiritualization of our mental, 40\n\n\n  P\n\n  Papyrus, a hieratic, 44\n\n  Parallel-postulate and the surface of a sphere, 70\n\n  -- as basis of non-Euclidean geometry, 54\n\n  -- -- stated by Euclid, 55\n\n  -- -- -- -- Lobachevski, 55\n\n  -- -- -- -- Manning, 82\n\n  -- failures to demonstrate, 48, 82-3\n\n  -- in the Elements of Euclid, 16\n\n  -- Saccheri's proof of, 47\n\n  Parallels, convergence of, 80\n\n  -- infinity of, through a given point, 70\n\n  -- meet at infinity, 88\n\n  -- theory of, 47-9\n\n  Passage from mechanics to biogenetics, 357\n\n  -- from mechanics to dynamics, 326\n\n  -- from three-space to four-space, 292-3\n\n  -- of mathetic contrivances, 241\n\n  -- of space, 16, 297\n\n  Pathway of life, sinuosities in, 330\n\n  Patterns of the intellect, ready-made, 166\n\n  Pentagrammaton of space, 237\n\n  Perceptibility in the Thinker, the faculty of, 188\n\n  Perception, domain of, and the sensible world, 19\n\n  -- inner organs of, 352\n\n  -- replaced by conception, 255\n\n  Percepts, and the ego, 244\n\n  Period, chaomorphogenic, 208, _ff_\n\n  -- determinative, 61-5\n\n  -- kathekotic, 210\n\n  Perisophical nature of analytics, 320, _ff_\n\n  Perisophism, a, 300\n\n  Permissibility of mathetic license, unlimited, 38\n\n  Perpendicularity, 132, 177\n\n  -- of lines in the four-space, 130\n\n  Phantasy and reality, line of demarkation between, 173\n\n  -- the world of, 146\n\n  Phantom-ideal, the, 277\n\n  Phase of the world age, evolutionary, 209\n\n  Phenomena, efforts to explain, on the basis of the fourth dimension, 129\n\n  -- fashioned by the intellect, 199\n\n  -- fragmentary interpretation of, 302\n\n  -- mind's method of apprehending, 186\n\n  -- physical phenomena amply explained by scientific methods, 154\n\n  Phenomenal, the inverse of realism, 278\n\n  Philosophy and the criterion of truth, 323-4\n\n  -- regeneration of, 358\n\n  -- systems of, and zones of affinity, 124\n\n  Phosphorescence, dim, added to the unilluminated pool of\n   sense-consciousness by the intuition, 26\n\n  Phylogeny and ontogeny, 334\n\n  -- represented in ontogeny, 27\n\n  Physical as embodiment of all possibilities, 227\n\n  -- phenomena as space-activities, 229\n\n  Physicality, the principle of, 210, _ff_\n\n  Physics of the brain, 292\n\n  _Pi_ proportion, significance of, 311-2\n\n  Pickering on space-curvature, 279\n\n  Pineal gland, energized by the pituitary body, 348-9\n\n  -- -- the, 344, _ff_\n\n  -- -- not a vestigal organ, 354\n\n  Pituitarial awakening, general results of, 355-8\n\n  Pituitary body and the Thinker, 351\n\n  -- -- as an organ, 344-352\n\n  -- -- C. De M. Sajous on, 345\n\n  -- -- hypertrophy of, 347\n\n  -- -- location of, 246\n\n  -- -- the, 344, _ff_\n\n  Plane, as generating element, 144\n\n  -- movement of, into the fourth dimension, 147\n\n  -- realism appears to be divided into, 164\n\n  -- rotation about a, 130, 146-151\n\n  -- the seven planes of matter, 212\n\n  Planes, a cube as a succession of, 148-9\n\n  Plasm, kosmic, strivings of, 215\n\n  -- psychic, 260\n\n  -- the differentiation of, 214\n\n  Plato, 322\n\n  -- and the divine geometry, 193, 237\n\n  -- -- -- shadow consciousness, 281\n\n  -- on God as geometrizer, 181\n\n  -- -- ideas, 34\n\n  Play, sensuous, of the intellect, 295\n\n  Plenum, space as a, 230\n\n  -- the universe as a, 107\n\n  Poincair\u00e9, 90\n\n  Point, an infinity of parallels through a given, 70\n\n  -- as generating element, 144\n\n  -- position of, 132\n\n  -- succession of points, a line as, 148\n\n  Polarity between concrete and abstract, 294-5\n\n  Ponderability, as property of hyperspace, 63\n\n  Popularization of the fourth dimension, 128\n\n  -- of the non-Euclidean geometry, 66-7\n\n  Possibilities in the world of hyperspace, 129\n\n  -- of four-space, the marvelous, 130\n\n  Postulate-systems, the multiplication of, 85\n\n  Powers of mind, not attained at one and the same time, 334\n\n  Pralaya, 213\n\n  -- as kosmic quietude, 17\n\n  Predicates, all mathematical, not justified by the phenomenal, 128\n\n  Principle, ensouling, 214-5\n\n  Printing press, the invention of, and the laws of psychogenesis, 30\n\n  Problem, the fourth dimension as a transcendental, 140\n\n  Procedure, the involved, of arriving at hyperspace notions, 73\n\n  Process, evolution a continuous, 327\n\n  -- space as a dynamic, 307\n\n  Proclus, 83\n\n  Profundities, kosmic, and hyperspace, 156\n\n  Progression eastward terminates at the west, 306\n\n  Proof of rotation about a plane, 146-151\n\n  Propagation of intuitographs, difficulty of, 315\n\n  Propositions, geometric, subjunctive quality of, 38\n\n  Pseudosphere and the shape of space, 70\n\n  -- as basis of Beltrami's calculations, 17\n\n  -- the nature of, 64-5\n\n  Psychics of consciousness, 292\n\n  -- transmuted, 260\n\n  Psychogenesis, alchemy of, 225\n\n  -- kosmic, 272\n\n  -- laws of, and the printing press, 30, 116\n\n  Psychogenesis, mind's place in, 123\n\n  -- outgrown phase of, 302\n\n  -- outstanding facts of, 131\n\n  Psychogeny, defined, 17\n\n  Ptolemy, 45, 52, 83\n\n  Publicist, the mathematical, and the duodim, 128\n\n  -- the non-Euclidean, 146\n\n  Pyknon as a kosmic principle, 212\n\n  -- -- basis of space-genesis, 17\n\n  Pyknosis, 212\n\n  -- and involution, 210, 231\n\n  -- as a metamorphotic process, 214\n\n  -- the seven processes of, 17, 294\n\n\n  Q\n\n  Quality, dimension as commensurable, 140\n\n  Quantities, algebraic, cannot represent space, 125-7\n\n  Quantity, consciousness as a tri-space, 172\n\n  Quartodim, 18, 134\n\n  Quartodimensionality, 292\n\n  Quartopyknosis, 17, 208-10\n\n  Quartopyknotic, principle as basis of spirituality, 228\n\n  Quintopyknon, 220\n\n  Quintopyknosis and the _a priori_, 223\n\n  -- meaning of, 17, 208-10\n\n  Quintopyknotic principle as basis of mentality, 228\n\n\n  R\n\n  Race, humanity to evolve a new, 358\n\n  -- resplendent, 338\n\n  Rajah-Tamas, 220\n\n  Rajah Yoga, 357\n\n  Rational, valueless when unsanctioned by the intuition, 187\n\n  Reaction, as a mystery, 167-8\n\n  Real, Thinker as a part of the, 201\n\n  Realism and its degrees, 164-5, 279\n\n  -- -- the Thinker, 315\n\n  -- as life, 286\n\n  -- consciousness as touch-stone of, 168\n\n  -- found in the direction inverse to the phenomenal world, 278\n\n  -- homogeneity of, 165\n\n  -- illusionary nature of, 36\n\n  -- infinite gamut of, 36\n\n  -- its degrees and the states of consciousness, 197\n\n  -- new scope of, bewilders the mind, 170-1\n\n  -- not of mathematical import, 290\n\n  -- of concepts and the sensible world, 40\n\n  -- -- the domain of mathesis, 33-4\n\n  -- psychological quality of, 285\n\n  -- the thread of, 169\n\n  -- three dimensional scope of, 172\n\n  Realities, abstractionizing, 144, 294\n\n  -- incomprehensible to the intellect, 126\n\n  -- innermost, of things, 356\n\n  -- lesser give way to greater, 195\n\n  -- natural symbolism of, 128\n\n  -- our, non-existent to beings on spirit-levels, 35\n\n  -- shaped upon conventional models, 294\n\n  -- supersensuous, 180\n\n  Reality, as life and consciousness, 173\n\n  -- barrier to the cognition of, 189, _ff_, 273\n\n  -- comprehension of, 165, 303\n\n  -- current of, 322\n\n  -- defined as life, 18\n\n  -- flow of, 319\n\n  -- flowing stream of, 124\n\n  -- myriad ways of presenting itself, 287\n\n  -- naked contact with, 193\n\n  -- not an inscrutable quantity, 194\n\n  -- obscured, 302\n\n  -- of the four-space glibly asserted, 154\n\n  -- thread of, 200\n\n  -- universum of, 127\n\n  Realms, supersensuous, 338\n\n  Re-becoming, life and consciousness as a, 257\n\n  Reciprocity of consciousness and realism, 197-8\n\n  -- of the manifest and the unmanifest, 217\n\n  Recoil of images upon us, 167-8\n\n  Reflexive development of the intellect, 254\n\n  Relativity of all knowledge, 101\n\n  Religion and realism, 170\n\n  -- the changes in, 358\n\n  Remains of life exhibited to the intellect, 283\n\n  Remaking of moods by the intellect, 201\n\n  Replica of consciousness as basis of judgments, 205\n\n  Representation, sensuous, compared with a shadow, 119-120\n\n  Research, improvement in the methods of, 125\n\n  Reservoir, psychic, of evolution, 260\n\n  Residuum, the unexplained, 157-8\n\n  Revelational impressions, 29\n\n  Riddle of spatiality and the intuition, 325\n\n  Riemann, G. F. B., 18, 19, 54, 66-7, 87\n\n  -- and a limited space, 108-9\n\n  -- -- the curvature of space, 5\n\n  -- -- -- determinative period, 65\n\n  -- -- finite space, 103\n\n  -- -- -- four-space, 27\n\n  -- -- -- manifold, 317\n\n  -- -- -- as inventor of the manifold, 86\n\n  -- on the bases of geometry, 61-4\n\n  Roentgen rays, 336\n\n  Rotation about a line, 147-8\n\n  -- about a plane, 146-151\n\n  -- -- -- -- illustrated, 148-50\n\n  -- intra-corpuscular, 152\n\n  Rudimentary organs, vague functions lying back of, 343\n\n  Rule in the evolution of faculties, 333\n\n\n  S\n\n  Saccheri, Girolamo, 47-8, 52, 83\n\n  Sajous, C. De M., on the pituitary body, 345\n\n  Schematism, the suitability of the present, 320-1\n\n  Schweikart, Ferdinand Karl, 19, 66, 69\n\n  -- on the non-Euclidean geometry, 58-60\n\n  _Science Absolute of Space_, 60\n\n  Science and reality, 170-1\n\n  -- no longer empirical, 358\n\n  -- skeptical attitude of, 349\n\n  Scientist, method of the, opposed to the analyst, 84-5\n\n  Scopographic impressions, 19, 190\n\n  Screen, the impregnated, of creation, 232\n\n  Sea, choppy, and egos, 256-7\n\n  Search, the path of, for spatial understanding, 278\n\n  Seb, the god, 215\n\n  Sects, coplanar, 79\n\n  Self and the immensity of space, 223\n\n  -- of the universe, 217\n\n  -- the kosmic, 278\n\n  Self-consciousness, 270\n\n  -- and space, 223\n\n  -- as I-making faculty, 243\n\n  -- determination of, 302\n\n  Self-consistency, 173\n\n  _Sella turcica_, the, 345\n\n  Semi-Euclidean geometry, the, 70\n\n  Sensationalists on the intellect, 26\n\n  Sense-data, spurned by the metageometrician, 73\n\n  Sense-delivery, one, justified by the other, 76\n\n  -- symbolism of, 203-4\n\n  Sense-knowledge, the sphere of, extended, 339-40\n\n  Senses, the answer of, to new needs, 332\n\n  Sensible world, as domain of perception, 19\n\n  Sensographic impressions, as perceptions, 19\n\n  Senso-mechanisms of the intellect, 165\n\n  Sentience, the principle of, 210, _ff_\n\n  Septopyknon, the, 227\n\n  Septopyknosis, as kosmic materiality, 18, 208-10\n\n  Septopyknotic, the, principle, 228\n\n  Seven planes of matter, 212\n\n  Sextopyknon, the, 209\n\n  Sextopyknosis, defined as kosmic sensibility, 18, 208-10\n\n  Sextopyknotic, as emotional principle, 228\n\n  Shadow likened to a sense presentation, 119\n\n  Shadowgraphs, 281\n\n  Silences, nature's great, 339\n\n  Similitude of agent and principle, 243-6\n\n  Sinuosities in the pathway of life, 330\n\n  Socrates, 322\n\n  Solitariness of intellectual testimony, 76\n\n  Space and edict of disorder, 230\n\n  -- -- four-dimensional entities, 128-130\n\n  Space and quartopyknons, 219-20\n\n  -- -- self-consciousness, 223\n\n  -- -- the idealized construction, 144\n\n  -- as a dynamic process, 216\n\n  -- -- all-mother, 229-30\n\n  -- -- an assemblage of spheres, 143\n\n  -- -- becoming, 195\n\n  -- -- boundary of another space, 49-50\n\n  -- -- consistence of the kosmos, 239\n\n  -- -- curved, 164\n\n  -- -- dynamic appearance, 228, 307\n\n  -- -- eldest born of kathekosis, 215\n\n  -- -- engendered product of life, 308, _ff_\n\n  -- -- evolution, 226, _ff_\n\n  -- -- finite extension, 62, 106\n\n  -- -- generable quantity, 103\n\n  -- -- infinite continuum, 70\n\n  -- -- intuitional concept, 225\n\n  -- -- kosmic order, 265, 267\n\n  -- -- nether pole of non-manifestation, 207-8\n\n  -- -- path of life, 293-4\n\n  -- -- pseudosphere, 70\n\n  -- -- sheer roominess, 115\n\n  -- -- system of co\u00f6rdinates, 99\n\n  -- -- unbounded extension, 61\n\n  -- best study of, the consciousness, 269\n\n  -- conceptual, as basis of non-Euclidean geometry, 72\n\n  -- -- fundamentally distinct from perceptual, 72-75\n\n  -- elliptical, 67\n\n  -- engenderment of, 260\n\n  -- essential nature of, 212\n\n  -- fabric of, the, lends itself to measurement, 99-100\n\n  -- four-, 8\n\n  -- fractionalized space, 296\n\n  -- foundations of, 221\n\n  -- four-dimensional, 41\n\n  -- Paul Carus on, 354\n\n  -- fourth dimension of, efforts at making it thinkable, 125\n\n  -- generability of, by lines, 143\n\n  -- genesis of, 227, _ff_\n\n  -- geometric, purely formal construction, 75-6\n\n  -- limits of and the Creative Logos, 193\n\n  -- manifoldness of, 77\n\n  -- mind and, the coevalism of, 240\n\n  -- mystery of, 278, _ff_\n\n  -- nature of, answered to by mind, 198-9\n\n  -- nether pole of non-manifestation, 207-8\n\n  -- non-Euclidean, the characteristics of, 72-3\n\n  -- not a pure form, 110\n\n  -- passage of, 16\n\n  -- path of search for, must be Thinker-ward, 278\n\n  -- perceptual, irreconcilable with the fourth dimension, 156\n\n  -- possible curvature of, and the non-Euclidean geometry, 76-7\n\n  -- primeval, and tridimensionality, 236\n\n  -- problem of devising a, 87-8\n\n  -- psychological nature of, 305\n\n  -- real, confounded with hyperspace, 91\n\n  -- Riemann on the curvature of, 5\n\n  -- the unity of, 224\n\n  -- -- -- -- with matter, 260\n\n  Space-activities, as chemism and physicism, 229-30\n\n  Space-center, the, 306-7\n\n  Space-consciousness, 225\n\n  -- as a direct process, 224\n\n  -- mergence of, with the individual consciousness, 195\n\n  Space-curvature, as an arbitrary construction, 262\n\n  -- as near-truth, 306\n\n  -- nature of, 279\n\n  -- no need for, 237\n\n  -- translation of, in terms of the intellect, 315\n\n  Space-forms, the construction of, 74\n\n  Space-genesis, 20, 232, 233\n\n  -- alphabet of, 237\n\n  -- completion of, 226\n\n  -- norm of, 174\n\n  -- pyknon as a basis of, 17\n\n  -- symbolism of, 208, _ff_, 235\n\n  Space-manifold, the, 279\n\n  Space-measurement, and reality, Euclid's system of, 96\n\n  -- systems of, 201\n\n  Space-mind and archeological evidences, 193\n\n  -- -- knowledge, 187\n\n  -- -- the Thinker, 253, 272\n\n  -- attainment of, 225, 231\n\n  -- granaries of, 235\n\n  -- the realization of, 238\n\n  Space-process, the, 305\n\n  Space-realities, hyperspace as a stepping stone to, 238\n\n  Spark, the omnipsyche as a, 258\n\n  Spatiality and the intellect, 263-4\n\n  -- consubstantial with materiality, 20, 331, _ff_\n\n  -- cut-off from non-spatiality, 266\n\n  -- interpenetrative with materiality, 236\n\n  -- nonconformity of, with logic, 164\n\n  -- riddle of, and the intuition, 325\n\n  --, rise of, 180\n\n  -- Thinker's outlook in, 316\n\n  -- world of nascent, 232\n\n  Spencer, Herbert, 322\n\n  Spinoza, 322\n\n  Spiritism, the phenomena of, 154\n\n  Spiritualists, the claim of, regarding the fourth dimension, 154-5\n\n  Spirituality, materiality transmuted into, 328\n\n  -- principle of, 210, _ff_\n\n  -- seeds of, 221\n\n  Spiritualization of the flesh, 227\n\n  -- -- man's mental outlook, 40\n\n  -- -- matter, the end of evolution, 327\n\n  Squares, the form of, in the non-Euclidean geometry, 59\n\n  St. Marie, Flye, 66\n\n  Stage of pyknosis, 208, _ff_\n\n  --, quartopyknotic, 219, _ff_\n\n  Stanley, Hiram M., on space as dynamism, 104-7\n\n  States, the two, of consciousness; one unaware of the other, 192\n\n  Straight, a, determined by two points, 79\n\n  Stuff, etherealized four-dimensional, 160\n\n  Sum, angular, 79\n\n  -- triangular, less than two right angles, 59\n\n  Superconcepts, as intuitographs, 255\n\n  Superconceptual, as the intuitional, 20\n\n  Superperception and automatism of the intellect, 20\n\n  Superperceptual, the, 74\n\n  Superunodim, the, 177\n\n  Swedenborg, 180, 184\n\n  Symbol, psychic, and the brain, 190\n\n  -- words as symbols of ideas, 126-7\n\n  Symbolism of life, 286\n\n  -- -- neurographical impressions, 186\n\n  -- -- realities, 128\n\n  -- -- sense-deliveries, 204\n\n  -- -- space-genesis, 208, _ff_\n\n  Symbology, depicting planes of consciousness, 270\n\n  -- of mathematical knowledge, 203, _ff_\n\n  Synchronous evolution of faculties and properties of matter, 332\n\n  Syncretization of mind and matter, 299\n\n  Synthesis and the norm of reality, 300\n\n  Synthetic school of geometry, 56\n\n\n  T\n\n  Tactographic impressions, 190\n\n  Tartrate and a component in the fourth dimension, 156-7\n\n  Task of mental evolution, the supreme, 169\n\n  Taurinus, 56-7, 69\n\n  Telescope, its effect upon consciousness, 298\n\n  Tendence, intellect as scientific, 165-6\n\n  Tendencies, destined to flower as faculties, 89-91, 332\n\n  Tendency to fragmentate, the intellect and, 296\n\n  Tesseract as represented by Carus, 136-8\n\n  -- illustrated, 133\n\n  -- the, 20, 145\n\n  -- the elements of, 135\n\n  -- the four co\u00f6rdinates of, 132\n\n  Testimony of the intellect, incapable of comparison, 76\n\n  Tetragrammaton, the, 308\n\n  Tetrahedron, the, 135\n\n  Thalamencephalon, the, 345\n\n  Thales, 44\n\n  Theory of knowledge, 290\n\n  Thingness of objects and the impossibility of comprehending, 122\n\n  Things, the dual nature of, 119\n\n  -- -- value of, 161\n\n  Things-in-themselves, the identity of, 353\n\n  Thinkability of the fourth dimension, 125\n\n  Thinker as a pure intelligence, 184, 248\n\n  -- -- engenderer of the intellect, 291\n\n  -- -- principal to the ego, 243, _ff_\n\n  -- -- six times removed from the sensible world, 201\n\n  -- -- the source of the intuition, 26\n\n  -- and concepts, 244\n\n  -- -- his approach to the kosmic mind, 202, 223\n\n  -- -- -- larger life, 196\n\n  -- -- -- treatment of the ego, 246\n\n  -- -- -- of sense-impressions, 74-5\n\n  -- -- realism, 315\n\n  -- -- space, 193\n\n  -- -- a segment of reality, 171\n\n  -- the, and the amplitude of the degrees of realism, 198, 270\n\n  -- and the barriers of consciousness, 207\n\n  -- -- -- brotherhood of man, 301\n\n  -- -- -- degrees of realism, 175\n\n  -- -- -- manipulation of concepts, 255\n\n  -- -- -- nature of space, 268\n\n  -- -- -- neurograph, 190\n\n  -- -- -- new faculty, 335\n\n  -- -- -- perceptive faculty, 188\n\n  -- -- -- pituitary body, 351\n\n  -- defined as the spiritual man, 21\n\n  -- his consciousness and the ether, 280\n\n  -- -- dependence upon the ego, 245\n\n  -- -- method of contacting the external world, 189-191\n\n  -- -- outlook upon the universe of spatiality, 242\n\n  Thinker, his schematism of cognitive powers, 206\n\n  -- -- sphere of awareness, 283\n\n  -- the difference between, and the intellectuality, 256\n\n  -- -- evolutionary needs of, 336\n\n  -- -- upreaching yearnings of, 336\n\n  Thinker-ward, space must be sought in a direction, 278\n\n  Thinking, abstract, elevating influence of, 33\n\n  Thought, prolonged abstract, benefits of, 249\n\n  Time, as an aspect of consciousness, 224\n\n  -- divested of timeliness, 279\n\n  Todhunter, 46\n\n  Tool, fashioning, life as a, 264\n\n  Topography of the intellect, 312\n\n  Touchstone, consciousness as a, 168\n\n  Tracery of connection between ideas and objects, 126\n\n  Trafficking in mental constructions, 154\n\n  Transcendental, the realm of, and mathematicians, 142\n\n  Transfinite as a limit, 122, 216\n\n  Transfinity, 21-2\n\n  Tree-reality, 126\n\n  Triadic phase of evolution, 210-1\n\n  Triangles, angular sum of, 78, 88\n\n  Tridim, the, 22, 129, 134, 147, 177\n\n  Tridimension, the, 147\n\n  Tridimensionality and primeval space, 236\n\n  -- and the space-mind, 193\n\n  -- a quality of perceptual space, 22\n\n  -- mastery of the phenomena of, 172\n\n  -- the sufficiency of, 198\n\n  Tripyknon, the, 212, 213\n\n  Tripyknosis, 17, 208-10, 219\n\n  Truth compared with facts, 288\n\n  -- criterion of, 5, 323-4\n\n  -- facets of, 284\n\n  -- kinds of, 324\n\n  -- logic does not illuminate, 287-8\n\n  Tuitive, the, and the intuitive faculties, 192-3\n\n\n  U\n\n  Ultima Thule, the, 207\n\n  Unbounded, the, as a finite extension, 70\n\n  Undulations, three, in the current of life, 329\n\n  Unfoldment, mental, and metageometrical discoveries, 131\n\n  Uni-circle, the, 308\n\n  Unification of all knowledge, 256\n\n  Uniqueness of real space, 95\n\n  Unitariness of all existences, 241\n\n  Unity, as end of analysis, 42\n\n  -- death as a fourth dimensional, 159\n\n  -- kosmic, integers as fractional parts of, 41\n\n  -- of mind and space, 224, 230\n\n  -- the new realization of, 225\n\n  Universality, geometric, based upon the formal character of\n   assumptions, 77-8\n\n  Universe, a glorified, 226, 268\n\n  -- as a full, 308\n\n  -- -- -- plenum, 107\n\n  -- and the seven planes of matter, 212\n\n  -- character of, fixed by consciousness, 162\n\n  -- limited and conditioned, 127\n\n  -- not a pure form, 108\n\n  -- the perfected, and the circle, 310\n\n  -- -- theory of the mechanical origin of, due to analytics, 40\n\n  -- unity in the, 301-2\n\n  Universum of appearance, 187\n\n  -- of life and consciousness, 257\n\n  Universum of reality, 127\n\n  -- -- space, 269\n\n  Unknowable as a symbol, 194\n\n  -- the darkness which shuts out the, 207\n\n  Unodim, the, 129, 134, 176-7\n\n  -- -- consciousness, 163\n\n  -- defined, 22\n\n  Up-raisement of matter, 329\n\n  Upspringing of a new faculty, 333\n\n\n  V\n\n  Vacuum-stillness of nature, 339\n\n  Validity of mathematical conclusions, 101\n\n  Value assigned by the intellect to the sensible world, 167\n\n  -- assumptional, vitiating influence of, 163\n\n  -- -- of the ideal, 275\n\n  Vanity of fragmentary evidences, 204\n\n  -- of intellectual method, 325, _ff_\n\n  -- -- segmenting space into many dimensions, 299\n\n  Vassal, life as the, of materiality, 329\n\n  Vehicle of life, form as, 330\n\n  -- -- -- mind as, 286\n\n  Vicissitudes of kosmic evolution, 215\n\n  Vision, the inner, 356\n\n\n  W\n\n  Wachter, 69\n\n  Wallis, John, 46, 83\n\n  Weissman, _note_, 260\n\n  Words, as symbols of ideas, 126-7, 204\n\n  World and the child mind, 121\n\n  -- as instrument of consciousness, 199, 298\n\n  -- fabric of, and geometrism, 261\n\n  -- impossibility of objectifying the fourth dimension in the\n   perceptual, 124\n\n  -- of phantasy, conditions of, identified with sensible realm, 146\n\n  -- phenomenal, and hyperspace entities, 128-130\n\n  -- sensible, as a carpet, 196\n\n  -- the domain of a perception, 19\n\n  -- -- genesis of, 167\n\n  World-plasm, kathekotic, 267\n\n  -- as elemental essence, 329\n\n\n  Y\n\n  Yoga, Rajah, 357\n\n  Youth of the earth, the nebulous, 335\n\n\n  Z\n\n  Zollner, and the claims of the spiritualists, 154\n\n  Zones of Affinity, 22, 124\n\n  Zoometer, the, 297\n\n\n\n\n  Transcriber's Notes\n\n  Variations in spelling, punctuation and hyphenation have been retained\n  except in obvious cases of typographical errors. Inconsistencies\n  between the text and index have been resolved in favour of the text.\n\n  Duplication of the sub title (The Mystery of Space) on consecutive\n  pages has been removed.\n\n  Italics are shown thus _italic_.\n\n  In the five cases where illustrations could be realistically rendered\n  using only text symbols this has been done.","title":"The mystery of space : $b a study of the hyperspace movement in the light of the evolution of new psychic faculties and an inquiry into the genesis and essential nature of space","tradition":"scripture-revelation","translated_text":null,"word_count":108580}