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Table of Contents

Historical Sketch of the Hyperspace Movement

Egypt the Birthplace of Geometry—Precursors: Nasir-Eddin, Christoph Clavius, Saccheri, Lambert, La Grange, Kant—Influence of the Mecanique Analytique—The Parallel-Postulate the Root and Substance of the Non-Euclidean Geometry—The Three Great Periods: The Formative, Determinative and Elaborative—Riemann and the Properties of Analytic Spaces.

The evolution of the idea of a fourth dimension of space covers a long period of years. The earliest known record of the beginnings of the study of space is found in a hieratic papyrus which forms a part of the Rhind Collection in the British Museum and which has been deciphered by Eisenlohr. It is believed to be a copy of an older manuscript of date 3400 B. C., and is entitled "Directions for Knowing All Dark Things" The copy is said to have been made by Ahmes, an Egyptian priest between 1700 and 1100 B. C. It begins by giving the dimensions of barns; then follows the consideration of various rectilineal figures, circles, pyramids, and the value of pi ([Greek: p]). Although many of the solutions given in the manuscript have been found to be incorrect in minor particulars, the fact remains that Egypt is really the birth-place of geometry. And this fact is buttressed by the knowledge that Thales, long before he founded the Ionian School which was the beginning of Greek influence in the study of mathematics, is found studying geometry and astronomy in Egypt.

The concept of hyperspace began to germinate in the latter part of the first century, B. C. For it was at this date that Geminos of Rhodes (B. C. 70) began to think seriously of the mathematical labyrinth into which Euclid's parallel-postulate most certainly would lead if an attempt at demonstrating its certitude were made. He recognized the difficulties which would engage the attention of those who might venture to delve into the mysterious possibilities of the problem. There is no doubt, too, but that Euclid himself was aware, in some measure at least, of these difficulties; for his own attitude towards this postulate seems to have been one of noncommittance. It is, therefore, not strange that the astronomer, Ptolemy (A. D. 87-165), should be found seeking to prove the postulate by a consideration of the possibilities of interstellar triangles. His researches, however, brought him no relief from the general dissatisfaction which he felt with respect to the validity of the problem itself.

For nearly one thousand years after the attempts at solving the postulate by Geminos and Ptolemy, the field of mathematics lay undisturbed. For it was at this time that there arose a strange phenomenon, more commonly known as the "Dark Ages," which put an effectual check to further research or independent investigations. Mathematicians throughout this long lapse of time were content to accept Euclid as the one incontrovertible, unimpeachable authority, and even such investigations as were made did not have a rebellious tendence, but were mainly endeavors to substantiate his claims.

Accordingly, it was not until about the first half of the thirteenth century that any real advance was made. At this time there appeared an Arab, Nasir-Eddin (1201-1274) who attempted to make an improvement on the problem of parallelism. His work on Euclid was printed in Rome in 1594 A. D., about three hundred and twenty years after his demise and was communicated in 1651 by John Wallis (1616-1703) to the mathematicians of Oxford University. Although his calculations and conclusions were respectfully received by the Oxford authorities no definite results were regarded as accomplished by what he had done. It is believed, however, that his work reopened speculation upon the problem and served as a basis, however slight, for the greater work that was to be done by those who followed him during the next succeeding eight hundred years.

About twenty years before the printing of the work of Nasir-Eddin, Christoph Clavius (1574) deduced the axiom of parallels from the assumption that a line whose points are all equidistant from a straight line is itself straight. In his consideration of the parallel-postulate he is said to have regarded it as Euclid's XIIIth axiom. Later Bolyai spoke of it as the XIth and later still, Todhunter treated it as the XIIth. Hence, there does not seem to have been any general unanimity of opinion as to the exact status of the parallel-postulate, and especially is this true in view of the uncertainty now known to have existed in Euclid's mind concerning it.

Girolamo Saccheri (1667-1733), a learned Jesuit, born at San Remo, came next upon the stage. And so important was his work that it will perpetuate the memory of his name in the history of mathematics. He was a teacher of grammar in the Jesuit Collegio di Brera where Tommaso Ceva, a brother of Giovanni, the well-known mathematician, was teacher of mathematics. His association with the Ceva brothers was especially beneficial to him. He made use of Ceva's very ingenious methods in his first published book, 1693, entitled Solutions of Six Geometrical Problems Proposed by Count Roger Ventimiglia.

Fig. 1.

Saccheri attacked the problem of parallels in quite a new way. Examining a quadrilateral, ABCD, in which the angles A and B are right angles and the sides AC and BD are equal, he determined to show that the angles C and D are equal. He also sought to prove that they are either right angles, obtuse or acute. He undertook to prove the falsity of the latter two propositions (that they are either obtuse or acute), leaving as the only possibility that they must be right angles. In doing so, he found that his assumptions led him into contradictions which he experienced difficulty in explaining.

His labors in connection with the solution of the problems proposed by Count Ventimiglia, including his work on the question of parallels, led directly into the field of metageometrical researches, and perhaps to him as to no other who had preceded him, or at least to him in a larger degree, belongs the credit for a continued renewal of interest in that series of investigations which resulted in the formulation of the non-Euclidean geometry.

The last published work of Saccheri was a recital of his endeavors at demonstrating the parallel-postulate. This received the "Imprimatur" of the Inquisition, July 13, 1733; the Provincial Company of Jesus took possession of the book for perusal on August 16, 1733; but unfortunately within two months after it had been reviewed by these authorities, Saccheri passed away.

All efforts which had been made prior to the work of Saccheri were based upon the assumption that there must be an equivalent postulate which, if it could be demonstrated, would lead to a direct, positive proof of Euclid's proposition. Although these and all other attempts at reaching such a proof have signally failed and although it may correctly be said that the entire history of demonstrations aiming at the solution of the famous postulate has been one long series of utter failures, it can be asserted with equal certitude that it has proven to be one of the most fruitful problems in the history of mathematical thought. For out of these failures has been built a superstructure of analytical investigations which surpasses the most sanguine expectations of those who had labored and failed.

In 1766 John Lambert (1728-1777) wrote a paper upon the Theory of Parallels dated Sept. 5, 1766, first published in 1786, from the papers left by F. Bernoulli, which contained the following assertions:[2]

1. The parallel-axiom needs proof, since it does not hold for geometry on the surface of the sphere.

2. In order to make intuitive a geometry in which the triangle's sum is less than two right angles, we need an "imaginary" sphere (the pseudosphere).

3. In a space in which the triangle's sum is different from two right angles there is an absolute measure (a natural unit for length).

At this time Immanuel Kant (1724-1804), the noted German metaphysician, was in the midst of his philosophical labors. And it is believed that it was he who first suggested the idea of different spaces. Below is given a statement taken from his Prolegomena[3] which corroborates this view.

"That complete space (which is itself no longer the boundary of another space) has three dimensions, and that space in general cannot have more, is based on the proposition that not more than three lines can intersect at right angles in one point.... That we can require a line to be drawn to infinity, a series of changes to be continued (for example, spaces passed through by motion) in indefinitum, presupposes a representation of space and time which can only attach to intuition."

His differentiation between space in general and space which may be considered as the "boundary of another space" shows, in the light of the subsequent developments of the mathematical idea of space that he very fully appreciated the marvelous scope of analytic spaces. His conception of space, therefore, must have had a profound influence upon the mathematic thought of the day causing it to undergo a rapid reconstruction at the hands of geometers who came after him.

Under the masterly influence of La Grange (1736-1813) the idea of different spaces began to take definite shape and direction; the geometry of hyperspace began to crystallize; and the field of mathesis prepared for the growth of a conception the comprehension of which was destined to be the profoundest undertaking ever attempted by the human mind. Unlike most great men whom the world learns tardily to admire, La Grange lived to see his talents and genius fully recognized by his compeers; for he was the recipient of many honors both from his countrymen and his admirers in foreign lands. He spent twenty years in Prussia where he went upon the invitation of Frederick the Great who in the Royal summons referred to himself as the "greatest king in Europe" and to La Grange as the "greatest mathematician" in Europe. In Prussia the Mecanique Analytique and a long series of memoirs which were published in the Berlin and Turin Transactions were produced. La Grange did not exhibit any marked taste for mathematics until he was 17 years of age. Soon thereafter he came into possession of a memoir by Halley quite by accident and this so aroused his latent genius that within one year after he had reviewed Halley's memoir he became an accomplished mathematician.

He created the calculus of variations, solved most of the problems proposed by Fermat, adding a number of theorems of his own contrivance; raised the theory of differential equations to the position of a science rather than a series of ingenious methods for the solution of special problems and furnished a solution for the famous isoperimetrical problem which had baffled the skill of the foremost mathematicians for nearly half a century. All these stupendous tasks he performed by the time he reached the age of nineteen.

The Mecanique Analytique is his greatest and most comprehensive work. In this he established the law of virtual work from which, by the aid of his calculus of variations, he deduced the whole of mechanics, including both solids and liquids. It was his object in the Analytique to show that the whole subject of mechanics is implicitly embraced in a single principle, and to lay down certain formulae from which any particular result can be obtained. He frequently made the assertion that he had, in the Mecanique Analytique, transformed mechanics which he persistently defined as a "geometry of four dimensions"[4] into a branch of analytics and had shown the so-called mechanical principles to be the simple results of the calculus. Hence, there can be no doubt but that La Grange not only completed the foundation, but provided most of the material in his analyses and other "abstract results of great generality" which he obtained in his numerous calculations, for the superstructure subsequently known as the geometry of hyperspace, and in which the fourth dimensional concept occupies a very fundamental place.

It is as if for nearly seventeen hundred years workmen, such as Geminos, of Rhodes, Ptolemy, Saccheri, Nasir-Eddin, Lambert, Clavius, and hundred of others who struggled with the problem of parallels, had made more or less sporadic attempts at the excavation of the land whereon a marvelously intricate building was to be constructed. There is no historical evidence to show that any of them ever dreamed that the results of their labors would be utilized in the manner in which they have been used. Then came Kant with the wonderfully penetrating searchlight of his masterful intellect who from the elevation which he occupied saw that the site had great possibilities, but he had not the mathematical talent to undertake the work of actual, methodical construction. Indeed his task was of a different sort. However, he succeeded in opening the way for La Grange and others who followed him. La Grange immediately seized upon the idea which for more than a thousand years had been impinging upon the minds of mathematicians vainly seeking lodgment and began the elaboration of a plan in accordance with which minds better skilled in the pragmatic application of abstract principles than his could complete the work begun. Unfortunately, on account of his intense devotion and loyalty to the study of pure mathematics, and when he had reached the summit of his greatness where he stood "without a rival as the foremost living mathematician," his health became seriously affected, causing him to suffer constant attacks of profound melancholia from which he died on April 10, 1813.

We come now to one of the most remarkable periods in the history of mental development. During the six hundred years between the birth of Nasir-Eddin and the death of La Grange the entire world of mathesis was being reconstituted. Since there had been gradually going on an internal process which, when completed, forever would liberate the mind from the narrow confines of consciousness limited to the three-space, it is not surprising that we should find, in the mathematical thought of the time, an absolutely epoch-making departure. The innumerable attempts at the solution of the parallel-postulate, all failures in the sense that they did not prove, have intensified greatly the esteem in which the never-dying elements of Euclid are held to-day. And despite the fact that there may come a time when his axioms and conclusions may be found to be incongruent with the facts of sensuous reality; and though all of his fundamental conceptions of space in general, his theorems, propositions and postulates may have to give way before the searching glare of a deeper knowledge because of some revealed fault, the perfection of his work in the realm of pure mathematics will remain forever a master piece demanding the undiminished admiration of mankind.

The parallel-postulate, as stated by Euclid in his Elements of Geometry, reads as follows:

"If a straight line meet two straight lines so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles."

On this postulate hang all the "law and the prophets" of the non-Euclidean Geometry. In it are the virtual elements of three possible geometries. Furthermore, it is both the warp and the woof of the loom of present-day metageometrical researches. It is the golden egg laid by the god Seb at the beginning of a new life cycle in psychogenesis. Its progeny are numerous—hyperspaces, sects, straights, digons, equidistantials, polars, planars, coplanars, invariants, quaternions, complex variables, groups and many others. A wonderfully interesting breed, full of meaning and pregnant with the power of final emancipations for the human intellect!

When the conclusions which were systematically formulated as a result of the investigations along the lines of hypotheses which controverted the parallel-postulate were examined it was found that they fell into three main divisions, namely: the synthetic or hyperbolic; the analytic or Riemannian and the elliptic or Cayley-Klein. These divisions or groups are based upon the three possibilities which inhere in the conception taken of the sum of the angles referred to in the above postulate as to whether it is equal to, greater or less than two right angles.

The assumption that the angular sum is congruent to a straight angle is called the Euclidean or parabolic hypothesis and is to be distinguished from the synthetic or hyperbolic hypothesis established by Gauss, Lobachevski and Bolyai and which assumes that the angular sum is less than a straight angle. The elliptic or Cayley-Klein hypothesis assumes that the angular sum is greater than a straight angle. Lobachevski, however, not satisfied with the statement of the parallel-postulate as given by Euclid and which had caused the age-long controversy, substituted for it the following:

"All straight lines which, in a plane, radiate from a given point, can, with respect to any other straight line, in the same plane, be divided into two classes—the intersecting and the non-intersecting. The boundary line of the one and the other class is called parallel to the given line."

This is but another way of saying about the same thing that Euclid had declared before, and yet, curiously enough it afforded just the liberty that Lobachevski needed to enable him to elaborate his theory.

For the purposes of this sketch the field of the development of non-Euclidean geometry is divided into three periods to be known as: (1) the formative period in which mathematical thought was being formulated for the new departure; (2) the determinative period during which the mathematical ideas were given direction, purpose and a general tendence; (3) the elaborative period during which the results of the former periods were elaborated into definite kinds of geometries and attempts made at popularizing the hypotheses.

The Formative Period

Charles Frederich Gauss (1777-1855) by some has been regarded as the most influential mathematician that figured in the formulation of the non-Euclidean geometry; but closer examination into his efforts at investigating the properties of a triangle shows that while his researches led to the establishment of the theorem that a regular polygon of seventeen sides (or of any number which is prime, and also one more than a power of two) can be inscribed, under the Euclidean restrictions as to means, in a circle, and also that the common spherical angle on the surface of a sphere is closely connected with the constitution of the area inclosed thereby, he cannot justly be designated as the leader of those who formulated the synthetic school. And this, too, for the simple reason that, as he himself admits in one of his letters to Taurinus, he had not "published anything on the subject." In this same letter he informs Taurinus that he had pondered the subject for more than thirty years and expressed the belief that there could not be any one who had "concerned himself more exhaustively with this second part (that the sum of the angles of a triangle cannot be more than 180 degrees)" than he had.

Writing from Göttingen to Taurinus, November 8, 1824, and commenting upon the geometric value of the sum of the angles of a triangle, he says:

"Your presentation of the demonstration that the sum of the angles of a plane triangle cannot be greater than 180 degrees does, indeed, leave something to be desired in point of geometrical precision. But this could be supplied, and there is no doubt that the impossibility in question admits of the most rigorous demonstration. But the case is quite different with the second part, namely, that the sum of the angles cannot be smaller than 180 degrees; this is the real difficulty, the rock upon which all endeavors are wrecked.... The assumption that the sum of the three angles is smaller than 180 degrees leads to a new geometry entirely different from our Euclidean—a geometry which is throughout consistent with itself, and which I have elaborated in a manner entirely satisfactory to myself, so that I can solve every problem in it with the exception of the determining of a constant which is not a priori obtainable."

It appears from this correspondence that Gauss had in the privacy of his own study elaborated a complete non-Euclidean geometry, and had so thoroughly familiarized himself with its characteristics and possibilities that the solution of every problem embraced within it was very clear to him except that of the determination of a constant. He concluded the above letter by saying:

"All my endeavors to discover contradiction or inconsistencies in this non-Euclidean geometry have been in vain, and the only thing in it that conflicts with our reason is the fact that if it were true there would necessarily exist in space a linear magnitude quite determinate in itself; yet unknown to us."

Judging from the correspondence between Gauss and Gerling (1788-1857), Bessel (1784-1846), Schumacher and Taurinus, the nephew of Schweikart, and that between Schweikart and Gerling, there had grown up a general dissatisfaction in the minds of mathematicians of this period with Euclidean geometry and especially the parallel-postulate and its connotations. Bessel expresses this general discontent in one of his letters to Gauss, dated February 10, 1829, in which he says:

"Through that which Lambert said and what Schweikart disclosed orally, it has become clear to me that our geometry is incomplete, and should receive a correction, which is hypothetical, and if the sum of the angles of the plane triangle is equal to 180 degrees, vanishes."

The opinion of leading mathematicians at this time seems to have been crystallizing very rapidly. Unconsciously the men of this formative period were adducing evidence which would give form and tendence to the developments in the field of mathesis at a later date. They appear to have been reaching out for that which, ignis fatuus-like, was always within easy reach, but not quite apprehensible.

A bolder student than Gauss was Ferdinand Carl Schweikart (1780-1857) who also has been credited with the founding of the non-Euclidean geometry. In fact, if judged by the same standards as Gauss, he would be called the "father of the geometry of hyperspace"; for he really published the first treatise on the subject. This was in the nature of an inclosure which he inserted between the leaves of a book he loaned to Gerling. He also asked that it be shown to Gauss that he might give his judgment as to its merits.

Schweikart's treatise, dated Marburg, December, 1818, is here quoted in full:

"There is a two-fold geometry—a geometry in the narrower sense, the Euclidean, and an astral science of magnitude.

"The triangles of the latter have the peculiarity that the sum of the three angles is not equal to two right angles.

"This presumed, it can be most rigorously proven: (a) That the sum of the three angles in the triangle is less than two right angles.

"(b) That this sum becomes ever smaller, the more content the angle incloses. (c) That the altitude of an isosceles right-angled triangle indeed ever increases, the more one lengthens the side; that it, however, cannot surpass a certain line which I call the constant."

Squares have consequently the following form:

Fig. 2.

"If this constant were for us the radius of the earth (so that every line drawn in the universe, from one fixed star to another, distant 90° from the first, would be a tangent to the surface of the earth) it would be infinitely great in comparison with the spaces which occur in daily life."

The above, being the first published, not printed, treatise on the new geometry occupies a unique place in the history of higher mathematics. It gave additional strength to the formative tendencies which characterized this period and marked Schweikart as a constructive and original thinker.

The nascent aspects of this stage received a fruitful contribution when Nicolai Lobachevski (1793-1847) created his Imaginary Geometry and Janos Bolyai (1802-1860) published as an appendix to his father's Tentamen, his Science Absolute of Space. Lobachevski and Bolyai have been called the "Creators of the Non-Euclidean Geometry." And this appellation seems richly to be deserved by these pioneers. Their work gave just the impetus most needed to fix the status of the new line of researches which led to such remarkable discoveries in the more recent years. The Imaginary Geometry and the Science Absolute of Space were translated by the French mathematician, J. Hoüel in 1868 and by him elevated out of their forty-five years of obscurity and non-effectiveness to a position where they became available for the mathematical public. To Bolyai and Lobachevski, consequently, belong the honor of starting the movement which resulted in the development of metageometry and hence that which has proved to be the gateway of a new mathematical freedom.

Gauss, Schweikart, Lobachevski, Wolfgang and Janos Bolyai were the principal figures of the formative period and the value of their work with respect to the formulation of principles upon which was constructed the Temple of Metageometry cannot be overestimated.

The Determinative Period

This period is characterized chiefly by its close relationship to the theory of surfaces. Riemann's Habilitation Lecture on The Hypotheses Which Constitute the Bases of Geometry marks the beginning of this epoch. In this dissertation, Riemann not only promulgated the system upon which Gauss had spent more than thirty years of his life in elaborating, for he was a disciple of Gauss; but he disclosed his own views with respect to space which he regarded as a particular case of manifold. His work contains two fundamental concepts, namely, the manifold and the measure of curvature of a continuous manifold, possessed of what he called flatness in the smallest parts. The conception of the measure of curvature is extended by Riemann from surfaces to spaces and a new kind of space, finite, but unbounded, is shown to be possible. He showed that the dimensions of any space are determined by the number of measurements necessary to establish the position of a point in that space. Conceiving, therefore, that space is a manifold of finite, but unbounded, extension, he established the fact that the passage from one element of a manifold to another may be either discrete or continuous and that the manifold is discrete or continuous according to the manner of passage. Where the manifold is regarded as discrete two portions of it can be compared, as to magnitude, by counting; where continuous, by measurement. If the whole manifold be caused to pass over into another manifold each of its elements passing through a one-dimensional manifold, a two-dimensional manifold is thus generated. In this way, a manifold of n-dimensions can be generated. On the other hand, a manifold of n-dimensions can be analyzed into one of one dimension and one of (n-1) dimensions.

To Riemann, then, is due the credit for first promulgating the idea that space being a special case of manifold is generable, and therefore, finite. He laid the foundation for the establishment of a special kind of geometry known as the "elliptic." Space, as viewed by him, possessed the following properties, viz.: generability, divisibility, measurability, ponderability, finity and flexity.

These are the six pillars upon which rests the structure of hyperspace analyses.[5]

Generability is that property of geometric space by virtue of which it may be generated, or constructed, by the movement of a line, plane, surface or solid in a direction without itself. Divisibility is that property of geometric space by virtue of which it may be segmented or divided into separate parts and superposed, or inserted, upon or between each other. Measurability is that property by virtue of which geometric space is determined to be a manifold of either a positive or negative curvature, also by which its extent may be measured. Ponderability is that property of geometric space by virtue of which it may be regarded as a quantity which can be manipulated, assorted, shelved or otherwise disposed of. Finity is that property by virtue of which geometric space is limited to the scope of the individual consciousness of a unodim, a duodim or a tridim and by virtue of which it is finite in extent. Flexity is that property by virtue of which geometric space is regarded as possessing curvature, and in consequence of which progress through it is made in a curved, rather than a geodetic line, also by virtue of which it may be flexed without disruption or dilatation.

Riemann who thus prepared the way for entrance into a veritable labyrinth of hyperspaces is, therefore, correctly styled "The father of metageometry," and the fourth dimension is his eldest born. He died while but forty years of age and never lived long enough fully to elaborate his theory with respect to its application to the measure of curvature of space. This was left for his very energetic disciple, Eugenio Beltrami (1835-1900) who was born nine years after Riemann and lived thirty-four years longer than he. His labors mark the characteristic standpoint of the determinative period. Beltrami's mathematical investigations were devoted mainly to the non-Euclidean geometry. These led him to the rather remarkable conclusion that the propositions embodied therein relate to figures lying upon surfaces of constant negative curvature.

Beltrami sought to show that such surfaces partake of the nature of the pseudosphere, and in doing so, made use of the following illustration: