The Mystery of Space

tch of the Hyp

fluence of the Mecanique Analytique-The Parallel-Postulate the Root and Substance of the Non-Euclidean Geometry-Th

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

nized 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

menon, 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 acc

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

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 doe

. 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 brothe

g.

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

tly 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, bel

e "Imprimatur" of the Inquisition, July 13, 1733; the Provincial Company of Jesus took possession of the book for perusal on

ttempts 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 t

rallels dated Sept. 5, 1766, first published in 1786, from the paper

, since it does not hold for geom

the triangle's sum is less than two right angles

different from two right angles there is an

philosophical labors. And it is believed that it was he who first suggested the idea of differen

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

lopments of the mathematical idea of space that he very fully appreciated the marvelous scope of analytic spaces. His conception of space, therefore, must h

e 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 Mecani

quations 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 p

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, t

nderfully 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

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

ted by Euclid in his Elements

of it taken together less than two right angles, these straight lines being continually produced,

-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, pola

und 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 th

shed 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

r straight line, in the same plane, be divided into two classes-the intersecting and the non-int

ad declared before, and yet, curiously enough it afforded just the l

tical thought was being formulated for the new departure; (2) the determinative period during which the mathematical ideas were given direction, purpose and a general tendenc

rmativ

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

1824, and commenting upon the geometric value

orous 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 t

y, and had so thoroughly familiarized himself with its characteristics and possibilities that the solution of every problem em

n in vain, and the only thing in it that conflicts with our reason is the fact that if it were true th

n Schweikart and Gerling, there had grown up a general dissatisfaction in the minds of mathematicians of this period with Euclidean geometry and especially t

to me that our geometry is incomplete, and should receive a correction, which is hypothet

ative period were adducing evidence which would give form and tendence to the developments in the field of mathesis at a later date.

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

ted Marburg, December, 18

ry in the narrower sense, the Euclidea

peculiarity that the sum of the three

oven: (a) That the sum of the three angles i

e altitude of an isosceles right-angled triangle indeed ever increases, the more one leng

nsequently the

g.

from one fixed star to another, distant 90° from the first, would be a tangent to the surface of

place in the history of higher mathematics. It gave additional strength to the formative tendenc

ion 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

the formative period and the value of their work with respect to the formulation of pr

rminativ

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 c

herefore, finite. He laid the foundation for the establishment of a special kind of geometry known as the "elliptic." Space, as vie

pon which rests the structu

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

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 standp

ake of the nature of the pseudosphere, and in d

g.

g.

surface will turn outward with ever-increasing flexure till it becomes perpendicular to the axis and ends at the edge with one curvature infinite. Or, the half of a pseudospherical surface may be rolled up into the shape of

iemann and Beltrami are chief among those whose labors characterize the scope of this period. Their work gave direction and general outline for later developments and all subsequent researches along these lines

ent of the hypothesis that space is unbounded, though finite, is really the first time in the history of human thought that expression was ever given to the idea that space

borativ

y them. Among those whose investigations have greatly multiplied the applications of hyperspace conceptions are Hoüel (1866) and Flye St. Marie (1871) of France; Helmholtz (1868), Frischa

debt of gratitude for the work that they have done in the matter of rendering the conceptions which constitute the fabric of metageometry understandable and thinkable. A glance at the bibliography appended at the end of this volu

which follows. For there is no doubt but that unheard of possibilities of thought have been revealed by investigations into the nature

decessor. Klein held that there are only two kinds of Riemannian space-the elliptical and the spherical. Or in other words, that there are only two possible kinds of space in which the propositions announced by Riemann could apply. Sophus

true descriptions of the spaces concerned, are, nevertheless, incompatible. All of them cannot be valid. It will perhaps be found that none of them are valid, especially objectively so. The only true view, therefore, of these systems of hyperspaces is that which assigns them to their rightful place in the infinitely vast world