that, if the difference can be discovered in this manner, it is less than the errors of observation of the present day. Riemann in 1867 published a paper, which he had worked out some years before, in which he took the ground that our notions of space were of a general type, and that by experience we have learned that the geometry of space in nature should be, at least to a high degree of approximation, the Euclidean geometry. He also introduced a third geometry, which permits no parallels, nor similar figures, and which is finite, just as a sphere permits infinite motion, but is yet finite. It seems that no one previously had thought of this geometry which permits two lines to enclose a space, and which was the necessary complement of the Lobatchevskian. Indeed, the formulae of trigonometry in this geometry are the usual formulae of spherical trigonometry, if we make the argument or angle equal to the side of the triangle multiplied by the curvature, while in Lobatchevskian geometry we need corresponding hyperbolic functions. Whether we live in a three-dimensional point-space with a fourdimensional curvature which is positive, zero, or negative, is a question. Indeed, the curvature might not even be constant. As to whether we may ever settle the question, opinions differ. Poincaré took the position that the question had no sense. It was equivalent, he said, to asking which is the true measure for space, a yard or a meter, or whether rectangular or polar coordinates are the true co-ordinates for space. The definitive conclusion, however, for our purposes is easily seen. If geometry is derived intuitively from experience, then we should know instinctively which is the geometry applicable to the world in which we live. If we have not yet ascertained the answer to this fundamental question, then we do not derive our geometry intuitively. Neither does it come from a hypothetical world of universals, which themselves are derived from experience, like composite photographs, or even appear as invariants of experience. We are forced back again upon the conclusion that geometry is the direct creation of the human intellect, drawing its sustenance from the world of phenomena, but wonderfully transforming it, just as the plant transforms the water, the air, the carbon dioxide, into a flower. Kant had based his philosophy upon the objective certainty of Euclidean geometry and his philosophy had to go through a revision, for space was no longer a necessary form imposed upon the world when it took the clothing of the mind, but the mind was free to impose what form it liked. The transcendental character of Kant's philosophy went down into ruins, though his contention that the mind supplied its share to the content of experience was most astonishingly vindicated. Indeed, it turns out that without the creative co-operation of the mind there would be no experience. So great an importance is thus attached to the working of the mind that Bergson takes occasion to warn us that the intellect is merely one of the active agencies of life, whose products are produced for specific ends, but are not sufficient for all the ends of life. Other researches also lead to the same conclusion. We need mention only the developments which were not of an analytical nature, although they may have been first suggested in that way. This was the creation of projective geometry. Of this Keyser says: "Projective geometry-a boundless domain Monge (1746-1818)-Poncelet (1788–1868). 2 Columbia University Lectures (1908), p. 2. of countless fields where reals and imaginaries, finites and infinites, enter on equal terms, where the spirit delights in the artistic balance and symmetric interplay of a kind of conceptual and logical counterpoint, an enchanted realm where thought is double and flows in parallel streams.” We find, indeed, here the common ground for the union of all geometries, ordinary or N-dimensional, parabolic, hyperbolic, or elliptic. Starting from this foundation, we may be led to take the view of Klein and others that geometry essentially is only the theory of the invariants of different groups. For instance, the geometry of Euclid is the theory of the invariants of a certain group called the group of Euclidean movements, the ordinary group of translations and rotations. We find as another development the geometry of reciprocal radii, with such points of union as this: the geometry of reciprocal radii is equivalent to a projective geometry on a quadric properly chosen. We may study other groups, as that of rational transformations, indeed, all of Lie's (1842-99) transformation-group theory. We come back to the usual space again in the group of all continuous transformations, giving us analysis situs. We have thus finally created a very general geometry which may be illustrated as follows. If we were to undertake to study the geometry of the plane in its reflection in a very crooked and twisted mirror, we might not for a long time find out the usual theorems. But there would nevertheless be certain theorems that would remain true, however much distorted the image might be. This kind of geometry is very general and is independent of the Lobatchevskian, Euclidean, or Riemannian postulates. One feature of it, for example, is the three dimensionality of space.1 1 Poincaré, Revue de mét. et mor., 20 (1912), pp. 483-504. We have from these investigations the definite result that even in a world of continual flux, where forms dissolve into others, point becoming point, or point becoming line, or point becoming circle or sphere, yet the intellect has created a mode of handling its problems of existence. We find, in other words, that an infinity of relativities are possible and of the most curious types, and even though the physicist is unable to locate any special point, line, plane, or configuration in space as an absolute point of departure, even though he must use changing scales of measurement, yet mathematics is superior to the world of sense and dominates it in all its forms. Whatever problems the ages may bring forth as to space or its measurement, or, indeed, as to its companion-timewe know to a certainty that mathematics will meet the situation, create a set of notions and relations sufficient to explain and manage the problems. If the Minkowski four-dimensional world of a mingled time and space becomes the most rational way to think of phenomena, we will find it just as easy as to think of the Copernican astronomy or the rotation of the earth. REFERENCES Mach, Space and Geometry, tr. by McCormack. Hinton, Fourth Dimension. Manning, The Fourth Dimension Simply Explained. Darboux, "Development of Geometrical Methods," Bull. Kasner, "Present Problems in Geometry," Bull. Amer. Math. Soc. (2), 11 (1905), pp. 283–314. Klein, "Erlanger Programme," Bull. New York Math. Soc., 2 (1893), pp. 215-249. CHAPTER IV ARRANGEMENTS AND MATHEMATICAL TACTIC There is a charm for most persons in the arrangement of a group of objects in symmetrical designs. Three objects placed at the vertices of an equilateral triangle, or four at the vertices of a square, or three at the vertices of an equilateral triangle and one at the center, and other more complicated arrangements, which, however, preserve similar symmetries, appeal to the aesthetic sense as beautiful. The fact that in certain arrangements of objects under the action of physical forces we find them at the vertices of regular polygons, as in the experiments of J. J. Thomson on the arrangements of small magnets, or the arrangements of molecules in crystals, leads some philosophers to a view of the universe not very remote from that of Pythagoras, for the integer dominates these forms. The arrangement of the integers in various designs, such as squares, stars, crosses, and other forms, so that the sums of certain selected lines are all equal, has even been supposed to have magic power, and we find "magic squares" used as talismans against misfortune, and other mystic diagrams ascribed with great power. The study of magic squares has fascinated many persons, the underlying harmonies and mathematical laws furnishing the incentive to prolonged study. We may quote the statement of MacMahon :1 What was at first merely a practice of magicians and talisman makers has now for a long time become a serious study for 1 Proceedings of the Royal Institution of Great Britain, 17 (1892), pp. 50-61. |