Manifold

“ But there is nothing in the mind's capacity of relating the manifold by way of figure to determine it to combine a given manifold into a figure of one kind rather than into a figure of any other kind; for to combine the manifold into a particular shape, there is needed not merely the thought of a figure in general, but the thought of a definite figure. ”

“ We maintain, therefore, that the presentation of sense, the continuum or manifold, or what you will, consists in the transmutations of a real substance itself unextended and unperceived; that the laws of these transmutations are what constitute the geometric all-containing Space; that at a point in this real energetic system organically related to the intelligent self, the transmutations occurring there constitute the individual's sensible experience ”

“ But measurement demands superposition, and consequently some magnitude independent of its place in the manifold. In passing, in a continuous manifold, from one element to another in a determinate way, we pass through a series of intermediate terms, which form a one-dimensional manifold. If this whole manifold be similarly caused to pass over into another, each of its elements passes through a one-dimensional manifold, and thus on the whole a two-dimensional manifold is generated. In this way we can proceed to n dimensions. ”

“ The totality of event-particles will form a four-dimensional manifold, the extra dimension arising from time—in other words—arising from the points of a timeless space being each a class of event-particles. ”

“ 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 [Pg 63] which geometric space is determined to be a manifold of either a positive or negative curvature, also by which its extent may be measured. ”

“ Now the bodies themselves may be different in kind, in nature, may be of various material, may differ widely in structure, weight, and shape, and still, since they all belong to the same type of motion, they are, in spite of their manifold diversity, expressed in one general formula, in one law, namely, that the spaces traversed are proportional to the square of times. ”

“ We cannot cogitate a geometrical line without drawing it in thought, nor a circle without describing it, nor represent the three dimensions of space without drawing three lines from the same point perpendicular to one another. We cannot even cogitate time, unless, in drawing a straight line (which is to serve as the external figurative representation of time) , we fix our attention on the act of the synthesis of the manifold, whereby we determine successively the internal sense, and thus attend also to the succession of this determination. ”

“ One can almost always rightfully take it for granted that several, indeed manifold interpretations are correct. It is best to think of a stratified structure. In the most superficial layer lies the most obvious explanation, in the second a somewhat more hidden one, and in yet deeper strata broader and more remote relationships and all have their part more or less in the manifested phenomenon. ”

“ Take, for instance, the manifoldness which embraces all the circles that can lie in a given plane. The points of such a manifoldness are the circles. It is three- dimensioned, for two coordinates will be required for the centre of each circle and one for its radius. It is obviously a continuous manifoldness, for one circle may by infinitely graduated modifications pass into any other. Yet from this manifoldness also the conception of distance is absent. There is no intelligible relation of one circle to another which is analogous to the distance which we require to determine. ”

“ The more deeply we perceive, the more striking becomes the evidence that a uniform plan links every form in manifold nature.” ”

“ Modern geometry bases itself not on space but on multiply-infinite number-manifolds—amongst which the three-dimensional is simply the undistinguished special case—and within these groups investigates functional formations with reference to their structure ”

“ It is important to observe that the elements of the manifoldness may be themselves objects of no little complexity. Thus, for instance, the conies forming a confocal group constitute the elements of a singly ex tended manifoldness. ”

“ In this an extremely important scientific procedure is indicated which consists in giving to the experience-content of a given field a formal manifoldness to which we impart the same manifoldness-character as that possessed by the former. Every science thus develops a formula language of its own, perfect in proportion to the accuracy with which the manifoldness-character of the object has been recognized, and the fitness of the formulas selected. ”

“ If two similar mediums be separated from each other by a space terminated on both sides by parallel planes, and a body in its passage through that space be attracted or impelled perpendicularly towards either of those mediums, and not agitated or hindered by any other force; and the attraction be every where the same at equal distances from either plane, taken towards the same hand of the plane; I say, that the sine of incidence upon either plane will be to the sine of emergence of the other plane in a given ratio. ”

“ The system of Physics which I have for many years inculcated, in the hope of removing from Philosophy the equivocal word attraction, supposes that space is filled with an elastic medium,—that this medium permeates bodies in proportion to their quantities of matter,—that resistance or re-action takes place between the universal medium of space and the novel arrangements of matter in bodies,—that this action and re-action diverge in the medium of space from the surfaces of bodies,—and that, like all diverging forces, they act inversely as the squares of the distances. ”

“ Dynamics, then, or the science of force and motion, is thus placed at the head of all the sciences; and, happily for human knowledge, it is one in which the highest certainty is attainable, a certainty no way inferior to mathematical demonstration. As its axioms are few, simple, and in the highest degree distinct and definite, so they have at the same time an immediate relation to geometrical quantity, space, time, and direction, and thus accommodate themselves with remarkable facility to geometrical reasoning. ”

“ We get it when we pass from a short vertical line to a long horizontal one, from a small square to a large circle, as well as when we pass between those figures whose shapes are congruous. But when the shapes are congruous our consciousness of the relation is a good deal more distinct, and it is most distinct of all when, in the exercise of our analytic attention, we notice, first, a part, and then the whole, of a single line or shape. ”

“ But every right line infinitely produced cuts a spiral in an infinite number of points; and the equation by which any one intersection of two lines is found at the same time exhibits all their intersections by as many roots, and therefore rises to as many dimensions as there are intersections. Be cause two circles mutually cut one another in two points, one of those intersections is not to be found but by an equation of two dimensions, by which the other intersection may be also found. ”

“ I. And has this Creature sides, as well as angles or what you call "terminal Points?"
Sphere. Of course; and all according to Analogy. But, by the way, not what you call sides, but what we call sides. You would call them solids.
I. And how many solids or sides will appertain to this Being whom I am to generate by the motion of my inside in an "upward" direction, and whom you call a Cube?
Sphere. How can you ask? And you a mathematician! The side of anything is always, if I may so say, one Dimension behind the thing. Consequently, as there is no Dimension behind a Point, a Point has 0 sides ”

“ So also we imagine space, the place of all motion, with its parts immovable, the field of all separation, with its parts inseparable; but we are treating of division, not of separation. If there is true extension, it is divisible; we conceive space with its parts inseparable, but still divisible; for we measure them, count them, and it is by relation to them that we form an idea of the size, distance, and motion of bodies. ”

“ A line having no balance or symmetry in itself may become balanced. The line may be regarded as if it were a series of dots close together. The line is then a relation of positions indicated by dots. It is a composition of attractions corresponding and equal. It is only necessary, then, to find what I have called the center of equilibrium, the balance-center of the attractions, and to indicate that center by a symmetrical inclosure. The line will then become balanced.
Fig. 93
Here is a line. ”

“ We suppose such two-dimensional beings to be absolutely limited to a certain surface. They could have no conception of space except as of two dimensions. The movement of a point would for them form a line, the movement of a line would form a surface. They could conduct their measurements and form their geometrical theories. They would be able to draw the shortest lines between two points, these lines being what we would call geodesies. ”

“ A circle may be described with any given point as a center and any given line as a radius. This involves the use of the second of the two geometric instruments, the compasses.
5. Any figure may be moved from one place to another without altering the size or shape. This is the postulate of the homogeneity of space, and asserts that space is such that we may move a figure as we please without deformation of any kind. It is the basis of all cases of superposition. ”

“ In solid geometry the elementary figures are the point, the line and the plane; we have, moreover, first, that which under one aspect is the curve and under another aspect the developable (or torse) , and which may be regarded as a singly infinite system of points, of lines or of planes; and secondly, the surface, which may be regarded as a doubly infinite system of points or of planes, and also as a special triply infinite system of lines. (The tangent lines of a surface are a special complex.) ”

“ The truths of geometry are summed up and embodied in its definitions and axioms.... Let us turn to the axioms, and what do we find? A string of propositions concerning magnitude in the abstract, which are equally true of space, time, force, number, and every other magnitude susceptible of aggregation and subdivision. Such propositions, where they are not mere definitions, as some of them are, carry their inductive origin on the face of their enunciation.... ”

“ There may be many different forms of energetic transmutation which may determine many other forms of space besides that form of tridimensional space in which our Activity is involved. For such, a different geometry may and will be applicable; but for the tridimensional conditions of our activity the proposition is necessary and absolute. No measurement of any stellar parallax, however minute and whatever the result might be, could have any bearing on its truth. Geometry is the science of the pure forms of our motor activity amidst corporeal bodies. ”

“ But whatever notions of three-dimensional space might have been developed from such rudiments, the perception of cubic existence which we actually possess and employ, is undeniably based upon the incomparably more important data afforded by locomotion, under which term I include even the tiny pressure of a finger against a surface, and the exploration of a hollow tooth by the tip of the tongue. ”

“ My dear sir, geometry deals with facts of space, space itself is taken as an ultimate datum. The work of geometry is not to ask what space is in itself, but what the relations are of spatial forms, space itself being postulated. ”

“ Here we have a perfect balance between variety and unity. The curvature is varied to an infinite degree, at no point is its curving at the same ratio as at any 139other point; perhaps the maximum amount of variety that can be got in a symmetrical figure, preserving, as it does, its almost perfect continuity, for it approaches the circle in the even flow of its curvature. ”

“ But if a body, which is moved, be considered as long, and be supposed to be so moved, as that all the several parts of it be understood to make several lines, then the way of every part of that body is called breadth, and the space which is made is called superficies, consisting of two dimensions, one whereof to every several part of the other is applied whole. ”

“ The only answer the geometrician could give is that we must assume that space is uniform, so that wherever we form our triangles we obtain the same results. The same is true in mechanics. The laws of motion and inertia hold good of the pebble on the roadside, of the dust grains dancing in the sunbeam, and of distant stars in the milky way. ”

“ Supossing the sine of incidence upon any superficies to be in a given ratio to the sine of emergence; and that the inflection of the passs of those bodies near that superficies is performed in a very short space which may be confirmed as a point, it is required to determine such a superficies as may cause all the corpuscles issuing from any one given place to converge to another given place. ”

“ Imagine curve lines to be described on any plane, and to revolve about any given axes passing through the centre of force, and by that revolution to describe curve superficies; and that the bodies move in such sort that their centres may be always found in those superficies. If those bodies reciprocate to and fro with an oblique ascent and descent, their motions will be performed in planes passing through the axis, and therefore in the curve lines, by whose revolution those curve superficies were generated. ”