It will facilitate the conception of this somewhat difficult subject to consider the case of hypothetical reasoning beings which Sylvester described as being infinitely attenuated bookworms confined to infinitely thin sheets of paper. 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 geodesics. To these two-dimensioned geometers geodesics would possess many of the attributes of straight lines in ordinary space. If the surface to which the beings were confined were actually a plane, then the geometry would be the same as our geometry. They would find that only one straight line could be drawn between two points, that through a point only one parallel to a given line could be drawn, and that the ends of a line would never meet even though the line be prolonged to infinity.

We might also suppose that intelligent beings could exist on the surface of a sphere. Their straightest line between two points would be the arc of the great circle joining those two points. They would also find that a second geodesic could be drawn joining the two points, this being of course the remaining part of the great circle. A curious exception would, however, be presented by two points diametrically opposite. An infinite number of geodesics can be drawn between these points and all those geodesics are of equal length. The axiom that there is one shortest line between two given points would thus not hold without exception. There would be no parallel lines known to the dwellers on the sphere. It would be found by them that every two geodesics must intersect, not only in one, but even in two points. The sum of the three angles of a triangle would for them not be constant. It would always be greater than two right angles, and would increase with the area of the triangle. They would thus have no conception of similarity between two geometrical figures of different sizes. If two triangles be constructed which have their sides proportional, the angles of the larger triangle would be greater than the corresponding angles of the smaller triangle.

It is thus plain that the geometrical axioms of the spheredwellers must be very different from those of the planedwellers. The different axioms depend upon the different kinds of space which they respectively inhabit, while their logical powers are identical In one sense, however, the dwellers on the sphere and on the plane have an axiom in common. In each case it will be possible for a figure to be moved about without alteration of its dimensions. A spherical triangle can be moved on the surface of a sphere without distortion just as a plane triangle may be moved in a plane. The sphere-dwellers and the plane-dwellers would be equally able to apply the test of congruence. It is, however, possible to suppose reasoning beings confined to a space in which the translation of a rigid figure is impossible. Take, for instance, the surface of an ellipsoid or even a spheroid such as the surface of the earth itself. A triangle drawn on the earth at the equator could not be transferred to the surface of the earth near the pole and still preserve all its sides and all its angles intact.

If a surface admits of a figure being moved about thereon so as still to retain all its sides and all its angles unaltered, then that surface must possess certain special properties. It can be shown that, if a surface is to possess this property, a certain function known as the “measure of curvature” is to be constant. The measure of curvature is the reciprocal of the product of the greatest and least radii of curvature. We do not now enter into the proof, but it is sufficiently obvious that a sphere of which the radius is the geometric mean between the greatest and least radii of curvature at each point will to a large extent osculate the surface, so that a portion of the surface in the neighbourhood of the point will, generally speaking, have the same curvature as the sphere. If tire sphere thus determined be the same at all the different points of the surface, then the curvature of the different parts of the surface will on the whole resemble that of the sphere, and therefore we cannot be surprised that the surface possessing this property will admit the displacement of a rigid figure thereon without derangement of its form.

We are thus conducted to a kind of surface the geometry of which is similar to that of the plane, but in which the axiom of parallels does not hold good. In this surface the radii of curvature at every point have opposite signs, so that the measure of curvature which is zero for the plane and positive for the sphere is negative for the surface now under consideration. This surface has been called the “pseudosphere,” and its nature has been investigated by Beltrami. In the geometry of two dimensions we can thus have either a plane or a sphere or a pseudosphere which are characterized by the property that a surface may be moved about in all directions without any change either in the lengths of its lines or in the magnitudes of its angles. The axiom which assumes that there is only one geodesic connecting two points marks off the plane and the pseudosphere from the sphere. The axiom that only one parallel can be drawn through a given point to a given line marks off the plane from the pseudosphere. The geometry of Euclid is thus specially characterized among all conceivable geometries of two dimensions by the following three axioms—(1) the mobility of rigid figures, (2) the single geodesic between two points, (3) the existence of parallels.

A very interesting account of this theory will be found in Clifford’s Lectures and Essays, vol. i. p. 317. We shall follow to some extent the method employed by him in order to obtain an idea of the important conception which is called the “curvature of space.” Suppose a geodesic be drawn on a surface of constant curvature. Then a piece of the surface adjoining this geodesic can be slid along the curve so as all the time to fit in close contact therewith. If the piece of surface be turned to the other side of the geodesic it will still fit along this side. A line possessing this property is called by Leibnitz a straight line. It can be easily shown that a geodesic drawn on a figure will also be a geodesic when the figure is transferred to any other position. Suppose that the figure be divided into two parts A and B by the geodesic; then the part B can be moved round so as to lie upon A, and the boundary lines of the two portions will be coincident. Now let the two parts while superposed be translated to any other position, then the piece B may be slid off and back to its original position with regard to A. It must still fit, because the whole figure might have been translated before the subdivision took place. It follows that the division between A and B having been a geodesic in its original position will continue to be a geodesic however the figure may be translated.

In a similar way we obtain the conception of a plane. According to Leibnitz’s definition a plane is a surface such that if a portion of the space contiguous thereto be slid along the surface it will continuously fit, and if the portion of space be transferred to the other side of the surface it will fit also. This definition has no meaning except we assume that the bodies may be translated in space without derangement of their dimensions. From any point we can imagine a doubly infinite number of geodesics radiating in all directions; if a plane be drawn through the point, then all the geodesics touching the plane at that point form what may be called a “geodesic surface.” It is shown that geodesic surfaces of this description can alone fulfil the conditions by which planes are to be defined. A doubly infinite number of geodesic surfaces can be drawn through every point. If a rigid body be divided into two parts by a geodesic plane, then no matter how the body be displaced the plane of section will still be geodesic. The plane of section may be made to pass through any point, and the body may then be given such an aspect as shall cause the section to coincide with any geodesic surface through the point, but this necessarily involves that the section shall fit each geodesic surface, in other words, that all the geodesic surfaces shall have a constant curvature.

The point which we have now gained is one of very great importance. In our ordinary conceptions of space the geodesic surfaces are of course our ordinary planes, and the common curvature they possess is zero, but the condition that rigid bodies shall be capable of translation with unaltered features does not require that the curvatures shall be zero, it merely requires that the curvatures shall be constant. If we add, however, the postulate of similarity, then the curvatures must be zero. The postulate of similarity requires that it shall be possible to construct a figure on any scale and anywhere similar to a given figure. This practically includes the ordinary doctrine of parallels. Lobatchewsky developed the system of geometry on the supposition that the space had a constant curvature different from zero. In this geometry the parallels can be drawn through a given point to a given line, and, to quote Clifford—

“The sum of the three angles of a triangle is less than two right angles by a quantity proportional to the area of the triangle. The whole of this geometry is worked out in the style of Euclid, and the most interesting conclusions are arrived at, particularly in the theory of solid space, in which a surface turns up which is not plane relatively to that space but which for the purpose of drawing figures upon it is identical with the Euclidean plane.”

The most comprehensive mode of viewing the whole theory is that adopted by Riemann in his celebrated memoir “Ueber die Hypothesen welche der Geometrie zu Grunde liegen,” 1854 (Abhandl. der königl. Gesellsch. zu Göttingen, vol. xiii.). The analytical treatment of this subject possesses one obvious advantage. The use of symbols only admits of deductions on purely logical principles. There is not therefore the risk of tacitly introducing other axioms in addition to those with which we started.

Magnitudes which have only one dimension present the theory of measurement in its simplest form. The length of a straight line may be taken as an illustration of a one-dimensioned magnitude. The velocity of a moving particle, the temperature of a heated body, the electric resistance of a metal, all these and many others are instances of one-dimensioned magnitude, the measure of which is to be expressed by a single quantity. But there may be magnitudes which require more than a single measurement for their complete specification. Take, for instance, a four-sided field which has been duly surveyed. Of what is the measurement of this field to consist? If the number of acres in the field be all that is required then the area is expressed by a simple reference to a number of standard acres. If, however, the entire circumstances of the field are to be brought into view, then a simple statement of the area is not sufficient. It can be easily shown that the surveyor must ascertain five independent quantities before the details of the shape of the field can be adequately defined. Four of these quantities may naturally be the lengths of the four sides of the field, the fifth may be one of the angles, or the area, or the length of one of the diagonals. Speaking generally, we may say that five distinct measurements will be necessary to define the field adequately. The actual choice of the particular measurements to be made is to a great extent arbitrary. The only condition absolutely necessary is that they shall be all independent and free from ambiguity. Once these five quantities are ascertained then all the other features of the figure are absolutely determined. For instance, the four sides and the diagonal being ascertained by measurement, then the other diagonal, the four angles, and the area can all be computed by calculation. The five quantities would determine everything about the field except its actual position on the surface of the earth. If we further desired to have the field exactly localized certain other quantities must be added. The latitude and the longitude of one specified corner of the field would completely indicate that corner, while the azimuth of one side from that corner would complete the definition of its position. We are thus led to see that for the complete delineation of every circumstance relating to the shape of the field and its locality eight different measurements have been required. Two sets of eight measurements differing in any particular can never indicate the same field. It is very important to notice that the number of quantities required is quite independent of the particular nature of the measurements adopted. We might for instance have simply measured the latitude and the longitude of each of the four corners of the field. Once these quantities are known, then the shape of the field, its area, its angles, and its diagonals have all been implicitly determined. Here again we see that as two quantities are required to localize each of the four corners, so eight quantities will be required to fully determine the whole field.

In the operations of analytical geometry we are accustomed to specify the position of a point by the relation which it bears to certain fixed axes. By means of certain quantities, either altogether linear or partly linear and partly angular, we are enabled to specify the position of the point with absolute definiteness. These quantities are called the coordinates of the point. In a similar though more extended sense we may use the word “coordinates” to express the group of eight magnitudes which we have found to be adequate to the complete specification of the field. By the measurement of the field in the most complete sense of the term we mean the measurement of its eight coordinates.

Suppose that an object is completely specified by n coordinates, then every different group of n coordinates will specify a different object. The entire group of such objects will form what is called a continuously extended manifoldness. The singly extended manifoldness may be most conveniently illustrated by the conception of time, the various epochs of which are the elements in the manifoldness. Space is a triply extended manifoldness whereof the elements are points. All conceivable spheres form a quadruply extended manifoldness. All conceivable triangles in space form a manifoldness of nine dimensions. The number of coordinates required to specify the position of an element in a manifoldness is thus equal to the order of the manifoldness itself. It is important to observe that the elements of the manifoldness may be themselves objects of no little complexity. Thus, for instance, the conics forming a confocal group constitute the elements of a singly extended manifoldness.

The essential feature of a singly extended manifoldness is that a continuous progress of an element can take place only in two directions, either forwards or backwards. But a singly extended manifoldness may be regarded as itself an element in a manifoldness of a higher order. Thus the points on a circle form a singly extended manifoldness, while the circle itself is one element of the manifoldness which consists of a series of concentric circles. The system of concentric circles may in like manner be regarded as an element in the manifoldness which embraces all systems of concentric circles whose centres lie along a given line. We are thus led to conceive of a multiply extended manifoldness as made up by the successive composition of singly extended manifoldnesses.

It follows from the conception of a manifoldness that in the case of a singly extended manifoldness the position of every element must be capable of being completely specified by a single quantity. It becomes natural to associate with each element of the manifoldness a special numerical magnitude. These magnitudes may vary from - ∞ to + ∞; to each magnitude will correspond one element of the manifoldness, and conversely each element of the manifoldness is completely specified whenever the appropriate number has been assigned. It is quite possible to have this association of numerical magnitude with the actual position of an element independent of any ordinary metrical relations of the system; it will, however, most usually be found that the numerical magnitudes chosen are such as admit of direct interpretations for the particular manifoldness under consideration. Thus, for instance, in the case of the system of concentric circles it will be natural to associate with each circle its radius, and the position of each circle in the manifoldness will thus be completely defined by the radius. So also in the case of that singly extended manifoldness which consists of colours, it will be natural to employ as the number which specifies each particular colour the wave-length to which that particular colour corresponds.

If the elements of such a manifoldness can receive a simultaneous displacement, then it is plain that to each element in the original position will correspond an element in the second position. Let x and y be the numerical magnitudes correlated to these two elements. Then, since the relation must be of the one-to-one type, it is necessary that the magnitudes x and y must be connected by an equation of the type axy + bx+cy + d≈O.

It follows from this that there are a pair of elements which are common to both systems, for if x=y we have the equation oλ 2 + (δ + c}x ÷ d = 0.

The original equation may be written in the form axy + (δ - ω)x + (c + ω)y + d = 0, and whatever value ω may have this equation will lead to the same quadratic for the two common elements. We thus have a singly infinite number of displacements which are compatible with the condition that the two fundamental elements shall remain unaltered, and it is displacements of this kind which express the movements of a rigid system.

The position of a point is to be defined by three coordinates. In our ordinary conception of coordinates the position of the point is defined by certain measurements, and thus it would seem that the very mention of coordinates had already presupposed the idea of distance. This, however, need not be the case. We can assume a point in space to be completely defined by three purely numerical quantities. It will be supposed that to each group of three coordinates corresponds one point, and that conversely to one point will correspond three coordinates and no ambiguity is to be present. This latter consideration will exclude from our present view such cases as those where the position of a point is defined by a line and two angles, because angles are subject to a well-known ambiguity amounting to any even multiple of τr. In this case it would not be true that to one point corresponds one set of coordinates, although the converse may be correct.

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