For the purposes of an abstract conceptual scheme time as empirically measured must be idealized. For this purpose Newton suggested the conception of absolute time as that which flows uniformly. Such a definition however can only be taken as generally descriptive and not as really defining a precise conception because the expression “flows uniformly” implies already an underlying conception of time. Thus his definition is open to the fatal objection of circularity. In abstract Dynamics and in fact in any purely abstract conceptual scheme the role of time is played by an independent variable which on account of the function it has in a scheme to be applied to the description of actual physical processes may be spoken of as the time-variable. The field of this time-variable is taken to be the arithmetic number-continuum; the aggregate of all real numbers. Thus a particular time in the abstract scheme of Dynamics is simply a number; a particular value of the time-variable; and an interval of time is the difference of the numbers which represent the two ends of the interval. In this completely abstract conception of time the generic distinction between future and past which exists in perceptual time has disappeared; there exists only an ordered aggregate of a particular type of which the elements are conceptual instants the relation of elements of lower rank in the order to those of higher rank being all that corresponds to the original distinction of past from future. When the conceptual scheme is applied to describe actual physical processes as for example the motions of bodies the time-variable is correlated with the public time as measured by the standard physical process in such a manner that equal intervals of the arithmetic continuum which is the field of the time-variable correspond with equal measurable amounts of the standard process. This procedure adopted by Science of representing abstract time by the arithmetic continuum has been spoken of by Bergson as the specialization of time no doubt on account of the fact that the linear spatial continuum can also be correlated with the arithmetic continuum. It should however be observed that in doing this a process of abstraction is applied to the space of perception of a character similar to that which has been employed in the case of time. Thus a more accurate account of the procedure of Science would be to say that both time and (linear) space are represented conceptually by one and the same abstract scheme that of the arithmetic continuum. Both have in fact been represented as ordered aggregates of the same type. It must be remembered that the arithmetic continuum is a construction not arrived at by idealizing the notion of linear space¹ but that it is logically independent of the notion of spatial magnitude. One of the chief applications of the arithmetic continuum is to provide an ideally exact theory of the measurement both of spatial and of temporal magnitudes.

As in the case of the temporal perception of an individual his spatial perception is only separable by abstraction from his whole physical experience. His spatial perceptions are dependent on a variety of sensations partly visual and partly tactile and motor. The spatial perceptions which arise from these two sources considered separately differ notably from one another in character; the representative space of the individual is a synthesis of the two in which habit that is past spatial experience plays a large part. The geometrical space which we employ when we reason about the spatial properties of ideal bodies is of a conceptual character and differs in important respects both from visual space and from tactile and motor space. Geometrical space is of three dimensions infinite or at least unbounded homogeneous and isotropic.

It is impossible for me to analyse in detail the characteristics of visual space and of tactile and motor space. It must suffice to say that visual space is neither homogeneous nor isotropic and that in it distance is only indirectly appreciated by convergence of the two eyes with muscular sensations due to accommodation between them. This space has three dimensions and the particular kind of Geometry adapted to the conceptual description of it is what is known as projective Geometry. In tactile and motor space we make direct experimental estimates of distances and measures. Each muscle gives rise when it is contracted to a special sensation so that motor space may be said to have as many dimensions as we have muscles. The conceptual Geometry which corresponds to tactile and motor space is what is called Metric Geometry. The private or representative space of an individual percipient like his private time is finite but with an indistinct boundary. Private space like private time is sensibly continuous as any portion of it is conceived to contain lesser portions without any definite limit of smallness. As in the case of time it is not atomic for it does not consist of points without extension: the notion of a point of space is like an instant of time a pure abstraction. In spatial as in temporal presentations there are qualitative distinctions of direction. The frame of spatial reference of the individual is provided by his own body and the qualitative distinctions of up and down right and left depend upon this frame of reference. The private space of the individual has a certain absoluteness because the sense of effort which he has when he moves relatively to his material environment is lacking when he himself remains quiescent and the external objects are moved relatively to his body; the relative motions of his body and the objects being the same in the two cases.

The space employed in social intercourse sometimes called physical space is a construction of such a character that the private spaces of individuals may be systematically correlated with portions of it. The possibility of such correlation depends upon the fact that an identity is attributed to a physical object of such a kind that it is regarded by all percipients as one and the same object. Thus physical space is the complex of the spatial relations of physical objects regarded as a system of objects capable of being perceived by any percipient in suitable circumstances. It is in physical space that all the phenomena with which Natural Science is concerned are regarded as happening. The frame of reference of the single individual having been eliminated the only meaning that can be ascribed to the rest or motion of a body in physical space is conservation or change of spatial relations with other bodies. For any measurement of position or of the motion of a body some standard frame of reference in some standard body such as the earth or the sun or the walls and floor of the room must be assigned. Thus all spatial relations in physical space are relations of extension of perceptual objects and between different objects; physical space cannot be regarded as empty space because with the disappearance of perceptual objects the whole scheme of relations which constitutes physical space would disappear.

The opinion has been prevalent that external objects are localized by us in a geometrical space of a unique character with definite properties; this geometrical space has been frequently regarded as a kind of ready-made framework in which we localize our physical perceptions. When I discuss the possible forms of geometrical space it will I think appear that this is by no means the case. To give a systematic scheme descriptive of the relations in physical space is the first object of the Science of Geometry although in some of its developments the Science has so extended itself as to transcend this primary object. In order to attain this object the spatial relations are idealized and transformed into a precise form by means of a system of definitions and postulations. By this process of abstraction and idealization conceptual space the space of abstract geometry has been created. It is in this conceptual space that all the ideal objects of Geometry are regarded as situated and as subject to a scheme of relations specified by a system of postulates. In accordance with the general method of scientific procedure such a scheme must satisfy the condition of logical coherence; that is of freedom from contradiction. It must also satisfy the test of applicability to the description of the spatial relations which are observed to hold in physical space. As a result of the efforts of Mathematicians prolonged through many centuries and supplemented by the critical examination to which the foundations of the subject have been subjected during the last few decades the Science of Geometry has attained to a degree of coherence which may be held to justify the designation of it that I made in an earlier lecture as a model to which other departments of Science may tend to conform. Geometry is better fitted than Arithmetic to be regarded as a model in this sense because although Arithmetic as we saw in the last lecture has been completely conceptualized it did not originate from the necessity of describing conceptually any one particular class of physical properties; whereas Geometry had its origin in the effort to describe one particular class of physical relations namely the extensional properties of perceptual objects.

I propose to give some account necessarily brief of the process of growth by which Geometry in its present highly developed state has come into being. A remarkable illustration of the fact that a valid conceptual scheme adequate for the purposes of description for which it was devised is not necessarily unique is afforded by the fact that different conceptual schemes of Geometry have been constructed. These systems are all logically coherent; they are inconsistent with one another as regards their several postulations; and yet several of them are adequate at least when they are suitably restricted for the purpose of description of percepts. The nature of our knowledge of spatial relations has been a problem for Epistemology and Psychology which has been very widely discussed. In this connection the history of Geometry is of great interest especially since as a result of modern investigations a decisive refutation has been provided from within the Science of Kant's celebrated views as to our spatial intuition. These views are still maintained in some philosophical circles often owing to an insufficient comprehension of the nature of the modern developments of abstract Geometry. I reserve a discussion of the latest theory of spatial and temporal relations that connected with the names of Minkowski and Einstein for a later lecture. It is sufficient here to remark that if that theory be finally established it does not affect the validity of the previously existing theories of Geometry or their applicability as descriptive schemes sufficient for ordinary purposes. The older Geometry will accordingly never be completely superseded by the more comprehensive theory to which I have referred.

The earliest rigorous treatment of the Science of Geometry was that of the Greek Mathematicians; of this we possess a systematic account in that great textbook of the subject Euclid's Elements of Geometry. This contains an account of the current Geometry in Euclid's time; it is the most ancient text-book of Science that we possess; and the fact that it has been used for many centuries is a testimonial to its excellence. To a large extent the Greek Geometry exhibits in dealing with physical extension the true scientific method. The simplest regularities and uniformities observable in the shapes and spatial relations of actual bodies were singled out and then conceptualized. Thus points straight lines planes rectilineal figures circles spheres pyramids and other objects transformed and idealized from percepts into concepts are the geometrical objects with which Euclid deals in his abstract Geometry. Later writers especially Apollonius and Archimedes treated of the Geometry of cones and conic-sections; an example of non-utilitarian scientific investigation which bore useful fruit in the hands of Kepler. The type of Geometry developed by the Greeks known as Euclidean Geometry and for a long time regarded as the only possible type of Geometry still remains the standard Geometry for practical purposes and for ordinary scientific purposes. It forms the conceptual basis for all our actual spatial measurements and notwithstanding the increased generality of modern geometrical schemes it will certainly continue to be employed for the more ordinary purposes of Science even if it be superseded for the purposes of certain general theories relating to gravitation and Electrodynamics. In order to distinguish the Euclidean Geometry from other more modern Geometrical schemes of a divergent character it is frequently spoken of as the Geometry of Euclidean space. In view of the fact that geometrical space cannot properly be regarded as an entity endowed with special properties since it in reality represents a mere possibility of spatial determinations it is more accurate to speak of Euclidean Geometry as a Euclidean system of spatial relations or as Geometry with a Euclidean Metric. In the nominal definitions of the geometrical objects with which his scheme deals Euclid gives a somewhat elaborate descriptive account of characteristics of those objects. In his deductive treatment many of these characteristics are not in any way employed in the argument. In some of the definitions postulates lie hidden which later criticism has brought to light and stated explicitly. An example of this is the fourth definition of the fifth Book which implicitly contains the postulate known as the axiom of Archimedes. What we now regard as the postulated or hypothetical scheme of relations in Euclidean Geometry appears in part in Euclid's elements in his axioms or common notions which he regards as self-evident; and in part in his postulates which are taken to be facts that are unproved but the assumption of the truth of which is necessary for the purposes of his theory. Other postulates derived from intuition are made implicitly in the course of his deductions. In Euclid's own form although his Geometry has many of the characteristics of a valid conceptual scheme the treatment is far from exhibiting that example of a flawless deductive scheme for which it has frequently been accepted. From Euclid's own time onwards there has been much discussion and criticism relating to the true character of the definitions axioms and postulates of the scheme. This has resulted during recent decades in a restatement of the foundations of the subject of such a kind that synthetic Geometry may now be regarded as conforming to all the requisites of a conceptual scientific scheme.

The objects with which Euclid deals points straight lines planes etc. are obtained as the result of idealization of actual percepts in which some constituents of the percept are removed by abstraction. Thus a point is the ideal limit postulated as existing of an object from which its extension is completely abstracted. It retains spatial relations with other objects that is position. A line is an object in which we abstract from the thickness of perceptual lines; and a straight line is an idealized object which arises from our observation of empirical straightness. Euclid gives no complete conceptual representation of relations of magnitudes. Thus equality of magnitude and the relations of congruence of segments of straight lines of angles and of areas are not made to depend upon a logically complete system of postulations; but recourse is had to intuition of magnitude in the region of percepts. This defect is exhibited prominently in Euclid's use of the method of superimposition employed in the theorems relating to congruent figures. The apparent reluctance with which Euclid employs this method restricting it as he does to cases in which its use seemed unavoidable would appear to indicate that he had misgivings as regards its logical validity. As it stands no complete defence is possible against the charge of circularity which has been made against this reasoning. In the modern form of synthetic Geometry a system of definitions and postulations relating to congruence is introduced into the foundations of the conceptual scheme itself rendering the intuitional method of superimposition unnecessary.

Euclid's theory of parallels has formed the starting point from his own day onwards of discussions which have ultimately led to a generalization of the whole theory of Geometry. It is remarkable that the so-called axiom of parallels was given by Euclid himself not as an axiom or self-evident truth but as a postulate an assumption necessary to his scheme; and thus it may be interpreted as having a hypothetical character. Since every actual observation of relations in physical space is confined to some region which is necessarily finite whereas abstract Geometry deals with relations extended into indefinitely great regions it is clear that any postulation relating to parallels must be incapable of complete empirical verification of a direct kind. But the postulate may have consequences such as that the sum of the angles of a triangle is two right angles which can subject to inevitable errors of measurement and with certain physical assumptions be regarded as capable of empirical verification. From an early time onwards attempts have been made to dispense with the use of the postulate relating to parallels as an independent assumption by showing that it can be proved as a deduction from the rest of Euclid's scheme. Proclus (410–485 A.D.) in his commentary on Euclid's Elements gave a detailed account of attempts made by Ptolemy and by himself to effect this deduction. These attempts and many others made in modern times were as we now know doomed to inevitable failure. The first Geometer who appears to have contemplated the possibility of an hypothesis relating to parallels inconsistent with that of Euclid was Girolamo Saccheri (1667–1733) a Professor at Pavia. But his invincible prejudice in favour of the Euclidean hypothesis as a necessary constituent of the only possible Geometry prevented him from recognizing the true implications of his investigations. Lambert (1728–1777) went further in the same direction as Saccheri; he showed that the area of a triangle is proportional to the difference between the sum of its three angles and two right angles in the two cases corresponding to what we now call hyperbolic and spherical Geometry. John Wallis (1616–1703) remarked that Euclid's postulate may be replaced by the equivalent assumption that similar triangles of different magnitudes exist. It is in fact sufficient to assume the existence of only two similar triangles with different magnitudes; as was observed by Laplace and by Carnot. Legendre (1752–1833) carried out investigations of permanent value in connection with various long-continued attempts to prove the truth of Euclid's postulate. These investigations prepared the way for the great change in the whole position of Euclidean Geometry which resulted from the labours of Lobachewsky Bolyai and Gauss; the last of whom was the first Mathematician to express a definite conviction that the postulate is incapable of proof.

The first publication of a synthetic geometrical theory in which a postulate was employed that differs essentially from Euclid's postulate relating to parallels was in the form of a treatise by Lobachewsky which appeared in 1829; this was followed in 1832 by the publication of a similar theory discovered independently by Bolyai. In accordance with Euclid's postulate through a point outside a straight line one and only one coplanar straight line can be constructed which does not intersect the first straight line but in the theory of Lobachewsky and Bolyai a whole sheaf of such straight lines can be constructed. This sheaf is bounded by two straight lines said to be parallel to the one considered. It was shown that when this postulate is substituted in the Euclidean scheme for that of Euclid it is possible to build up a systematic Geometry in which the properties of figures will be in some respects very different from those in the Euclidean Geometry. For example the sum of the angles of a triangle is less than two right angles the amount of the difference depending upon the size of the triangle so that in this system there can exist no relation of similarity between two rectilineal figures of different dimensions. This Geometry is a species of what is called non-Euclidean Geometry and in view of later discoveries this particular non-Euclidean Geometry developed by Lobachewsky and Bolyai is now called hyperbolic Geometry. That Geometry in which no postulate relating to parallels is employed and which therefore includes what is common to both Euclidean and non-Euclidean Geometry is frequently called absolute or general Geometry. A later discovery was made by Riemann of which I shall presently have more to say that a Geometry is possible in accordance with which all coplanar straight lines intersect one another so that no parallel straight lines exist. In this Geometry of which two distinct forms are now recognized called respectively spherical and elliptic a straight line is always a closed figure of finite length. In place of the well-known Pythagorean theorem relating to the sides of a right-angled triangle which is fundamental in Euclidean Geometry more general metrical relations were developed by Lobachewsky and Bolyai for the hyperbolic case. In spherical and elliptic Geometry the corresponding metrical relations are identical with those expressed by the ordinary formulae of spherical Trigonometry. The two important questions which arise as regards the systems of non-Euclidean Geometry sometimes called meta-Geometry are first that of the logical validity of the schemes and secondly that of their applicability to the description of actual spatial relations in physical space.

Before however I discuss these questions it is convenient to refer to two developments of Geometry different in kind but both of the most far-reaching importance and both of such a character that as a result of them the whole Science of Geometry can be viewed from standpoints much more general than that of the older traditional scheme of synthetic Geometry. The first of these new departures in the Science was the introduction of Analytical Geometry by Descartes in his method of employing coordinates to represent the positions of points of space. An essential element in Descartes' coordinate Geometry is that sets of three numbers (or in plane Geometry two numbers) are correlated uniquely with points of space. This method of correlation taken in conjunction with the conception of current coordinates in which variables are employed each of which is capable of taking up the values of the elements of a number-continuum has the effect of reducing the statement of all geometrical relations and properties of figures to purely algebraical statements. Thus all Geometry—in Descartes' original scheme this is restricted to Euclidean Geometry—is reduced to arithmetical Algebra. Every theorem of synthetic Geometry is correlated with a corresponding theorem of a purely arithmetic nature that is one in which the objects to which the theorem relates are sets of elements each of which consists of a triplet of numbers. We possess a modern Mathematical theory known as the Theory of Aggregates in which when an aggregate of elements is taken as fundamental the properties of selected sub-aggregates or portions of the fundamental aggregate are classified and developed. If the fundamental aggregate be taken to be the set of all triplets of numbers each of which numbers may be any number whatever of the arithmetic continuum this fundamental aggregate may be taken as corresponding to all the points of geometrical space. All geometrical constructs such as straight lines planes circles spheres etc. correspond to particular sub-aggregates or parts of the fundamental aggregate; and their geometrical relations correspond to properties of such sub-aggregates. It thus appears that Geometry is capable of a further stage of abstraction in which synthetic Geometry is replaced by a scheme of relations between sets of objects each of which is merely a triplet of numbers. Geometry thus becomes in fact a purely arithmetic scheme. The purely technical advantage of the consequent reduction of the ascertainment of particular geometrical properties to calculation by means of equations is for most purposes enormous. But besides this simplification other consequences follow from the substitution of relations of variables and numbers for the objects and relations of synthetic Geometry. In plane Geometry a curve is represented by an equation involving as variables the coordinates of a point on the curve; this equation is a relation which is satisfied whenever the variables take the values of the coordinates of any particular point whatever on the curve. When however we consider the arithmetic number-system to be extended so as to become the system of complex numbers we find in general that there exist pairs of complex numbers which satisfy the equation of a curve and thus the equation of a curve determines relations between complex numbers besides those of real numbers which alone have reference to the curve as originally conceived. It is convenient so to extend the language of synthetic Geometry as to take account of the extended interpretation of which the equations employed in the analytical scheme are capable. We speak therefore of a pair of complex numbers as the coordinates of an imaginary point in the plane. Thus besides the original curve that is correlated with its equation there exists in general a set of imaginary points in the plane the coordinates of which satisfy the equation as do those of the real points. Another extension of the use of geometrical language to denote arithmetical or algebraical facts concerning the equations of analytical Geometry is that involving the employment of infinite numbers pairs of which in a certain sense may satisfy the equation of a curve. We then say that the curve contains points at infinity which may be real or imaginary. The great advantage of this last extension of the use of geometrical language to denote arithmetical facts is apparent in the greater generality of form which it enables us to give to the statement of geometrical properties. To take a very simple case instead of the statement that two circles in a plane may either not intersect one another or may touch at one point or may intersect one another in two points we may assert that any two circles as also is the case for any two conics intersect one another in four points. The correctness of this statement implies that two points may be coincident and that any point may be real or imaginary and either in the finite part of the plane or at infinity. The comprehensive statement is in reality an expression of a property of the equations of the circles or conics. Without this kind of generalization of the use of geometrical language it is impossible to make a general statement about the spatial relations of figures of certain classes without reference to a number of special cases which may arise some of them exceptional in character. Thus the statement that two straight lines in a plane intersect one another holds good for Euclidean Geometry without mentioning the exceptional case of parallels because on the algebraical ground I have referred to parallels intersect one another at a point at infinity.

The full import of these extensions of the elements with which Geometry deals appears however only in connection with the second great development to which I have referred that of Projective Geometry. Any precise description of the scope of this kind of Geometry would necessarily be of so technical a character that I cannot attempt to give one here. It may however be observed that a property which is distinctive of Projective Geometry is that two coplanar straight lines always intersect one another. Euclidean Geometry is accordingly not projective but when the new entities which I have spoken of as points at infinity are introduced into it it can then be expressed in the projective form. In the general sense of the term Geometry that is now employed the fundamental elements in each Geometry consist of points considered as a class of primitive elements; particular sub-classes of this fundamental class defining straight lines planes etc. One Geometry differs from another one in accordance with the nature of the relations that are postulated to exist between the fundamental elements such as those relating to the order of points on a straight line. A method of introducing coordinates independent of the ordinary Euclidean notions of congruence was introduced by von Staudt; and this forms the basis of the analytical treatment of Projective Geometry which is essentially independent of metrical considerations. But metrical relations of distance and of angles have been introduced into Projective Geometry in the form of purely descriptive relations by Poncelet and later in a very general form by Cayley in connection with the theory of invariants. The method of Cayley has an important bearing on the question of the validity of the non-Euclidean Geometries which we call Lobachewskyan and Riemannian. It has in fact been shown that all the relations in either of these systems are capable of being represented as relations within the Euclidean scheme. It follows that if there arises any logical contradiction from the postulations made in hyperbolic or in elliptic Geometry there must be exhibited a corresponding contradiction in Euclidean Geometry. Thus if Euclidean Geometry be assumed to be a scheme free from contradiction it is demonstrable that this is also true of non-Euclidean Geometry whether hyperbolic or elliptic. All three Geometries stand therefore on the same footing as conceptual schemes free from internal contradiction.

Before I consider the question whether the non-Euclidean Geometries have the same applicability as the Euclidean for the purpose of describing relations in physical space it is necessary to refer to a mode of considering the matter which was devised by Riemann and Helmholtz and which has produced an epoch making effect upon the whole theory of the foundations of Geometry. This new development was explained in two memoirs the one by Riemann the Mathematician bears the title On the hypotheses which lie at the base of Geometry the other by Helmholtz the Physicist is entitled On the facts which lie at the base of Geometry. Thus what Riemann regards as hypotheses in a scheme of conceptual space relations correspond to what Helmholtz regards as facts of observation in physical space. The fact of experience which is regarded by Helmholtz as of significance in relation to the theory of Geometry is that freely movable rigid bodies exist in physical space; their dimensions remaining unaltered during the motion. This may be stated in the more precise form that if a pair of particles A B of any one such body can be brought into coincidence with a pair A′ B′ of another such body then the coincidence of congruency remains unaltered when the pair of bodies are moved in any manner. Of course it is assumed that certain conditions as regards temperature and absence of strains are satisfied so that approximately rigid bodies which are the only ones that exist may be regarded as perfectly rigid. The statement is equivalent to the one that a rigid body is freely movable and so that the measurable distance of any pair of points of the body remains unaltered the measure of distance being estimated by means of some standard body. In order that a system of abstract Geometry may be applicable to describe actual spatial relations in which these facts of experience are taken into account the measure of distance between any two points of the geometrical space should be such as to be an invariant for a certain set of transformations which shall represent mobility. In any transformation of this set points P Q are made to correspond to other points P′ Q′ respectively and the metric system of the Geometry should be such that the measures of the distances PQ P′ Q′ should be identical for every pair of points and for every transformation of the set. Any Geometry founded upon a metric system in which this condition is satisfied can be applied to the representation of spatial relations in physical space in such wise that the numerical measure of the distance between any pair of particles of a rigid body remains unaltered as the body is freely moved without strains or changes of temperature.

Riemann's theory is based upon an extension of Gauss' general theory of curved surfaces in Euclidean space that is when the ordinary Euclidean metric system is employed. In Gauss' theory the position of a point on a given surface is specified by two numbers the values of two variables which may be regarded as the coordinates of the point in a widely generalized sense of the word. There is a large degree of arbitrariness in the choice of these variables; when to either of these variables a constant value is assigned the other one remaining variable the point represented by the coordinates lies on a curve upon the surface. Thus a particular coordinate-system on a surface is defined by a mesh-system formed by two families of curves lying upon the surface; this mesh-system being arbitrary but subject to certain conditions of continuity. The element of distance of a point from a neighbouring point of the surface can then be shown to be the square root of a quadratic function of the differentials of the coordinates of the point the coefficients depending in general upon the particular point. Gauss established the existence at each point of a certain function of these coefficients which is invariant for all systems of meshes and depends only on the nature of the surface in the neighbourhood of the point; it is called the absolute curvature of the surface at the point. He further showed that in order that a certain set of transformations might exist which represent the motion of a portion of the surface into a new position in which every element of length in a first position corresponds to an equal element in a second position the necessary and sufficient condition is that the absolute curvature of the surface should at all points have the same value. Interpreted in physical language this means that a portion of a material surface can be freely shifted along the surface without stretching any part of the material bending being however permissible. This condition is for example realized by the surface of a sphere on which all figures are movable without alteration of dimensions; it would however not be satisfied for example by the surface of an ellipsoid for which the property of free-mobility is accordingly lacking. In Niemann's abstract theory a manifold of elements is taken as fundamental each of which is specified by a set of n real numbers. Each of these n numbers may be any number in the arithmetic continuum or in some specified portion of it; thus the fundamental manifold is ordered and has an n-fold order. The particular case which provides a Geometry applicable to physical space is that in which n has the value 3. The manifold of elements is purely abstract and free from any conception directly dependent upon spatial intuition as is emphasized by the fact that n may have any integral value. The question was considered by Riemann as also in the case n = 3 by Helmholtz what system of metric relations must be introduced into the manifold in order that the system which consists of the manifold subject to this metric system may be regarded as the space of a Geometry capable when n = 3 of affording a conceptual representation of the physical space in which measurements of physical bodies are made. Denoting an element of the manifold by the term point and the numbers which specify that point by the term coordinates both Riemann and Helmholtz consider the question what is the most general form of an element of distance between two points of the manifold expressed as an integral of which the integrand or element of distance involves the differentials of coordinates in order that continuous transformations may exist in which the distances between corresponding pairs of points remain unaltered in any of these transformations. The answer to this question is that in the first place the square of the element of length must be defined as a quadratic function of the differences of coordinates of its extremities the coefficients in this quadratic function being continuous functions of the coordinates of the point from which the element of length is measured. Further there exist a certain number of functions (in the case n = 3 this number is 6) of the coefficients in the quadratic function which must have one and the same value invariant for continuous transformations of the coordinates. It can be proved that (in case n > 2) if this holds good at one point it holds at all points. Thus there must exist a certain constant which has one and the same value at all points of the manifold. This constant may be either positive negative or zero. In the last case the metric system introduced into the manifold is Euclidean and the manifold is then when n = 3 said to be the space of Euclidean Geometry. When the constant is negative the space for n = 3 is that of the hyperbolic geometry of Lobachewsky and Bolyai. When it is positive the space is of the new kind discovered by Riemann and may be either spherical or elliptic. It is unbounded but in a certain sense finite and Riemann has drawn special attention to the fact that a space being unbounded is quite consistent with its being in the sense referred to finite. This is for example the case with that two-dimensional space which consists of the surface of a sphere.

On account of the analogy with Gauss' case of the two-dimensional space which forms a surface in ordinary Euclidean Geometry Riemann termed the constant which I have referred to the curvature of the particular space. This term has proved a somewhat unfortunate one as it has led to much misunderstanding. It has given rise to the idea that non-Euclidean three-dimensional space is itself curved and this notion has been strengthened by the usual illustration of the case of a non-Euclidean two-dimensional space regarded as embedded in a three-dimensional space for example in that of the spherical surface to which I have already referred. The fact is however that the so-called curvature is not curvature of the space but represents only a property of the metric system introduced with a considerable degree of arbitrariness into the manifold. It is for this reason better to speak of this constant as the space-constant of the particular geometrical space. The manifold as such has no curvature and no metric properties; these latter are introduced into it when it becomes a space for the purpose of representing conceptually our system of actual measurements in physical space which depend upon physical properties of objects in that space. Instead of speaking of Euclidean or non-Euclidean three-dimensional space it is more accurate to speak of geometrical space with an imposed Euclidean or with an imposed non-Euclidean metric system with an assigned space-constant positive or negative. Moreover it is unnecessary to regard such space as a section of a space of higher dimensions since the metrical scheme introduced into it does not require the consideration of a manifold with more dimensions than that of the space considered. The element of length being assigned as the square root of a quadratic function of the differentials of coordinates the distance between any two points of the space is determined as the stationary value of the integral of the element of length taken from one of the points to the other. In the case taken for purposes of illustration only of the two-dimensional Geometry of a surface in ordinary Euclidean space the distance between two points is the length of a geodesic which passes through the two points. The term geodesic has been frequently used in connection with the distance between two points in space with a non-Euclidean metric; but again this use of the term although convenient is apt to be misleading as it suggests that the space must necessarily be regarded as a “surface” in space of four dimensions.

From the point of view that the difference between so-called Euclidean and so-called non-Euclidean geometrical space does not refer to any distinction of property of the point-manifold itself—that manifold being regarded as a mere field of possible metrical relations to be imposed upon it—the question whether our physical space is Euclidean or non-Euclidean in itself would appear to have no immediate meaning. The real question can only be taken to be whether or under what restrictions if any the actual observed relations in physical space can be described by means of an abstract geometrical scheme with a non-Euclidean metric.

Euclidean Geometry and also non-Euclidean Geometry with either a positive or a negative space-constant are all self-consistent conceptual schemes. The crucial question of very considerable theoretical interest which arises in connection with them is that of their applicability to the description of actual spatial relations in physical space. Are they all so applicable or is there anything in physical phenomena which compels us to assign to Euclidean Geometry a position in relation to such applicability which does not attach to non-Euclidean Geometry? No doubt Euclidean Geometry is the simplest for the purpose but is non-Euclidean Geometry a possible system for application to physical space if we are prepared to sacrifice simplicity? It should be observed that the question as here discussed is considered from the pre-Einstein point of view in which no attempt is made to include gravitational or electromagnetic phenomena in the geometrical scheme.

In the first place it may be observed that in sufficiently small portions of physical space the results of adopting one or other of these schemes will be in view of the approximative character of all our measurements indistinguishable from one another. It would thus appear that all our ordinary spatial measurements are consistent with a non-Euclidean geometrical scheme provided the numerical measure of the space-constant when correlated with our ordinary scales is sufficiently small; no deviations being then observable from the results of applying a Euclidean metric. For example in either hyperbolic or elliptic Geometry the sum of the angles of a triangle of sufficiently small dimensions is indistinguishable from two right angles.

It has however been maintained by Poincaré and many other Geometers that the applicability of non-Euclidean Geometry is theoretically possible independently of any such restriction on the value of the space-constant; that it is in fact fundamentally a matter of convention and convenience not of absolute necessity which system we may employ. It is held by these Geometers and supported by cogent reasoning that no crucial experiment consisting of the measurement of lengths and angles can be made which whatever its results may be is inconsistent with a non-Euclidean scheme of representation provided a requisite readjustment of the statement of physical laws be made especially of the laws of Optics. If points of physical space be suitably correlated with points of geometrical space whose coordinates are assigned in accordance with either of the systems in question the measurement of lengths by means of measuring rods will be consistent with the assumed principle of the existence of rigid bodies freely movable with unaltered numerical dimensions. If angles are determined indirectly by means of formulae connecting them with measured lengths these formulae will differ with the system adopted. When angles are measured directly by methods which involve the use of rays of light the determinations will depend upon the assumptions made as to the paths of such rays. For example as we have already seen in either hyperbolic or elliptic Geometry the sum of the angles of a triangle of sufficiently small dimensions measured in physical space will be indistinguishable from two right angles. In order to measure the angles of larger triangles we have to make use of rays of light; and the comparative simplicity of the Euclidean metric system for purposes of application arises from the fact that if we employ it we can assert that the path of a ray of light is a straight line.

It has frequently been suggested that astronomical observation might be employed to decide the question which kind of Geometry is the true one as representative of our physical space; in particular by the measurement of the angles of a triangle with very long sides. We might suppose for example that by such measurement a triangle was discovered for which the sum of the angles differed from two right angles by an amount which could not be accounted for by instrumental errors. We should then have a choice of two interpretations of the observed fact. We might either say that physical space is only describable by a non-Euclidean Geometry or we might say that it is Euclidean but that the paths of rays of light are not strictly straight lines but curved paths of such a character that the triangle with curvilinear sides was such as to explain the observed amount of the deviation of the sum of the angles from two right angles. Again if we found that the sum of the angles was two right angles we might either affirm that physical space is Euclidean or else that it is non-Euclidean but that the path of a ray of light is not a straight line. As an illustration we may take the fact that on a spherical surface the sum of the angles of a triangle of which the sides are geodesics exceeds two right angles; but that there exist triangles of which the sides are not geodesics for which the sum of the angles is equal to two right angles.

Whilst admitting the strength of the case in favour of the view that physical laws are capable of being so stated that our actual spatial measurements are capable of being described by means of a Geometry with a non-Euclidean metric it would certainly be more satisfying as a confirmation of this view if we were in possession of a detailed statement of the precise modification of physical laws of our habits in relation to spatial intentions and of our practical modes of measurement which would be rendered necessary by the adoption of an abstract non-Euclidean Geometry as the mode of description of actual spatial relations.

That the Euclidean metric is the simplest for all ordinary applications to physical space because it admits of greater simplicity in the statement of physical and dynamical laws is abundantly clear. For all ordinary purposes it will not be superseded but the possibility of the employment of other schemes as theoretically applicable is also clear. The interest in the development of non-Euclidean Geometries apart from their technical interest for Mathematicians lies in the distinction it has laid bare between those elements in our Geometry which are introduced as definitions and conventions admitting variety of detail and those which are fixed by facts of observation.

It appears to have been held by Kant that our Euclidean system of Geometry is present in the mind a priori as a necessary presupposition of physical experience. The development of systems of Geometry of logical validity equal to that of the Euclidean and capable of being applied although with great loss of simplicity to describe our actual experiences of spatial relations would appear to provide a definite refutation of the Kantian view of space as an a priori form at least as regards so precise a form as that of the Euclidean scheme. It has frequently been suggested that the study of non-Euclidean Geometry and still more general schemes of Geometry which have occupied the attention of Mathematicians during more than half a century is of purely technical interest and can lead to nothing which has any relation to physical phenomena. The rise of the Einstein theory of relativity which essentially depends upon the ideas developed by Riemann Helmholtz Minkowski and others affords a sufficient proof of the hazardous character of all prophecies as to the non-applicability of abstract theories and their generalizations for the purpose of representing physical phenomena.