Riemann, Bernhard - On the Hypotheses which lie at the Bases of Geometry

On the Hypotheses which lie at the Bases of Geometry. Bernhard Riemann Translated by William Kingdon Clifiord [Nature, Vol. VIII. Nos. 183, 184, pp. 14{17, 36, 37.] Transcribed by D. R. Wilkins Preliminary Version: December 1998 On the Hypotheses which lie at the Bases of Geometry. Bernhard Riemann Translated by William Kingdon Clifiord [Nature, Vol. VIII. Nos. 183, 184, pp. 14{17, 36, 37.] Plan of the Investigation. It is known that geometry assumes, as things given, both the notion of space and the flrst principles of constructions in space. She gives deflnitions of them which are merely nominal, while the true determinations appear in the form of axioms. The relation of these assumptions remains consequently in darkness; we neither perceive whether and how far their connection is necessary, nor a priori, whether it is possible. From Euclid to Legendre (to name the most famous of modern reform- ing geometers) this darkness was cleared up neither by mathematicians nor by such philosophers as concerned themselves with it. The reason of this is doubtless that the general notion of multiply extended magnitudes (in which space-magnitudes are included) remained entirely unworked. I have in the flrst place, therefore, set myself the task of constructing the notion of a multiply extended magnitude out of general notions of magnitude. It will follow from this that a multiply extended magnitude is capable of difierent measure-relations, and consequently that space is only a particular case of a triply extended magnitude. But hence °ows as a necessary consequence that the propositions of geometry cannot be derived from general notions of magnitude, but that the properties which distinguish space from other con- ceivable triply extended magnitudes are only to be deduced from experience. Thus arises the problem, to discover the simplest matters of fact from which the measure-relations of space may be determined; a problem which from the nature of the case is not completely determinate, since there may be several systems of matters of fact which su–ce to determine the measure-relations of space|the most important system for our present purpose being that which Euclid has laid down as a foundation. These matters of fact are|like all 1 matters of fact|not necessary, but only of empirical certainty; they are hy- potheses. We may therefore investigate their probability, which within the limits of observation is of course very great, and inquire about the justice of their extension beyond the limits of observation, on the side both of the inflnitely great and of the inflnitely small. I. Notion of an n-ply extended magnitude. In proceeding to attempt the solution of the flrst of these problems, the development of the notion of a multiply extended magnitude, I think I may the more claim indulgent criticism in that I am not practised in such under- takings of a philosophical nature where the di–culty lies more in the notions themselves than in the construction; and that besides some very short hints on the matter given by Privy Councillor Gauss in his second memoir on Biquadratic Residues, in the Go˜ttingen Gelehrte Anzeige, and in his Jubilee- book, and some philosophical researches of Herbart, I could make use of no previous labours. x 1. Magnitude-notions are only possible where there is an antecedent general notion which admits of difierent specialisations. According as there exists among these specialisations a continuous path from one to another or not, they form a continuous or discrete manifoldness; the individual special- isations are called in the flrst case points, in the second case elements, of the manifoldness. Notions whose specialisations form a discrete manifoldness are so common that at least in the cultivated languages any things being given it is always possible to flnd a notion in which they are included. (Hence mathematicians might unhesitatingly found the theory of discrete magni- tudes upon the postulate that certain given things are to be regarded as equivalent.) On the other hand, so few and far between are the occasions for forming notions whose specialisations make up a continuous manifoldness, that the only simple notions whose specialisations form a multiply extended manifoldness are the positions of perceived objects and colours. More fre- quent occasions for the creation and development of these notions occur flrst in the higher mathematic. Deflnite portions of a manifoldness, distinguished by a mark or by a boundary, are called Quanta. Their comparison with regard to quantity is accomplished in the case of discrete magnitudes by counting, in the case of continuous magnitudes by measuring. Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which 2 case also we can only determine the more or less and not the how much. The researches which can in this case be instituted about them form a general division of the science of magnitude in which magnitudes are regarded not as existing independently of position and not as expressible in terms of a unit, but as regions in a manifoldness. Such researches have become a necessity for many parts of mathematics, e.g., for the treatment of many-valued analytical functions; and the want of them is no doubt a chief cause why the celebrated theorem of Abel and the achievements of Lagrange, Pfafi, Jacobi for the general theory of difierential equations, have so long remained unfruitful. Out of this general part of the science of extended magnitude in which nothing is assumed but what is contained in the notion of it, it will su–ce for the present purpose to bring into prominence two points; the flrst of which relates to the construction of the notion of a multiply extended manifoldness, the second relates to the reduction of determinations of place in a given manifoldness to determinations of quantity, and will make clear the true character of an n-fold extent. x 2. If in the case of a notion whose specialisations form a continuous manifoldness, one passes from a certain specialisation in a deflnite way to another, the specialisations passed over form a simply extended manifold- ness, whose true character is that in it a continuous progress from a point is possible only on two sides, forwards or backwards. If one now supposes that this manifoldness in its turn passes over into another entirely difierent, and again in a deflnite way, namely so that each point passes over into a deflnite point of the other, then all the specialisations so obtained form a doubly extended manifoldness. In a similar manner one obtains a triply extended manifoldness, if one imagines a doubly extended one passing over in a deflnite way to another entirely difierent; and it is easy to see how this construction may be continued. If one regards the variable object instead of the deter- minable notion of it, this construction may be described as a composition of a variability of n + 1 dimensions out of a variability of n dimensions and a variability of one dimension. x 3. I shall show how conversely one may resolve a variability whose region is given into a variability of one dimension and a variability of fewer dimen- sions. To this end let us suppose a variable piece of a manifoldness of one dimension|reckoned from a flxed origin, that the values of it may be compa- rable with one another|which has for every point of the given manifoldness a deflnite value, varying continuously with the point; or, in other words, let us take a continuous function of position within the given manifoldness, which, moreover, is not constant throughout any part of that manifoldness. 3 Every system of points where the function has a constant value, forms then a continuous manifoldness of fewer dimensions than the given one. These man- ifoldnesses pass over continuously into one another as the function changes; we may therefore assume that out of one of them the others proceed, and speaking generally this may occur in such a way that each point passes over into a deflnite point of the other; the cases of exception (the study of which is important) may here be left unconsidered. Hereby the determination of position in the given manifoldness is reduced to a determination of quantity and to a determination of position in a manifoldness of less dimensions. It is now easy to show that this manifoldness has n ¡ 1 dimensions when the given manifold is n-ply extended. By repeating then this operation n times, the determination of position in an n-ply extended manifoldness is reduced to n determinations of quantity, and therefore the determination of position in a given manifoldness is reduced to a flnite number of determinations of quantity when this is possible. There are manifoldnesses in which the deter- mination of position requires not a flnite number, but either an endless series or a continuous manifoldness of determinations of quantity. Such manifold- nesses are, for example, the possible determinations of a function for a given region, the possible shapes of a solid flgure, &c. II. Measure-relations of which a manifoldness of n dimensions is capable on the assumption that lines have a length independent of position, and consequently that every line may be measured by every other. Having constructed the notion of a manifoldness of n dimensions, and found that its true character consists in the property that the determina- tion of position in it may be reduced to n determinations of magnitude, we come to the second of the problems proposed above, viz. the study of the measure-relations of which such a manifoldness is capable, and of the condi- tions which su–ce to determine them. These measure-relations can only be studied in abstract notions of quantity, and their dependence on one another can only be represented by formul‰. On certain assumptions, however, they are decomposable into relations which, taken separately, are capable of geo- metric representation; and thus it becomes possible to express geometrically the calculated results. In this way, to come to solid ground, we cannot, it is true, avoid abstract considerations in our formul‰, but at least the results of calculation may subsequently be presented in a geometric form. The foun- dations of these two parts of the question are established in the celebrated memoir of Gauss, Disqusitiones generales circa superflcies curvas. x 1. Measure-determinations require that quantity should be independent of position, which may happen in various ways. The hypothesis which flrst 4 presents itself, and which I shall here develop, is that according to which the length of lines is independent of their position, and consequently every line is measurable by means of every other. Position-flxing being reduced to quantity-flxings, and the position of a point in the n-dimensioned manifold- ness being consequently expressed by means of n variables x1; x2; x3; : : : ; xn, the determination of a line comes to the giving of these quantities as functions of one variable. The problem consists then in establishing a mathematical expression for the length of a line, and to this end we must consider the quan- tities x as expressible in terms of certain units. I shall treat this problem only under certain restrictions, and I shall conflne myself in the flrst place to lines in which the ratios of the increments dx of the respective variables vary continuously. We may then conceive these lines broken up into elements, within which the ratios of the quantities dx may be regarded as constant; and the problem is then reduced to establishing for each point a general expression for the linear element ds starting from that point, an expression which will thus contain the quantities x and the quantities dx. I shall sup- pose, secondly, that the length of the linear element, to the flrst order, is unaltered when all the points of this element undergo the same inflnitesimal displacement, which implies at the same time that if all the quantities dx are increased in the same ratio, the linear element will vary also in the same ratio. On these suppositions, the linear element may be any homogeneous function of the flrst degree of the quantities dx, which is unchanged when we change the signs of all the dx, and in which the arbitrary constants are continuous functions of the quantities x. To flnd the simplest cases, I shall seek flrst an expression for manifoldnesses of n ¡ 1 dimensions which are everywhere equidistant from the origin of the linear element; that is, I shall seek a continuous function of position whose values distinguish them from one another. In going outwards from the origin, this must either increase in all directions or decrease in all directions; I assume that it increases in all directions, and therefore has a minimum at that point. If, then, the flrst and second difierential coe–cients of this function are flnite, its flrst difierential must vanish, and the second difierential cannot become negative; I assume that it is always positive. This difierential expression, of the second order remains constant when ds remains constant, and increases in the duplicate ratio when the dx, and therefore also ds, increase in the same ratio; it must therefore be ds2 multiplied by a constant, and consequently ds is the square root of an always positive integral homogeneous function of the second order of the quantities dx, in which the coe–cients are continuous functions of the quantities x. For Space, when the position of points is expressed by rectilin- ear co-ordinates, ds = qP (dx)2; Space is therefore included in this simplest 5 case. The next case in simplicity includes those manifoldnesses in which the line-element may be expressed as the fourth root of a quartic difierential ex- pression. The investigation of this more general kind would require no really difierent principles, but would take considerable time and throw little new light on the theory of space, especially as the results cannot be geometrically expressed; I restrict myself, therefore, to those manifoldnesses in which the line element is expressed as the square root of a quadric difierential expres- sion. Such an expression we can transform into another similar one if we substitute for the n independent variables functions of n new independent variables. In this way, however, we cannot transform any expression into any other; since the expression contains 1 2 n(n+1) coe–cients which are arbitrary functions of the independent variables; now by the introduction of new vari- ables we can only satisfy n conditions, and therefore make no more than n of the coe–cients equal to given quantities. The remaining 1 2 n(n¡ 1) are then entirely determined by the nature of the continuum to be represented, and consequently 1 2 n(n¡1) functions of positions are required for the determina- tion of its measure-relations. Manifoldnesses in which, as in the Plane and in Space, the line-element may be reduced to the form pP dx2, are therefore only a particular case of the manifoldnesses to be here investigated; they re- quire a special name, and therefore these manifoldnesses in which the square of the line-element may be expressed as the sum of the squares of complete difierentials I will call °at. In order now to review the true varieties of all the continua which may be represented in the assumed form, it is necessary to get rid of di–culties arising from the mode of representation, which is ac- complished by choosing the variables in accordance with a certain principle. x 2. For this purpose let us imagine that from any given point the system of shortest limes going out from it is constructed; the position of an arbitrary point may then be determined by the initial direction of the geodesic in which it lies, and by its distance measured along that line from the origin. It can therefore be expressed in terms of the ratios dx0 of the quantities dx in this geodesic, and of the length s of this line. Let us introduce now instead of the dx0 linear functions dx of them, such that the initial value of the square of the line-element shall equal the sum of the squares of these expressions, so that the independent varaibles are now the length s and the ratios of the quantities dx. Lastly, take instead of the dx quantities x1; x2; x3; : : : ; xn proportional to them, but such that the sum of their squares = s2. When we introduce these quantities, the square of the line-element is P dx2 for inflnitesimal values of the x, but the term of next order in it is equal to a homogeneous function of the second order of the 1 2 n(n ¡ 1) quantities (x1 dx2 ¡ x2 dx1), (x1 dx3 ¡ x3 dx1) : : : an inflnitesimal, therefore, of the fourth order; so that 6 we obtain a flnite quantity on dividing this by the square of the inflnitesimal triangle, whose vertices are (0; 0; 0; : : :), (x1; x2; x3; : : :), (dx1; dx2; dx3; : : :). This quantity retains the same value so long as the x and the dx are included in the same binary linear form, or so long as the two geodesics from 0 to x and from 0 to dx remain in the same surface-element; it depends therefore only on place and direction. It is obviously zero when the manifold represented is °at, i.e., when the squared line-element is reducible to P dx2, and may therefore be regarded as the measure of the deviation of the manifoldness from °atness at the given point in the given surface-direction. Multiplied by ¡3 4 it becomes equal to the quantity which Privy Councillor Gauss has called the total curvature of a surface. For the determination of the measure- relations of a manifoldness capable of representation in the assumed form we found that 1 2 n(n ¡ 1) place-functions were necessary; if, therefore, the curvature at each point in 1 2 n(n¡1) surface-directions is given, the measure- relations of the continuum may be determined from them|provided there be no identical relations among these values, which in fact, to speak generally, is not the case. In this way the measure-relations of a manifoldness in which the line-element is the square root of a quadric difierential may be expressed in a manner wholly independent of the choice of independent variables. A method entirely similar may for this purpose be applied also to the manifoldness in which the line-element has a less simple expression, e.g., the fourth root of a quartic difierential. In this case the line-element, generally speaking, is no longer reducible to the form of the square root of a sum of squares, and therefore the deviation from °atness in the squared line-element is an inflnitesimal of the second order, while in those manifoldnesses it was of the fourth order. This property of the last-named continua may thus be called °atness of the smallest parts. The most important property of these continua for our present purpose, for whose sake alone they are here investigated, is that the relations of the twofold ones may be geometrically represented by surfaces, and of the morefold ones may be reduced to those of the surfaces included in them; which now requires a short further discussion. x 3. In the idea of surfaces, together with the intrinsic measure-relations in which only the length of lines on the surfaces is considered, there is al- ways mixed up the position of points lying out of the surface. We may, however, abstract from external relations if we consider such deformations as leave unaltered the length of lines|i.e., if we regard the surface as bent in any way without stretching, and treat all surfaces so related to each other as equivalent. Thus, for example, any cylindrical or conical surface counts as equivalent to a plane, since it may be made out of one by mere bend- ing, in which the intrin