[Nature, Vol. VIII. Nos. 183, 184, pp. 14--17, 36, 37.]
From Euclid to Legendre (to name the most famous of modern reforming 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 first 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 different measure-relations, and consequently that space is only a particular case of a triply extended magnitude. But hence flows 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 conceivable 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 suffice 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 matters of fact - not necessary, but only of empirical certainty; they are hypotheses. 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 infinitely great and of the infinitely small.
In proceeding to attempt the solution of the first 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 undertakings of a philosophical nature where the difficulty 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 Göttingen Gelehrte Anzeige, and in his Jubilee-book, and some philosophical researches of Herbart, I could make use of no previous labours.
§ 1. Magnitude-notions are only possible where there is an antecedent general notion which admits of different 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 specialisations are called in the first 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 find a notion in which they are included. (Hence mathematicians might unhesitatingly found the theory of discrete magnitudes 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 frequent occasions for the creation and development of these notions occur first in the higher mathematic.
Definite 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 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, Pfaff, Jacobi for the general theory of differential 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 suffice for the present purpose to bring into prominence two points; the first 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.
§ 2. If in the case of a notion whose specialisations form a continuous manifoldness, one passes from a certain specialisation in a definite way to another, the specialisations passed over form a simply extended manifoldness, 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 different, and again in a definite way, namely so that each point passes over into a definite 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 definite way to another entirely different; and it is easy to see how this construction may be continued. If one regards the variable object instead of the determinable 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.
§ 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 dimensions. To this end let us suppose a variable piece of a manifoldness of one dimension - reckoned from a fixed origin, that the values of it may be comparable with one another - which has for every point of the given manifoldness a definite 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. Every system of points where the function has a constant value, forms then a continuous manifoldness of fewer dimensions than the given one. These manifoldnesses 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 definite 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 finite number of determinations of quantity when this is possible. There are manifoldnesses in which the determination of position requires not a finite number, but either an endless series or a continuous manifoldness of determinations of quantity. Such manifoldnesses are, for example, the possible determinations of a function for a given region, the possible shapes of a solid figure, &c.
Having constructed the notion of a manifoldness of n dimensions, and found that its true character consists in the property that the determination 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 conditions which suffice 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 formulæ. On certain assumptions, however, they are decomposable into relations which, taken separately, are capable of geometric 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 formulæ, but at least the results of calculation may subsequently be presented in a geometric form. The foundations of these two parts of the question are established in the celebrated memoir of Gauss, Disqusitiones generales circa superficies curvas.
§ 1.
Measure-determinations require that quantity should be
independent of position, which may happen in various ways. The
hypothesis which first 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-fixing being
reduced to quantity-fixings, and the position of a point in the
n-dimensioned manifoldness 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 quantities x as
expressible in terms of certain units. I shall treat this
problem only under certain restrictions, and I shall confine
myself in the first 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 suppose,
secondly, that the length of the linear element, to the first
order, is unaltered when all the points of this element undergo
the same infinitesimal 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 first 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 find the simplest cases, I shall seek first
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 first and second differential coefficients of this function
are finite, its first differential must vanish, and the second
differential cannot become negative; I assume that it is always
positive. This differential 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 coefficients are continuous
functions of the quantities x. For Space, when the position of
points is expressed by rectilinear co-ordinates,
ds = ;
Space is therefore included in this
simplest case. The next case in simplicity includes those
manifoldnesses in which the line-element may be expressed as the
fourth root of a quartic differential expression. The
investigation of this more general kind would require no really
different 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 differential
expression. 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
½ n (n + 1) coefficients which are arbitrary
functions of the independent variables; now by the introduction
of new variables we can only satisfy n conditions, and
therefore make no more than n of the coefficients equal to
given quantities. The remaining ½ n (n - 1) are then
entirely determined by the nature of the continuum to be
represented, and consequently ½ n (n - 1) functions
of positions are required for the determination of its
measure-relations. Manifoldnesses in which, as in the Plane and
in Space, the line-element may be reduced to the form
,
are therefore only a particular case of the
manifoldnesses to be here investigated; they require 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 differentials I will call flat. 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
difficulties arising from the mode of representation, which is
accomplished by choosing the variables in accordance with a
certain principle.
§ 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
for infinitesimal values of the x, but the term of next order in it
is equal to a homogeneous function of the second order of the
½ n (n - 1) quantities
(x1 dx2 - x2 dx1),
(x1 dx3 - x3 dx1),...
an infinitesimal, therefore, of the fourth order; so that we
obtain a finite quantity on dividing this by the square of the
infinitesimal 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
flat, i.e., when the squared line-element is reducible to
,
and may therefore be regarded as the measure of the
deviation of the manifoldness from flatness at the given point in
the given surface-direction. Multiplied by -¾ 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
½ n (n - 1) place-functions were necessary; if,
therefore, the curvature at each point in ½ 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 differential 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
differential. 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 flatness in the squared
line-element is an infinitesimal 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 flatness 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.
§ 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 always 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 bending, in which the intrinsic measure-relations remain, and all theorems about a plane - therefore the whole of planimetry - retain their validity. On the other hand they count as essentially different from the sphere, which cannot be changed into a plane without stretching. According to our previous investigation the intrinsic measure-relations of a twofold extent in which the line-element may be expressed as the square root of a quadric differential, which is the case with surfaces, are characterised by the total curvature. Now this quantity in the case of surfaces is capable of a visible interpretation, viz., it is the product of the two curvatures of the surface, or multiplied by the area of a small geodesic triangle, it is equal to the spherical excess of the same. The first definition assumes the proposition that the product of the two radii of curvature is unaltered by mere bending; the second, that in the same place the area of a small triangle is proportional to its spherical excess. To give an intelligible meaning to the curvature of an n-fold extent at a given point and in a given surface-direction through it, we must start from the fact that a geodesic proceeding from a point is entirely determined when its initial direction is given. According to this we obtain a determinate surface if we prolong all the geodesics proceeding from the given point and lying initially in the given surface-direction; this surface has at the given point a definite curvature, which is also the curvature of the n-fold continuum at the given point in the given surface-direction.
§ 4. Before we make the application to space, some considerations about flat manifoldness in general are necessary; i.e., about those in which the square of the line-element is expressible as a sum of squares of complete differentials.
In a flat n-fold extent the total curvature is zero at all
points in every direction; it is sufficient, however (according
to the preceding investigation), for the determination of
measure-relations, to know that at each point the curvature is
zero in
½ n (n - 1) independent surface directions.
Manifoldnesses whose curvature is constantly zero may be treated
as a special case of those whose curvature is constant. The
common character of those continua whose curvature is constant
may be also expressed thus, that figures may be viewed in them
without stretching. For clearly figures could not be arbitrarily
shifted and turned round in them if the curvature at each point
were not the same in all directions. On the other hand, however,
the measure-relations of the manifoldness are entirely determined
by the curvature; they are therefore exactly the same in all
directions at one point as at another, and consequently the same
constructions can be made from it: whence it follows that in
aggregates with constant curvature figures may have any arbitrary
position given them. The measure-relations of these
manifoldnesses depend only on the value of the curvature, and in
relation to the analytic expression it may be remarked that if
this value is denoted by
,
the expression for the
line-element may be written
§ 5. The theory of surfaces of constant curvature will serve for a geometric illustration. It is easy to see that surface whose curvature is positive may always be rolled on a sphere whose radius is unity divided by the square root of the curvature; but to review the entire manifoldness of these surfaces, let one of them have the form of a sphere and the rest the form of surfaces of revolution touching it at the equator. The surfaces with greater curvature than this sphere will then touch the sphere internally, and take a form like the outer portion (from the axis) of the surface of a ring; they may be rolled upon zones of spheres having new radii, but will go round more than once. The surfaces with less positive curvature are obtained from spheres of larger radii, by cutting out the lune bounded by two great half-circles and bringing the section-lines together. The surface with curvature zero will be a cylinder standing on the equator; the surfaces with negative curvature will touch the cylinder externally and be formed like the inner portion (towards the axis) of the surface of a ring. If we regard these surfaces as locus in quo for surface-regions moving in them, as Space is locus in quo for bodies, the surface-regions can be moved in all these surfaces without stretching. The surfaces with positive curvature can always be so formed that surface-regions may also be moved arbitrarily about upon them without bending, namely (they may be formed) into sphere-surfaces; but not those with negative-curvature. Besides this independence of surface-regions from position there is in surfaces of zero curvature also an independence of direction from position, which in the former surfaces does not exist.
§ 1. By means of these inquiries into the determination of the measure-relations of an n-fold extent the conditions may be declared which are necessary and sufficient to determine the metric properties of space, if we assume the independence of line-length from position and expressibility of the line-element as the square root of a quadric differential, that is to say, flatness in the smallest parts.
First, they may be expressed thus: that the curvature at each point is zero in three surface-directions; and thence the metric properties of space are determined if the sum of the angles of a triangle is always equal to two right angles.
Secondly, if we assume with Euclid not merely an existence of lines independent of position, but of bodies also, it follows that the curvature is everywhere constant; and then the sum of the angles is determined in all triangles when it is known in one.
Thirdly, one might, instead of taking the length of lines to be independent of position and direction, assume also an independence of their length and direction from position. According to this conception changes or differences of position are complex magnitudes expressible in three independent units.
§ 2. In the course of our previous inquiries, we first distinguished between the relations of extension or partition and the relations of measure, and found that with the same extensive properties, different measure-relations were conceivable; we then investigated the system of simple size-fixings by which the measure-relations of space are completely determined, and of which all propositions about them are a necessary consequence; it remains to discuss the question how, in what degree, and to what extent these assumptions are borne out by experience. In this respect there is a real distinction between mere extensive relations, and measure-relations; in so far as in the former, where the possible cases form a discrete manifoldness, the declarations of experience are indeed not quite certain, but still not inaccurate; while in the latter, where the possible cases form a continuous manifoldness, every determination from experience remains always inaccurate: be the probability ever so great that it is nearly exact. This consideration becomes important in the extensions of these empirical determinations beyond the limits of observation to the infinitely great and infinitely small; since the latter may clearly become more inaccurate beyond the limits of observation, but not the former.
In the extension of space-construction to the infinitely great, we must distinguish between unboundedness and infinite extent, the former belongs to the extent relations, the latter to the measure-relations. That space is an unbounded three-fold manifoldness, is an assumption which is developed by every conception of the outer world; according to which every instant the region of real perception is completed and the possible positions of a sought object are constructed, and which by these applications is for ever confirming itself. The unboundedness of space possesses in this way a greater empirical certainty than any external experience. But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite provided this curvature has ever so small a positive value. If we prolong all the geodesics starting in a given surface-element, we should obtain an unbounded surface of constant curvature, i.e., a surface which in a flat manifoldness of three dimensions would take the form of a sphere, and consequently be finite.
§ 3. The questions about the infinitely great are for the interpretation of nature useless questions. But this is not the case with the questions about the infinitely small. It is upon the exactness with which we follow phenomena into the infinitely small that our knowledge of their causal relations essentially depends. The progress of recent centuries in the knowledge of mechanics depends almost entirely on the exactness of the construction which has become possible through the invention of the infinitesimal calculus, and through the simple principles discovered by Archimedes, Galileo, and Newton, and used by modern physic. But in the natural sciences which are still in want of simple principles for such constructions, we seek to discover the causal relations by following the phenomena into great minuteness, so far as the microscope permits. Questions about the measure-relations of space in the infinitely small are not therefore superfluous questions.
If we suppose that bodies exist independently of position, the curvature is everywhere constant, and it then results from astronomical measurements that it cannot be different from zero; or at any rate its reciprocal must be an area in comparison with which the range of our telescopes may be neglected. But if this independence of bodies from position does not exist, we cannot draw conclusions from metric relations of the great, to those of the infinitely small; in that case the curvature at each point may have an arbitrary value in three directions, provided that the total curvature of every measurable portion of space does not differ sensibly from zero. Still more complicated relations may exist if we no longer suppose the linear element expressible as the square root of a quadric differential. Now it seems that the empirical notions on which the metrical determinations of space are founded, the notion of a solid body and of a ray of light, cease to be valid for the infinitely small. We are therefore quite at liberty to suppose that the metric relations of space in the infinitely small do not conform to the hypotheses of geometry; and we ought in fact to suppose it, if we can thereby obtain a simpler explanation of phenomena.
The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metric relations of space. In this last question, which we may still regard as belonging to the doctrine of space, is found the application of the remark made above; that in a discrete manifoldness, the ground of its metric relations is given in the notion of it, while in a continuous manifoldness, this ground must come from outside. Either therefore the reality which underlies space must form a discrete manifoldness, or we must seek the gound of its metric relations outside it, in binding forces which act upon it.
The answer to these questions can only be got by starting from the conception of phenomena which has hitherto been justified by experience, and which Newton assumed as a foundation, and by making in this conception the successive changes required by facts which it cannot explain. Researches starting from general notions, like the investigation we have just made, can only be useful in preventing this work from being hampered by too narrow views, and progress in knowledge of the interdependence of things from being checked by traditional prejudices.
This leads us into the domain of another science, of physic, into which the object of this work does not allow us to go to-day.
Plan of the Inquiry:
I. Notion of an n-ply extended magnitude.
§ 1. Continuous and discrete manifoldnesses. Defined parts of a manifoldness are called Quanta. Division of the theory of continuous magnitude into the theories,(1) Of mere region-relations, in which an independence of magnitudes from position is not assumed;
(2) Of size-relations, in which such an independence must be assumed.
§ 2. Construction of the notion of a one-fold, two-fold, n-fold extended magnitude.
§ 3. Reduction of place-fixing in a given manifoldness to quantity-fixings. True character of an n-fold extended magnitude.
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.
§ 1. Expression for the line-element. Manifoldnesses to be called Flat in which the line-element is expressible as the square root of a sum of squares of complete differentials.
§ 2. Investigation of the manifoldness of n-dimensions in which the line element may be represented as the square root of a quadric differential. Measure ofits deviation from flatness (curvature) at a given point in a given surface-direction. For the determination of its measure-relations it is allowable and sufficient that the curvature be arbitrarily given at every point in ½ n (n - 1) surface directions.
§ 3. Geometric illustration.
§ 4. Flat manifoldnesses (in which the curvature is everywhere = 0) may be treated as a special case of manifoldnesses with constant curvature. These can also be defined as admitting an independence of n-fold extents in them from position (possibility of motion without stretching).
§ 5. Surfaces with constant curvature.
III. Application to Space.
§ 1. System of facts which suffice to determine the measure-relations of space assumed in geometry.
§ 2. How far is the validity of these empirical determinations probable beyond the limits of observation towards the infinitely great?
§ 3. How far towards the infinitely small? Connection of this question with the interpretation of nature.