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\Large\bfseries
On the Hypotheses which lie at the Bases of Geometry.\\[12 pt]
Bernhard Riemann\\[12 pt]
Translated by William Kingdon Clifford\\[12 pt]
[\emph{Nature}, Vol.~VIII.  Nos.~183, 184, pp. 14--17, 36, 37.]\\[24 pt]
\large\mdseries
Transcribed by D. R. Wilkins\\[12 pt]
Preliminary Version: December 1998
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\title{On the Hypotheses which lie at the Bases of Geometry.}
\author{Bernhard Riemann\\
Translated by William Kingdon Clifford}
\date{[\emph{Nature}, Vol.~VIII.  Nos.~183, 184, pp. 14--17, 36, 37.]}
\maketitle

\begin{center}
\emph{Plan of the Investigation.}
\end{center}

It is known that geometry assumes, as things given, both the
notion of space and the first principles of constructions in
space.  She gives definitions 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 \emph{a priori}, whether it is possible.

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.

\begin{center}
\emph{I. Notion of an $n$-ply extended magnitude.}
\end{center}

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 \emph{G\"{o}ttingen
Gelehrte Anzeige}, and in his Jubilee-book, and some
philosophical researches of Herbart, I could make use of no
previous labours.

\bigbreak

\S~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 \emph{continuous} or
\emph{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
\emph{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 \emph{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, \emph{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.

\bigbreak

\S~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.

\bigbreak

\S~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 \emph{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.

\begin{center}
\emph{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.}
\end{center}

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 formul{\ae}.
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 formul{\ae}, 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,
\emph{Disqusitiones generales circa superficies curvas}.

\bigbreak

\S~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
$x_1, x_2, x_3,\ldots, x_n$,
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 $ds^2$ 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,
$ds = \sqrt{\sum (dx)^2}$; 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
$\frac{1}{2} n (n + 1)$ 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 $\frac{1}{2} n (n - 1)$ are then
entirely determined by the nature of the continuum to be
represented, and consequently $\frac{1}{2} n (n - 1)$ 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
$\sqrt{\sum dx^2}$, 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 \emph{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.

\bigbreak

\S~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 $dx_0$ of the
quantities $dx$ in this geodesic, and of the length~$s$ of this
line.  Let us introduce now instead of the $dx_0$ 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
$x_1, x_2, x_3,\ldots, x_n$ proportional to them, but such that
the sum of their squares $= s^2$.  When we introduce these
quantities, the square of the line-element is $\sum dx^2$ 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
$\frac{1}{2} n (n - 1)$ quantities
$(x_1 \, dx_2 - x_2 \, dx_1)$,
$(x_1 \, dx_3 - x_3 \, dx_1) \, \ldots$
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
$(0,0,0,\ldots)$, $(x_1, x_2, x_3,\ldots)$,
$(dx_1, dx_2, dx_3,\ldots)$.
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, \emph{i.e.}, when the squared line-element is reducible to
$\sum dx^2$, 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 $-\frac{3}{4}$ 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
$\frac{1}{2} n (n - 1)$ place-functions were necessary; if,
therefore, the curvature at each point in $\frac{1}{2} n (n - 1)$
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 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, \emph{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.

\bigbreak

\S~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---\emph{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.

\bigbreak

\S~4.
Before we make the application to space, some
considerations about flat manifoldness in general are necessary;
\emph{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
$\frac{1}{2} n (n - 1)$ 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 $\alpha$, the expression for the
line-element may be written
\[ \frac{1}{1 + \frac{1}{4} \alpha \sum x^2}
      \sqrt{\textstyle \sum dx^2 }.\]

\bigbreak

\S~5.
The theory of \emph{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 \emph{locus in quo} for surface-regions
moving in them, as Space is \emph{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 \emph{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 \emph{direction} from position, which in the
former surfaces does not exist.

\begin{center}
\emph{III. Application to Space.}
\end{center}

\S~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.

\bigbreak

\S~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 \emph{unboundedness} and
\emph{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, \emph{i.e.}, a surface which in a \emph{flat}
manifoldness of three dimensions would take the form of a sphere,
and consequently be finite.

\S~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.

\newpage

\begin{center}
\emph{Synopsis.}
\end{center}

\begin{description}
\item[]
Plan of the Inquiry:

\item[\textmd{I.}]
Notion of an $n$-ply extended magnitude.

\begin{description}
\item[\textmd{\S~1.}]
Continuous and discrete manifoldnesses.  Defined parts of a
manifoldness are called Quanta.  Division of the theory of
continuous magnitude into the theories,
\begin{description}
\item[\textmd{(1)}]
Of mere region-relations, in which an independence of magnitudes
from position is not assumed;
\item[\textmd{(2)}]
Of size-relations, in which such an independence must be assumed.
\end{description}

\item[\textmd{\S~2.}]
Construction of the notion of a one-fold, two-fold, $n$-fold
extended magnitude.

\item[\textmd{\S~3.}]
Reduction of place-fixing in a given manifoldness to
quantity-fixings.  True character of an $n$-fold extended
magnitude.

\end{description}

\penalty-1000

\item[\textmd{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.

\begin{description}
\item[\textmd{\S~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.

\item[\textmd{\S~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 $\frac{1}{2} n (n - 1)$ surface directions.

\item[\textmd{\S~3.}]
Geometric illustration.

\item[\textmd{\S~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).

\item[\textmd{\S~5.}]
Surfaces with constant curvature.

\end{description}

\penalty-1000

\item[\textmd{III.}]
Application to Space.

\begin{description}
\item[\textmd{\S~1.}]
System of facts which suffice to determine the measure-relations
of space assumed in geometry.

\item[\textmd{\S~2.}]
How far is the validity of these empirical determinations
probable beyond the limits of observation towards the infinitely
great?

\item[\textmd{\S~3.}]
How far towards the infinitely small?  Connection of this
question with the interpretation of nature.

\end{description}
\end{description}

\end{document}