May 8, 1873] 
about the measure-relations of space in the infinitely small are 
not therefore superfluous questions. 
Tf 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 determi- 
nations 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 
— 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 reaiity which 
underlies space must form a discrete manifoldness, or we must 
seek the ground 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 notallow us to go to- 
day. 
: : Synopsis 
PLAN of the Inquiry : 
I. Notion of an x-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 -fold extended 
magnitude, 
II. Measure-relations of which a manifoldness of 7 dimensions 
is capable on the assumption that lines have a length inde- 
pendent 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 #-dimensions in 
which the line-element may be represented as the square 
root of a quadric differential. Measure of its deviation 
from flatness (curvature) at a given point in a given sur- 
face-direction. For the determination of its measure- 
relations it is allowable and sufficient that the curvature 
n-I 
surface 
be arbitrarily given at every point in 
(flixections. : 
.§ 3. Geometric illustration. 
§:4. Elat manifoldnesses (in which the curvature is every- 
' where =.0) may be treated as a special case of manifold- 
jhesses.with constant curvature. These can also be defined 
{ 
Ly 
NATURE 
37 
as admitting an independence of x-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 measures 
relations of space assumed in geometry. 
§ 2. How far is the validity of these empirical determina- 
tions 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. 
THE DEVELOPMENT THEORYINGERMANY* 
III. 
Chorology: or, the Geographical Distribution of Living Beings 
THE importance of the theory of Evolution does not consist 
in its accounting for this or that particular fact, but in its 
explaining all biological facts collectively. It is found to be 
confirmed in every detail by the mode of distribution of the 
various organisms on the surface of the earth. This distribution 
had already been studied by Alexander von Humboldt and 
Fr. Schouw for plants, by Berghaus and Schmarda for animals. 
But previous to Darwin and Wallace, this study had produced 
only a collection of unsystematised facts ; Haeckel has attempted 
to create out of it a special science under the name of Chorology. 
With the exception of the monocellular protozoa, which, on 
account of their simplicity, have been able to appear at the 
same time or at several times in different places ; with the ex- 
ception also of species which owe théir origin to a hybrid or 
bastard generation, and which it has been possible to reproduce 
in different circumstances wherever the parent species have pre- 
viously spread, it must be admitted that each of the other species 
has only been originated a single time and in a single place. 
But, once produced, they must, as a consequence of the struggle 
for existence, and in virtue of the laws of population, or rather 
of excess of population, tend to spread to the widest possible 
extent. Animals and plants migrate as well as man, both 
actively and passively. 
In the case of animals, which have, more than plants, freedom 
of movement, active migration plays the principal part. The 
more easy locomotion is in the case of any species, the more rapidly 
is the species bound to spread. This is why birds and insects, 
furnished with wings, although referable to a less number of 
orders or natural groups than other animals, yet present a very 
great diversity of species slightly distinguishable from one an- 
other; this is to be ascribed to the fact that the facility with 
which they can move from place to place has subjected them to 
the modifying influences of the most varied localities. After 
birds and insects the swiftest runners among the denizens 
of the land, the best swimmers among the inhabitants of the 
water have been subject to the widest extension, With regard 
to animals which are fixed or immoyable while being developed, 
corals, tubicolz, tunicata, crinoids, &c., they usually enjoy 
during their youth so much of the power of movement as admits 
of their displacement. A great number of floating plants are 
also transported to great distances by water. 
But the spread of a large number of plants and of certain 
animals can be explained only by a passive migration. The 
wind sweeps to great distances, sometimes over seas, eggs of 
small animals, seeds, and sometimes even minute organisms ; 
this explains the well-known phenomena of showers of frogs. 
These eggs, these seeds, these small organisms, sometimes 
fall into the water, which transports them to still greater 
distances. Trunks of trees, which traverse the ocean under. 
the direction of the currents, and those which the tempest 
hurls from the mountain tops, can carry with them, hidden in 
their interstices, in the moss or the parasitical plants with which 
they are covered, in the earth which adheres to their roots, in- 
numerable germs to be developed in new regions. The icebergs 
of the polar sea have landed foxes and bears even on the shores 
of Iceland and Britain. Birds, insects, mammals which are 
removed, carry with them thousands of parasites, microscopic 
beings, eggs or germs, Man himself carries them about more 
abundantly still along with the varied materials he employs for 
kis works and his industry. 
The fact of the distribution of certain species which cannot 
be explained by migration, either active or passive, may be 
accounted for by geological facts. In consequence of the im- 
* Continued from vol. vii. p. 434. 
