<= se 
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 #-fold continuum at the given point in the given surface- 
direction. 
§ 4.—Before we make the application to space, some con- 
siderations about flat manifoldnesses in general are necessary ; 7. ¢. 
about those in which the square of the line-element is expressible 
as a sum of squares of complete differentials. 
Ina flat #-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 
7 
w—T <p 3 
n =e 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 these continua whose curvature is constant may be also ex- 
pressed thus, that figures may be moved in them without stretch- 
ing. 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 mea- 
sure-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 a, 
the expression for the line-element may be written 
I 
a be 
§ 5.—The theory of surfaces of constant curvature will serve 
for a geometric illustration. It is easy to see that surfaces whose 
curvature is posuive 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 wiil then touch the sphere in- 
ternally, 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 less 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 
suriace 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 ofa ring. If we regard these surfaces as 
locus in quo for surface-regions moving in them, as Space is 
locus in guo 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 dending, 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 inde- 
pendence of direction from position, which in the former surfaces 
does not exist. 
(To be continued.) 
SCIENTIFIC SERIALS 
Zeitschrift fiir Ethnologie, No. 6.—The present number gives 
a compendium of useful suggestions, which might advantageously 
be acted on in other countries besides Germany, addressed by 
the Anthropological Society of Berlin to all persons engaged in 
exploring, or other expeditions to distant regions, In those 
directions for observing and collecting whatever is most adapted 
to extend and rectify ouractual knowledge, information is given 
in regard to the various races with whom travellers may come in 
contact, and the special geographical, linguistic, social and other 
conditions, which more particularly require further elucidation. 
—Prof. A. Bastian gives usin this number with his habitual | 
NATURE 
completeness an exposition of the worship of the heavenly 
bodies among different nations, and the extent to which local 
conditions of climate and ethnological differences have influenced 
the character of the adoration offered to the sun and the moon 
and the stars. According to him a ¢rue worship of the sun— 
except in the polar regions—is only to be found on elevated 
plateaux, where the return of the orb of day was welcomed with | 
gratitude after the colder night, while in low-lying tropical lands 
the aborigices looked with dread at the glowing ball of fire 
which each summer seemed to threaten their world with annihi- 
lation. We can strongly commend this paper as a most com- 
prehensive, although not specially novel exposition of Aryan and 
other mythological systems.—The German engineer, Herr H. 
Keplin, has drawn attention to the mussel-hills (Casguecros 
sambaguis) of Brazil in the district of the Rio do San Francisco 
do Sol. The position of these deposits appears to refute the 
idea of their being mere Kjokkenmédings, while the great 
respect shown by the natives for the dead, and their care to pro- 
vide them proper sepulture, would seem to afford further evidence 
that these elevations, which often rise to a height of 50 feet, 
cannot be due to the hand of man, In relerence to the 
above, it may interest our own archeologists to know that Herr 
Walter Kauffman draws attention in the same number to his 
discovery in the neighbourhood of Hull, at a spot known as 
Castle Hill, near Holderness, of a burial place belonging, as he 
conjectures, to the transition period between the Stone and 
Bronze ages. Herr Kauffman found on the western side of the 
hill, where the ground had been cut for building purposes, a 
fragment of some loam vessel, a compact mass of oys:er shells, 
some flint flakes, andahuman rib. After carefully removing the 
earth, Herr K. discovered at from 4 to 44 feet below the surface 
the vertebrze of another skeleton, and finally collected nearly all the 
bones of two skeletons, completely enclosed in a mass of oyster 
shells—Dr. A. B. Meyer, of Manilla, in the course of a short visit 
in the Philippines, found skulls which presented that peculiar 
appearance of sharpening or filing of the teeth, described by the 
old traveller, Thévenot, and the accuracy of which has often been 
called in question. The Negrito skulls from the Philippines, 
examined by Dr. Meyer, also exhibited the artificial flattening 
of the heads noticed by Thévenot —Herr Virchow drew atten- 
tion last summer to the fact that occasional deviations present 
themselves from the normal cranial configuration of a race, which 
ought to teach us extreme caution in regarding any single 
specimen as a typical form. He was led to make this remark 
by his observation in the Anatomicil Museum of Copenhagen 
of the skull of Kay Lykke, a man of the noblest Danish descent, 
who had flourished two hundred years ago, and been celebrated 
in his day for his personal beauty, his effeminacy, and the sensual 
bias of his disposition. Yet the skuil of this once elegant, 
accomplished, and self-indulgent courtier of the 17th century, 
belonging to an otherwise brachycephalic race, is more strikingly 
dolichocephalic and depressed than the Neanderthal head, and 
might readily be supposed to have belonged to an Australian 
savage. The cranial capacity which is givea by Professor 
Panoum, of Copenhagen, as 1,250 cubic centrm., is, moreover, 
below the amount that is conjecturally assurned for the Nean- 
derthal skull. 
The supplement to the vol. of the ‘‘Zeits. f. Ethnologie,” for 
1872, is exclusively occupied with the Linguistic Notes of Dr. 
G. Schweinfurth, drawn up as the result of his travel in Central 
Africa, and gives numerous vocabularies and specimens of the 
languages of the different tribes who occupy the district of the 
Bahr-el-Ghasal, among whom Dr. Schweinfurth lived more 
than two years. 
Nuovo Giornale Botanico Italiano, vol. iv. Nos. 1—4, Jan.— 
Dec., 1872. The volume for 1872 of this journal, edited by one 
of the most accomplished of Italian botanists, Prof. Caruel, con- 
tains evidence of considerable scientific activity in the Peninsula, 
A large space of these four numbers is devoted to cryptogamic 
botany; we have papers on the mosses of Abyssinia, by De 
Venturi, and of Ceylon and Borneo, by Hampe; on the tungi of 
Parma, by Passerini ; on Diatoms, by Ardissone, and on a new 
classification of cryptogams, proposed by Prof. Cohn. Besides 
several papers on systematic, descriptive, and geographical 
botany, one of the most interesting on physiological and histo- 
logical subjects is by Saccardo, on the amyloid corpuscles 
contained within the fovilla of pollen, illustrated by a 
plate. Prof. Caruel contributes a very valuable biographical 
notice of the Italian botanist, Andrea Cesalpino, born at 
Arezzo in 1519, and a summary of the contents of his great 
