36 
NATURE 
[May 8, 1873 
souri Valley, and possibly from Oregon and British Columbia, 
eastward, over the lakes to Canada or New England; nine 
passed from ~the south-west, north and- eastward, to the 
Middle or Eastern States; three passed from the south- 
west, eastward, to the South Atlantic States, and thence 
north-eastward; and two passed up north-eastward some 
distance off the Atlantic coast. Several of these storms divided 
into two portions, pursuing separate routes ; and, with but one 
or two exceptions, they all increased in severity as they advanced 
eastward. The rainfall returns show a general deficiency on the 
Pacific coast ; that, however, which was reported in the States 
east of the Rocky Mountains in March is probably compensated 
by the excess during January and February. During the entire 
three months the temperature has been colder than usual—at 
least for the country east of the Rocky Mountains. 
WE have received the programme of the Leeds Naturalist’s 
Field Club for the quarter April to June, from which we see 
that alternately with “ exhibition of specimens and conversation,” 
which takes place once a fortnight, papers on subjects of scientific 
interest are to be read. Excursions also take place on an average 
once a fortnight, the first object of the Club being “ the minute 
investigation of the natural history, in all its branches, of the 
immediate neighbourhood of Leeds, and a more general investi- 
gation of the whole of the West Riding.” This Society was 
founded in 1870, and was reorganised on a broader basis in 
March 1872, and seems to be doing good work. 
A CORRESPONDENT writes, asking information with reference 
to the etymology of the word aphis. 
THE following additions to the Brighton Aquarium have been 
made during the past week :—Picked Dogfish ( Acanthias vulgaris), 
Larger Spotted Dog-fish (Scy//ium stellare), Lesser do. (Scyllium 
canicula), Monkfish (Rina squatina), Spotted Rays (Raja macu- 
lata), Sharp-nosed do. (Raja lintea), Streaked Gurnards ( Zrigla 
lineata), Grey Gurnards (Triglia gurnardus), Greater Weevers 
(Trachinus draco), Lesser do. (T7vachinus vipera), Gemmeous 
Dragonets (Callionymus lyra), Lump Fish (Cyclopterus lumpus), 
Sea Snail (Zifaris vulgaris), Yarrell’s Blenny (Blenniops ascani), 
Sand Smelts (A¢herina presbyter), Turbot (Rhombus maximus), 
Brill (Rhombus levis), Sail Fluke (Rhombus punctatus), Plaice 
(Pleuronectes platissa), Flounders (Pleuronectes flesus), Soles (Solea 
vulgaris), Minnows (Leuciscus phoxinus), Tench ( Tinca vulgaris), 
Masked Crab (Corystes cassivelanus), Tube Worms (Serpula 
contortuplicata), Sea Mice (Aphrodite aculeata), Sun Starfish 
(Solaster papposa), Mediterranean Corals (Balanophyllia verru- 
cari), Golden Cup Coral (Balanophyllia regia), Devonshire Cup 
Coral (Caryophyllia smithii), Sea-fingers (Alcyonium digitatum), 
Sea-anemones (various). 
THE additions to the Zoological Society’s Gardens during 
the past week include an Indian leopard (Felis pardus), two 
Indian jackals (Canis aureus), presented by Capt. Henry; a 
Malabar Squirrel (Sciusus maximus), presented by Mr. White- 
side ; three Egyptian cats Fé/is chaus (?) from Cashmere, pre- 
sented by Capt. J. J. Bradshaw ; two Egyptian geese (Chena- 
lopex egyptiaca), presented by Mr. H. W. Thornton ; a haw- 
finch (Coccothraustes vulgaris), from the British Isles, presented 
by the Viscountess Downe; four European Terrapins (Zmys 
lufaria) and a green lizard (Lacerta viridis, var. chloronotus), 
presented by Lord A. Russell ; two black-handed spider monkeys 
Ateles melanochir) ; a white-throated Capuchin (Cebus hypoleucus) ; 
a blue-fronted Amazon (Chrysotis estiva); a yellow-fronted 
Amazon (C. ochrocephala), and an orange-winged Amazon (C. 
amazonica), from Cartagena; a crested agouti (Dasyfrocta 
cristata) from Colon ; an alligator, and a red and yellow macaw 
(Ara chloroptera), from Barauquilla; a golden eagle (Aguila 
chrysaétus), purchased ; a bladder-nosed seal (Cystophora cris- 
tata), from the North Atlantic, deposited. 
i ee 
eS 
ON THE HYPOTHESES WHICH LIE AT 
THE BASES OF GEOMETRY* 
IIl.— Application to Space. 
§ 1.—By means of these inquiries into the determination of 
the measure relations of an 7-fuld extent the conditions may be 
declared which are necessary and suffitient 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 independ- 
ience 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 dis- 
tinguished 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 sinple 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 isa 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 declara- 
tions 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 
thatit 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 mea- 
sure-relations. ‘That space isan unbounded three-fold manifold- 
ness, 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, z.¢., a surface which in a /la¢ mani- 
foldness of three dimensions would take the form of a sphere, 
and consequently be finite. 
§ 3. The questions about the infinitely great are for the inter- 
pretation 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 con- 
struction which has become possible through the invention of 
the infinitesimal calculus, and through the simple principles dis- 
covered 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 
(Continued from page 17.) 
