ba =" 
rete ox 
hs. 
% % 
Feb. 1, 1883] 
called right-handed. His examples are chiefly taken 
from vegetable spirals, such as those of the tendrils of 
the convolvulus, the hop, the vine, &c., some from fir- 
cones, some from snail-shells, others from the “snail” 
in clock-work. He points out in great detail the confu- 
sion which has been introduced in botanical works by the 
want of a common nomenclature, and finally proposes to 
found such a nomenclature on the forms of the Greek 
6 and 2. 
The consideration of double-threaded screws, twisted 
bundles of fibres, &c., leads to the general theory of 
paradromic winding. From this follow the properties of 
a large class of knots which form “clear coils.” A special 
example of these, given by Listing for threads, is the 
well-known juggier’s trick of slitting a ring-formed band 
up the middle, through its whole length, so that. instead 
of separating into two parts, it remains in a continuous 
ring. For this purpose it is only necessary to give a strip 
of paper one Aa/f-twist before pasting the ends together. 
If three half-twists be given, the paper still remains a 
continuous band after slitting, but it cannot be opened 
into a ring, it isin fact a trefoil knot. This remark of 
Listing’s forms the sole basis of a work which recently 
had a large sale in Vienna:—showing how, in emulation 
of the celebrated Slade, to tie an irreducible knot on an 
endless string ! ; 
Listing next gives a few examples of the application of 
his method to knots. It is greatly to be regretted that 
this part of his paper is so very brief; and that the 
opporiunity to which he deferred farther development 
seems never to have arrived. The methods he has given 
are, as is expressly stated by himself, only of limited 
application, There seems to be little doubt, however, 
that he was the first to make any really successful attempt 
to overcome even the preliminary difficulties of this 
unique and exceedingly perplexing subject. 
The paper next gives examples of the curious problem: 
—Given a figure consisting of lines, what is the smallest 
number of coztinuous strokes of the pen by which it can 
be described, no part of a line being gone over more 
than once? Thus, for instance, the lines bounding the 
64 squares of a chess-board can be drawn at 14 separate 
pen-strokes. The solution of all such questions depends 
at once on the enumeration of the points of the complex 
figure at which an odd number of lines meet. 
Then we have the question of the “area” of the pro- 
jection of a knotted curve on a plane; that of the number 
of iaterlinkings of the orbits of the asteroids ; and finally 
some remarks on hemihedry in crystals. This paper, 
which is throughout elementary, deserves careful trans- 
lation into English very much more than do many 
German writings on which that distinction has been 
conferred. 
We have left little space to notice Listing’s greatest 
work, Der Census raiimlicher Complexe (G6ttingen 
Abhandlungen, 1861). This is the less to be regretted, 
because, as a whole, it is far too profound to be made 
popular; and, besides, a fair idea of the nature of its 
contents can be obtained from the introductory chapter 
of Maxwell’s great work on Electricity. For there the 
importance of Listing’s Cyclosis, Periphractic Regions, 
&c., is fully recognised. 
One point, however, which Maxwell did not require, we 
may briefly mention. 
In most works on Trigonometry there is given what is 
called Euler's Theorem about polyhedra :—viz. that if § 
be the number of solid angles of a polyhedron (not 
self-cutting), # the number of its faces, and £ the 
number of its edges, then 
SEF =E--42, 
The puzzle with us, when we were beginning mathe- 
matics, used to be “ What is this mysterious 2, and how 
came it into the formula?” Listing shows that this is a 
NATURE 
317 
! mere case of a much more general theorem in which 
corners, edges, faces, and regions of space, have a homo- 
geneous numerical relation. Thus the mysterious 2, in 
Euler’s formula, belongs to the two regions of space :— 
the one inclosed by the polyhedron, the other (the A7- 
plexum, as Listing calls it) being the rest of infinite 
space. The reader, who wishes to have an elementary 
notion of the higher forms of problems treated by Listing, 
is advised to investigate the modification which Euler’s 
formula would undergo if the polyhedron were (on the 
whole) ring-shaped :—as, for instance, an anchor-ring, or 
a plane slice of a thick cylindrical tube. Pi Gea; 
CLAUDE BERNARD 
1) NDER the title of ‘‘ Notes et Souvenirs sur Claude 
3ernard,” Prof. Jousset de Bellesme, of the School 
of Medicine of Nantes, has published an interesting 
sketch of the life and labours of the great French physio- 
logist, his master, which those who are admirers of 
Claude Bernard will be glad to have their attention called 
to. The essay was meant for the opening address to be 
delivered at the commencement of the present session of 
the Nantes School.. It seems to have been a little too 
outspoken to meet with the approbation of the director 
of the school. On the representation of a majority of 
the professors of the school, it was forbidden to be de- 
livered ex cathedrd by the Minister of Public Instruction, 
in an Order dated October 28, 1882. In the pages of the 
November number of the Revue Internationale des Sct- 
ences biologigues, the address appeals in type to a wider 
audience than the assembled professors and pupils of the 
School of Nantes. Commencing with an extremely 
graphic account of the author’s first introduction to 
Claude Bernard, which concludes as follows :—“‘ With a 
kind gesture of his head he bid meattend his laboratory ; I 
thanked him, and was retiring. Just as I was about to close 
the door, he, taking his attention off his experiment, turned 
his eyes upon me and said, ‘ Have you read Descartes’ 
“‘Discours de la Méthode?” Read it, and read it again.” 
At the time of this interview Claude Bernard was in his 
forty-fifth year, and a great number of his striking works 
had been achieved. Havingassisted for many years with 
astonishment at the apparently inexhaustible series of 
discoveries, Bellesme ventured to ask him one day, what 
was the secret which enabled him to penetrate so easily 
into things hidden from others. ‘‘Do not seek for a 
mystery,” said Bernard, “ nothing can be simpler, or less 
mysterious. My secret is open to all. When I was a 
young man, I lived greedily on the writings of Descartes. 
His ‘ Discourse’ always completely satisfied my soul, and 
I was passionately fond of it. His rules appeared to me 
so just, that I came to the conclusion that by a strict ob- 
servance of them all questions might be solved. That is 
all.”’? The most important of these rules, Bellesme reminds 
his readers, is as follows :—‘‘ Ne recevoir jamais aucune 
chose pour vraie qu’on ne la connaisse évidemment étre 
telle, éviter soigneusement la Précipitation et la Prévention 
dans ses jugements.” The author, then, in avery striking 
manner, draws a series of comparisons between Descartes 
and Cl. Bernard. Passing from this, he criticises some- 
what severely the tendency of a modern school, which 
without taking notice of the complexitiness of biological 
phenomena, seem to have culminated in the idea that no 
contagious disease can be conceived of which has not 
some special microbe as its cause ; but the disciples of 
this school, he urges, have not meditated on the third 
rule of Descartes: ‘“ Conduire par ordre ses pensées, en 
commencant par les objets les plus simples et les plus 
aisés 2 connaitre, pour monter peu & peu comme par 
degrés jusqu’a la connaissance des plus composés.” 
We are afforded a little glimpse of the private life of 
the great French physiologist, which explains a sadness 
