548 
the year 1873 was madder from Radia tnctorum, an 
article of considerable profit to landowners. It was 
largely used in dyeing Turkey-red yarns. The discovery 
of coal-tar dyes seriously interfered with the demand for 
madder, so that its growth has much decreased. In con- 
sequence, however, of the mineral dyes being much in- 
ferior in fastness, madder is being again sought after, and 
should the demand continue, there is a prospect that 
madder will again assume its former importance. Among 
other interesting exhibits, the rude native cart of the form 
in use for over 2009 years, and still in use, attracts 
much attention, as does the threshing-board, the same as 
was in use in patriarchal times. It is studded with flints 
on the under-side, and is drawn by bullocks or horses over 
the grain, by which means the seed is separated from the 
ears and the straw reduced to small particles.. This is 
said to be the only system employed for threshing in 
Cyprus. 
A few well-known woods, such as Olive (Olea europea), 
Cedar of Lebanon (Cedrus Libanz), Bay Laurel (Laurus 
nobilis), Chian Turpentine, (Pistacta Terebinthus) are 
exhibited, as well as the concrete resin of the. latter, or 
crude Chian turpentine, under the name of Trimithia 
gum, in curious small greenish-coloured pots. 
Malta.—The Maltese exhibits will be best remembered 
by the fine show of lace and silver filigree wors. A good 
show is also made of preserved fruits, and tobacco of 
very fine quality and varied forms is exhibited; besides 
these there are very few other vegetable products. 
JOHN R. JACKSON 
GREEK GEOMETRY 
E have before us parts 6, 7(?), of Dr. Allman’s 
“Greek Geometry from Thales to Euclid,’? in 
which we are brought almost into touch with Euclid. 
There is then but little wanting to complete the task 
commenced by the author in 1877, in the performing of 
which so much light has been thrown upon the contribu- 
tions of the early Greek mathematicians to geometrical 
science. 
of Archytas and Eudoxus ; the present parts commence 
with a discussion of the claims of Menzchmus, “ pupil of 
Eudoxus, associate of Plato, and the discoverer of the 
conic sections.” In the forefront are placed translations 
of eleven fragments which contain what is known of 
Menezchmus. The various points which arise are most 
carefully reasoned out, with considerable detail, but we 
cannot attempt here to compress what is already con- 
cisely given. Thenotesare very valuable, and show over 
what a wide field of reading Dr. Allman’s researches 
have taken him. We note only the prominence given to 
M. Tannery’s papers, as we have frequently had occasion 
in these pages to draw attention to this mathematician’s 
valuable memoirs on Greek geometry. The last part 
(which we have numbered 7) opens with an account of 
Dinostratus, brother of Menzechmus, whose name occurs 
in connection with the quadratrix. Dr. Allman states 
the case of Dinostratus versus Hippias: “The result of 
the whole discussion seems to be that the quadratrix was 
invented, probably by Hippias of Elis, with the object of 
trisecting an angle, and was originally employed for that 
purpose; that subsequently Dinostratus used the curve 
for the quadrature of the circle, and that its name was 
thence derived.” Sporus (or Porus) comes in for a men- 
tion, and then we come to Aristzeus, who wrote on the 
conic sections, and is the author of the theorem, “ The 
same circle circumscribes the pentagon of the dodeca- 
hedron and the triangle of the icosahedron, these solids 
* The following references to the several parts may be of service :— 
Hermathena, part 1, vol. iii. No. 5, pp. 160-75; part 2, same number, 
PP- 175-207 ; part 3, vol. iv. No. 7, pp. 180-228; part 4, vol. v. No. 10, 
pp. 186-212; part 5, same number, pp. 212-235; part 6, vol. v. No. 11, 
pp- 403-32; part 7(?), vol. vi. No. 12, pp. 105-30. 
NATURE 
* Pp.223. 
[Oct. 7, 1886 
being inscribed in the same sphere.” This occurs in his 
“Comparison of the Regular Solids.” Bretschneider 
thinks the thirteenth Book of Euclid’s Elements is “a 
recapitulation, at least partial, of this work of Aristzeus” 
(cf. also Dr. C. Taylor’s “ Conics,” p. xxxiii.). 
Of Aristzus, in closing, our author writes : he “ may, 
therefore, be regarded as having continued and summed 
up the work, which, arising from the speculations of 
Philolaus, was carried on by his successors—Archytas, 
Eudoxus, and Menzchmus. These men were related to 
one another in succession as master and pupil, and it 
seemed to me important that the continuity of their work 
should not be broken in its presentation.” 
We hope another year will suffice to bring this sketch 
of Greek geometry to a close, and that then the author 
will collect these parts, whose appearances have been 
extended over nearly ten long years, in one volume, with 
such additional notes as his subsequent reading will 
enable him to append. 
We can only commend these two parts, as we have the 
previous ones, to the careful study of all who are inter- 
ested in these researches: they have taken a high place 
in the estimation of foreign mathematicians, even in cases 
where the author’s conclusions have not been unhesitat- 
ingly accepted. 
THE HYGIENE OF THE VOCAL ORGANS * 
HE cultivation of the voice and the means of main- 
taining it in a state of excellence under the varying 
strain of daily life, are subjects of interest to us all, but 
become of paramount importance to those who are pro- 
fessionally brought before the public as speakers or 
singers. Although the laryngoscope is invaluable in the 
recognition and treatment of disease, it is surprising how 
little it has up to the present time added to our knowledge 
of the physiology of the larynx. 
The difficulties of examining the larynx during singing 
are so great that a large number of singers have to be 
examined to obtain a complete view of the whole pracess 
by even the most expert laryngoscopist. The results of 
the examination of some three or four hundred persons 
with fine voices, including most of the best singers of the 
day, form not the least interesting portion of the book. ; 
There is no question that the voice, whether the note 
be high or low, whether a chest or head note, whether 
bass or falsetto, is produced by vibration of the free edges 
of the vocal cords, which are two movable ligamentous ~ 
bands about half an inch long stretched from back to 
front of the larynx. In other words, the only place where © 
all notes, whatever their character may be, can be pro- — 
duced, is in the larynx. : 
These bands are attached anteriorly in contact with — 
one another, but their posterior fourth is attached to the © 
small pyramidal-shaped arytenoid cartilages, which can — 
move laterally. The glottis, the space between them, is 
thus divided into a ligamentous and a cartilaginous por- 
tion. There is the greatest difference of opinion among 
authorities as to the position of the cords and arytenoid 
cartilages, and as to how much of the cord vibrates in the — 
production of the various sounds. : 
Dr. Mackenzie divides the range of the voice into two 
registers, viz. one (chest) in which the pitch is raised, by 
means of increasing tension and a (consequent trivial) H 
lengthening of the cords, as the voice sings upwards ; 
the other (head), by which a similar result is brought — 
about by gradual shortening of the vibrating reed, which 
is still tense, though less so than in the chest register. 
These fundamental divisions are the so-called chest and 
head modes of production, and the falsetto corresponds 
to the head register of the female voice, of which it is an 
imitation. 
wore 
ay 
yao 
1 ‘* The Hygiene of the Vocal Organs.” 
By Morell Mackenzie, M.D. 
(London : Macmillan & Co., 1886.) 
