196 
NATORE 
| DECEMBER 31, 1896 
The fundamental principles of chemistry, and the 
nature of chemical action, are laid down in the first 
twenty pages of the book, after which the non-metals 
and some of their common compounds are described. 
As a companion in the laboratory, containing details of 
many instructive experiments, the book should find 
favour. 
On page 8 we read : “‘ Quite recently it has been found 
that Helium, one of the bodies which had already been 
observed in the corona of the sun, occurs in the gases 
extracted from certain minerals by heating them in 
vacuo.” Helium is a constituent of the solar promin- 
ences, but not of the corona. 
Mr. Trotman’s book follows very much the same lines 
as that of Dr. Bailey ; but it is more suitable for use in 
connection with elementary classes than for the labora- 
tory. It is an attractive little volume, simply worded, 
clearly printed, and plainly illustrated. We regret to 
notice the absence of an index. 
flygtiene for Beginners. By Ernest S. Reynolds, M.D. 
Pp. xiv + (London: Macmillan and Co., Ltd., 
1896.) 
THERE are a number of good elementary books on 
hygiene, but this one will find a place among the best of 
them. The author’s “ Primer of Hygiene” is very well 
known, being widely used in Evening Continuation 
Schools, Technical Institutes, and County Council courses. 
A knowledge of elementary anatomy and physiology is, 
however, essential before the main principles of hygiene 
can be intelligently grasped. Recognising this, the 
author has introduced chapters on the structures and 
functions of the various parts of the human body, and has 
considerably enlarged his “ Primer” in other directions. 
The first hundred pages of the present volume comprise 
nine chapters on elementary anatomy and physiology ; the 
remaining nine chapters are devoted to that extensive 
and varied knowledge concerned in the prevention of 
disease.. The book. is thus thoroughly in touch with 
the syllabus of elementary hygiene of the Department 
of Science and Art. We are not given to praising books 
moulded to particular syllabuses, but the present volume 
does not slavishly follow the lines laid down by the 
examiner in the subject with which it deals, and the 
independence is a sign of the author’s ability to judge 
for himself the. best. arrangement. and scope of the 
matter. It would be to the advantage of the community 
if every individual had to pass an examination in the 
subjects dealt with ; and we venture to say that every 
householder, and every mother having the care of children, 
should be acquainted with as much of. the elementary 
principles of hygiene as is contained in this volume. 
As to teachers of South Kensington classes in hygiene, 
they only need to see the book to appreciate its admirable 
qualities. 
The Parasitic Diseases of Poultry. By Fred. ‘V. 
Theobald, M.A., F:E.S. Pp. xv + 120. (London: 
Gurney and Jackson, 1896.) 
POULTRY are subject to many parasitic diseases, and the 
object of this manual is-to inform poultry-keepers of the 
life-histories of these pests, so that means of prevention 
may be successfully carried out. Mr. Theobald is zoo- 
logist to the Agricultural College at Wye, while his know- 
ledge of the characteristics and habits of the parasites 
he describes has been gained from observation of many 
diseased birds. Poultry-breeders and fanciers may, 
therefore, safely trust themselves to be guided by him ; 
and they will learn from his book how to distinguish and 
cope with the animal and vegetable parasites which 
often cause them such, serious loss. Entomologists will 
discover-in the work some new points on the life-histories 
of the parasitic forms dealt with, as well as a list of the 
parasites found upon fowls. 
NO. 1418, VOL. 55 
22c 
235: 
LETTERS TO THE EDITOR. 
[The Editor does not hold himself responsible for opinions ex- 
pressed by his correspondents. Neither can he undertake 
to return, or to correspond with the writers of, rejected 
manuscripts intended for this or any other part of NATURE. 
No notice ts taken of anonymous communications. | 
The Letters of Charles Darwin. 
I am preparing to publish a supplementary series of Charles- 
Darwin’s letters. My projected volume will include a full 
selection from those letters of a purely scientific interest which 
I was unable to print in the “ Life and Letters,” as well as from 
any fresh material that may now be entrusted to me. 
I would, therefore, ask those of my father’s correspondents 
who have not already done so to allow me to make copies of 
any letters of his which they possess. I venture to remind those 
who may be inclined to help me, that letters of apparently slight 
or restricted interest are often of value. FRANCIS DARWIN. 
Wychfield, Cambridge, December 26. 
On the Goldbach-Euler Theorem regarding Prime 
Numbers. 
IN the published correspondence of Euler there is a note from 
him to Goldbach, or, the other way, from Goldbach to Euler, in 
which a very wonderful theorem is stated which has never been 
proved by Euler or any one else, which I hope I may be able to 
do by an entirely improved method that I have applied with 
perfect success to the problem of partitions and to the more general 
problem of demonstration, z.e. todetermine the number of solutions 
m positive integers of any number of linear equations with any 
number of variables. In applying tnis method I saw that the 
possibility of its success depended. on the theorem named being 
frue in a stricter sense than that used by its authors, of whom 
Euler verified but without proving the theorem by innumerable 
examples. As given by him, the theorem is this: every ever 
number may be broken up in one or more ways into two primes. 
My stricter theorem consists in adding the words ‘‘ where, if 27 
F ' : 4 n 
is the given number, one of the primes will be greater than a and 
the other less than sm. This theorem I have verified by 
innumerable examples. Such primes as these may be called 
mid-primes, and the other integers between 1 and 27 —1 extreme 
primes in regard to the range I, 2,3... , 27-1. 
I have found that with the exception of the number 10, Euler’s 
theorem is true for the resolution of 272 into two extreme primes 5. 
but this I do not propose to consider at present, my theorem 
being that, with exception of 27 = 2, every even number 277, 
may be resolved into the sum of two mid-primes of the range 
(5 25 3} - 22-1). AS, ex. 27. 
4-= 2 2 OSes 8 = 5 Oey 
= 5 + 7 ae 7 16 = |S =e 
Wsyiaes We Gas 13) Ss Jaen) © 20° +7) Eins, : 
40>, “11 + 29 = iget23) 50 = 13 7 BV Ost 
roo = 29+ 7I1= 41 + 59 
200 — 61 + 149 = Sy7arrii27 — &c 
500 = 127 + 373 = 193 + 307 = Kc 
1000 = 257 + 743 = &e: 
And so on. 
My method of investigation is as follows, I prove that the 
number of ways of solving the equation « + y = 22, where x 
and 7 are two mid-primes to the range 27-1, z.e. twice the 
number! of ways of breaking up 27 into two mid-primes + zero. 
or unity, according as 7 is a composite or a prime number, is. 
exactly equal to the coefficient of .v*” in the series 
(222 2 EE oS: ty 
ete A 
where 2, 9, , Zare the mid-primes in question. This co- 
efficient, we know @ grior?, is always a positive integer, and 
therefore if we can show that the coefficient in question is not 
zero, my theorem is proved, and as a consequence the narrower 
one of Goldbach and Euler. By means of my general method 
. nun . 
1 This number may be shown to be of the order -— and a very fair 
i (log »)* 
s 4 : L is i 
approximate value of it is = where p is the number of mid-primes corres- 
ponding to the frangible number 2% 
