438 NATURE 
to have daughters has no resource but to find a still | 
weaker husband. The thesis, if accepted, should beget 
humility in those male parents who have large families of 
lusty sons. Ives at 
RONTGEN RAYS IN MEDICINE AND 
SURGERY. 
The Rontgen Rays in Medicine and Surgery as an aid 
in Diagnosis and as a Therapeutic Agent. By 
Francis H. Williams, M.D. (Harvard). Pp. xxxii+704 ; 
4o1 illustrations. Second edition, with appendix. (New 
York : The Macmillan Company ; London : Macmillan 
and Co., Ltd.) Price 25s. net. 
HE second edition of this excellent work was called 
for because the first was unexpectedly exhausted 
within three months, and we congratulate the author 
upon his deserved success. Only those acquainted with 
the subject can appreciate how difficult it is for any 
author to give a correct view of the progress of sucha 
branch of science as the X-rays, because of the great 
advances made within a comparatively short period, the 
number of authors engaged in research and the nature of 
the subject itself. As might have been expected, Dr. 
Williams fully understands this, because in his preface 
he states that the work is rather a report of progress 
than a final presentation of a growing subject. Further, 
owing to the short time at his disposal for the preparation 
of a second edition, he has only been able to add some 
forty pages, chiefly on apparatus and the therapeutic uses 
of the X-rays. This will be found in the appendix. 
Dr. Williams very properly introduces his subject by 
reference to the principles of physical science, and, with- 
out overburdening the student, he tells what is necessary 
for their appreciation. Next he deals in the most prac- 
tical way with the equipment necessary for photographic 
and therapeutic work. Having thus prepared the way, 
he enters into a full description of the normal conditions 
of the cavities of the body so that the observer may be 
able to appreciate deviations from the normal, a principle 
which will be thoroughly appreciated by all those who 
are seeking for information from the clinical aspect. 
The pathological changes are well described by photo- 
graphic illustrations, diagrams and histories of selected 
cases, 
A noticeable feature of the work is the amount of atten- 
tion devoted to what might be called the medical aspect 
of the subject as opposed to the surgical. This is 
interesting, because for a long time many who believed 
in the value of X-rays in the detection of fractures, dis- 
locations of the hard structures and foreign bodies were 
inclined to think that the use of X-rays would be limited 
to these. If any are still of this opinion we commend 
them to a perusal of this work. 
The third great step in the development of X-rays in 
medicine was their application in diseased structures, and 
the present position of their therapeutic action is frankly 
and fairly stated in these pages. 
While it is true that the work gives a very strong 
representation of the methods employed in America— 
indeed, the illustrations themselves show that the work 
has not been produced in any European laboratory—still 
NO. 1714, VOL. 66] 
[SEPTEMBER 4, 1902 
the labours of others have not been neglected. In 
future editions the work might be enhanced in value by 
a reference to what has been done in this country and the 
European schools of medicine, a fact which is admitted 
by the author in his preface, because he states that he 
had intended to include as complete a list as possible of 
the publications on the subject. This was not found 
possible on account of its extent, so he adds that had he 
foreseen this he would have referred in the text to many 
other important papers. 
The work is well written by one thoroughly familiar 
with the subject, is profusely illustrated, and to those 
who desire a guide to the study of the subject the work 
may be thoroughly recommended; and this remark applies. 
to students and practitioners. 
OUR BOOK SHELF. 
Elementary Geometry. By W. C. Fletcher, M.A., Head 
Master of the Liverpool Institute; late Fellow of St. 
John’s College, Cambridge. Pp. 80. (London: Edward 
Arnold, n.d.) Price 1s. 6d. 
THIS is a very small book and avery good one. Its 
object is to teach geometry to boys without hindering 
and wearying them with metaphysical subtleties, or re- 
quiring them to express the proofs of propositions with 
that pedantic recitation of details—that parody of logical 
accuracy—which has long been identified with the study 
of Euclid. 
The author is perfectly correct when he says that his 
little book “contains the whole substance of Euclid 
i.-iv. and vi. except the elegant but unimportant proposi- 
tion, iv. 10.” 
The branches of the subject are taken in the following 
order:—Angles, triangulation (7.2. the discussion of the 
properties of triangles), quadrilaterals, loci, propor- 
tionals, circles, tangents, areas, maxima and minima, 
this last section being very short and merely illustrating. 
what is meant by a maximum ora minimum. There is 
no formality whatever in the proofs, the most simple’ 
propositions being often left to the student with a hint 
sufficient for the solution. Each section, besides ter- 
minating with a number of simple exercises (well within 
the power of the beginner), contains a number of 
numerical illustrations to be worked by actual drawing 
with instruments. This is precisely the kind of teaching 
which is now being advocated by those who have taken 
up the question of the reform of mathematical teaching. 
In propositions relating to proportion—as, for example, 
that a line drawn parallel to the base of a triangle divides 
the sides in the same ratio—the author explicitly states 
that he assumes two magnitudes to have a common 
measure, and that the difficulty which arises in the case 
in which they have not “had better be disregarded for 
the present.” The reason for thus making an essen/ial 
difference between “ commensurable” and ‘ incommen- 
surable” quantities of the same kind is not obvious, since 
any proposition which holds for the former will be ad- 
mitted, even by the beginner, to hold for the latter when 
it is pointed out that the unit magnitude may be taken so 
small that the distinction between commensurable and 
incommensurable quantities practically disappears. The 
proposition that the sum of two sides of a triangle is 
greater than the third is proved by the definition of a 
right line as the shortest distance between two points. 
The nature of a tangent as the limiting position of a 
chord is that which the author adopts. This also is in 
accordance with modern notions, and it offers no difficulty 
whatever even to the merest beginner. In p. 42, line 4, 
for “place them so that two pairs of sides are parallel,” 
