604 
NATURE 
[APRIL 25, 1907 
as forming a point of separation between rational 
numbers of two classes, numbers of one class 
being less than those of the other. This definition 
appears to involve the assumption (pp. 7, 10, &c.) that 
the point of separation is unique, in other words, that 
there cannot be two irrational numbers which have 
not some rational number separating them. Perhaps 
this assumption may be regarded as a definition of 
equality of irrational numbers; in any case, the 
inquiring reader would find it necessary to examine 
more fully the references to Dedekind’s and Cantor’s 
writings given on p. 56. Once the assumption or 
definition is made, the representation of numbers by 
sequences readily follows. The theory of limfts is 
discussed on p. 57, and in the following chapter the 
notion of a sequence is shown to give rise to that 
of a series. The remaining portion of the book is 
mainly devoted to the study of convergence, and in- 
cludes the well-known multiplication theorem and 
applications to the still better-known binomial and 
exponential series. 
Prof. Schubert is rightly regarded as an authority 
on the teaching of mathematics, but if this descrip- 
tion leads the English reader to expect that the pre- 
sent selection of lecture notes will consist of a mere 
repelition of. the ‘“‘ school geometry ’’ and ‘‘ graphs ”’ 
which are being ridden to death in England to the 
exclusion of other equally important reforms, that 
reader will be greatly disappointed. Dr. Schubert 
has rather shown us what can be done by any teacher 
who will endeavour to make himself ‘‘a snapper up of 
unconsidered trifles.’’ He finds, in the first place, 
that the determination of centres of gravity is not 
well treated in text-books either on mechanics or on 
the calculus; accordingly, this problem forms the sub- 
ject of the first section. The discussion includes 
curves, areas, and figures of revolution, and we 
notice the three- and four-cusped hypocycloids, the 
lemniscate, the kissoid, and other well-known curves 
figuring among the worked-out examples. Next 
follows a chapter on Snellius’s law of refraction. 
Some properties of the parabola deduced from the 
equation of the tangent are next discussed. Then 
follow certain stereometric problems, and in particular 
an extension of Simpson’s rule for the volume of a 
frustum. Each of these sections deals with points 
which are not satisfactorily treated in existing text- 
books. The book concludes with some interesting 
problems in spherical trigonometry, in particular 
the ‘*‘ Heronic’’ triangle, in which the sines and 
cosines of the sides and angles are rational fractions. 
The book. is interesting reading, and quite easy for 
anyone with an elementary knowledge of the subjects 
discussed, to follow. 
““Lecons de Géométrie supérieure ’’ consists of a 
collection of lecture notes on a course delivered in 
1905-6, and transcribed by M. Anzemberger. The 
notes are type-written, not printed, and we can only 
the 
wish that a similar method of procedure could be } 
adopted with the mass of dry, uninteresting, super- 
fluous, and wholly irrelevant details which so often 
occupy pages of printing in modern published “ re- 
searches.’? The course can be precisely described to 
NO. 1956, VOL. 75 | 
English readers as ‘‘solid geometry of curves, sur- 
faces and complexes.’’ It deals mainly with the large 
subject of curvature, but, in addition to considering 
systems of lines, the author gives some elegant dis- 
cussions of systems of spheres and circles. The 
present reviewer has for some time past given a 
course of lectures on solid geometry in which the 
curvature of curves is treated kinematically. It is 
interesting to see this most useful and suggestive 
method adopted in the present notes, for example, in 
defining the osculating plane as the plane containing 
the tangent and the acceleration. 
M. Laurent’s book also deals with analytical geo- 
metry, mainly solid geometry, but treats principally 
those portions of the subject which are studied 
before curvature. It has for its object the develop- 
ment of geometry from a purely abstract point of 
view independently of any preconceived notions re- 
garding space. It is thus based on the study of 
orthogonal transformations and quadratic forms, and 
an instance of the spirit of the book is afforded by 
the preliminary note, in which the periodicity of the 
circular functions is derived from their definition as 
exponentials apart from any consideration of their 
geometrical properties. The subject-matter includes 
the study of tangents and envelopes, the properties 
of surfaces of the second degree, their diameters and 
polars, the principle of duality, and a final chapter on 
the non-Euclidean spaces of Riemann and Bolyai. 
The author at the outset advises his readers to make 
a clean sweep of all their previously acquired geo- 
metrical notions. It is pointed out that in order to 
pass from the abstract to. the concrete one defini- 
tion is required, namely, the definition of rigid-body 
displacement. ‘This definition is to be regarded as 
fundamental, and as superseding Euclid’s axiom 
of parallels. Among the applications we notice 
Abel’s theorem and an important theorem of 
Chasles. 
The story of Abel’s life has been told recently in 
more than one book, yet it is a story that well bears 
re-telling, if for no other reason because it ought to 
be read as widely as possible. It is natural that M. 
de Pesloiian should give considerable attention to the 
part of Abel’s life which was spent in Paris, and in a 
concluding chapter he offers some reflections as to the 
causes which led to Abel’s great memoir being 
neglected at the time it was offered to the academy. 
To understand these causes, M. de Peslotian considers 
it is only necessary to study the trend of mathematical 
thought in Paris about the year 1826. At that time 
French mathematicians were too much engrossed 
with applied mathematics—such as dynamics and 
electricity—to give heed to a paper dealing with a 
property of transcendental functions, and thus nobody 
understood or appreciated the value of Abel’s work. 
The author further cites the parallel case of Galois 
as another unappreciated mathematical genius who 
interested himself greatly in Abel’s work. It might 
be easy to cite other examples, such as Grassmann. 
The misfortune is that there is nothing to prevent a 
recurrence at the present time of the circumstances 
which led to Abel’s dying in poverty without obtain- 
