314 
NATURE 
i 
[Aucust 3, 1899 
PROJECTIVE GEOMETRY. 
Premiers Principes de Géométrie Moderne, Par E. 
Duporeq. Pp. vili + 160. (Paris: Gauthier- Villars, 
1899.) 
T isa curious fact that while projective geometry is 
becoming better appreciated in England it seems 
to be going out of favour in France. M. Duporcq, in 
his introduction, pathetically deplores the predominant 
place assigned to analysis in the syllabuses of the official 
examinations ; and in France, as with ourselves, most 
teachers are compelled to neglect a subject that does not 
pay. It will be sad indeed if, in the fatherland of Monge, 
Poncelet and Chasles, pure geometry is to be deposed 
from her former high estate, and made a kind of Cinder- 
ella, called in to do odd jobs for Her Serene Highness 
the Princess Avalyse, or to amuse the children with 
tricks of the triangle. 
M. Duporcq’s book itself helps us to realise the danger 
that is threatened. One cannot help feeling that his 
attitude is apologetic, and that his exposition is a half- 
hearted one. At the very outset we are confronted with 
homogeneous coordinates ; homography is based on an 
algebraic relation ; points at infinity lie in a plane “by 
definition ” ; imaginary elements have no real existence, 
and the introduction of them, due to analysis, is a mere 
facon de parler, vaguely justified by the “ Principle of 
Continuity.” With all respect to Poncelet, it may be 
doubted whether his “ principle of continuity,” apart from 
algebraical considerations, has any real working value ; 
on the other hand, von Staudt elaborated, forty years 
ago, a theory of imaginary elements which, so far as 
curves and surfaces of the second order are concerned, 
gives a consistent geometrical theory (quite independent 
of analysis) in which the principle of continuity has a 
real meaning, and is at the same time practically self- 
evident, as one would expect it to be. Von Staudt’s 
name does not appear to be mentioned in M. Duporcq’s 
book, and the reader might not unreasonably infer that 
the author was ignorant of v. Staudt’s existence. 
It would, of course, be absurd to advocate the ex- 
clusive use of pure, as opposed to analytical, geometry, 
even in problems of a strictly geometrical character. 
The ideal geometrician should be equally expert in both 
methods, and apply one or the other or both combined 
according as circumstances may require. But it may 
fairly be urged that a treatise on the first principles of 
projective geometry should avoid the introduction of co- 
ordinates except by way of illustration, and for the pur- 
pose of showing the points of contact between the two 
methods. It is right to teach an apprentice the use of 
a saw as well as that of a plane; but you will not attain 
this end by giving him a tool that is neither a saw nor a 
plane, but contains something of both. 
Thus to give an explicit example, M. Duporcq fre- 
quently infers homography from a one-to-one relation 
established, not from an equation, but from the inspection 
of a figure. Thus (p. 49): 
“Si donc m et m’ désignent les deux points ot une 
droite quelconque A coupe une conique circonscrite au 
quadrangle aéc¢ d, on voit qu’a tout point 7 de A ne 
correspond ainsi qu’un point 7’. Comme, d’ailleurs, ces 
points sont évidemment réciproques, ils déterminent donc 
une involution sur A,” &c. 
NO. 1553, VOL. 60] 
These statements are doubtless correct, but are they 
sufficiently justified ? How is the beginner to distinguish 
the argument from the following : j 
“Two points S, H are taken on a tangent to an 
ellipse, and any ellipse with foci 5, H cuts the given 
ellipse in the points M, M’: then to each point M cor- 
responds one point M’ and wzce versa, hence we have a | 
system in involution, and MM’ goes through a fixed” 
point ” ? 
It is not a sufficient answer to say that M, M’ are 
only a pair of four associated points, because this is not 
geometrically evident. Again, we have cases of Cremona 
correspondence with the fixed points imaginary: how is 
the untrained student to distinguish them from homo- 
graphic correspondences ? 
We are far from wishing to suggest that M. Duporcq’s 
work is devoid of interest and value. Considering its 
size it is remarkable for the range and variety of its 
contents ; it comprises a very attractive and, indeed, 
brilliant sketch of homography, poles and polars, in- 
volution, quadratic transformation (including inversion), 
together with an outline of Lie’s line-sphere corre- 
spondence. For a reader prepared by previous study, 
it affords an excellent and suggestive vésemé ; it is rather 
when it is examined as a methodical text-book for 
students that it seems to us to fall short of perfection. 
To the student we would still say : Read Reye, work his 
exercises, and then, if you like the subject, gird up your 
loins and tackle von Staudt. Foy it is a truth past gain- 
saying that v. Staudt’s “ Geometrie der Lage” and the 
immortal “ Beitraége” contain, as no other books do, the 
essentials of projective geometry. G. B. M. 
A SYSTEM OF PHYSICS. 
Kanon der Physik. By Felix Auerbach. Pp. xil + 522. 
(Leipzig : Viet and Co., 1899.) 
— CIENTIFIC books may be divided into two groups, 
those which are written because the author has 
something to teach, and those which are written because 
he has something to learn. It is no reproach to a writer 
if his book is classed with the second group, for there 
may be as much originality in learning as in teaching, 
and his autodidactic efforts will often prove a source of 
instruction to others. It is not possible to say whether 
Prof. Auerbach has been consciously writing his “ Kanon” 
of physics to clear up his own ideas on scientific prin- 
ciples, but the book he has produced gives the impression 
that this has been one of his principal motives; and ] 
would even go a step further and say that, if life were 
long enough, every physicist ought, when he gets to the 
age of fifty, to spend three years in putting his ideas into 
shape and write a similar treatise. It would serve as a 
kind of ‘‘Abiturienten Examen” to his state of crystal- 
lisation. 
It is easier to talk about this book in vague and 
general terms than to give an account of what it is and 
what it contains. I am afraid of becoming definite in 
my own words, for fear of giving a wrong impression, 
and must content myself with the translation of a few 
sentences taken out of the preface. 
“ A comprehensive book is still wanting—and not only 
in Germany—in which the conceptions, principles, 
