NoveMBER 12, 1903] 
NATURE 
2h: 
of.many minor points of interest in the volume before | 
us, but we hope to refer to the subject again when 
the second portion of Mr. Thompson’s work appears. 
Meanwhile, the book may be commended to all 
students ef follk-lore, as well as to the Semitic philo- 
logist and anthropologist. 
LINE GEOMETRY. 
A Treatise on the Line Complex. By C. M. Jessop. 
Pp. xv+364. (Cambridge: University Press, 1903.) 
Price tos. net. 
HE systematic study of what is now called line 
geometry was begun by Pliicker in his ‘‘ Neue 
Geometrie des Raumes.’’. There was, it is true, a 
fairly complete theory of the linear complex due to 
Chasles and others before the publication of Pliicker’s 
work, and the geometry of systems of lines (con- 
gruences) has in some respects not progressed very 
much since the date of Kiimmer’s celebrated memoir, 
but it was only after the idea of line coordinates had 
been introduced that the essential qualities of the new 
geometry were recognised. 
In his treatise on the line complex, Mr. Jessop has 
aimed at presenting the extensive investigations of 
German and Italian geometers in a form easily 
accessible to the English student of mathematics. It 
has been found necessary to write an introductory 
chapter on the elementary methods of synthetic 
geometry that are used in the book; then the next four 
chapters contain the theory of line coordinates, in the 
general form introduced by Klein, and the linear com- 
plex. A great portion of these chapters will be familiar 
to anyone who has studied the theory of forces in three 
dimensions with attention; the only omission that 
occurs to us is a number of easy exercises on the use 
of line coordinates, particularly in the ordinary 
Cartesian form, but this defect is partly remedied by 
an excellent collection of examples at the end of the 
book. 
The theory of the quadratic complex is naturally the 
most important part of the book—the linear complex 
is too well known, and the higher complexes too 
difficult to deal with, to illustrate the methods of the 
subject. The author has devoted six ample chapters 
(vi.—xi.) to this theory. Chapter vi. deals mainly with 
the singular surface, which is remarkable in all com- 
plexes as being both the locus of singular points and 
the envelope of singular planes. Two proofs of the 
identity of this locus and envelope are given, one de- 
pending on von Staudt’s theorem concerning a tetra- 
hedron, and the other on infinitesimal properties. The 
first is particularly interesting although peculiar to 
the quadratic complex, because a tetrahedron being 
the simplest form of the singular surface, von Staudt’s 
theorem is a particular case of a property of Kiimmer’s 
quartic from which the result follows; the other proof 
can be extended readily to any complex (chapter xvii.). 
The discussion of Kiimmer’s quartic is the author’s 
own, and will be very welcome to the beginner as being 
both elementary and direct. 
It is curious that an infinite number of quadratic 
complexes have the same singular surface, the theory 
NO. 1776, VOL. 69] 
being similar to that of confocal quadrics. Such co- 
singular complexes are discussed in chapter viii., and 
by developing the idea of corresponding lines in 
cosingular complexes Mr. Jessop has obtained some 
very interesting and novel proofs. Another chapter 
deals with the beautiful classification of quadratic com- 
plexes, and contains an exposition of Darboux’s proof 
of the fundamental theorem of Weierstrass on the 
equivalence of quadratic forms. 
In chapter vii. an account of some special complexes 
is given, the greater part of the space being devoted 
to the tetrahedral complex; this complex was studied 
long before the introduction of line coordinates, and 
lends itself readily to synthetic treatment. 
In another part of the book it is shown that a tetra- 
hedral complex can always be found which contains the 
complete intersection of a quadratic complex and a 
linear complex. Substantially this important result is 
due to Kiimmer, but the first complete account of it 
we owe to Caporali. 
Only two chapters on congruences appear in the 
work; this part of the subject is difficult, because the 
analytical methods are clumsy when applied to such 
congruences as are not complete intersections of com- 
plexes, and the purely synthetic methods of Sturm and 
others are extremely tedious. Mr. Jessop follows 
Kiimmer on the general principles, and only gives a 
detailed account of the simplest congruence, namely, 
that of the second order and the second class. 
The latter portion of the book does not strike us as 
being so attractively arranged as the earlier parts, but 
the final chapter on the connection of line geometry 
and differential equations is valuable as an introduction 
to Lie’s theories. 
There is no doubt that the book will be a boon to 
a student of the subject, and that anyone with a taste 
for geometry will find much that is interesting and 
something that is new in it. oda (es 
OUR BOOK SHELF. 
Geological Rambles in East Yorkshire. By Thomas 
Sheppard, F.G.S. Pp. xi+235; 53 illustrations and 
geological map. (London: A. Brown and Sons, 
1903-) 
Tuts is a pleasantly written and attractive guide to 
the geology of east Yorkshire, the work of a sturdy 
local geologist who shows himself to be master of 
his subject and of the literature past and present. 
Under his enthusiastic leadership we are taken from 
Hull to the out-of-the-world promontory of Spurn 
Head, where we learn many lessons about recent geo- 
logical changes. Thence we are conducted north- 
wards to Withernsea and Hornsea, examining some 
of the finest sections in the Boulder-clay of Yorkshire ; 
successive beds of drift with transported mollusca and 
Scandinavian rocks, deposits with local detritus, and 
others with rocks from the Cheviots and elsewhere. 
We see also lacustrine deposits and peat beds, remains 
of old lakes, of which Hornsea Mere alone appears 
to survive. Then we are taken on to Bridlington, 
noted for its shelly ‘‘ Crag,’ really a part of the base- 
ment Glacial drift, which is now hidden behind a 
strong sea-wall. The buried cliff of Sewerby, with its 
basement clay and older mammaliferous deposit yield- 
ing Elephas antiquus, hippopotamus, rhinoceros, &c., 
claims attention. From this we pass on to the fine 
