JUNE 19, 1902 ] 
ceasing to be. A careful perusal has disclosed nothing 
that can give a well-intentioned critic occasion to say 
“*this is a happy idea—that is capitally put—this is some- 
thing to help us.” On the contrary, if this book were to 
be reviewed in detail, it would be necessary to write 
columns of complaint. One feature of novelty appears 
in the book in the form of full-page illustrations of 
apparatus and materials used in all the experiments. 
These pictures are reproduced from photographs, and 
show three tiers of apparatus arranged as if for sale. 
In many cases it is not easy for an experienced chemist 
to recognise the individual pieces, and in plate xx. we 
reach a climax. It represents on the top shelf two tin 
canisters, a stoppered bottle, a Bunsen burner, a beaker, 
a tin dish, a blowpipe and another stoppered bottle. On 
the next shelf are three stoppered bottles, a hammer, four 
tin canisters, a small structure like a dog kennel, and a 
rack of twelve test-tubes. On the bottom shelf are two 
developing trays, a beaker, a stoppered bottle, a sugar 
basin, a stone gingerbeer bottle, a pocket handkerchief 
and apparently a bank-note ora shirt cuff. The plate 
bears the legend “The Metals.” By the use of a lens 
one word of two of the labels can be deciphered. 
SOLID GEOMETRY. 
The Elements of Euclid, Book XT. By R. Lachlan, Sc.D. 
Pp. 51. (London: Edward Arnold, n.d.) Price ts. 
T is to be hoped that some of the scientific committees 
which are now dealing with the improvement of 
mathematical teaching, and more especially with that of 
the teaching of elementary geometry, will, in the process 
of pruning Euclid, direct attention to this little-read 
Book xi. As in other books of the Elements, many of the 
propositions are of the trivial, or even ludicrous, character, 
while some of the definitions lack precision. For example, 
can prop. I—‘‘one part of a straight line cannot lie in a 
plane and another part without the plane ”—be seriously 
cegarded as necessary? Indeed, the proof assumes the 
thing which it seeks to prove; let A B C be the given 
straight line ; let a part of it, A B, lie in the plane, and 
a part, B C (if possible), out of the plane ; produce A B 
in the plane to any point, D, &c. To this several other 
instances might be added. 
Then as regards definition, the descriptions of di- 
hedral, trihedral and (generally) polyhedral angles leave 
something to be desired. Possibly some better term 
than angle can be found in such cases. We are told 
that “when two planes meet and are terminated at their 
line of intersection, they are said to form a dihedral 
angle” ; “ when several planes meet in a point, they are 
said to form a polyhedral angle.” All that such planes 
visibly “form” is a certain figure ; the “angle” which 
they form (as it is employed in subsequent mathematics) 
is, in reality, an avea on a sphere of unit radius. It is 
true that Book xi. is not concerned with this precise 
quantitative definition of (so-called) sold angles—better 
called conical angles—but merely with certain plane, or 
face, angles connected with them ; nevertheless, it may 
be desirable to give the student, who when he reaches 
Book xi. can scarcely be called a degénner, this quanti- 
tative notion. 
NO. 1703, VOL. 66] 
NAAT ORE 
171 
In the small compass of this book there is little oppor- 
tunity for anything strikingly original or novel. Dr. 
Lachlan finishes it with an appendix which contains a 
large number of propositions, examples, &c., and this 
appendix will be found much more valuable than Book xi. 
itself. 
A few criticisms of a minor character may not be out 
of place. We notice that in the enunciation of each 
proposition, Dr. Lachlan always uses the simple word 
“is” or “are” when the proposition states a fact which 
can be proved ; thus, “if two planes intersect, their line of 
intersection is a straight line.” The typical editor of a 
modern Euclid would say “their line of intersection 
shall be a straight line,” employing a ridiculous com- 
pulsory form of expression. There is now the beginning 
of a revulsion against this style, which has been con- 
sidered for some curious reason to be appropriate and 
essential to Euclid, but to no other subject of study or 
conversation. So far, Dr. Lachlan is in agreement with 
common sense ; but why does he, when setting out on 
the proof of the proposition, re-state the fact with a 
“shall be”? Twice he forgot his rule—in prop. 1, 
where “must be” is employed, and _ prop. 14, where the 
simple and sensible ‘“‘are” of the formal enunciation 
remains “are” in the re-statement. 
The proof of prop. 20 would avoid a tendency to 
mislead the student if it stated that the point C is first 
taken (arbitrarily), then E, and finally B and D by draw- 
ing avy line, E B D, through E. 
In the third line of the proof of prop. 21, the proof is 
rendered very much more clear by the insertion of the 
word “all” before the words “the ds,” the statement 
then being the very obvious one that if there are two 
sets of fifty plane triangles, the sum of all the angles in 
the first set is equal to the sum of all those in the second 
set. 
Finally, the employment of the word “ power” in the 
definition (p. 536) “the square on the distance between 
a point and the centre of a sphere less the square on the 
radius of the sphere is called the power of the point with 
respect to the sphere” does not seem justifiable or neces- 
sary, although it has been employed by a geometer of 
high repute. The word ower is already employed in 
science for something quite different from the square of 
a tangent. Indeed, a student of electricity might be 
tempted to think that this geometrical “‘ power of points ” 
is a mere pun on the well-known term used in connec- 
tion with frictional machines. Everything must not be 
sacrificed to brevity ; if new terms are wanted in science, 
they should be appropriate and expressive. 
BELGIAN BOTANICAL INVESTIGATIONS. 
Recueil de Institut Botanique (Université de Bruxelles), 
Par L. Errera. Tomev. Pp. xii + 357. (Bruxelles: 
Henri Lamertin, 1902.) 
N this book there are brought together recent papers 
by botanists of the Royal Academy of Belgium, which 
have already been published in different journals during 
the last two years. Although this is the first volume to 
be published, it appears as vol. v., since the first 
four volumes will be given up to earlier papers. Thus 
