564 
Professor Thompson’s text-book on Elemen- 
tary Solid Geometry will be received with 
pleasure by American teachers of elementary 
mathematics. It fills an almost unoccupied 
place by confining itself to a narrow field. Col- 
leges that do not require solid geometry for en- 
trance will find it especially useful. 
There are six chapters devoted to those parts 
of the elementary solid geometry ordinarily 
taught in our colleges and secondary schools, 
one appropriate chapter on the conic sections 
and one on mensuration. Each chapter, ex- 
cepting the fifth, includes a large and well 
selected set of exercises. 
Of course plane geometry is assumed but 
the first seven pages are given up to a careful, 
though designedly not exhaustive, considera- 
tion of certain fundamental notions. It is well 
stated that postulates are propositions ‘taken 
without proof and upon which a train of reason- 
ing is to be built,’ and ‘that it is no part of 
geometry to justify their use except in so far 
as their form is concerned.’ 
The sequence of propositions is developed in 
a scholarly and logical though decidedly con- 
servative manner. The assumed construc- 
tion is rigorously excluded. Many of the demon- 
strations are informal or left entirely to the 
student. The treatment of mensuration, apart 
from the geometry proper, is a good feature. 
Considering the completeness of the work asa 
whole, the proof on pages 122 and 123 is notice- 
able. The theorem is: ‘‘ The arc ofa great circle 
less than a semicircle is the shortest line on the 
surface of a sphere between two given points 
not diametrically opposite.’? This proposition 
can only be proved by the use of some such 
postulate as the following: ‘‘The magnitude 
of a curved line is the limit toward which a 
broken line made up of consecutive chords of 
that curved line approaches when the number 
of chords is increased in such a manner that 
the chords are all diminished without limit.’’ 
(Thomas 8. Fiske, ScrrENcE, Vol. IV., p. 724.) 
The words ‘curved line’ and ‘broken line’ 
are to be understood to mean respectively ‘a 
line no part of which is a great circle are’ and 
“a line made up of great circle ares.’ It seems 
unfortunate that such a postulate was not ex- 
plicitly stated. 
SCIENCE. 
[N. S. Von. VI. No. 145. 
The terminology used is, on the whole, that of 
the average text-book, but the author has ren- 
dered a genuine service to the American 
geometric vocabulary by the introduction of 
Mr. Hayward’s term ‘cuboid’ in place of the 
clumsy expression ‘rectangular parallelepipid.’ 
The pages have a different appearance from 
those of the majority of our text-books, for they 
are solidly printed in the English style and no 
abbreviations are used. 
C. B. WILLIAMS. 
Thirty Years of Teaching. LL. C. MraAtu. Lon- 
don and New York, The Macmillan Com- 
pany. 1897. Pp. viiit250. $1.00. 
There is at present in the educational systems 
of all countries a circle—to call it a vicious cir- 
cle would be over-emphatic—discriminating in 
favor of the classical languages and against the 
sciences. Those having a classical education 
at college and university find positions in the 
schools and in turn prepare boys for the classical 
course at college. The circle tends to remain 
unbroken. Teachers of the classics, being a 
great majority of all teachers, are apt to 
write most of the educational books. But from 
the point of view of the man of science a new 
era has begun when students of biology, such 
as Huxley, Morgan and Miall, begin to write 
on educational topics. The circle is broken 
and adjustment will follow in accordance with 
the physical principle of gravitation or the bio- 
logical principle of the survival of the fittest. 
Professor Miall’s papers, reprinted with some 
additions from the Journal of Education (Lon- 
don), cover a wide range of subjects. He does 
not hesitate to write on the teaching of history, 
of geometry and of Latin grammar, as well as on 
nature study and school musuems, but through- 
out he urges by precept and by example the 
methods of natural science, of nature. Treat 
the child as a child, speak plainly, be interest- 
ing—such maxims are sufficiently trite, but 
they carry weight and influence when put in 
a book that treats the teacher as a teacher, and 
addresses him in a plain and interesting man- 
ner. Professor Miall’s book will repay reading 
by the teacher, whether of the classics or of 
science, whether in the kindergarten or in the 
university. 
