A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

From the "l EDUCATIONAL TIMES."
"The editor has been very happy in some of the changes he
has made. The combination of the general and particular enunciations of each proposition into one is good; and the shortening
of the proofs, by omitting the repetitions, so common in Euclid, is
another improvement. The use of the contra-positive of a proved
theorem is introduced with advantage, in place of the reductio ad
absurdum; while the alternative (or, in some cases, substituted)
proofs are numerous, many of them being not only elegant but
eminently suggestive. The notes at the end of the book are of
great interest, and much of the matter is not easily accessible.
The collection of exercises, 'of which there are nearly eight
hundred,' is another feature which will commend the book to
teachers. To sum up, we think that this work ought to be read
by every teacher of Geometry; and we make bold to say that no
one can study it without gaining valuable information, and still
more valuable suggestions."
Prom the "JOURNAL OF EDUCATION," Sept. 1, 1883.
( In the text of the propositions, the author has adhered, in all
but a few instances, to the substance of Euclid's demonstrations,
without, however, giving way to a slavish following of his occasional verbiage and redundance. The use of letters in brackets
in the enunciations eludes the necessity of giving a second or
particular enunciation, and can do no harm. Hints of other
proofs are often given in small type at the end of a proposition,
and, where necessary, short explanations. The definitions are
also carefully annotated. The theory of proportion, Book V., is
given in an algebraical form. This book has always appeared to
us an exquisitely subtle example of Greek mathematical logic,
but the subject can be made infinitely simpler and shorter by a
little algebra, and naturally the more difficult method has yielded
place to the less. It is not studied in schools; it is not asked for
even in the Cambridge Tripos; a few years ago, it still survived
in one of the College Examinations at St. John's; but whether
the reforming spirit which is dominant there has left it, we do
not know. The book contains a very large body of riders and
independent geometrical problems. The simpler of these are
given in immediate connexion with the propositions to which
they naturally attach; the more difficult are given in collections
at the end of each book. Some of these are solved in the book,
and these include many well-known theorems, properties of orthocentre, of nine-point circle, &c. In every way this edition of
Euclid is deserving of commendation. We would also express a
hope that everyone who uses this book will afterwards read the
same author's ' Sequel to Euclid,' where he will find an excellent
account of more modern Geometry."
Second Edition, Price 6s.
A KEY TO THE EXERCISES IN THE FIRST SIX BOOKS OF
CASEY'S "ELEMENTS OF EUCLID."
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