This edition of the Elements of Euclid, undertaken at the request of the prin-
cipals of some of the leading Colleges and Schools of Ireland, is intended to
supply a want much felt by teachers at the present day -- the production of a
work which, while giving the unrivalled original in all its integrity, would also
contain the modern conceptions and developments of the portion of Geometry
over which the Elements extend. A cursory examination of the work will show
that the Editor has gone much further in this latter direction than any of his
predecessors, for it will be found to contain, not only more actual matter than
is given in any of theirs with which he is acquainted, but also much of a special
character, which is not given, so far as he is aware, in any former work on the
subject. The great extension of geometrical methods in recent times has made
such a work a necessity for the student, to enable him not only to read with ad-
vantage, but even to understand those mathematical writings of modern times
which require an accurate knowledge of Elementary Geometry, and to which it
is in reality the best introduction.
In compiling his work the Editor has received invaluable assistance from the
late Rev. Professor Townsend, s.f.t.c.d. The book was rewritten and con-
siderably altered in accordance with his suggestions, and to that distinguished
Geometer it is largely indebted for whatever merit it possesses.
The Questions for Examination in the early part of the First Book are in-
tended as specimens, which the teacher ought to follow through the entire work.
Every person who has had experience in tuition knows well the importance of
such examinations in teaching Elementary Geometry.
The Exercises, of which there are over eight hundred, have been all selected
with great care. Those in the body of each Book are intended as applications of
Euclid's Propositions. They are for the most part of an elementary character,
and may be regarded as common property, nearly every one of them having
appeared already in previous collections. The Exercises at the end of each
Book are more advanced; several are due to the late Professor Townsend, some
are original, and a large number have been taken from two important French
works -- Catalan's Th´eor`emes et Probl`emes de G´eom´etrie El´ementaire, and
the Trait´e de G´eom´etrie, by Rouch´ e and De Comberousse.
The second edition has been thoroughly revised and greatly enlarged. The
new matter includes several alternative proofs, important examination questions
on each of the books, an explanation of the ratio of incommensurable quantities,
the ?rst twenty-one propositions of Book XI., and an Appendix on the properties
of the Prism, Pyramids, Cylinder, Sphere, and Cone.
The present Edition has been very carefully read throughout, and it is hoped
that few misprints have escaped detection.
The Editor is glad to ?nd from the rapid sale of former editions (each 3000
copies) of his Book, and its general adoption in schools, that it is likely to accomplish the double object with which it was written, viz. to supply students
with a Manual that will impart a thorough knowledge of the immortal work
of the great Greek Geometer, and introduce them, at the same time, to some
of the most important conceptions and developments of the Geometry of the
present day.

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