“This book contains a large number of elementary geometrical propositions not given in Euclid, which are required by every student of Mathematics. Here are such propositions as that the three bisectors of the sides of a triangle are concurrent, needed in determining the position of the centre of gravity of a triangle; propositions in the circle needed in Practical Geometry and Mechanics; properties of the centres of similitudes, and the theories of inversion and reciprocations so useful in certain electrical questions. The proofs are always neat, and in many cases exceedingly elegant.” The \Educational Times.” “We have certainly seen nowhere so good an introduction to Modern Geometry, or so copious a collection of those elementary propositions not given by Euclid, but which are absolutely indispensable for every student who intends to proceed to the study of the Higher Mathematics. The style and general get up of the book are, in every way, worthy of the ‘Dublin University Press Series,’ to which it belongs.” The \School Guardian.” “This book is a well-devised and useful work. It consists of propositions supplementary to those of the first six books of Euclid, and a series of carefully arranged exercises which follow each section. More than half the book is devoted to the Sixth Book of Euclid, the chapters on the ‘Theory of Inversion’ and on the ‘Poles and Polars’ being especially good. Its method skilfully combines the methods of old and modern Geometry; and a student well acquainted with its subject-matter would be fairly equipped with the geometrical knowledge he would require for the study of any branch of physical science.” The \Practical Teacher.” “Professor Casey’s aim has been to collect within reasonable compass all those propositions of Modern Geometry to which reference is often made, but which are as yet embodied nowhere.…We can unreservedly give the highest praise to the matter of the book. In most cases the proofs are extraordinarily neat.…The notes to the Sixth Book are the most satisfactory. Feuerbach’s Theorem (the nine-points circle touches inscribed and escribed circles) is favoured with two or three proofs, all of which are elegant. Dr. Hart’s extension of it is extremely well proved.…We shall have given sufficient commendation to the book when we say, that the proofs of these (Malfatti’s Problem, and Miquel’s Theorem), and equally complex problems, which we used to shudder to attack, even by the powerful weapons of analysis, are easily and triumphantly accomplished by Pure Geometry. “After showing what great results this book has accomplished in the minimum of space, it is almost superfluous to say more. Our author is almost alone in the field, and for the present need scarcely fear rivals.” The \Academy.”