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THIS is the fourth edition of a book which has been received with warm approval by English and Continental geometers. The first eight sections present no notable changes from the corresponding sections in the last edition. In our previous notice (NATURE, vol. xxix. p. 571) we remarked that the author was “not so well up in the literature of the modern circles as he might be.” This reproach is quite removed in the present edition. Indeed in this direction the author has himself now done excellent yeoman's service. The “supplementary chapter” of fifty-eight pages gives an admirable account of this modern branch in six sections. The first section states and illustrates the theory of isogonal and isotomic points, and of anti-parallel and symmedian lines. The second discusses “two figures directly similar” in homothetic figures. The third section is headed “Lemoine's and Tucker's circles.” The fourth discusses the “general theory of a system of three similar figures.” The fifth gives “special applications of the theory of figures directly similar,” more particularly with reference to Brocard's circle and triangles. In the sixth section on the “theory of harmonic polygons,” the author, starting from Mr. Tucker's extension of the Brocard properties to the harmonic quadrilateral, and Prof. Neuberg's continuation of the same, gives his own beautiful generalisations to the harmonic hexagon and other allied polygons. This latter extension has been made the subject of a communication by MM. Tarry and Neuberg to the French Association meeting at Nancy in August of the present year. The paper, which is not expected to be published until April 1887, contains a complete generalisation of points of Lemoine and Brocard, and the modern circles cited above for polygons and polyhedra.A Sequel to the First Six Books of the Elements of Euclid; containing an Easy Introduction to Modern Geometry (with numerous Examples).By John Casey (Dublin: Hodges, 1886.)