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THIS is a work which we have read with considerable interest. As the author states, “it is not an edition of Euclid's ‘Elements,’ in fact it has little relation to that celebrated ancient work except in the subject matter.” Its alliances are with such treatises as Casey's “Sequel to Euclid” and McDowell's “Exercises on Euclid and in Modern Gometry,” but it appears to, us to be better adapted in some respects than either of these works to the use of junior students. These are too condensed for some readers, whereas the book before us, without being too diffuse, enters into greater detail, and leads the pupil up, by a course of sound teaching, so as to enable him to attack with success the subject of modern analytical geometry. Mr. Dupuis looks at a triangle not as “the three-cornered portion of the plane inclosed within its sides, but the combination of the three points and three lines forming what are usually termed its vertices and its sides and sides produced.” His object is to lead up to the idea of a figure as a locus, with a view to preparing the way for the study of Cartesian geometry. Here the necessity for a careful distinction between congruence and equality arises. He introduces freely the principle of motion in the transformation of geometric figures, and devotes some space to the principle of continuity. Further, he connects geometry with algebraic forms and symbols, “(1) by an elementary study of the modes of representing geometric ideas in the symbols of algebra, and (2) by determining the consequent geometric interpretation which is to be given to each interpretable algebraic form.” The subject of proportion is treated on the method of measures, and the term tensor is freely used. The first part (pp. 1-90) traverses the point, line, parallels, the triangle and circle. The second part (pp. 91-146) considers the measurement of lengths and areas: each part closes with a section devoted to illustrative matter drawn from constructive geometry. The third part (pp. 147-177) consists of two sections—the first on proportion amongst line-segments, and the second on functions of angles and their applications in geometry. Some instruments are described, as the proportional compasses, the sector, the pantagraph, and the diagonal scale. In the fourth part (pp. 178-251) there are seven sections, which are taken up with such matters as the centre of mean position, inversion and inverse figures, pole and polar, radical axis, and centres and axes of perspective. The closing part (pp. 252-290) introduces the student to harmonic and anharmonic properties, polar reciprocals and reciprocation, and to homography and involution. The author discusses all these points in a lucid style, and illustrates them with full store of carefully selected solutions: in addition there are a great number of unworked exercises in all the subjects. These good results are the outcome of many years' teaching of geometry to the junior classes in the University of Queen's College, Kingston, Canada. The book is closed with a full index, and clearly drawn figures accompany the text.

Elementary and Synthetic Geometry of the Point, Line, and Circle in the Plane.

By N. F. Dupuis. (London: Macmillan, 1889.)