BY a recent regulation for the Cambridge Mathematical Tripos provision is made for the introduction of a paper on “Pure Geometry”: this to include, in addition to Euclid, the simple properties of lines and circles, the elementary properties of conic sections treated geometrically, for which a place has already been found, such questions as may be treated by inversion, reciprocation, and by harmonics. It has been for some time a reproach that pure geometry has not occupied a more prominent position in the University curriculum. The University has never lacked able geometers, and amongst the present generation our author has won for himself a good name. He has put together an excellent manual complete enough to meet present wants, and doubtless in subsequent editions he will bring the present work even more up to date than it is. Some of our best text-books are overloaded with corollaries and much other matter which it is difficult for the student to retain clearly in his mind. Mr. Lachlan appears to us to have steered a most judicious course, and avoided overloading his book in this way. Mr. Pinto (in “Lothair”) speaking of the limited range of the English language (which, however, he admitted to be expressive), said it consists of four words. If this be so, the word we should select to characterise Mr. Lachlan's essay is that it is “charming.” It treats of the subject in sixteen chapters, in which, after devoting the first three to an introduction, measurement of geometrical magnitudes and fundamental metrical propositions, he starts from harmonic ranges and pencils, and carries the student at once to the theory of involution. He then discusses properties of the triangle (giving an account here of the recent additions to this branch of the subject, from which we infer that it has at length got a footing in the University) and of rectilinear figures. The reader then has laid before him a clear account of the theories of perspective, of similar figures (previously introduced to English readers in Casey's “Sequel”), and of reciprocation. The properties of the circle are discussed under the heads of the circle as a figure by itself, and then in relation to one or more circles. In this division of the subject our author gives account of his own discoveries and of the many interesting additions contributed by Mr A. Larmor. In two remaining chapters the theories of inversion and of cross ratio are unfolded. The treatment in the text is strictly confined to the line and circle. We believe that a further volume extending the methods herein employed to the conic sections is in course of preparation. A few slips have caught our eye, viz. p. 53, ex. 4; § 97 ex.; p. 55, ex. 7; § 116; § 262; § 268, and one or two other easily corrected mistakes. In such a mass of mathematical work there may well be others. References are given to the sources whence many of the questions are taken. We note that an oversight, which we have had occasion to point out twice before in NATURE in reviewing the late Dr. Casey's “Sequel,” is perpetuated here. On pp. 68, 71, a question is cited from a “Trin. Coll., 1889” paper, whereas it was given many years previously in the Educational Times (Feb., 1865, and April), and was then by a correspondent carried back to Steiner (Crelle, vol. liii.). Tha figures illustrate the text very clearly, and there is a full index at the end.
An Elementary Treatise on Modern Pure Geometry.
By R. Lachlan (Macmillan, 1893.)
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