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LTV. Notices respecting New Books, 



A Treatise on the Analytical Geometry of the Point, Line, Circle, and 

 Conic Sections, containing an account of its most recent extensions, 

 with numerous examples. By Dr. J. Casey, F.B.S. Second 

 Edition. London: Longmans, 1893. (Pp. xxxii + 564.) 

 HPITE mathematical world was informed, on the occasion of 

 -*- Dr. Casey's death in 1891, th?,t he had himself passed 400 pages 

 of this second edition through the press, and that the concluding 

 portion was in manuscript. Dr. Casey received considerable 

 assistance in the preparation of this eaition from Prof. Neuberg, 

 of Liege ; and we learn from the preface, supplied by the editor, 

 Prof. Dowling, that the final proof-sheets, which had not received 

 the author's revision, have been submitted to the same gentleman 

 for correction and approval. The book brought out under such 

 auspices is, as might naturally be expected, quite up to date, and 

 will amply repay a careful study. How greatly the work has been 

 enlarged will be seen when we mention that the first edition 

 extended to 331 pages as against the 564 pages of this volume. 



The first seven chapters have identical headings in the two 

 editions. The subject-matter has grown from 227 pages to 279. 

 There are new articles on biradial and biangular coordinates ; the 

 treatment of anharmonic and harmonic ratio is advanced and now 

 concludes the section on Cartesian Coordinates (applied to the 

 line), and in place of the section on Trihnear Coordinates we have 

 an enlarged one on " Systems of Three Coordinates." This section 

 gives a clear account of many of the modern terms which have 

 come into vogue with the rise of the modern geometry of the tri- 

 angle. In fact this is the book to which a student will naturally 

 turn who wishes to read in English what has been done in this 

 direction. A glance at the " Contents " will put in evidence how 

 deeply Dr. Casey had explored this new field, and how fully he 

 had adopted the ways of the " surname " geometers. Chapter viii. 

 is still headed " miscellaneous investigations," but its sections are 

 now devoted to figures inversely similar, pencils inversely equal, 

 twin-points (the Zwillingspunkte of Artzt), triangles derived from 

 same triangle and tripolar coordinates. Chapter ix. discusses 

 special relations of the Conic Sections, and much of the matter of 

 chapter Tin. of the first edition is now rearranged and rewritten 

 in chapters x.-xni. A long chapter (xit.) is devoted to the recent 

 geometry, and in it will be found almost all that is known at 

 present of the numerous circles which stud the firmament of the 

 modern geometer. 



It is to this subject, as well as to the theories of the Mean 



Centre, of Anharmonic Ratios, and of Homographic Division and 



Involution, that the principal additions have been made. But 



perhaps the fullest and most important additions are contained in 



Phil. Mag. S. 5. Vol. 36. No. 223. Dec. 1893. 2 



