Notices respecting New Books. 309 



Geometry." Prof. Genese then devotes a chapter to " Biangular 

 Coordinates :" i. e. take a triangle PAB, ZPAB = 0, ZPBA = </>; 

 0, (p are the coordinates of P. Several neat results are obtained ; 

 the method, however, does not seem likely to be of general utility. 

 Pages 99-1S4 are devoted to a very full and interesting account 

 by the Rev. T. C. Simmons, of what has been called the " Recent 

 Geometry of the Triaugle." There is nowhere to be found such a 

 handy account, so that we can recommend those of our readers who 

 want to know what it is all about to read this article. If, further, 

 they procure Dr. Casey's 'Conies,' and the fourth edition of his 

 ' Sequel to Euclid,' they will be able to learn all, or nearly all, that 

 has been printed on the subject in Euglish. The writer has treated 

 the question in an independent manner, and has contributed much 

 of novelty to it. Some of our old " Diary " friends would have 

 revelled in this portion of the geometrical field : some, indeed, did a 

 little in this direction, before Brocard laid his hands upon his points, 

 and his circle, and compelled more close study of these matters. 

 There is a short historical Note founded upon a communication 

 by M. E. Lemoine : some corrections upon the early history of the 

 subject are being supplied by Dr. Emmerich to the columns of the 

 ' Educational Times ' (see July 1888). Chapter vi. has four proofs 

 of Feuerbach's theorem, by Eeuerbach himself, Messrs. Davis, 

 Langley, aud Genese. The last has given, perhaps, a simpler proof 

 still in a Note read before the London Mathematical Society at its 

 March meeting. Chapter vii., by Mr. Langley, treats of the 

 " Theory of Inversion '' and " Pedals." The remaining chapters 

 contain " Geometrical and Mechanical Constructions," " Theory of 

 Elimination," " Summation of Series," " Binomial Series," and 

 " Algebraical and Trigonometrical Identities," " Miscellaneous 

 Articles," and specimens of recent scholarship-papers. There is 

 much useful matter furnished for junior students which they will 

 find it difficult to come across elsewhere, for the matters treated of 

 do not come into the ordinary text-books. 



A Treatise on Plane Trigonometry, containing an account of Hyper- 

 bolic Functions, with numerous examples. By J. Casey, LL.D., 

 F.R.S. Dublin: Hodges. 1888. 

 The theme of the treatise before us is now a somewhat hackneyed 

 one as it is presented in the numerous manuals which follow one 

 another in rapid succession ; but Dr. Casey, with a master's touch, 

 has imparted a wonderful amount of freshness to the details he has 

 grouped together. He has not only brought out of his treasure- 

 house things new and old, but he has deftly clothed each small or 

 great detail, and assigned it a niche which seems to be the fittest 

 for it. He does not follow on the lines suggested by Hoiiel, in his 

 Bemarques sur VenseMjnement de la Trigonometric, and run-a-muck 

 against tedious calculations*, but he explains at some length, and 



* " Au lieu d'occuper durant tant de mortelles heures les pauvres 

 ecoliers a transcrire de nombres a sept figures (le nombre sept est, parait-il, 



