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XXIV. Notices respecting New Books. 



Lehrbuch der Algebra. By H. Webeb. Vol. II. 2nd edition. 

 (Brunswick, 1899: pp. xvi + 856.) 



TT is hardly necessary to do more than call attention to the fact that 

 -*- this splendid work, the first edition of which appeared in 1896, 

 has been carefully revised and has grown by the addition of some 

 60 pages. In our notice of the first volume we were struck with 

 the care and exactness with which the foundations of the subject 

 were laid. It has been remarked that the original second volume 

 represented very fairly the progress Algebra has made in the last 

 25 years. The present work comes even more nearly up to the 

 high- water mark of present knowledge. The first three, out of 

 four, Books contain an admirable account of the group-theory. 

 Chapter 2, in the discussion of the Abelian groups, is a revised 

 version of the Author's memoir in vol. viii. of the Acta Mathematical 

 and Chapter 3 also is an important reproduction of much of the 

 original memoir and discusses fully the groups of a cyclotomic 

 corpus. In Chapter 5 we have a full account of what has been 

 done by Tylow, Frobenius, Cole, and others ; in a footnote on 

 p. 121 we are glad to see a reference to Prof. W. Burnside's 

 recent work, for which he was awarded the DeMorgan Medal by 

 the London Mathematical Society. 



Book II. is on Linear groups, especially polyhedral and 

 congruence groups, with which Klein's name is so intimately 

 associated. 



Book III. gives applications of the Group Theory, to the config- 

 uration of the inflexional tangents of a plane cubic and to that of 

 the 28 double tangents of a quartic, as well as to many other 

 interesting properties of Analytical Geometry. A solution of 

 the general quintic comes into Chapter 14. The last Book occupies 

 nine chapters, treats of Algebraical numbers, and is very interesting. 

 The concluding chapter contains a proof of the transcendence of 

 e and ir. English readers will find an excellent translation of the 

 whole of this Chapter, by Prof. W. W. Beman, in the Feb. 

 (1897) No. of the Bulletin of the American Mathematical Society 

 (pp. 174-195). Students who desire an exhaustive analysis of the 

 (first edition of the) work can read such in the Feb. No. (1898) of 

 the said Bulletin (pp. 200-234) by Prof. J. Pierpont. Is it too 

 much to hope that an English translation will soon make its 

 appearance ? 



