October 17, 1913] 



SCIENCE 



551 



there is probably none wHcb is more favorably 

 known tban Weber's " Lehrbucli der Algebra " 

 in three large volumes. The great extent of 

 the work doubtless discouraged many begin- 

 ners as well as those who have only time to 

 learn the fundamental principles of this vast 

 subject. Hence the small volume before us 

 should find a hearty welcome among many 

 students of mathematics who understand the 

 German language. 



The present book begins with a study of the 

 elementary properties of determinants and 

 their applications in the solution of a system 

 of linear equations. The remaining fourteen 

 chapters bear the following headings, in order : 

 Numbers and integral functions, symmetric 

 functions, roots, cubic and biquadratic equa- 

 tions, Sturm's theorem, approximation of the 

 roots, groups, the Galois theory, cyclic equa- 

 tions, divisions of the circle, solution of the 

 cyclotomie equation, algebraic solution of equa- 

 tions, numbers and functions of an algebraic 

 realm, applications to cyclic realms. 



From these chapter headings it is evident 

 that the book under review is not confined to 

 the most elementary matters, which can be 

 found in nearly all the text-books on this sub- 

 ject. On the other hand, it does not presup- 

 pose very much, but develops from the begin- 

 ning most of the subjects which it treats. As 

 the book is a final effort, on the part of a 

 great scholar and excellent writer, to present 

 the main subjects of advanced algebra, it has 

 a peculiar interest, both as regards the choice 

 ■of material and the methods of treatment. 



Although most students who are in position 

 to profit much by the study of such a work can 

 read German, yet there is doubtless a consider- 

 able number to whom an English translation 

 vfould be very helpful, since there is no algebra 

 in the English language which covers the same 

 ground. The excellent " Introduction to Mod- 

 ern Algebra," by Professor Bocher, for in- 

 stance, does not enter into the Galois theory 

 of equations and the theory of algebraic num- 

 hers — theories which occupy a prominent 

 place in the present work. 



In the preface it is stated that the author 

 ■was assisted by his colleagues, especially by 



Messrs. Lowy, Epstein and Levi, while cor- 

 recting the proof. These names, together with 

 that of H. Weber, are a sufficient guarantee 

 that no important errors appear in the book. 

 Among the minor errors the statement that 

 Dedekind first divided a group into double 

 co-sets, which appears as a foot-note on page 

 196, is of especial interest. It is well known 

 that Frobenius developed this method exten- 

 sively in an article which appeared in Crelle's 

 Journal in 1887, while Dedekind's article ap- 

 peared seven years later. 



G. A. Miller 

 Universitt of Illinois 



Measures of Proper Motion Stars Made with 

 the JfO-inch Refractor of the Yerhes Olserv- 

 atory in the Years 1907 to 1912. By S. W. 

 BuRNHAM. Washington, D. 0. Published 

 by the Carnegie Institution of Washington. 

 1913. 



This handsome volume of iv + 311 quarto 

 pages is so fully described by its title, given 

 above, that comment upon it may be brief. To 

 the astronomers of old time the stars were 

 '■' fixed," i. e., abiding eternally in the same 

 celestial place without any trace of motion 

 relative to their fellows. Less than two cen- 

 turies ago, it was found that a few of the 

 brighter stars appeared to be exceptional in 

 this respect. Since increasing refinement of 

 observation indicated a slow but continuous 

 progression across the sky, peculiar or 

 " proper " to a few stars that were forthwith 

 assumed to be nearer than the others. The 

 search for and determination of these proper 

 motions has been one of the standard prob- 

 lems of astronomy since the time of Halley 

 and the present volume is a contribution to 

 that end. Its fundamental idea is that per- 

 ceptible motion, being an unusual stellar 

 attribute, may be assumed limited to the 

 brighter stars and may be determined by 

 measuring the change in the position of these 

 exceptional stars by reference to any of the 

 fainter ones about them. Possibly some sus- 

 picions with regard to the assumed fixity of 

 the fainter stars finds expression in the au- 

 thor's introductory words, " It goes without 



