664 



SCIENCE 



[N. S. Vol. XXVI. No. 672 



detic Institute. Leipzig and Berlin, B. G. 



Teubner. 1907. Second edition. Pp. xviii 



+ 578. 



In preparing the well-known first edition of 

 this work Professor Helmert had in view the 

 needs of the physicist, the astronomer and 

 the geodesist rather than those of the mathe- 

 matician; and, though the treatment of the 

 subject was of necessity mathematical, the 

 emphasis was not placed upon the more in- 

 tricate parts of the mathematical theory. As 

 a result the book gave a clear presentation of 

 the method of least squares and supplemented 

 it by a mathematical discussion which was 

 ample for all ordinary purposes and which 

 in some particulars went beyond the range 

 of the ordinary texts on the subject. Numer- 

 ous well-chosen problems furnished illustra- 

 tions of the details of the use of the method 

 in the chief cases. 



The plan of the earlier part of the new 

 edition is substantially that of the former one, 

 though minor changes have been made. Nor 

 does this adherence to the plan of a book 

 thirty-five years old necessarily imply a defect 

 in the new work. For the method of least 

 squares is one of the few advanced branches of 

 mathematical science in which such a proceed- 

 ing is not inappropriate. 



Certain features common to both editions 

 deserve notice, and of these one is the treat- 

 ment of the law of error. No conclusive argu- 

 ment in favor of this law has been given and 

 the author has chosen to base it upon its 

 accord with the results of observation. This 

 is commendable, for it tends to clear a state of 

 affairs which some one has characterized by 

 'saying of the law of error that both mathe- 

 maticians and physicists accept it, the former 

 because they believe the latter have obtained 

 sufficient experimental evidence and the latter 

 because they believe that it has been mathe- 

 matically demonstrated. It is true that in 

 the second edition one of the numerous 

 mathematical arguments in favor of the law is 

 included, but it is given a secondary place. 

 Moreover, the author expressly considers 

 several possible laws of error. 



Clear explanations of the most important 

 ideas of the subject are given early in the 



work and accompanying them are illustrations 

 of their practical use. Then follows the de- 

 velopment of the subject along standard lines 

 from the discussion of direct observations of 

 equal weight to that of indirect determinations 

 of the values of quantities which are not inde- 

 pendent. 



Of the improvements made in preparing the 

 new edition, one notes an increase in the 

 amount of space devoted to pure theory, par- 

 ticularly in regard to the relations to each 

 other of various kinds of errors of observation 

 and in regard to the application of the method 

 to interpolation. The size of the volume has 

 been increased from 348 to 578 pages, and a 

 large part of this increase is made up of the 

 last three chapters, which deal with technical 

 problems of physical, astronomical and geo- 

 detic work. 



Pleasing are the frequent references to orig- 

 inal sources and the excellence of the 

 examples by means of which the theory is 

 illustrated. A detailed table of contents and 

 an index make all of the matter in the book 

 accessible to the reader, and the publishers 

 have made the book attractive in appearance. 

 An occasional sacrifice of mathematical rigor 

 for the sake of brevity will not prevent even 

 an exacting reader from regarding the text as 

 an excellent treatise on the subject. 



George H. Ling. 



Columbia Univeksity 



SCIENTIFIC JOURNALS AND ARTICLES 

 The opening (October) number of volume 

 14 of the Bulletin of the American Mathe- 

 matical Society contains the following articles: 

 " Application of a Definite Integral involving 

 Bessel's Functions to the Self-Inductance of 

 Solenoids," by A. G. Webster; "On the 

 Apsidal Angle in Central Orbits," by F. L. 

 Grifiin ; " The Maximum Value of a Deter- 

 minant," by E. W. Davis ; " The Invariant 

 Substitutions under a Substitution Group," by 

 G. A. Miller; Shorter Notices (Tannery's 

 Legons d'Algebre et d' Analyse a I'Usage des 

 Eleves des Classes de Mathematiques spe- 

 ciales. Tome Premier, by F. Cajori; Tannery's 

 LcQons d'Algebre et d' Analyse, Tome Second, 

 by G. W. Myers; Pionchon's Mathematiques. 



