December 5, 1902.] 



SCIENCE. 



90i 



Princeton University this book is sold at less 

 than the cost of publication. English-speak- 

 ing mathematicians will be certainly grateful 

 for the public spirit shown by Princeton Uni- 

 versity and its alumni. While it is true that 

 most mathematicians can read memoirs in 

 foreign languages, yet its difficulty often 

 deters them from doing so when they are not 

 directly interested in the subject. There is 

 an ease in one's own idiomatic forms of ex- 

 pression which makes the reading much pleas- 

 anter, and if to such translations are added 

 notes of interest and bibliographies of value, 

 then their usefulness is unquestionable. One 

 likes to add such works to one's own library 

 where they can be an incentive to a broader 

 knowledge. Similar translations to the one 

 in hand, such as of the work of Lie which leads 

 up to and includes his theory of transforma- 

 tion groups, would be equally valuable and 

 acceptable. 



The translators present us first with Gauss's 

 paper of 1827, and his own abstract of the 

 same. Here, in 47 pages, is the original de- 

 velopment of the theory of surfaces, relating 

 principally to questions of curvature, treated 

 mainly by Gauss's own method of curvilinear 

 coordinates, which formed the source of many 

 remarkable thorems, such as that the measure 

 of curvature of a surface remains unchanged 

 by bending it without stretching or breaking. 

 The notes to this paper occupy 28 pages, and 

 give historical information, explanations and 

 omitted figures and proofs of many theorems. 



Next follows the paper of 1825, which was 

 not published until after the death of Gauss. 

 It is his less finished and incomplete first 

 paper on the subject. Curvilinear coordi- 

 nates are not used; there is an introduction 

 which treats of curvature in a plane; and, 

 altogether, it shows the manner in which 

 many of the ideas of the more coiuplete paper 

 were evolved. There are 29 pages in this 

 paper, followed by 4 pages of notes. Then 

 comes a bibliography of 11 pages, containing 

 343 titles, which is limited to works which 

 use Gauss's methods in the subjects of curvi- 

 linear coordinates, geodesic and isometric 

 lines, curvature, deformation, orthogonal sys- 

 tems, and the general theory of surfaces, but 



not including minimal surfaces, congruences, 

 etc. A few corrections of misprints and an 

 additional note appear on the last page. 



It seems umiecessary to give this review a 

 learned appearance for the readers of Science, 

 by entering into a discussion of details of 

 theorems and formulas. The work of Gauss 

 is of primary importance in the theory of 

 surfaces, and these papers are classical in the 

 subject. What I wish is to note the useful- 

 ness and importance of this translation of the 

 work of an original master to all who desire 

 to study the subject, and to express what I 

 conceive to be the general obligations of 

 American mathematicians to the translators 

 for their careful labors and to Princeton 

 University and its alumni for their thought- 

 fulness and generosity in its publication. 

 Arthur S. H.\thaway. 



Acht Vortrdge iiber Physihalische Gliemie, 

 gehalten auf Einladwng der Universitdt 

 Chicago. By Professor J. H. van't Hoff. 

 Braunschweig, F. Vieweg and Sohn. 1902. 

 Nothing written by the great master of 

 modern physical chemistry can fail to be of 

 interest and value. The excellently lucid 

 treatment of the subject to be seen in these 

 lectures will imdoubtedly assist in dispelling 

 that remnant of distrust concei-ning the new 

 chemistry which still sometimes lurks in con- 

 servative minds. To those conversant with the 

 author's other works, these lectures will bring 

 nothing new except the details of their presen- 

 tation, which covers a wide field with the help 

 of a few typical examples. The lectures treat 

 in succession the relation of physical chem- 

 istry to piire chemistry (especially inorganic), 

 to technical chemistry, to physiological 

 chemistry and to geology. They call atten- 

 tion in a striking manner to the far-reaching 

 influence of the new ideas. Among other ex- 

 amples the phase relations of iron and steel, 

 and of carnallite, are discussed in detail in 

 their appropriate places, and the fundamental 

 importance of osmotic phenomena and of 

 enzymes is especially emphasized in the two 

 chapters upon physiological chemistry. To 

 Americans these lectures are especially inter- 

 esting because of their having formed one of 



