Juke 20, 3913] 



SCIENCE 



947 



The algebraical character of the paper needs 

 a word. Instead of using a coordinate system 

 and ordinary algebra, the authors develop a 

 vector-algebra whose expressions represent di- 

 rectly the geometrical entities under discus- 

 sion, and which in itself is unchanged by the 

 changes in the axes of reference. This alge- 

 bra is based upon the notions of Gibbs, and is 

 the same as was developed by Lewis.' A 

 rather complete development is given, includ- 

 ing the analysis, or differential calculus of 

 these vectors. In terms of the constancy of 

 one of the vectors defined, the vector of ex- 

 tended momentum, the laws of conservation 

 of mass, energy and momentum, are deduced, 

 as well as fields of gravitational force and po- 

 tential. It is not possible to enter into detail, 

 as the technical character of the developments 

 would demand a large amount of space to do 

 them justice. However, any one desiring a 

 complete and elegant account of the relativity 

 theory, as it is seen in a geometric setting, 

 will find it here. The laws of electromagnet- 

 ics and mechanics are seen to be theorems in 

 this geometry, which means of course that the 

 representation as a non-Euclidean geometry 

 of four dimensions is not only a fair repre- 

 sentation, but is a complete representation of 

 all the facts. It is not to be concluded, how- 

 ever, that it is the only representation; others 

 have been suggested, which do not introduce 

 the notion of a four-dimensional space in the 

 sense it has above.^ It should be pointed out, 

 however, that the eleetrodynamic equations 

 remain unaltered if we substitute a distance 

 X for ci and at a time for x given by cT. So 

 that if the universe is four-dimensional and 

 we are moving with the velocity of light in 

 one of the four directions of the fundamental 

 axes, we can not tell which one it is, and in- 

 deed it makes no difference. Which means in 

 the end (does it not?) that as we assumed in 

 the beginning that the only thing we could 

 measure absolutely is velocity, therefore, all 

 distances must be expressed as velocities, that 

 is. as times, or conversely, that time as we view 



^ Froc. Amer. Acad. Arts and Sci., 46: 163-182. 



"Timerding, JaJiresb. d. Math. Ver., 21: 274- 

 285, 1913. 



it is a distance. Indeed this is the funda- 

 mental assumption of the whole theory, that 

 we may never know correctly absolute distance 

 (if there be such a thing) nor absolute time, 

 but we do know correctly absolute velocity. 



The memoir is interesting also to mathema- 

 ticians as a study of a particular non-Eucli- 

 dean space and the corresponding vector alge- 

 bra. It illustrates in a very happy way the 

 great simplification introduced into a problem 

 when we apply the proper symbolic analysis. 

 James Byrnie Shaw 



Introduction into Higher Mathematics for 



Scientists and Physicians. By Dr. J. 



Salpeter. Jena, Verlag von Gustav 



Fischer. Pp. 3-36. 



This book has the advantage — as compared 

 with similar previous works — of being written 

 in a very elementary and yet thoughtful fash- 

 ion. The author has succeeded very well in 

 explaining the principles of higher mathe- 

 matics in an exceedingly plain way, yet so 

 that he gives all the essential points. For in- 

 stance, the first three chapters of the book 

 (32 pages or about one tenth of the whole 

 book) are exclusively devoted to a most de- 

 tailed and elaborate explanation of the three 

 fundamental conceptions upon which higher 

 mathematics are based. These are: (1) the 

 conception of the limiting value of an infinite 

 series of figures ; (2) the conception of a func- 

 tion; and (3) the conception of the derivation 

 of a function. To explain the importance and 

 real meaning of these fundamentals the au- 

 thor uses much space, and especially cites a 

 great number of examples from difPerent do- 

 mains of natural science. In view of the pur- 

 pose of this work, however, this explanation 

 is not too long. After this introduction only, 

 the technique of differentiating is discussed, 

 also very clearly. Maxima and minima of 

 functions, differential equations, integration, 

 etc., are then explained thoroughly and clearly. 

 At the end of each chapter numerical examples 

 are given, as well as applications to scientific 

 problems. The graphic method is extensively 

 used. As a whole, the book can be recom- 

 mended to such experimental investigators 



