REVIEWS 519 



find such a very different branch of their subject brought within easy reach. 

 With charming clearness and brevity, Prof. Carslaw sets forth the history and 

 outlines of non-Euclidean geometry, for readers who possess only an elementary 

 knowledge of mathematics. The book is well printed, but some of the figures 

 should have been repeated overleaf. 



The first two chapters are historical : we could, perhaps, have spared even 

 such passing insistence on the human weaknesses of the great inventors. Then 

 the different parallel postulates are explained, and it is shown that on each of 

 them there is built up a different geometry, consistent with itself but contradicting 

 the others. There are three chapters on hyperbolic geometry, trigonometry, and 

 infinitesimal calculus ; these are in turn carried just far enough for the newcomer 

 to begin to feel at home among the ideas and notations, and to see that there is a 

 great deal more to be learnt. Much use is made of Saccheri's symmetrical 

 quadrilateral, with two right angles and two acute angles ; and Bolayai's classical 

 construction, for a parallel to a given line through a given point, is the most 

 prominent proposition. No use is made of the principle of continuity, nor of 

 solid geometry. Elliptic geometry, with its two varieties, is treated rather more 

 shortly in the next two chapters ; the analogy with spherical trigonometry is clearly 

 brought out, so that the ideas are more familiar than in the section before. The 

 last chapter is devoted to philosophy ; it establishes the truth of non-Euclidean 

 geometry by interpreting its postulates in terms of a family of circles and spheres 

 in Euclidean space, so showing that any contradiction in either system must be 

 accompanied by a corresponding contradiction in the other. " One geometry 

 cannot be more true than another ; it can only be more convenient " is one of the 

 apt quotations with which the volume closes. 



H. P. H. 



Exercices et Lecons de Mecanique Analytique. Par R. de Montessus, Pro- 

 fesseur a la Faculte libre des Sciences de Lille. [Pp. vi + 334.] (Paris.: 

 Gauthier-Villars et Cie., 191 5. Price 12 francs.) 



This book consists of problems, arranged in order of difficulty and solved by 

 analytical methods, and supplements those courses of lectures on mechanics in 

 which, owing to the place which must be given to certain fundamental questions 

 such as the pendulum and the gyroscope, little attention has been paid to 

 educationally valuable problems. There are also numerous exercises. The first 

 part deals with the calculation of centres of gravity of curves and surfaces, and 

 attractions, and then treats of potential and theorems on it. The second part 

 deals with moments of inertia, the principle of virtual work, Lagrange's equations, 

 the motion of a rigid body, stability, small oscillations of a system about a position 

 of stable equilibrium, and impact. A long' note, of more than sixty pages, on 

 elliptic integrals in the real domain terminates the book. Elliptic functions have 

 many applications in mechanics, and, as the author says, there is great difficulty in 

 applying the theory as usually given in lectures on the theory of functions. 



There does not seem to be anything strikingly novel in this useful collection of 

 problems. I think that a clear discussion of the modifications introduced into 

 Lagrange's equations by cases of rolling with friction (non-holonomy) would impress 

 on the student's mind the nature of the tacit assumptions made on p. 117. 



Philip E. B. Jourdain. 



