486 



NA TURE 



[February 25, 1909 



of those and later times. Perhaps the latter spirit is 

 still effective, as cricket is apparently never played on 

 Sunday. 



The neglect of physical education up to the time of 

 Rousseau is sketched by Prof. Welton, and its advance 

 since then in secondary schools. He tells us with 

 regard to elementary schools that the conception of 

 education that guided the Education Act of 1S70 was 

 essentially the scholastic tradition, that education and 

 instruction are synonymous, and he affirms the most 

 crying need in English education of to-day to be 

 adequate provision for physical training. H. R. B. 



Bathy-o/ographical Miip of the Britisli Isles. Natural 

 .Scale I : 875,300, or 14 miles to an inch. Balhy- 

 orographical Map of South America. Natural 

 Scale I : 6,150,000, or 97 miles to an inch. Con- 

 structed and engraved by W. and A. K. Johnston, 

 Ltd. Prices not stated. 



Handbook to accompany the Map of tlic Britisli Isles. 

 Pp. 32. Price 6rf. net. 



No more convincing indication could be found of 

 the improvement which has taken place in recent 

 years in the methods of geographical instruc- 

 tion in schools than the enterprise shown by 

 publishers in the production of good orographical 

 maps, both in atlases and on a large scale for class- 

 teaching purposes. The present wall-maps are good 

 examples of the excellent aids which are available to 

 assist teachers in demonstrating the fundamental 

 importance of the distribution of the highlands and 

 lowlands of the areas being studied. In the map of 

 the British Isles six shades of brown are employed to 

 show grni>hically the course of important contours on 

 the land, and two shades of blue indicate the 20- and 

 50-fathom lines in the surrounding seas. In the case 

 of South .\nierica the varving heights of the land 

 above sea level are depicted by five shades of brown 

 and two of green, while the 100-, 1000-, and 2000- 

 fathom lines are shown on the oceans. Care has 

 been taken to avoid crowding, and the maps . are 

 models of clearness. 



The " Handbook " should prove a great help to those 

 teachers of geography who have had little experience 

 in teaching their subject by modern practical methods. 



Invariants of Oiiadratic Differential Forms. By J. E. 

 Wright. Pp. vi + 90. Cambridge Tracts in iMathe- 

 matics and Mathematical Physics, No. 9. (Cam- 

 bridge : University Press, 190S.) Price 2s. 6d. net. 

 This number of the Cambridge Tracts deals with a 

 clear and definite problem, the simplest case of which 

 may be stated as follows. Let a, b, c be given func- 

 tions of the inde]jendent variables, x, y, and let 



adjL- + bd.vdy + cdf 

 become 



a<l^' + 0ydiAi + rf?)- 



by a change of variables from (.v, r) to (f, rj) ; what 

 functions of a, b, c and their differential coefficients 

 transform into the same functions of o, yS, y and their 

 differential coefficients? The importance of this 

 inquiry begins to appear in Gauss's celebrated memoir 

 on the deformation of surfaces ; and a very large part 

 of what is called the differential geometry of surfaces 

 is, from another point of view, the invariant theory 

 of a quadratic differential form in two variables. In 

 the general theory there are n variables, and the first 

 great step in this direction was taken by Rieniann ; 

 references to his principal successors are given by 

 Prof. Wright (pp. 5-8). The methods explained in 

 the tract are tho.se of Christoffel, Lie, and Maschke; 



NO. 2052, VOL. 79] 



the last, which is symbolical, and quite recent, is only 

 very briefly summarised, but enough is done to show 

 its interesting character. Another special calculus 

 applied to the subject is that of Levi-Civiti and Ricci 

 (pp. 20-S) ; and other manipulative devices may 

 doubtless be discovered. So far as one can see at 

 present, the essential elements of the theory are the 

 Riemann-Christoffel four-figure symbols; while the 

 broadest aspect of it is presented by Lie. 



Pp. 51-90 give various geometrical and dynamical 

 applications, concluding with the representation of one 

 manifold on another with correspondence of geodesies. 

 Besides being a useful guide to the analytical theory, 

 this tract will be of service to readers of Darboux's 

 and Bianchi's works on the theorv of surfaces. 



G. B. M. 



A Course of Plane (ieonietry for .ldva)iced Students. 

 Part I. By C. V. Durell. Pp. xi-l-219. (London : 

 Macniillan and Co., Ltd., 1909.) Price 55. net. 

 This is a really capital book for students of what may 

 be called scholarship standard. It contains, among 

 other things, sections on similarity, transversals, 

 vector geometry, inversion, and coaxal circles. As 

 examples of the author's choice of elegant methods, 

 and his clearness of exposition, may be taken the 

 proof (due to Mr. Hillyer) that the centres of the 

 diagonals of a complete quadrilateral are collinear 

 (p. 118), and the proof of Feuerbach's theorem by 

 inversion (p. 149). In the latter example, as in many 

 others, teachers will notice the excellence of the 

 diagrams, which give, without confusion, all that is 

 required and no more. There is a practically 

 inexhaustible stock of examples, with a very wide 

 range of difficulty. Mr. Durell is a master at 

 Winchester College, and those who remember the 

 late Mr. Richardson's success in making his boys like 

 and learn geometry will be glad to see that there is 

 no risk of the subject being neglected now that he 

 is gone. 



The Contents of the Fifth and Sixth Book.': of Euclid. 



Bv .M. j. M. Hill. .Second edition. Pp. xx-l-167. 



(Cambridge : Iniversitv Press, 190S.) Price 6s. 



net. 

 This is a new work rather than a new edition. 

 Prof. Hill has now completely abandoned Euclid's 

 treatment of proportion as given in his fifth and sixth 

 books, and replaced it by an arithmetical theory. 

 Two commensurable quantities, ^A, q.\, are defined 

 as having the ratio piq. Equal ratios are defined 

 as those between which no rational fraction lies. The 

 theory is now made rigorous by means of Dedekind's 

 treatment of irrational numbers, the Cantor-Dedekind 

 axiom, and the axiom of Archimedes. It is a foolish 

 man that never changes his mind; and Prof. Hill's 

 deliberate change of method after eight more years 

 oi teaching is a fact to which special attention 

 should be directed. 



Tlw Elementary Dynamics of Solids and Fluids. By 

 Prof. W. Peddie. With Sectional and General 

 Examples by J. D. Fulton. Pp. xii4-i88. (Edin- 

 burgh and London : Oliver and Bovd, 1909.) Price 

 2S. 6d. 

 This little book is intended for use by junior students 

 in university classes, and for boys in the higher forms 

 of secondary schools. The treatment is very 

 elementary, and fluids are disposed of in the conclud- 

 ing three of the thirteen chapters. The wisdom of 

 printing answers immediately after the exercises 

 throughout the book may be doubted. .As an intro- 

 duction to dynamics, the book should prove useful. 



