August i8, 192 i] 



NATURE 



775 



One illustration may be given. Comparing the 

 climate of the Lofoten Isles with that of Verkho- 

 yansk in the same latitude, "and no farther apart 

 than Portland, Maine, and Portland, Oregon," 

 the authors ascribe the diiferences solely to the 

 influence of the ocean. One cannot but ask, if 

 that is so : How are we to explain the great differ- 

 ences especially between the winter climates of 

 Portland, Maine, and Portland, Oregon, or the 

 fact that the average mean January temperature 

 of Cape Hatteras (46° F.) is just the same as that 

 of the Scilly Isles 15° farther north, although the 

 Gulf Stream proper passes close by the cape with 

 a mean winter temperature of 72° F., while the 

 winter temperature of the water round the Scilly 

 Isles is only about 50° F. ? One has to go well 

 into the book to find any recognition of the agency 

 of the winds as an intermediary influence on tem- 

 perature- 

 One of the excellent features of the book 

 is the number of ingenious and thought- 

 provoking exercises at the end of each 

 chapter, but a good teacher might find it 

 profitable also to supplement those exercises 

 by asking his students to fill up the gaps in the 

 generalisations which are, without doubt, com- 

 plete in the minds, but not in the text, of the 

 authors. 



The smaller book, by Dr. Brown, has no illus- 

 trations, but the other has, in addition to views, 

 numerous instructive diagrams and maps. 



Geo. G. Chisholm. 



Calculus for Students. 



An Elementary Course of Infinitesimal Calculus. 

 By Prof. H. Lamb. Third edition, revised. 

 .Pp. xiv-f-530. (Cambridge: At the University 

 Press, 1919.) 205. net. 



THE merits of Prof. Lamb's text-books are 

 so well known and appreciated that it is 

 unnecessary to analyse or commend the present 

 one, especially as it appears in its third edition. 

 As the work of an experienced teacher, revised 

 in the light of modern mathematics, the book 

 affords a model, and suggests a few observations. 

 Naturally, the influence of recent research is most 

 evident in the first chapter, "Continuity." This 

 contains a discussion of sequences, upper and 

 lower limits, limiting values and infinitesimals, 

 which we may presume to be the author's idea of 

 Avhat is suitable for the average student before 

 starting upon the infinitesimal calculus. In the 

 light of present knowledge it is a kind of indis- 

 pensable minimum ; but it will probably be found 

 as much as can be comprehended by the type of 

 NO. 2703, VOL. 107] 



student for whom the book is mainly designed. 

 In any case, the inclusion of such a chapter is 

 a significant mark of progress in the practical 

 aims of mathematical teachers. 



There are two points to which the author him- 

 self directs attention. The first of these is that, 

 in dealing with series, he has confined himself 

 mainly to power-series, and omitted the discus- 

 sions of uniform convergence previously included. 

 Remembering that this is an elementary course, 

 we may acquiesce, if with some reluctance, in the 

 author's judgment. The second point is that 

 exp X is defined as a particular solution of the 

 differential equation dy/dx=y. This is Clifford's 

 procedure in his "Elements of Dynamic," and has 

 everything to be said in its favour — assuming that 

 the student begins the calculus at the proper time 

 in his general course. All the properties of the 

 function and its inverse can be deduced with 

 great ease, and in a way that needs no amend- 

 ment when the variable is complex. We rather 

 regret that Prof, Lamb has omitted the complex 

 variable altogether; Clifford's graph of exp i, and 

 its connection with the radian, do, in fact, 

 interest quite average students when they are 

 working at de Moivre's theorem and its conse- 

 quences. 



It would be easy to point out many features of 

 the book which show the advantages accruing 

 from the fact that the writer is an applied as 

 well as a pure mathematician. One of these is 

 the excellence and instructiveness of the dia- 

 grams; another is the variety of the examples; 

 and as a third we may take the discussion of the 

 linear differential equation y" + ay' + hy = o and 

 those closely associated with it. It is possible to 

 make the discussion as dull and mechanical as the 

 most old-fashioned solution of a quadratic by 

 completing the square ; here we have a treatment 

 which is really instructive, and illustrated by the 

 right sort of examples. 



It is curious to notice that nobody seems to have 

 suggested a " standard " sequence of theorems in 

 elementary differentiation, though every argu- 

 ment urged for such a thing in elementary geo- 

 metry applies here with at least equal force. 



G. B. M. 



Our Bookshelf. 



Map Reading. By G. H. C. Dale. Pp. vii-t- 

 170-1- XX plates. (London: Macmillan and Co., 

 Ltd., 192 1.) 75. 6d. net. 

 It is stated in the introduction to this book that 

 a map is at times as valuable as a rifle. One 

 might go further and say that a knowledge of 

 local topography may be as valuable as a bat- 



