390 



NATURE 



[December i8, \'^v^ 



for as the result of his intentional omission of 

 details of the sense-organs in general, which 

 would have made the book too unwieldy. 



Since a review is scarcely complete without a 

 word of criticism, it may be pointed out that the 

 spelling of "edema" and "hemoglobin" is un- 

 welcome to British eyes, and that " neuron " 

 should have a final e. 



But these are trivialities, and the work as a 

 whole can be highly commended. Circulation, 

 respiration, digestion, and metabolism are espe- 

 cially well discussed. There are, of course, some 

 aspects of these questions on which our knowledge 

 is more complete than at the time the book was 

 written. These are matter for a future edition. 

 In such an edition a short account of the pheno- 

 mena of immunity, from a general point of view, 

 would be valuable, although this would not be a 

 particularly easy task. The illustrations and dia- 

 grams are by no means the least useful part of 

 the book. W. M. Bayliss. 



MATHEMATICS, FARTKTI.AK AND 

 GENERAL. 

 (i) Analytic Geometry. By Prof. Maria M. 

 Roberts and Prof. Julia T. Colpitis. Pp. x-t- 

 245. (New York : John Wiley and Sons, Inc. ; 

 London: Chapman and Hall, Ltd., 1918.) Price 

 ys. 6d. net. 



(2) Elementary Mensuration, Coustnulive T'lane 

 Geometry, and Numerical Trigonometry. By 

 P. Goyen. Pp. viii-fi6g. (London: Macmillan 

 and Co., Ltd., .1919.) Price 3.s-. 6d. 



(3) Lectures on the Philosophy of Mafhen}atics. 

 By James Byrnie Shaw. Pp. vii + 206. 

 (Chicago and London : The Open Court Pub- 

 lishing Co., 1918.) Price 6s. net. 



(i) OROFS. ROBERTS AND COLPITTS 

 -•- have wisely included in a single course 

 on "Analytic Geometry" the most important 

 parts of the theory of conic sections, the theory 

 of curves in Cartesian and polar co-ordinates, and 

 the elements of solid geometry treated analytic- 

 ally, thus representing as a connected whole the 

 parts of co-ordinate geometry in which every uni- 

 versity student of mathematics should be' tho- 

 roughly grounded. The book is very sound peda- 

 gogically, the treatment being based largely on 

 the intuitive use of geometrical constructmns. 

 The figures are well and neatly drawn, and many 

 of the curves are accurately plotted. 



One must, however, point out some of the more 

 serious faults, with a view to their elimination 

 from a future edition. Some of the figures are 

 unsatisfactory — e.g. those on pp. 117, 136, 1^2, 

 159, 168, 178, and 188. A ve-y important point 

 is missed on p. 38. The .student should be in- 

 formed that in a family of curves like 



ay- = x{x — 2aY, 



which can be written in the form 



(y/a)2=(.v/a)(.v/a-2)2, 



the value of the parameter a is to indicate the 

 NO. 2616, VOL. 104] 



dimension of any curve of the family, so that the 

 family consists of similar curves with the origin 

 as centre of similitude. The method of ? 70 is 

 clumsy. 



An interesting feature is the discussion on 

 pp. 83-4 of the difficulties arising from the mul- 

 tiple representation of points in the polar system. 

 A definite point on a plane has definite Cartesian 

 co-ordinates if the axes are given, but it has 

 more than one pair of polar co-ordinates even 

 when the pole and original line are given. Thus the 

 points {p, 0), {-p,e + -!r), (/>, 6 + 2-tt), {-p, »+T,-!r), 

 etc., are all really one and the same point. This 

 may cause confusion and error in the practical use 

 of polar co-ordinates. L'nfortunately, the authors 

 do not indicate clearly the steps to be taken in 

 order to avoid the danger. This is done bv writ- 

 ing any equation p-f{H) In the more general 

 form /> = ( — )"/(^ + 'it), in which n is any positive 

 or negative integer. 



The book is one that can be unhesitatingly 

 commended to the notice of teachers and 

 students. There are numerous exercises, amply 

 illustrative of the principles taught. 



(2) There is no doubt that Mr. Goyen is right 

 in claiming for the experimental method of geo- 

 metrical teaching the advantage of preparing " the 

 way for such subsequent abstract proofs as will 

 enable learners to reinforce the test of experience 

 by the test of reason." The present book depends 

 somewhat on the "test of experience," and its 

 value lies principally in the information given, 

 and not in the process of proof employed to 

 establish the results. Occasionally, indeed, state- 

 ments are made without attempt at any justifica- 

 tion. 



Mr. Goyen's book thus caters for the student 

 who wishes to acquire geometrical knowledge 

 rather than geometrical training. It includes the 

 usual geometry of rectilinear figures and circles, 

 with application to ordinary mensuration ; the 

 quadrilateral and regular and irregular polygons; 

 and the use of similar figures and some three- 

 dimensi( nal mensuration. .\ welcome chapter is 

 that on numerical trigonometry, a subject too- 

 much neglected in elementary mathematical 

 teaching. Every student who claims to have 

 "done mathematics" should be ab'e to deal suc- 

 cessfully with easy problems in "heights and dis- 

 tances." 



There are numerous examples. The tables and 

 the answers add to the usefulness of a verv useful 

 book. 



(3) One of the great defects of scientific teach- 

 ing, due no doubt to the high degree of special- 

 isation that recent progress has made necessary, 

 is the restricted outlook of the presentation. The 

 pass student learns a certain amount of two or 

 three subjects ; the honours man studies one sub- 

 ject to a higher pitch of excellence ; but both the 

 pass and the honours men are generally ignorant 

 of two important aspects of their studies : they 

 know little of the historical development of their 

 subjects, and they rarely acquire a view of the 

 whole of any subject, the correlation of its parts, 



