170 



NATURE 



[October 30, 19 19 



P. 236. The exponential series is defined as the 

 Umit of (x^xjnY, and is denoted by e* ; on 

 p. 237 it is taken for granted that e*, so defined, 

 obeys the laws of indices. 



P. 241. The proof of Euler's exponential ex- 

 pressions for the sine and cosine is new to the 

 reviewer, but he fails to see why the variable 

 must be expressed in radians rather than in any 

 other unit of angular measure. The fact that the 

 authors tell us on three occasions (pp. 108, 147, 

 243) that angles must be given in radians scarcely 

 seems an adequate reason. 



We are told twice (pp. 22 and 29) that feet and 

 inches are denoted by the symbols ' and ", but it 

 is apparently considered superfluous to define a 

 degree (until p. 106, though degrees are used on 

 p. 41) or to give the details of sexagesimal 

 measure, and the student is referred to the tables 

 for the values of the trigonometrical functions of 

 30°, 45°, and 60°. 



P. 63. A definition of "variable" is given, but 

 no definition of "constant." 



P. 91, ex. 10. To solve sin (% — 25°) = o-6 by 

 using the addition theorem is a method which 

 seems unnecessarily cumbrous. 



P. 91, ex. II. The equation arc tan a: = 

 arcsec^; — 45° seems to lead to a cubic equation. 

 Methods for solving cubics are not given until 

 p. 144. 



P. 182. The student should not be asked to 

 prove that, in the hyperbola, F'P — FP = 2a, with- 

 out being told that the equation is true for one 

 branch only. 



P. 269. In a book which does not define even 

 hyperbolic functions, it is going rather far to ask 

 the student to find the length of y = sin:x: from 

 x~o io rv = 7r. 



Chap. xvii. The notations//(%) and //(x)d.-v: seem 

 to be used indifferently. The object of the former 

 notation is not apparent. The notation " cot " 

 used hitherto is here replaced by " ctn " without 

 explanation. 



Misprints and other minor errors have been 

 noted at p. 35, ex. 17; p. 50, ex. 17; p. 69, 1. 9; 

 p. 106, 1. 7 up; p. 116, ex. 9; p. 120, 1. 2 up; 

 p. 123, 1. 2 up; p. 136, 1. 21; p. 173, ex. 12; 

 p. 180, ex. 6; p. 232, 11. 3, 4; p. 243, II. 5, 7, and 

 8; p. 253, ex. 22; p. 263, 1. i; p. 271, ex. 2;. 

 p. 274, exs. 14, 21; p. 275, ex. 25; and p. 277, 

 ex. 15. 



(2) This work, which was first published in 

 191 1, has now been revised by the author, with 

 the assistance of six of his colleagues. It forms 

 an admirable introduction to the subject for the 

 stydent, and deserves very high commendation. 

 The mode of presentation has been carefully 

 thought out, with the result that the style is clear 

 and lucid, and any student of ordinary intelligence 

 should be able to get from the book a sound 

 knowledge of the subject without the aid of a 

 teacher. 



The first chapter contains a synopsis of the 

 notations used in the book ; then follow four 

 chapters on the representations of points and lines 



NO. 2609, VOL. 104] 



by elevation and plan, and of planes by their 

 traces ; next there are four chapters on curved 

 surfaces — mainly cones, cylinders, and spheres — 

 a useful chapter on shadows, and a brief account 

 of perspective. These chapters contain numerous 

 practical problems, each worked out in full with 

 enunciation, discussion, analysis, and construc- 

 tion. The book concludes with a collection of 

 eight long papers of problems and a good index. 

 The diagrams are clear and well-proportioned, 

 though a few of them would have been improved 

 by being made rather larger. 



The reviewer would like to make a few minor 

 suggestions for the improvement of future 

 editions. In the first place, the student may be 

 a little puzzled at finding that the " profile-plane " 

 plays a subordinate part compared with the other 

 two co-ordinate planes {e,.g. it is not mentioned 

 in § 19 on the "alphabet" of a point), and the 

 explanation of this would be useful. Also, the 

 terms "profile ground-line " (§ 13) for a line which 

 is not horizontal, and "vertical of a plane " (§ 42) 

 for a line which is not vertical, seem somewhat 

 misleading. 



Two omissions must be mentioned. The first is 

 that no use is made of the method of changing 

 the co-ordinate planes — a method which gives an 

 elegant solution of such a problem as finding the 

 true length of a line, by taking a new vertical 

 plane parallel to the plan of the line. The second 

 omission is of rather more importance to the 

 student ; he would find the subject much more 

 interesting and concrete if some work (possibly 

 in the form of examples) on solids with plane 

 faces were included. The reviewer well remembers 

 how fascinated he used to be by drawing cubes 

 and pyramids in fantastic positions, particu- 

 larly if a section of the solid had also to be 

 drawn. 



The book would have been enhanced in value to 

 the student of crystallography if some account of 

 isometric projection had been given, and the 

 reviewer would have been glad to see some 

 developments of the theory of perspective — e.g. 

 the theorem that plane figures in perspective 

 remain in perspective when rotated about their 

 axis of collineation ; but possibly the author con- 

 siders that such additions would have unduly 

 increased the size of the book. 



(3) In this, an interesting and suggestive work, 

 the author (an engineer) discusses the theory of 

 the quadrilateral after the manner in which 

 various modern geometers have discussed the 

 triangle. The treatment is quite elementary, and 

 the object of the author is not to give a complete 

 discussion of the subject, but to encourage and 

 facilitate research. The book, "public au moment 

 oil la France vient de reconquerir les provinces 

 qui lui ont 6te arrachees en 1871," is dedicated to 

 the memory of Joseph Pruvost, professor of 

 mathematics at Strasbourg until the annexation ; 

 it contains a useful bibliography — a feature 

 hitherto somewhat rare in mathematical works 

 published in France. 



