152 SCIENCE PROGRESS 



encourage individual work, and to this end have presented the detailed methods 

 and formulas rather as suggestions than as rules necessarily to be followed." 

 This book is very simply written, and is, on the whole, a far better book than the 

 same author's Differential Calculus. The examples and both geometrical and 

 physical applications are very good and numerous. Beginning in the traditional 

 way by defining integration as the inverse of differentiation, and hence developing 

 the usual formulae and methods, definite integrals are then introduced as limits of 

 sums. Then come various geometrical and physical applications, a treatment 

 of approximate methods, and chapters on double and triple integration. The last 

 chapter is on differential equations, and it is both pleasant and interesting to see 

 differential equations treated in a textbook on the integral calculus — where they 

 used to be treated and where it seems that they still should be treated. The rest 

 of the book is made up by supplementary exercises, answers, tables of integrals 

 and natural logarithms, and an index. A very useful book, especially for those 

 who are going to devote their attention to applied mathematics. 



Philip E. B. Jourdain. 



Solid Geometry : Including the Mensuration of Surfaces and Solids. By R. S. 

 Heath, M.A., D.Sc, Professor of Mathematics and Vice-Principal of the 

 University of Birmingham. [Fourth Edition. Pp. iv + 123.] (London : 

 Rivingtons, 1919. Price 4s.) 

 Some of the propositions in that part of this useful little work which is devoted to 

 lines and planes in space and polyhedra are some of the propositions of the 

 eleventh and twelfth Books of Euclid which are rearranged and given abbreviated 

 proofs. Besides other propositions, Dr. Heath has added a large number of 

 exercises and a chapter on spherical triangles and mensuration of the sphere, 

 cylinder, cone, and polyhedra — " particularly the very general formula known as 

 Simpson's Rule" (pp. iii, 64a, 104^). Also there is (p. 71) Euler's relation between 

 the number of faces, vertices, and edges of any polyhedron. It may be remarked 

 that the author refers back by name to propositions of Euclid even when they are 

 reproduced in a shortened form in this book ; thus, on p. 22 he refers to Euclid XI. 

 10, whereas it would be more convenient for a reader to refer to the version of 

 this proposition on p. 15. 



In the preface to the fourth edition the author refers to the great importance of 

 solid geometry in the training of surveyors as well as of students of mathematics. 

 It may perhaps be pointed out that an introduction to the study of the regular 

 polyhedra, such as is given on pp. 30, 65-70, seems to be of the greatest import- 

 ance in teaching the elements of geometry. It was, perhaps, chiefly the study of 

 polyhedra that brought about the birth of geometry, and that subject shows at 

 once the power that geometry has of adding to a student's knowledge of space. 

 It may be mentioned that Desargues's name is not " Desargue " (p. 107). 



Philip E. B. Jourdain. 



A First Course in the Calculus. Part I : " Powers of .1." By William P. 

 Milne, M.A., D.Sc, Mathematical Master, Clifton College, Bristol ; 

 formerly Examiner in Mathematics, University of St. Andrews ; late 

 Scholar of Clare College, Cambridge ; and G. J. B. Westcott, M.A., Head 

 of the Department of Mathematics and Physics, Bristol Grammar School ; 

 formerly Scholar of Queen's College, Oxford, andUniversity Mathematical 

 Exhibitioner. [Pp. xx + 196.] (London: G. Bell & Sons, Ltd., 1918. 

 Price 3-y. 6d.) 



