738 



NA TURE 



[June 



1922 



vincing narrative." Prof. Jessop's " Elementary 

 Analysis " is so short that on occasion it seems to 

 err in the direction of the second extreme. Yet it is 

 a very clear and useful account of what the average 

 student needs to know if he is to benefit by further 

 work, where the calculus and the methods of analytical 

 geometry are required. University courses continue 

 to become more and more inclusive, and many students 

 will be grateful to the author for the brief presentation 

 he offers them. The straight line and circle are dis- 

 cussed in about fifty pages, differential and integral cal- 

 culus occupy about a hundred, and about twenty pages 

 are devoted to the general equation of the second degree 

 and the properties of the ellipse, parabola, and hyper- 

 bola. The last chapter is almost tabloid in character. 



(3) This, like the other publications of the University 

 Tutorial Press, seems to be just the book required for 

 " learning up " for an examination. Nothing, appar- 

 ently, is left to chance, and the student who has 

 mastered its contents should be able to defy any 

 examiner to do his worst. On reading this book one 

 feels inclined to exclaim : Did I know all this when I 

 passed the matriculation examination ? For a class 

 text-book a shorter book with more emphasis on 

 principles and less on the examination spectre would 

 be preferable : but for private students — and they are 

 more numerous than many suppose — this book has 

 its obvious advantages. In the second edition chapters 

 have been added on indices and logarithms. 



(4) and (5) Elementary mathematics is gradually 

 being released from the burden of manipulative skill 

 and the bogey of the examiner. Students are no 

 longer expected to do jig-saw puzzles with mosaics 

 of simple, square, and double brackets, with pluses 

 and minuses peppered about like the charges in the 

 modern chemist's atom. This release is well symbolised 

 in Messrs. Durell and Arnold's two books on algebra, 

 written for American schools. Each book represents 

 a year's course, developed with skill and knowledge 

 of pedagogical methods. In the first book each chapter 

 is divided into two parts, the first for the beginner in 

 his first half-year, the second for revision during the 

 second half-year. It is doubtful, however, whether 

 revision by complete repetition is an ideal educational 

 process. The second book is in two parts, the second 

 being " a reservoir of extra work for bright pupils." 

 The complete course is very suitable for the standard 

 of matriculation. 



(6) Messrs. Durell and Arnold's " Geometry " con- 

 tains a full and competent account of all the pure 

 geometry that is required by pupils of higher schools. 

 The sequence is reasonable, and the treatment is 

 practical although the book is essentially a course on 

 formal geometry. 



NO. 2745, VOL. 109] 



The subject-matter comprises the usual plane 

 geometry and a rather extensive course on solid 

 geometry. The sphere is dealt with in some detail, 

 and an interesting feature of the book is the brief 

 account of the properties of spherical triangles as 

 regards congruence and area. This is a very desirable 

 innovation that English books might copy with 

 advantage. Spherical figures are of importance in 

 many branches of knowledge and the student must 

 somehow pick up a little knowledge about them : but, 

 like hydrostatics, spherical trigonometry is a step- 

 child of the modern mathematical teacher. 



As the authors use algebraic symbols, it would have 

 been an advantage to introduce numerical trigono- 

 metrical methods. The historical sketches are rather 

 dull. 



(7) Books on practical geometry usually give con- 

 structions for a number of geometrical exercises with 

 little justification, if any. Books on formal geometry • 

 aim at giving a systematic and logical course on the 

 subject : occasionally a lapse into real life takes 

 place, but the main object is to build up a structure 

 of reasoning based on a few fundamental notions and 

 postulates. Where the theoretical and practical are 

 combined, one usually has the two more or less dis- 

 sociated. Mr. Foster's idea is to combine the advan- 

 tages of both the practical and the formal by working 

 them into an organic whole ; he tries to inculcate the 

 geometry of the class-room by means of the observa- 

 tion of outdoor and home life. He has achieved con- 

 siderable success. The separate formal propositions 

 are reduced to a very small number — and this is 

 an advantage. In the two parts already issued 

 the ordinary plane geometry is covered, up to and 

 including proportionality and similar figures. A 

 third part is promised on solid geometry. 



(8) Like Mr. Foster, Messrs. Beckett and Robinson 

 make it their aim to combine deductive with practical 

 geometry. They commence with a number of practical 

 constructions without proofs, even such complicated 

 problems as the drawing of common tangents to two 

 unequal circles. The student is then introduced to 

 notions of area and to solid figures, thus completing the 

 preliminary section. Section I. gives the formal 

 geometry of triangles and parallelograms, with exercises 

 from life, mechanics, and physics. In Section II. are 

 given the properties of circles in formal order, while 

 Section III. deals with areas and Pythagoras 's theorem. 

 Cartesian co-ordinates are introduced, contours ex- 

 plained, and a brief course on numerical trigonometry 

 completes the first part. It will be interesting to see 

 the second part. Pythagoras's theorem is too late in the 

 book : the student usually feels quite excited about 

 this theorem, and the sooner he gets excited about 



