; r 6 



NA TURE 



[February 6, 1908 



(2) This book is written for schoolboys who have 

 had a preliminary Irainingf in practical geometry, and 

 is devoted ahiiost entirely to theoretical work. The 

 authors are not very fortunate in the first few pages, 

 but when once the reader is fairly started, he will 

 find very little to which he can take exception, pro- 

 vided he is in sympathy with the general arrange- 

 ment of the book. The authors adopt a con- 

 servative point of view, and give a very strong 

 Euclidean flavour to their treatise, but they show 

 themselves capable of appreciating the chief lessons 

 to be learnt from recent experiments in geometrical 

 teaching. Hypothetical constructions are allowed if 

 it can be proved that the construction is possible. 

 The theory of parallels based on Playfair's axiom is 

 deferred until after the principal properties of con- 

 gruent triangles have been proved. The book covers 

 the substance of Euclid i. to vi. ; those of Euclid's 

 theorems which are not included in the text are set 

 as riders together with a large number of well-chosen 

 examples. The treatise is very complete within the 

 limits chosen, and contains sections on loci, geo- 

 metrical dissections, Jhe nine-point circle, inscribed 

 and escribed circles, Ceva's theorems, &c. A teacher 

 who has conservative views could, on the whole, hardly 

 wish for a better text-book. 



(3) The treatise on analytical conies in this series 

 was undertaken by Mr. R. \A". H. T. Hudson, and 

 Miss Scott, while pursuing her own plan, has had 

 at her disposal the outline he drew up before his 

 death. The book will prove interesting to the teacher 

 on account of the extreme novelty of the arrangement. 

 The author claims to have shown deference to 

 existing conventions, but it is not so easy to see where 

 self-restraint has been exercised. Apart from the pro- 

 fessed innovation of introducing line-coordinates 

 concurrently with point-coordinates from the very first, 

 we have tlie novelty that the circle is taken as a special 

 case of the ellipse, change of a.xes is deferred until 

 necessary for the tracing of conies, and so on. 



The chief fault of the treatise is probablv that the 

 arrangement is far too confused. Properties of the 

 circle are spread over three or four chapters in various 

 parts of the book, interspersed with theorems on 

 conies and straight lines, which theorems are in their 

 turn introduced apparently incidentallv, then re- 

 capitulated further on, only to be extended in a still 

 later chapter. It seems very doubtful whether a pupil 

 brought up on this method would be able in any 

 way to systematise his knowledge. 



Introductory remarks and definitions are apt to be 

 a little obscure, but this is amply compensated by 

 excellent diagrams and very intelligible examples 

 worked at full length. It is a pity that no answers 

 to the exercises are given. 



(4) This book is somewhat on the lines of Casev's 

 sequel to Euclid, and covers a good deal of the same 

 ground. It is in many ways an improvement on 

 that standard treatise, and will probably replace it 

 with those students who are just beginning an honours 

 course in mathematics. The chief criticism we have 

 to make is that the contents are of too miscellaneous 

 a character; no one subject is treated quite fully 



NH. 1Q97, VOL. 77] 



enough, and the reader is led from one idea to another 

 with almost bewildering rapidity. Perhaps some im- 

 provement might have been effected by omitting the 

 chapter on "recent geometry," which contains very 

 little that is new except the nomenclature, and treat- 

 ing more important subjects at greater length. 



In a book of this kind the chief danger lies in the 

 insertion of artificial geometrical proofs of theorems- 

 best established by analytical or other methods. The 

 author is to be congratulated in having avoided this- 

 danger, on the whole, with marked success, though 

 perhaps it would be better to solve Fermat's problem 

 and other examples in chapter xi. by the more instruc- 

 tive methods of chapter xii. The reasoning adopted 

 is of a simple character, and in many cases alternative 

 proofs of equal elegance and simplicity are given. 

 There is a plentiful supply of well-chosen exercises; 

 in many cases concise but useful hints arc given n\ 

 the text for their solution, and a key to the remainder 

 of the examples is promised. The book will prove 

 very inspiring to the beginner, and give pleasure tO' 

 the more advanced reader. 



(5) Mr. Martin's book is intended for readers wh» 

 have a fair knowledge of differential calculus and are 

 beginning integral, and it covers the more elementary 

 portions of uniplanar dynamics of the particle and 

 rigid body. There is room for such a treatise, but 

 the present one is not entirely satisfactory. Much 

 of it is carelessly worded, e.^. a movement is called 

 the motion of a " point " and of a " particle " in 

 the same section, the definition of a radian is unin- 

 telligible, &-c. Some of the proofs, as in the case of 

 normal acceleration, are far too cumbrous, while others 

 are hardly rigid — an instance of this is the absence 

 of any mention of D'.Membert's principle or a sub- 

 stitute therefor. The 420 examples will be useful, 

 though no answers are given. 



(6) In this treatise an attempt is made to cover the 

 ground very thoroughly; for instance, three distinct 

 proofs are given of the resultant of two parallel forces 

 and the three requisites of a good balance are dis- 

 cussed, while chapters on work and energy, frame- 

 works, virtual work, elasticit\-, &c., are given. The 

 object aimed at is to include all that part of statics 

 which can be profitably discussed without the use 

 of the calculus. The result is a book which every 

 teacher should possess ; it contains all the bookwork 

 he is likely to want and more, while it is a most 

 useful mine of excellent examples. It is more doubtful 

 whether the book is equally suited to class use; it 

 is hardly simple enough for beginners, and the prac- 

 tical experiments are not described in sufllcicnt detail 

 to be of much use for such a purpose. Readers wh(» 

 have Borchardt and Perrott's " Trigonometry " will 

 have a very fair idea of the style and aim of this 

 " Statics." 



(7) -Mr. Hawkins's book is rather attractive; for 

 a boy who was learning trigonometry in order td 

 become a surveyor it would be ideal. It may b(- 

 doubted, however, whether the ordinary pupil will 

 take much interest in so many technical details of 

 land-measurement, even granting that practical appli- 

 cations have a fascination for beginners. With a 



