698 



NATURE 



[February 27, igi_ 



with some truth, that the majority of tearhers 

 aim at impressing a set of facts upon their pupils 

 rather than training them how to attack and 

 discuss mathematical problems. Unfortunately, 

 there are few teachers who are free agents ; the 

 requirements of the various examining boards 

 must first be satisfied before personal individuality 

 can be freely exercised, and much of the best work 

 of the first-rate teacher is of a character that 

 examinations can scarcely test. At the same time, 

 mathematical teachers should undoubtedly know 

 something of the science of teaching, and cannot 

 fail to profit by a knowledge of the experience of 

 others. In the present volume, there is much of 

 real interest and value. After preliminary general 

 discussions, the author examines in great detail the 

 theory of geometrical teaching and somewhat 

 In-iefly the elements of a suitable course in algebra 

 and trigonometry. Such a work as this should 

 find a place in the common-rocjm libraries of our 

 secondary schools. 



(4) This is a continuation of the author's 

 former treatise on algebra for secondary schools. 

 It opens with the binomial theorem and includes 

 all that usually finds a place in an advanced school 

 course. Among the chief features of the book 

 may be noted an excellent chapter on complex 

 quantity ; the geometry of vectors is developed, 

 and the use of trigonometric functions renders the 

 account reasonably complete. By introducing the 

 notation of the calculus, the treatment of limits 

 is simplified and the usual applications in the theory 

 of equations become possible. The work on con- 

 tinued fractions is put rather more briefly than 

 usual, but nothing of importance for any ordinary 

 purpose has been omitted. The volume has an 

 attractive appearance, the examples are really 

 good, and the essay questions at the end will be 

 of great assistance to scholarship candidates. 



(5) The study of non-Euclidean geometry has, 

 till recently, attracted the attention only of the 

 specialist ; no doubt this has been due principally 

 to the general belief that the difficulties were so 

 considerable, the philosophical problems so in- 

 tricate, and the subject so contrary to ordinary 

 experience that the ordinary mathematician vvoui3, 

 without prolonged study, make little of it. Time, 

 however, invariably lowers the levels and extends 

 the boundaries of the territory accessible to 

 ordinary students. There are now a number of 

 elementary text-books which make its pursuit a 

 comparatively easy task. There are two valuable 

 studies by Mr. Frankland — "The Theories of 

 Parallelism " and " Euclid I., with a Commentary " 

 — there is a primer by Prof. Manning, a more 

 elaborate treatise By Prof. Coolidge, and, for those 



NO. 2261, VOL. go] 



who read German, the works of Killing, Lieb- 

 mann, Hilbert, Vahlen, etc. 



The translation of Prof. Bonola's valuable 

 critical and historical summary will be of the 

 greatest assistance to students. The book opens 

 with an account of the attempts to prove Euclid's 

 parallel postulate which were made from the time 

 of the Greek geometers down to the sevente.enth 

 century. The next section deals with the period 

 when men were first beginning to inquire whether 

 a form of geometry could exist independently of 

 this postulate. This work is associated with the 

 names of Saccheri, Lambert, 'Wolfang Bolyai and 

 others ; but it was not until the time of Gauss, 

 Taurinus, Lobatschewsky, and Johann Bolyai that 

 the foundations of non-Euciidean geometry were 

 securely laid. .\ most interesting sketch is given 

 of the growth of thought in this period. The con- 

 cluding chapter discusses the later work of 

 Riemann, Helmholtz, Lie, Cayley, Klein, etc. 

 There are five appendices ; these deal with (a) the 

 fundamental principles of statics; (b) Clifford's 

 parallels ; (c) constructions ; (d) the independence 

 of projective geometry ; (e) a method of exhibiting 

 the impossibility of proving Euclid's postulate by 

 a consideration of the analogous geometry of a 

 system of circles orthogonal to a fixed circle. This 

 last appendix, which is due to Prof. Carslaw and 

 is based on Wellstein's work, establishes, by an 

 elementary and elegant method, a number of in- 

 teresting theorems in hyperbolic geometry. 



(6) These notes on the calculus, drawn up for 

 science and engineering students, are intended to 

 supplement the earlier parts of the ordinary text- 

 books. The first chapter gives the analytical 

 geometry of the straight line, the second illus- 

 trates the meaning' of differentiation by examples 

 from physics and geometry ; in the next three the 

 rules for differentiations are given ; and the last 

 two, after a few pages on the geometry of the 

 conic, give an account of integration and its 

 applications. 



01: R BOOKSHELF. 

 Kausalc und konditionale Weltanschauung. 



By Max Verworn. Pp. 46. (Jena : Gustav 



Fischer, 1912.) Price i mark. 

 Even when one profoundly disagrees, it is always 

 a pleasure to listen to Prof. Max Verworn, for 

 he has clear-cut convictions which he states vividly 

 and witli enthusiasm. The present essay is an 

 exposition of "conditionism " as contrasted with 

 "causalism," and it deals hard blows at vitalism, 

 dualism, entelechy, free will, and other naive and 

 uncritical assumptions, as Verworn thinks. 



It may be of interest to state the five proposi- 

 tions of conditionism: — (i) There are no isolated 



