NA TURE 



193 



THURSDAY, JUNE 30, 1904. 



MATWEUkllCAL BOOKS. 

 (1) .4n Introduction to the Study of Geometry. By 



A. J. Pressland, M.A., F.R.S.E. Pp. 40. (London: 



Rivingtons, 1904.) Price 15. 

 <2) Elementary Geometry. By Cecil Hawkins, M.A. 



Parti. Pp. viii+i6s. Part ii. Pp. 166-296. 



(London: Blacl<ie and Son, Ltd., 1904) Price 2^. 



eacli volume. 

 <3) Geometry for Technical Students. By E. H. 



Sprag-ue, .\ssoc.M.Inst.C.E. Pp. viii + 6o. (London: 



Crosby Lockwood and Son, 1904.) Price is. 



(4) Graphs and Imaginaries. By J. G. Hamilton, B.A., 

 and F. Kettle, B.A. Pp. 42. (London : Edward 

 Arnold, 1904.) Price is. 6d. 



(5) Five-figure Tables of Mathematical Functions. By 

 John Borthwick Dale, M.A. Pp. xvi + 92. (London : 

 Edward Arnold, 1903.) Price 3s. 6d. net. 



<6) Logarithms for Beginners. By Charles N. Pick- 

 worth. Pp. 47. (London: Whittaker and Co., 

 1904.) Price IS. 

 (7) Calculating Tables. By Dr. H. Zimmermann. 

 Translated from the German by L. Descroiz. Pp. 

 xxxi + 204. (London: Asher and Co., 1904.) Price 

 6s. net. 

 (i) TV /I R. Pressland adopts the heuristic method in 

 iVl this course of experimental geometry for 

 beginners. The first exercises only require the use of 

 a pencil and a graduated straight edge cut from ruled 

 school paper. With these the boy draws triangles and 

 quadrilaterals, bisects lines and erects perpendiculars. 

 Symmetrical figures, such as the square, rhombus, 

 kite, Sec, are made by paper folding. 



A ruler with two edges decimally subdivided into 

 inches and centimetres is then introduced, together 

 with two set squares and a protractor. Parallel and 

 perpendicular lines are now readily drawn, and the 

 work becomes quantitative, lengths, angles, and also 

 areas being measured. 



The pupil is next required to use compasses, and be- 

 comes acquainted with some properties of circles. 

 Two or three pages are then given to proportion and 

 graphic arithmetic. The book concludes with a set of 

 examples in practical geometry, and a table of general 

 properties of figures, only partially enunciated, and 

 intended to be completed by the pupil himself from 

 observation and discovery in the course of his ex- 

 perimental work. 



The aim has been to train the hand and eye, to 

 create interest, and to make the boy acquainted with 

 the groundwork of the subject. Deductive geometry 

 is not introduced, nor is there any attempt at a logical 

 >equence. The author states that many features of his 

 book are due to his experience as an inspector of 

 schools in Canton Zurich. The course seems a good 

 one within its limited sphere, but the experimental 

 work might with advantage have been somewhat more 

 extended and varied. 



(2) In so far as the subject is dealt with, Mr. 

 Hawkins's geometrical course is a very good one. It 

 NO. 1809, VOL. 70] 



is confined to plane geometry, part i. relating to 

 simple rectilineal figures and the circle, and part ii. 

 dealing more particularly with areas, proportion, 

 similar figures, and further properties of the circle. 

 The author follows the reform movement, the proposi- 

 tions required for the Previous Examination under the 

 new Cambridge syllabus being marked with an 

 asterisk. A prominent feature of the work is the very 

 large number of examples which are given, extending 

 to upwards of 1500, of varied character, and affording 

 ample choice for practice in experimental, practical, 

 and theoretical work. The only thing lacking is a 

 collection of the numerical answers. The examples are 

 appended to the successive propositions, and in addi- 

 tion, at the end of each volume is a set of miscellaneous 

 examples, carefully graduated, and covering the whole 

 of the previous ground. 



By omitting- any reference to simple functions of 

 angles, the author has deprived himself and his readers 

 of a very instructive and fruitful field for examples in 

 ratio and proportion, and of elementary calculations of 

 right angled triangles; and the omission of solid 

 geometry leaves the course incomplete. But the 

 general scheme is well planned and developed, and the 

 book cannot fail to give satisfaction to many readers. 



(3) The plan of Mr. Sprague's book is based largely 

 on experience gained by the author in the teaching of 

 engineering students for the Chinese Government, and 

 is intended more especially for those who take up 

 geometry as part of their professional training as 

 engineers. 



The more important fundamental properties of plane 

 figures, including the triangle and circle, and of 

 simple geometrical solids are established by deduc- 

 tive methods, comprised in forty-eight propositions 

 with corollaries, and accompanied by a few exercises, 

 and the book concludes with fourteen problems in 

 practical plane geometry. 



In comparison with many recent manuals, this text- 

 book seems to be deficient. It will not satisfy those 

 who require a good theoretical course of elementary 

 geometry, nor yet others who are more interested in the 

 e.xperimental and practical development of the subject, 

 and the work is not likely to be generally adopted by 

 any class of students. 



(4) In graphing the parabola y = ax^+bx + c, the ob- 

 ject being to solve the quadratic equation ax' + bx+c = o, 

 the authors show that when the roots are unreal, say 

 a + 1/3, the points a±/3 lie on a second parabola. The 

 latter is easily drawn, being the former turned into 

 a new position, and a and & can then be measured. 

 In like manner the coordinates of the imaginary points 

 w-here an external line cuts a parabola are shown to 

 be readily found by making use of the properties of a 

 certain companion or '' shadow " parabola. 



The authors then give the " circle method " of 

 solving a quadratic, and by means of a " shadow " 

 circle extend the solution to the case of unreal roots. 

 Further examples of shadow circles are given, applied 

 to the imaginary points of Intersection or contact of 

 lines and circles. The constructions are curious and 

 interesting, but of little or no value to young students, 

 whose time should not be employed on them. 



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