5/8 



NA TURE 



[April 23, 190^ 



ever may be added, however, should be carefully 

 selected, having regard to modern conditions. Very 

 little additional matter will be taken from Euclid. We 

 think the book would have been improved by a chapter 

 on the solution of right-angled triangles, using the 

 trigonometrical tables given at the end of the volume, 

 results obtained graphically being verified by calcula- 

 tion. In subsequent work graphical and numerical 

 computation would go on side by side. There are 

 calculations relating to right-angled triangles quite as 

 important as that of Euc. i. 47, and the drawing class 

 seems to be the proper place in which to teach them 

 to beginners. What better examples than the trigo- 

 nometrical functions are to be found of ratio and pro- 

 portion ? Consider what a satisfaction it must be to a 

 boy to find himself in possession of and familiar witli 

 this powerful modern weapon. And, moreover, the 

 knowledge gained is of the utmost importance. In 

 connection with this part of the subject, the radian 

 measure of an angle should not be neglected; it is 

 very desirable that a student should be trained so as to 

 be able to think in radians as well as in right angles 

 and degrees. 



Next, a course seems very incomplete without some 

 notion of projection, and how lengths and angles in 

 three dimensional space are measured and represented. 

 Following the author's plan, the principles of Euclid 

 xi. would be inculcated along with exercises in de- 

 scriptive geometry, involving quantitative measure- 

 ment. This can be rendered quite interesting. 



And lastly, one of the most fruitful additions that 

 could possibly be made would be to introduce the idea 

 of a vector, giving the triangle or parallelogram law, 

 with some of its consequences. Geometry is essentially 

 a vector subject, and an early knowledge of vectors 

 would have far-reaching effects. 



In the " Geometry " by Mr. S. O. Andrew we have 

 another text-book in which exercises in drawing and 

 deductive reasoning are carried on together, so that 

 the student acquires some practical acquaintance with 

 the subject-matter. But the work is not based suffici- 

 ently on accurate quantitative measurement, and the 

 author seems satisfied with drawing of an inferior 

 quality. We find no description of what sort of scales 

 • ire suitable for measuring lengths. There is no in- 

 formation as to the manner of using and testing 

 straight edges and squares. In the absence of any 

 guidance to the contrary, the student is sure to use 

 soft blunt pencils. There are no numerical answers 

 given to any of the exercises. 



But a teacher using the book could, to some extent, 

 supply these omissions, and would find the volume very 

 serviceable; it is the result of practical teaching ex- 

 perience. It covers substantially the same ground as 

 the book previously considered, with a chapter on solid 

 geometry and orthographic projection. Loci and 

 graphs are introduced, and trigonometrical tables are 

 given and explained, but are made very little use of in 

 the text. 



The text-books of Messrs. Allcock, Baker and 



Bourne, Lachlan and Fletcher, and Petch are alike in 



having for their main object the development of a 



system of formal geometry on Euclidean lines. The 



SO. 1747, VOL. 67] 



changes they introduce with the object of improving 

 geometrical teaching are such alterations as the re- 

 vision of the definitions and axioms, the rearrangement 

 and regrouping of the propositions, the employment of 

 arithmetic, algebra, loci, &c. Euclid's form of reason- 

 ing has in all cases been retained. Experimental geo- 

 metry is not made prominent; it is brought in rather 

 in connection with the examples which follow the pro- 

 positions. 



As it appears to us, these books are not sufficiently 

 free of the Euclidean tradition to make them suitable 

 for boys at school. They are more fitted for subsequent 

 study. The presentation of the substance of Euclid i. 

 by Allcock is excellent, and may well replace Euclid 

 when the time comes for taking up the philosophy of 

 the subject. In the volumes by Messrs. Baker and 

 Bourne there is an introductory chapter on experi- 

 mental geometry, extending over twenty pages, com- 

 prised of nearly two hundred exercises, ranging 

 over the whole subject up to the end of Euclid 

 vi., and intended to make the student practically 

 acquainted with the ground to be subsequently covered. 

 This chapter is a valuable and extremely suggestive 

 one, so far as it goes; if the material had been set out 

 in greater detail, and worked in alone with the de- 

 ductive geometry and accorded equal importance with 

 the latter, a geometry quite suitable for youths would 

 have been the result. As text-books of formal geometry 

 these manuals by Messrs. Baker and Bourne can be 

 strongly recommended. They cover the ground usually 

 studied, including Euclid xi., and there are chapters on 

 graphs and mensuration formula?. They are beauti- 

 fully printed and arranged, and contain many practical 

 exercises. 



Mr. Deakin's Euclid is written on strictly orthodox 

 lines ; it contains some useful notes and exercises by the 

 author. The only evidence of any influence of the re- 

 form movement is at the end of Euclid vi., where an 

 abstract is given of the recommendations of the com- 

 mittee of the Mathematical Association of 1902. 



The little book on " Graphical Algebra " by Mr. Hall 

 is intended to accompany the well-known " Elementary 

 Algebra " of Messrs. Hall and Knight. It is concerned 

 with graphs and squared paper work, and illustrates 

 some part of the service which geometry is rendering 

 to algebra. Some of the exampies are evidently taken 

 from previous publications, though the author forgets 

 to acknowledge their source. y. Harrison. 



SYSTEMATIC PETROGRAPHY. 

 Quantitative Classification of Igneous Rocks Based 01, 

 Chemical and Mineral Characters, with a Systematic 

 Nomenclature. By Whitman Cross, Joseph P. 

 Iddings, Louis V. Pirsson, and Henry S. Washing- 

 ton, with an Introductory Review of the Develop- 

 ment of Systematic Petrography in the Nineteenth 

 Century, by Whitman Cross. Pp. x + 286. 

 (Chicago: the University of Chicago Press; 

 London: Wm. Wesley and Co., 1903.) Price Si', 

 net. 

 T) V the very first page this book is defined as dealing 

 -L* with " the science of petrography." Petrology 

 is "the broad science or treatise of rocks"; petro- 



