June i8, 1903] 



NATURE 



147 



least I have not erred in supposing that a serious trea- 

 tise on these topics is nothing else than the inevitable 

 complement and conclusion of the slow process by which 

 man has brought under the domain of science every 

 group of attainable phenomena in turn — every group 

 save this." 



In the belief that this book marks an epoch in 

 the history of psychical science, and that it will ulti- 

 mately react with beneficial effect on the progress and 

 enlargement of the scope of science generally, 1 

 venture to introduce this life-work of my friend to the 

 readers of Nature, or at least to such of them as are 

 not already familiar with the subject. 



Oliver Lodge. 



SCHOOL GEOMETRY REFORM. 

 A School Geometry. Parts i. and ii. By H. S. 



Hall, M.A., and F. H. Stevens, M.A. Pp. x + 140. 



(London: Macmillan and Co., Ltd., 1903.) Price 



IS. 6d. 

 Experimental and Theoretical Course of Geometry. 



By A. T. Warren, M.A. Pp. viii + 248. (Oxford: 



the Clarendon Press, 1903.) Price 2s. 

 Elementary Geometry. By Frank R. Barrell, M.A., 



B.Sc. Section i., part i., pp. xi +116. Price is. Section 



i., part ii., pp. vii+ 117 to 168. Price is. (London : 



Longmans, Green and Co., 1903.) 

 Solid Geometry. By Dr. Franz Hocevar. Translated 



and Adapted by C. Godfrey, M.A., and E. A. Price, 



B.A. Pp. vii + 80. (London : Adam and Charles 



Black, 1903.) 



A PERSON may be a Cambridge Wrangler, and yet 

 unable to make a simple graphical construction 

 with accuracy. The ordinary schoolboy's knowledge 

 of practical geometry is generally worthless or nil, and 

 his knowledge of pure geometry, the result of his pre- 

 mature encounter with Euclid, is of like character. 



But this state of affairs is being rapidly changed. 

 As Messrs. Hall and Stevens say in the first volume 

 of their new geometry, " The working of examples 

 should be made as important a part of a lesson 

 in geometry as it is so considered in arithmetic and 

 algebra." 



The book contains an excellent collection of easy 

 graphical and deductive exercises, many of the 

 examples requiring numerical answers. The latter are 

 given at the end. A boy working through this course 

 should acquire a working knowledge of geometry, and 

 a fair insight into the methods of deductive logic. 



The volume contains the substance of Euclid book i., 

 and is based on the recommendations of the Mathe- 

 matical Association ; the sequence of Euclid is in the 

 main adhered to. There are two parts, the latter 

 dealing with areas. In this the experimental course is 

 incorporated with the deductive exercises, and assigned 

 equal importance with the latter. This is a good 

 feature, and is to be continued in a further volume 

 which the authors have in preparation. In the present 

 case, it seems to be a defect that the plan has not been 

 carried out to the same, or even a greater, extent in 

 |)art i., which is concerned with lines, angles, and 

 rectilineal figures. Here it would appear to be 

 NO. 1755, VOL. 68] 



especially necessary to make the experimental course 

 predominate. But the subject of school geometry is in 

 a state of transition, and the authors have probably 

 thought it well to proceed cautiously. 



Mr. Warren's volume is also based on the report 

 of the Committee of the Mathematical Association. 

 The course includes the fundamental properties of the 

 triangle and circle. Ratio and proportion, similar 

 figures, and polygons are likewise considered. The 

 experimental treatment occupies the first half of the 

 book, and in the second half the same ground is 

 covered, the propositions being formally established 

 by deduction. 



The two volumes by Mr. Barrell comprise the first of 

 three sections of a new school geometry which, when 

 complete, will extend to Euclid xi. and the mensuration 

 of the simple geometrical solids. It is written in ac- 

 cordance with the new syllabus of the Cambridge Local 

 Examinations, and the report of the Mathematical 

 Association. Part i. is intended to take the place of 

 Euclid, book i. Part ii. corresponds with Euclid, book 

 iii., 1-34, and also includes a portion of book iv. In 

 the treatment adopted, the experimental and practical 

 course is worked in along with the deductive geometry, 

 and is always made subordinate to the latter. We 

 should like to see the demonstrative geometry relatively- 

 less prominent. A feature to be noticed is that the 

 author gives three meanings of a plane angle, in the 

 last of which the angle is regarded as the plane space 

 swept out by a line of indefinite length (one way) turn- 

 ing about one end ; the amount of turning is not the 

 angle, but the measure of its magnitude. The author 

 is right in stating that this conception is implied in 

 many of Euclid's phrases. The numerical answers of 

 lengths and areas are given to three significant figures, 

 and of angles to the nearest ten minutes. In the 

 latter case decimals of a degree would perhaps have 

 been preferable. 



The actual personal use of mathematical instruments 

 for graphical computations is probably largely foreign 

 to many of the authors of the new text-books, and the 

 treatment suffers on this account. There must be 

 much future development before any text-book can 

 be allowed to become crystallised. 



Now that the study of pure geometry is to include 

 numerical as well as graphical computations, it may 

 become necessary, and it is certainly very desirable,^ 

 to introduce simple tables of functions of angles so as- 

 to be able to solve right angled triangles completely, 

 instead of being restricted as at present to the property 

 of complementary angles and the use of Euclid i., 47. 



The "Solid Geometry" by Dr. Hocevar will illustrate 

 how this branch of the subject is presented to youths 

 in Germany. Chapters i. and ii. deal with the pro- 

 perties of the line and plane in space, and the solid 

 angle, but in a much less formal manner than is the 

 case in Euclid xi. The remaining chapters relate to 

 the properties and mensuration of the prism, cylinder, 

 pyramid, cone, sphere and regular polyhedra. Exercises 

 are provided in great variety, chiefly of the numerical 

 type, and all necessary answers are collected at the 

 end of the volume, where the reader will also find a 

 useful index. 



