NA TURE 



433 



THURSDAY, MARCH lo, \\ 



TWO TEXT-BOOKS OF ELEMENTARY 

 GEOMETRY. 

 Geometry for Beginners. By G. M. Minchin, M.A., 

 F.R.S. Pp. xii + I02. (Oxford : at the Clarendon 

 Press, 1898.) 

 Euclid'' s Elements of Geometry, Books I. and II. Edited 

 for the use of Schools, by Charles Smith, M.A., and 

 Sophie Bryant, D.Sc. Pp. viii + 160. (London : Mac- 

 millan and Co., Ltd., 1897.) 



THE appearance of these two little books shows that 

 practical teachers have not yet agreed upon the 

 best method of teaching elementary geometry. That 

 this should be so is by no means a matter for regret ; 

 in the course of the controversy each party learns some- 

 thing from the criticism of the other ; examination 

 papers tend to become less stereotyped, and better 

 adapted to test the student's real knowledge of the 

 subject ; while an intelligent teacher is more and more 

 able to assert his right of freedom in giving geometrical 

 instruction according to the method which, after a fair 

 trial, he finds to be most efficient. 



Prof. Minchin's book is a very favourable specimen of 

 the methods of the reforming party. It is really what it 

 professes to be, a book for beginners ; it is obviously. the 

 result of long experience, and there is no reason to be 

 surprised at the author's statement that the book has 

 been used with boys of eight years with very great 

 success. A very welcome feature, which might be 

 adopted by all writers of introductory text-books, is the 

 description of the graduated scale, the compass, the 

 protractor, and set-squares, and of the way in which they 

 are used. Parallel rulers are also described ; it would 

 have been well, in our opinion, to add that, while much 

 may be learnt by handling a pair of parallel rulers, 

 they are of little use for practical purposes, and that, 

 generally speaking, set-squares should be used for 

 drawing parallels. It is probably an oversight that the 

 use of set-squares for drawing perpendiculars has not 

 been explained. We thoroughly agree with Prof. Minchin 

 in the opinion that the careful construction of figures, by 

 means of the proper instruments, should be insisted 

 upon for the sake of training the eye and hand of the 

 pupil ; and it may be added, that this exercise invariably 

 helps to maintain a boy's interest in the problem or 

 theorem upon which he happens to be engaged. 



The choice of propositions is very judicious ; they 

 include the congruence of triangles, the elementary 

 theory of parallels, exclusive of the so-called "axiom," 

 properties of parallelograms, easy theorems about areas, 

 and the theorem of Pythagoras. Besides this there are 

 a few miscellaneous propositions, as, for instance, that 

 the angle in a semicircle is a right angle. Definitions, 

 we are glad to see, are introduced when they are wanted, 

 and not before ; and when necftssary they are clearly 

 explained and illustrated. The exercises are simple and 

 well-chosen : it would be easy to add to their number 

 and variety, and this would be an advantage to the 

 book. 



Of course there are some points which provoke 

 NO. 1480, VOL. 57] 



criticism ; but in offering any remarks upon them it 

 is necessary to bear in mind the intention of the author ; 

 thus, for instance, however much the orthodox may 

 protest, we think it is quite legitimate to define a straight 

 line as the shortest distance between two points ; on the 

 other hand, it seems to us that the statement " a point is 

 the smallest dot that can be made or imagined" is 

 distinctly misleading, and that it is quite possible to give 

 a more accurate idea of what is meant. Then again 

 there is no objection, at this stage, to saying that a plane 

 is a perfectly flat surface : but the top of a table, a 

 school-slate, or the surface of water in a basin is a better 

 illustration than a sheet of paper. It would also be a 

 good thing to point out the reverse order of procedure, 

 where a surface appears as the boundary of a solid, 

 a line as that of a surface, and a point as that of a line, 

 or the crossing of two lines. A playing ball painted in 

 different colours makes a [good illustration ; so does a 

 cube, a ruler, or a piece of india-rubber of the ordinary 

 shape. 



In some cases only one figure has been given where 

 there ought to have been two or more : for example, in 

 propositions L, S and T ; and in the problem on p. 79 

 it might be pointed out that R may be on either side 

 of PQ, but that the construction fails if we take PR 

 equal to PO, and try to put R on the same side as O. 



In cases where circles are used in the construction of 

 figures, it is properly observed that it is not always 

 necessary to draw the complete circles. We would go 

 a little further than this, and in the later propositions, 

 at any rate, introduce in the figures quite small arcs, 

 such as those made by a practical draughtsman ; this 

 might be done, for instance, on pp. 78, 79. As the 

 student is supposed to use a pencil compass, there would 

 be four small arcs to indicate on p. 79. 



It would be a great mistake to overload a work of 

 this kind ; still it might be thought worth while, and 

 would not be inconsistent with the plan of the book, to 

 give the construction for dividing a line into a given 

 number of equal parts, and that for reducing a recti- 

 lineal polygon to a rectangle of equal area. The con- 

 struction of a regular hexagon and a regular octagon 

 might perhaps be included, either as propositions or 

 examples ; and it would be well to add some examples, 

 say, of plotting quadrilateral or pentagonal fields from 

 given data, or of drawing such figures as, for instance, 

 a square with an outward semicircle on each side of it 

 as diameter. 



These suggestions are made in the confident hope that 

 there will soon be a demand for another edition of Prof. 

 Minchin's excellent little book. In the hands of a com- 

 petent teacher it cannot fail to be successful ; and it 

 will do much, we hope, to convince the average parent 

 and schoolmaster that geometry is not dull mechanical 

 drudgery, and that its principles and results may be 

 comprehended by means of ordinary common-sense. 

 Its adoption as an introductory course will not hinder, 

 but greatly help, a more rigorous study of the subject 

 afterwards ; even the most questionable part of the book, 

 the sham proof (for such it is) of the sum of the angles 

 of a triangle being equal to two right angles, need not 

 be a permanent stumbling-block in the pupil's way. 



The edition of Euclid's first two books by Mr. C. 



U 



