434 



NATURE 



[March io, 1898 



Smith and Mrs. Bryant is of the moderately conserv- 

 ative type, preserving Euclid's order of propositions, but 

 admitting, especially in Book II., some simplifications of 

 construction and proof. It is not easy to see any de- 

 finite advantages of this edition over others already in 

 the market ; and in some respects it is certainly inferior. 

 The figures are poor ; not a few awkward for the pupil 

 to draw, e.g. those on pp. 76, 100 ; and some are grossly 

 inaccurate, such as those on pp. 80, 122. Then, con- 

 sidering the authors' evident desire for strict logic and 

 accurate statement, some of their obiter dicta are really 

 surprising. Thus we are told that " the unequal side of 

 an isosceles triangle is generally called the base " ; that 

 I. 44 is "sometimes enunciated in the form, 'Construct 

 a parallelogram equal to a given parallelogram, and 

 having one of its sides of given length' "; and that I. 17 

 is equivalent to " If a straight line intersects two other 

 straight lines which meet in a point, the two interior 

 angles which it makes with those straight lines are 

 together less than two right angles." Not to mention 

 the fact that the three lines may be concurrent, the 

 interior angles referred to ought to have been specified. 



The bad plan is followed of placing all the definitions, 

 axioms, and postulates at the beginning ; a wrong de- 

 finition oi postulate is given so as to confine it to pos- 

 tulates of construction, and the postulate required for the 

 theory of parallels appears in its time-honoured, but 

 non-Euclidean place as an axiom. The authors are of 

 opinion that axiom 9 ("magnitudes which can be made 

 to coincide are equal") is Euclid's definition of the 

 equality of geometrical magnitudes, and ought to be put 

 first, and then the other axioms i — 7 can be proved by 

 superposition. In this we feel sure they are mistaken. 

 The real meaning of axiom 9 is that magnitudes which 

 can be made to coincide are equal in a sense consistent 

 with the term as used in the preceding axioms ; thus, 

 for instance, if a figure A can be made to coincide with 

 a figure B, and if another figure C can be made to co- 

 incide with B, then A and C can be made to coincide. 

 That Euclid does not imply the congruence of equal 

 magnitudes is obvious from I. 35, &c. It is true that 

 either of the parallelograms in I. 35 may be cut up into 

 pieces which may be fitted together so as to make up the 

 other parallelogram ; but this fact does not appear in 

 Euclid's proof, and it is doubtful whether he was aware 

 of it. And it is clear that " equal " cannot mean simply 

 "congruent," because if congruent figures are taken 

 from congruent figures the remainders are not necessarily 

 congruent. 



This brings us to the statement (p. 66), "Whenever 

 there is equality of area " of two figures, " one of the 

 figures can theoretically be divided into parts which, 

 when properly fitted together, will coincide with the other 

 figure." If "parts" means "a finite number of parts," 

 we should like to know the proof of this assertion ; it 

 cannot be true in any sense except for figures drawn upon 

 surfaces which are applicable to each other. To take a 

 very simple case, can it be verified for three circles whose 

 diameters are the sides of a right-angled triangle ? ^ 



In the alternative proof of II. 12, no reason whatever is 

 given for the equality of the rectangles AY, AZ, so that 



i Unless the arguments of Rethy (/1/aM. Ann. xxxviii.) are unsound, this 

 question must be answered in the negative. 



NO. 1480, VOL. 57] 



the whole difficulty of this method of proof is shirked by 

 means of a " similarly " applied to dissimilars ; the same 

 imperfection occurs in the alternative proof of II. 13. 



In I. 24, figures to illustrate the different cases ought 

 surely to have been given ; and we should have thought 

 that the direct proof, by superposition, of the first case 

 of I. 26 might have been admitted as an alternative. 



Twenty-six abbreviations have been adopted ; the 

 definite article is expressed or omitted according to some 

 mystical principle which we have been unable to dis- 

 cover ; in some cases the construction and proof are kept 

 separate, and duly labelled in Clarendon type accord- 

 ingly, in others they are mixed up. 



We confess that, on the whole, the perusal of this book 

 has had a depressing effect ; it is like reading a treatise 

 on apologetics, and finding that it leaves you more 

 inclined to be sceptical than before. The most serious 

 objection made against text-books of the more modern 

 type is that young boys fail to really grip the essential 

 parts of some of the proofs, and thus, though they under- 

 stand them at the time, reproduce them in an imperfect 

 and shpshod manner. But' here we have two editors 

 of the orthodox text-book, brought up themselves, no 

 doubt, in the true Euclidean faith, and with scores of 

 school editions from which to take warning and example, 

 who nevertheless are by no means above reproach, in ^ 

 grammar, logic, or precision of statement. ' ■ 



Erroneous methods of teaching elementary geometry 

 are still so prevalent, and teachers are so apt to rely 

 entirely on their text-book, that every treatise, Euclidean 

 or not, which is intended for beginners, should contain a 

 description of simple instruments and their use, and some 

 hints on the proper way of learning the propositions. 

 Before a teacher sets a proposition to be learnt he should, 

 with a class of beginners, go through it with a black- 

 board explaining every point, and in particular every 

 technical term when it first occurs ; he should insist upon 

 the data of the figure, and these only, being first drawn, 

 and the rest put in as the steps of the construction are 

 stated (this should also be done, at first, by the pupil 

 when learning the proposition) ; and he should, from the 

 outset, avoid using the same letters as those in the book. 

 This, and the early introduction of very simple exercises, 

 will ensure that the pupil uses his brains and not merely ■■ 

 his memory ; unless this is the case, the study of geometry I 

 is about as improving as it would be to learn by heart 

 a page of the London Directory. G. B. M. 



OUR BOOK SHELF. 

 Whittaker's Mechanical Engineer's Pocket-Book. By 

 Philip R. Bjorling. Pp. 377 -H viii. (London : Whit- 

 taker and Co., 1898.) 

 A GOOD pocket-book is a necessity to the engineer ; it 

 supplies him with reference tables and constants for 

 facilitating calculations, and also the experience of other 

 engineers in a condensed and handy form for use. One 

 feature of this work is that a rather larger share than 

 usual is given to hydraulics and hydraulic machinery, 

 and also to mining plant. In the formula on p. 3, for i 

 the discharge over weirs, a too small coefficient of 1 

 discharge (apparently 0*45 only) has been adopted ; there 

 is also a misprint in the first line of the second column 

 on p. 7, it should be 8-025. I" the formula on p. 55, 

 for friction of the leather collars of rams, it is not stated 



