26 



NA TURE 



[NoVEMI!EF< lO, I J 



"Proposition IV. — The reciprocal of any quantity with 

 a positive exponent is equal to the same quantity with 

 an equal exponent taken negatively. 



" Proof : 



" So, too, 



= ■' ^ ,.l_2, 





and 



"This is a very useful principle, and should be carefully 

 attended to." This " proof" is repeated later on (p. 253), 

 and the rest of the chapter on indices, so far as theory 

 goes, is of a similar kind. 



To give anything like a full account of the contents of 

 this book, its utter absence of plan or proportion, its 

 preposterous and antiquated arrangement, its shallow 

 pretence of "theory," and its innumerable misleading 

 and even erroneous statements, would take up too much 

 space. It does not much matter where we look. " If 

 one root of a quadratic equation be imaginary, the other 

 will also be imaginary," nothing being said as to the 

 reality of the coefficients ; " now the greater the value of 

 «, the smaller does ;-" become. Hence (!), when n be- 

 comes indefinitely great, r" will become indefinitely 

 small." " Sum (i) to n terms, and (2) to infinity the 

 series i + 3.1; + 5.1-- + Tx^ + . . .," where neither in the 

 question, nor in the solution, is any restriction imposed 

 on the value of .v. " Let x = circumference of fore- wheel," 

 504- i:;6 



29 



*-=^ = 12 feet.' 



Find 



&c., concluding with ".r = 

 the values of x,y and s from the simultaneous equations 

 X = a{y + z); y = b[z + x); z = c (x + y), 



and prove that ab -\- ac + be + 2abc = i "; " the only four 

 factors of 30 are i, 2, 3, and 5," — so much for its accuracy 

 in detail. The primary definitions and the four funda- 

 mental rules are disposed of in fifty pages, " imaginary 

 quantities " in ten, two of which are taken up by playing 

 tricks with the complex cube roots of unity, while half 

 a page goes to a misleading " demonstration " that 

 J - a X ^ -b = - ijab ; fifty pages are occupied by 

 chapters on indices, surds, and simple equations involving 

 radicals, very inaccurate and choked by the usual fantastic 

 examples ; seventy pages are assigned to ratio, propor- 

 tion, variation, and progressions: such is Mr. O'Uea's 

 conception of the relative importance of the different 

 items of his programme. "Elementary Factoring" (all 

 jumbled up) will be found on pp. 57-73 ; quadratic 

 equations are deferred until p. 319:— logical sequence, 

 forsooth, requires the precedence of Fractions, Involution, 

 Evolution, Indices and Surds, and the substance is sacri- 

 ficed to the shadow accordingly. Of graphical methods 

 or illustrations there is not a trace : there is not a single 

 geometrical figure in the book ; the bearing of algebraical 

 sign upon the "sense" of geometrical and physical 

 quantities is absolutely ignored ; and nothing is said 

 about the nature or properties of logarithms. Not only 

 is it possible for a clever boy to work through the whole 

 of this book, and be as ignorant of the theory of algebra 

 as when he began ; he will be totally unacquainted 

 with those elementary practical things which are most 

 NO. 151 5. VOL. 59] 



important for him if he wishes to apply his knowledge to 

 engineering or to physics. 



The book is avowedly written to help teachers to ob- 

 tain " results ": that is, to exploit their unhappy pupils for 

 the purpose of scoring in examinations, and so getting 

 grants, or scholarships or some other kind of profit. A 

 method is recommended because it will impress an 

 examiner more favourably, and earn a greater number of 

 marks : the student's attention is directed to this or that, 

 not because it is important in itself, but because the 

 candidate is very likely to be asked a question about it ; 

 and an enormous amount of misplaced ingenuity is 

 wasted upon the solution of ridiculous and fantastic ques- 

 tions which ought never to have been set. 



It would not be fair to lay the whole blame for all this 

 upon Mr. O'Dea. Like the poor Irish schoolboys, he is 

 the victim of a most iniquitous system : that " payment 

 by results " which warps and corrodes every branch of 

 primary and intermediate education in Ireland. That 

 this plan was originally adopted with the best intentions 

 may be admitted ; but it is a disgraceful scandal that it 

 should be continued in Ireland, when it has been (re- 

 luctantly enough, it is true) abandoned in Great Britain. 

 The evils of it ha\e been exposed again and again ; it 

 has been denounced unanimously by all true teachers 

 who have seen how it works ; it puts a premium upon 

 w-rong methods, it encourages quackery and cruelty, it 

 destroys sympathy between master and pupils, and the 

 " results " which it produces are a delusion and a sham. 

 It is heart-breaking to think of whole generations of 

 clever, docile Irish lads condemned to the soul-destroying 

 slavery which this rotten system perpetuates. And the 

 evil is intensified when, as in the case of mathematics, 

 the examinations for which the pupils are prepared are 

 thoroughly unsatisfactory. Among the examples con- 

 tained in this book are the following : — 



(i) Simplify 



b-c- , fi<^ a-6- 



c)(b-a) U- a)[c~b) 



(Preparatory Grade.) 

 = 2, ^ = 8, find the value of 



.^ c^ - 2i 'IK-"' 

 40 



(2) If<J = 9,b-. 



sja isV 



\/ ,/■■' he 

 (Preparatory (Irade.) 



(3) Determine the value of 



(.«:=+ (a -1- b)x -f ab\ {x^ -I- aifi -^ abx -t- aH) - (a- -»• .\~){x - *> 

 when .V = - a. (Junior Grade.) 



(4) If f( -I- /' -t- f = o, find the value of 



ab 

 (First Class Teachers.) 



be 



(5) If 



show that 



ey-ar 





(Mill Jle Grade.) 



(6) Show that the expression 

 (j- + 2)'f 4 



(.V - 2)«\ (;r2 - \)(x' 

 is equivalent to the fraction 



4)' + (^' 



I -f-.r 



3)<- 



2-'} 



(Senior Grade.) 



