November 3, 1892] 



NATURE 



12) As one end of a bracket, of which the other end is ] . 

 This bracketing influence may he directed either forwards or 

 backwards, or both ways at once. 



Examples. 



Continued fractions also can readily be brought into one line 

 by this notation — 



^^ ' f = \alb^.cl{d+e)+flg\. 



b + d + e + ^ 

 0. Involution. 



a^ = d\b,a~'' = a\-d, 

 (a + by + d = \ a + 6\c + d, 

 a + b''^'^ = a + b\c + d\, 



{a + b)'^" = \a + b\c +d\, 

 c+d 



{a + b)'+J-= \a + b\\c + d/e + / 1 . 

 a\b/c and a/b\c are ambiguous, but 1 a\b/c = — , be- 

 cause I being unnecessary for \ in this case, can apply only 



(a + by 

 to/. ^ is II a + b\c/d, two vertical lines being required. 



Similarly | a/b\c = 



d 



a\b\c\d= a' . 

 y. Evolution — 



i/b\c I = jc, a\b/c I = a^' 



l> ■ sjc + d = a + b^c + d\ , 



■ + X/c + d = a + I bs^c + d. 



In some cases lines of differing thickness might be advisable ; 



for instance — 



c{d 



^3= l^\i/2| /c\d+e\3\. 

 + eY 



There are many other ways in which this notation might be 

 used ; but the above will suffice to illustrate the advantages of 

 it. And these advantages are substantial. It enables the work 

 to be printed in the same space as ordinary letterpress, and thus 

 avoids the special spacing, from which nine-tenths of the troubles 

 in mathematical printing arise. It requires no new types, except, 

 perhaps, \, and each of the signs used is suggestive of the 

 original mode of writing for which it is a substitute. It can be 

 used without confusion in conjunction with all ordinary brackets. 

 How far this notation would suit very complicated expressions, 

 is a point that would have to be determined by experience ; but 

 for printing mathematics of ordinary complexity it would be 

 useful in economizing space and diminishing the risk of 

 printers' errors without any sacrifice of clearness. 



Cambridge, October 27. W. Cassie. 



NO. I 20 1, VOL. 47] 



" Sunshine." 

 In acknowledging the courteous criticism and the kind re- 

 marks which " C. V. B." has been pleased to make about my 

 little book, may I be permitted to comment on one or two points, 

 which I think he has imperfectly understood from the text. We 

 all know that when "C. V. B." undertakes to review a book, 

 he does his work in a thorough and searching manner, and from 

 his critique it is evident that "Sunshine" has been well read. 

 Notwithstanding this, in one or two of the instances selected for 

 criticism the meaning, at once simple and obvious to a little 

 child, who neither knows nor suspects any other, seems to have 

 missed him, presumably because he knows all the bearings of 

 the subject. Thus it is sometimes a disadvantage to be learned. 

 Of this I propose presently to give an instance in the order which 

 it occurs. 



After poking fun at me, because, the " Sunshine " course being 

 ended. Tommy meets King Sol face to face and "has it out 

 with him," my critic proceeds to discuss the limits within which- 

 the imagination may be appealed to as a factor in scientific 

 education, and while I agree with him in the main, I am tempted 

 in passing to remind him of what Tyndall terms " the scientific 

 use of the imagination," to which the clearness and (to me) the 

 charm of his own lectures is largely due. Be that as it may, in 

 one of " Nature's Story Book? " I feel fully justified in employ- 

 ing, within the litnits of scientific accuracy, any or all of the 

 powers of the mind, which shall help children and others to realize 

 the relation they bear to their surroundings, assured that in a 

 course based upon some hundreds of experiments synthetically 

 worked out and deductions made — a course whose main object 

 is to lead children to go direct to Nature, via experiment, for 

 their knowledge, there is little danger that the imagination be 

 cultivated at the expense of the reasoning faculties. The ex- 

 perience of the writer is that the children attending the lectures 

 became extremely critical — a state of mind which, although of 

 inestimable value in acquiring knowledge, is not one of the 

 happiest in other respects. Therefore it was thought desirable 

 to provide them with some necessary ballast, and this is mydefence 

 of the hypnotic visit to the moon, and the other two chapters ta 

 which the critic alludes. 



Natural science apart, it seems to me that thetendency of the 

 school-teaching of to-day is calculated rather to make children 

 hard and matter of fact. For this reason I have endeavoured 

 in these Sunshine Stories to interest children io the poetry of 

 their common lives, myself playing somewhat the role of an 

 optical instrument, presenting images sometimes real, some- 

 times virtual of those physical beauties which touchthem at 

 every point. The fact that "C. V. B." recognizes in " Sun- 

 shine" the realism which the "picturesque language" was in- 

 tended to convey, disposes of the case of my Cape Town 

 reviewer, who mildly insinuates that I have been guilty of some 

 fraud xx^on little children in calling "Sunshine" a story-book. 

 Therefore I am the more glad that " C. V. B. " agrees with me 

 that the mathematical side of these questions should not be 

 obtruded. There are so many excellent text-books which 

 supply that information for older pupils. I need not say 

 that I shall be most happy to add the exception in the 

 case of the rainbow. I thank him also for pointing out a 

 passage in the notes where an additional clause is necessary, 

 owing to the transposition of a paragraph. But I take excep- 

 tion to the statement about the top, for it is evident that the 

 experiment is not made under the same conditions as that which 

 "C. V. B."has in mind, because my boys get green and he 

 gets (he says) white, or nearly white. The home experiment 

 reads : "I am giving each of you squares of coloured paper to 

 take home . . . then you may have the papers to put on your 

 tops — e.g., cover half blue and half yellow, spin the top and 

 you will see green." A note on page 341 refers to the kind of 

 paper. Now it seems to me from the expression "painted 

 disc," which " C. V. B." has made use of, that possibly he may 

 have had Clerk Maxwell's top in mind when he wrote. 



When I say to a boy, " Here are two squares of paper, one blue 

 and one yellow ; when you've done so and so, you can have the 

 paper to keep— cover your top, half yellow, half blue, &c.," the 

 lad understands me, and when I am not there he takes out his 

 halfpenny whip-top, tears a piece of the blue paper, and 

 rendering it slightly adhesive hammers it down on the top with 

 his right fist ; he tears a similar piece and treats it in the same 

 way, and so on until he has covered half Then he takes the 

 yellow paper and covers the other half with irregular patches of 

 yellow. He spins the top and sees green. 



