Nov. 11, 1881.] 



KNOWLEDGE 



37 



make thoasands of hats for heads with a circamference of aboat 

 twenty-one inches." " I have received similar statements," Sir. 

 Kesteven says, " from other members of the trade, both wholesale 

 and retail. The statement comes to me not only from men of ex- 

 ['L'rience in the trade, but from men of intelligence and observation 

 iNccrcised beyond the limits of the shop or the factory. It is, I am 

 informed, extensively believed among batters ; it may, nevertheless, 

 be merely a general impression. The diminution, it is said, is ob- 

 served mostly among grooms and men of that class in the social 

 scale. If this be really the case, the change shonld be noticeable 

 also among soldiers. The diminntion is possibly more apparent 

 than real, and may be traceable to alteration in the style of hair- 

 cutting, or of wearing the hat. It has been suggested to mo that 

 men of the present generation have from birth smaller heads, de- 

 pendent upon an alteration in the dimensions of the female pelvis, 

 in consequence of modem fashion in dress. Of this opinion, however, 

 I obtain no contirmation from eminent obstetricians of whom I have 

 made inquiries. The statement, then, as it stands, is wanting in 

 explanation, and calls for farther investigation." Mr. Kesteven 

 quotes the reply sent him by Professor Flower to his question as to 

 the statement made by the hatters, " that men's heads were smaller 

 than they were twenty years ago" : — 



" Before drawing any important conclusion from such a state- 

 ment, it would be necessary to know much about the authority upon 

 which it is made. Who, for instance, are the hatters that make it ? 

 Do all hatters concur in the same statement ? Is it a mere general 

 impression, or is it founded upon actual arithmetical data 't Does 

 it refer to any particular class of men, and does it refer to the same 

 class of men ? If it should be true, may it not arise from some 

 change of fashion, if only founded upon the size of the hat, and 

 not of the head) other even than the one you suggest of hair 

 being worn shorter — such as hats being worn more on the top of 

 the head than formerly (in old-fashioned prints one sees the hat 

 ■well down over the ears, which is certainly not the case now), or, 

 perhaps, hats of the kind specified being now worn by a different 

 (perhaps lower) class of the community, or by younger people ? 

 All these questions mnst be considered, and, perhaps, other sources 

 of error eliminated which may not occur at first, before the state- 

 ment can be accepted. If the evidence of the statement appears 

 to bear investigation, it wonld be well worth while following it up, 

 as, if true, it would be one of the most remarkable facts ivith 

 ■which I am acquainted, that in the space of twenty years a 

 material diminution in the average size of the heads of the same 

 population had taken place — a fact so contrary to all theory and 

 to all experience." 



Professor Flower's opinion seems to me very much to the point. 

 I may note, in addition, that the different material of which hats 

 were made thirty years ago may have something to do with the 

 supposed change. Those who remember the heavy beavers of that 

 period will hardly doubt, I think, that they must have been worn 

 more loosely-fitting than the lighter hats of the present time. 



Can any readers of K.nowxepge throw Hght on this subject ? Con- 

 sidering that the hope of the future lies much in our growing men 

 with larger heads than now, it would be a serious matter were the 

 hitters right. 



Are grooms and men of that kind drawn now from the same 

 classes as of old ? May not the jest of those classes now seek 

 better employment ? Or may not emigration have had something 

 to do with the supposed change ? — Tours, Ac, 



CESEBBtm. 



[The question raised by Mr. Kesteven seems to ns of considerable 

 interest, though it is utterly nnUkely that within so short a time, 

 any change, such as hatters suppose, can really have taken place in 

 the size of men's heads — even if, which is almost as unlikely, any 

 change in the direction suggested is going on at all. We may men- 

 tion one circumstance, which, however, would hardly affect grooms. 

 Wigs were certainly more commonly worn thirty years ago than 

 now, and wigs in those days were wigs indeed. TTie average size 

 of hats must have been quite appreciably greater in those times on 

 that account alone, wo should imagine. It is, however, really true 

 that hats of 23i inches are no longer kept in stock ? We should have 

 supposed, from our own observation, that in any 'good hat-shop 

 a hat of 2-t inches could generally be obtained. This leads us to 

 consider another point. Possibly hatters measured heads differently 

 in former times than at present. If they measured rotmd the head 

 then, instead of taking, as now, the two diameters of the cranial 

 oval, they wonld certainly have had a higher average for the circuit 

 of the head. Any one who has examined the head-shapes in 

 American hat-shops will know that nine heads out of ten are quite 

 irregularly shaped. We have seen some having an outline more 

 like a long oblong than the oval which a well-shaped head should 

 have. But taking the case of a regular oval (or egg-shape), or even 

 a truly elliptic head, the true circumference wauld be somewhat 

 greater than that inferred from the hatter's reckoning. Take, for 



instance, a head section having diameters 6 and 7 ; then, if I 

 remember rightly, the hatter would call the circumfcreuce 3 times 

 Ci inches +1 inch (i.e., an inch moro than three times the mean 

 between the two diameters), or20t inches. Now the actual circum- 

 ference would be in the case of an ellipse — 



l- * ao 6-1 (49)' ^ 



143303 



49 



31416 



21952 



or 20'58 inches ; that is, nearly a tenth of an inch longer. In the 

 case of an oval shape the difference would be about a tenth and a 

 half, while, in the case of an irregular head, it would be ftilly a 

 quarter of an inch. Where the section of the head is long 

 (dolichocephalic), the difference between the estimated and the 

 measured circumference would be much greater. — Ed.] 



THE FIFTEEN PUZZLE. 



[13] — 1 am told that in a magazine article which appeared some 

 time since, yon have attempted to show that there are positions in 

 the Fifteen Puzzle from which the won position can never be ob- 

 tained. As I believe that the won position can be obtained from 

 any position whatever, including that in which the numbers 13, 15, 

 14 appear in that order on the last line, I should like to know how 

 the reverse has, in your opinion, been demonstrated. — Boss. 



[I thought the Fifteen Puzzle was dead, and hoped I had had 

 some share in killing the time-absorbing monster. (It is an excel- 

 lent puzzle, by-the-way, except when the puzzled ones try to do 

 impossible things with it.) I have no doubt " Boss" has succeeded 

 in obtaining, from the losing position he names, or others of the 

 same class, what he regards as a won position. For instance, he 

 may have obtained the arrangement — 



or some other, which seems as satisfactory to him as the true won 

 position. But he cannot have obtained this last. The proof wonld 

 occupy too much space to be given here. But " Boss " may try 

 this. Taking any positions at random, let him take the fifteen 

 nimibers one after another as they occur, and for each let him 

 count how many come after it which ought to precede it (running 

 along the linesin the way in which wo read the lines of a book, or 

 as the numbers ruu in the won position). Let him add together all 

 the numbers he thus obtains, and call the result the " total displace- 

 ment." This number ■will be cither odd or even. The vacant 

 square ■will be either on an odd line (the first or third) or on an even 

 line (the second or fotirth), or, for convenience of expression, say 

 the vacant square ■svill be either odd or even. Now, he will find 

 that if the "total displacement" and vacant square are both odd 

 or both even, so they will remain after any change he may make by 

 sliding a block, after two such changes, after three, in fine, after 

 any number of legitimate sliding changes. If the " total displace- 

 ment" is odd and the vacant square even, or vice versi, so will 

 they be after any number of legitimate sliding changes. (" Boss " 

 will readily see the raisnn d'etre of this, after examining a few 

 cases.) No amount of changes, then, will cause the " total displace- 

 ment" and the vacant square to be both even or both odd, unless 

 they were so at the outset. As they have to be both even in the 

 won position (for which the total displacement isO, an even number, 

 and the vacant square on the fourth, an even line, whereas, when 

 the last line runs 18, 15, 14, the total displacement is odd and the 

 vacant square even) no amount of changing ivill bring the losing 

 position, mentioned by " Boss," to the true won position. 



In the article referred to I showed that what, as I have above said, 

 will be found on trial in any given cases, must be universally true. 

 I also showed, in a part of the article which most readers found 

 rather tough reading (there were several misprints, too, the article 

 having been written when I was in Australia), that from any posi- 

 tion any other of the same class, either losing or winning, can be 

 attained. As there are more than ten millions of each kind, it is not 

 wonderful that the proof of this general proposition was not 

 altogether simple. 



It is singular to think that though probably not fewer than 

 twenty milhons of persons tried the Boss Puzzle, probably not a 



