Some of these dear good creatures send incidents of real life which they are sure will be useful to dear Dick' for his next book-narratives of accidents in a hansom cab, of missing the train by the Underground, and of Mr. Jones being late for his own wedding, 'which, though nothing in themselves, actually did happen, you know, and which, properly dressed up, as you so well know how to do,' will, they are sure, obtain for him a marked success. 'There is nothing like reality,' they say, he may depend upon it,' for coming home to people.'

After all, one need not read these abominable letters. One's relatives (thank Heaven!) usually live in the country. The real Critics on the Hearth are one's personal acquaintances in town, whom one cannot escape.

'My dear friend,' said one to me the other day-a most cordial and excellent fellow, by the bye (only too frank) —‘I like you, as you know, beyond everything, personally, but I cannot read your books.'

My dear Jones,' replied I, 'I regret that exceedingly; for it is you, and men like you, whose suffrag I am most anxious to win. Of the approbation of all intelligent and educated persons I am certain; but if I could only obtain that of the million, I should be a happy man.'

But even when I have thus demolished Jones, I still feel that I owe him a grudge. What the infernal regions,' as our 'bus driver would say, 'is it to me whether Jones likes my books or not? and why does he tell me he doesn't like them?'

Of the surpassing ignorance of these good people, I have just heard an admirable anecdote. A friend of a justly popular author meets him in the club and congratulates him upon his last story in the Slasher [in which he has never written a line]. It is so full of farce and fun [the author is a grave writer]. Only I don't see why it is not advertised under the same title in the other newspapers.' The fact being that the story in the Slasher is a parody- and not a very good-natured oneupon the author's last work, and resembles it only as a picture in Vanity Fair resembles its original.

Some Critics on the Hearth are not only good-natured, but have rather too high, or, if that is impossible, let us say too pronounced, an opinion of the abilities of their literary friends. They wonder why they do not employ their gigantic talents in some enduring monument, such as a life of Alexander the Great' or a popular history of the Visigoths. To them literature is literature, and they do not concern themselves with little niceties of style or differences of subject. Others again, though extremely civil, are apt to affect more enthusiasm than they feel. They admire one's works without exception-they are all absolutely charming'-but they would be placed in a position of great embarrassment if they were asked to name their favourite: for as a matter of fact, they are ignorant of the very names of them. A novelist of my acquaintance lent his last work to a lady cousin because she 'really could not wait till she got it from the library;' besides, she was ill, and wanted some amusing literature.' After a month or so he got his

three volumes back, with a most gushing letter. It had been the comfort of many a weary hour of sleeplessness,' &c. The thought of having 'smoothed the pillow and soothed the pain' would, she felt sure, be gratifying to him. Perhaps it would have been, only she had omitted to cut the pages even of the first volume.

But, as a general rule, these volunteer censors plume themselves on discovering defects and not beauties. When any author is particularly popular, and has been long before the public, they have two methods of discoursing upon him in relation to their literary friend. In the first, they represent him as a model of excellence, and recommend their friend to study him, though without holding out much hope of his ever becoming his rival; in the second, they describe him as 'worked out,' and darkly hint that sooner or later [they mean sooner] their friend will be in the same unhappy condition. These, I need not say, are among the most detestable specimens of their class, and only to be equalled by those excellent literary judges who are always appealing to posterity, which, even if a le temporary success has crowned you to-day, will relegate you to your proper position to-morrow. If one were weak enough to argue with these gentry, it would be easy to show that popular authors are not 'worked out,' but only have the appearance of being so from their taking their work too easily. Those whose calling it is to depict human nature in fiction are especially subject to this weakness; they do not give themselves the trouble to study new characters, or at first hand, as of old; they sit at home and receive the congratulations of Society without paying due attention to that somewhat changeful lady, and they draw upon their memory, or their imagination, instead of studying from the life. Otherwise, when they do not give way to that temptation of indolence which arises from competence and success, there is no reason why their reputation should suffer, since, though they may lack the vigour or high spirits of those who would push them from their stools, their experience and knowledge of the world are always on the increase.

As to the argument with regard to posterity which is so popular with the Critic on the Hearth, I am afraid he has no greater respect for the opinion of posterity himself than for that of his possible great-greatgranddaughter. Indeed, he only uses it as being a weapon the blow of which it is impossible to parry, and with the object of being personally offensive. It is, moreover, noteworthy that his position, which is sometimes taken up by persons of far greater intelligence, is inconsistent with itself. The praisers of posterity are also always the praisers of the past; it is only the present which is in their eyes contemptible. Yet to the next generation this present will be their past, and, however valueless may be the verdict of to-day, how much more so, by the most obvious analogy, will be that of to-morrow. It is probable, indeed, though it is difficult to believe it, that the Critics on the Hearth of the generation to come will make themselves even more ridiculous than their predecessors.

JAMES PAYN, in Nineteenth Century.

CALCULATING BOYS.

In one of the essays of my Science Byways' I considered, in a paper On some Strange Mental Feats,' the marvellous achievements of Zerah Colburn, one of the most remarkable of the so-called 'calculating boys.' I advanced a theory in explanation of his feats which was in some degree based on experience of my own. I have since found reason to believe that the theory, if correct in his case, is certainly not generally applicable to cases of rapid mental calculation. I now propose to consider, in relation to that theory and also independently, the remarkable feats of calculation achieved by the late Mr. George Bidder in his boyhood. It may be remembered that, in my former paper, I had specially in view the possibility of ascertaining from the discussion of such achievements the laws of cerebral action, and especially of cerebral capabilities. It is with reference to this possibility that I wish now to examine some of the evidence afforded by the feats of Colburn, Bidder, and other 'calculating boys.'

And first, let me show reason for still retaining faith in the theory which I advanced in 1875 respecting Colburn's calculating powers. In so doing, a difference between his feats and Bidder's will be indicated. which appears to me important.

So far as the long and elaborate processes of computation are concerned, which Colburn achieved so rapidly and correctly, there may be no special reason for adopting any other explanation in his case than we are forced, as will presently appear, to adopt in Bidder's case. Thus, Colburn multiplied 8 into itself fifteen times, and the result, consisting of fifteen digits, was right in every figure. But Bidder could multiply a number of fifteen digits into another number of fifteen digits with perfect correctness and amazing rapidity, and we know he employed a process familiar to arithmeticians. Again, Colburn extracted the cube root of 268,336,125 before the number could be written down; and this feat was one which had seemed to me beyond the power of any computer employing the ordinary methods, or any modification of those methods. Yet I am inclined now to believe that Bidder would have obtained the result as quickly, simply through the marvellous rapidity with which he applied ordinary processes.

Where, however, we seem compelled in Colburn's case to recognise the employment of a method entirely different from those given in the books, is in cases resembling the following:-He was asked to name two numbers which, multiplied together, would give the number 247,483, and he immediately named 941 and 263, which are the only two numbers satisfying the condition. The same problem being set with respect to the number 171,395, he named the following pairs of numbers: 5 and 34,279, 7 and 24,485, 59 and 2,905, 83 and 2,065, 35 and 4,897, 295 and 581, and lastly, 413 and 415. Still more marvellous was

the next feat. He was asked to name a number which will divide 34,083 without remainder, and he immediately replied that there is no such number; in other words, he recognised this number as what is called a prime, or a number only divisible by itself and unity, as readily and quickly as most people would recognise 17, 19, or 25, as such a number, and a great deal more quickly than probably nine persons out of ten would recognise 53 or 59 as such." The last feat of this special kind was the most remarkable of all, but the length of time required for its accomplishment, even by this wonderful calculating boy, was such that the evidence does not appear altogether so striking as that afforded by the last case, which I must confess seems to me utterly inexplicable, save on the theory presently to be re-enunciated. Fermat had been led to the conclusion that the number 4,294,967,297, which exceeds by unity the number 2 multiplied fifteen times into itself, has no divisors. But the celebrated mathematician Euler, after much labour, succeeded in showing that the number is divisible by 641. The number was submitted to Zerah Colburn, who was, of course, not told of the result of Euler's researches into the problem, and after the lapse of some weeks the boy discovered the one divisor which Euler had only found with much greater labour.

My theory respecting achievements of this special kind—that is, cases in which a calculator rapidly finds the exact divisors of large numbers, if such divisors exist, or ascertains the non-existence of any exact divisor of such numbers-was based on the known fact that all good calculators have the power of picturing numbers not as represented by such and such digits, but as composed of so many things.' Having once this power in no inconsiderable degree myself, and knowing that, when I had it, I frequently used it in the special manner in question, I was led to believe that Colburn and other calculating boys would employ it in that manner, only with much greater rapidity, dexterity, and correctness. Let us suppose that the number 37 is thought of, taking it for convenience of illustration as a representative of some much larger number, whose real nature (as to divisibility by other numbers) is not known. Requiring to know whether 37 is a prime number or not, I would not, (in the time to which I now carry back my thoughts) divide the number successively by 2, 3, &c., but would see the number passing through the forms here indicated.

These various arrays would all be formed from the following mental presentation of the number 37:

which, it will be observed, is derived directly from the number as presented in the common notation. Thus 37 means three tens and seven units, and the grouping above (numbered 6, but really the first pictured grouping) shows three rows of ten dots and one row of seven. It is easily seen that groupings 2 and 3 are in a moment formed from 6. Grouping 2 is formed from 6 by imagining the lowest row of seven dots set into the form

and run over to the right of the three rows of ten dots. Grouping 3 is formed from 6, by imagining the little square of nine dots on the right set into the form