Besides, by thus presenting the cross-process we are able to see better what a task Bidder had to accomplish when he multiplied together mentally two numbers, each containing fifteen digits. The processes then stand thus:

It is to be observed that in the case of large numbers we do not get more troublesome products in the course of the work when cross-multiplying than in the case of small numbers, like those above dealt with. We get more such products, that is all. Thus in the middle of the above case of cross-multiplication we have three products of two digits each. In the middle of a case of cross-multiplication with two numbers of fifteen digits we should have fifteen such products-at least, products not containing more than two digits. We should also have, if working mentally, a large number carried over from the next preceding process. This we should have even if we were working out the result on paper, but not writing down the various products used in getting the result. To most persons this would prove an effectual bar to the employment of the cross-method, especially as there would be no way of detecting an error without going through the whole work again. It is true this has to be done when the common method is employed. But in this method if an error exists we can recognise it where it is. In the other, unless we recollect what our former steps were, we have no means of knowing where an error arose. And quite commonly it would happen that two different errors, one in the original process, and another in the work of checking, would give the same erroneous result, so that we should mistakenly infer that result to be correct. But to the men

1

1 This happens frequently in mercantile computations. Thus a clerk may add a column of figures incorrectly, then check his work by adding the same column in another way (say in one case from the top, in the other from the foot): yet both resalts will not uncommonly agree, though the incorrect result is obtained in the two several cases by different mistakes.

tal arithmetician, especially when long-continued practice has enabled him to work accurately as well as quickly, the cross-method is far the most convenient. We know that this was the method applied by Bidder. And to explain his marvellous rapidity we have only to take into account the influence of long practice combined with altogether excep tional aptitude for dealing with numbers.

Of the effect of practice in some arithmetical processes curious evidence was afforded by the feats of a Chinese who visited America in 1875. He was simply a trained computer, asserting that hundreds in China were trained to equal readiness in arithmetical processes, and that among those thus trained those of exceptional abilities far surpassed himself in dexterity. Among the various tests applied during a platform exhibition of his powers was one of the following nature. About thirty numbers of four digits each were named to him, as fast as a quick writer could take them down. When all had been given he was told to add them, mentally, while a practised arithmetician was to add them on paper. 'It is unnecessary for me to add them,' he said, 'I have done that as you gave them to me; the total is-so-and-so.' It presently appeared that the total thus given was quite correct.

At first sight such a feat seems astounding. Yet in reality it is but a slight modification of what many bankers' clerks can readily accomplish. They will take an array of numbers, each of four or five figures, and cast them up in one operation. Grant them only the power of as readily adding a number named as a number seen to a total already obtained, and their feat would be precisely that of the Chinese arithmetician. There can be no doubt that, with a very little practice, nine-tenths, if not all, of the clerks who can achieve one feat would be able to achieve the other feat also.

I do not know how clerks who add at once a column of four-figured numbers together accomplish the task. That is to say, I do not know the mental process they go through in obtaining their final result. It may be that they keep the units, tens, hundreds, and thousands apart in their mind, counting them properly at the end of the summation; or, on the other hand, they may treat each successive number as a whole, and keep the gradually growing total as a whole. Or some may

follow one plan, and some the other. When I heard of the Chinese arithmetician's feats, my explanation was that he adopted the former plan. I should myself, if I wanted to acquire readiness in such processes, adopt that plan, applying it after a fashion suggested by my method of computing when I was a boy. I should picture the units, tens, hundreds, and thousands as objects of different sorts. Say the units as dots, the tens as lines, the hundreds as discs, the thousands as squares. When a number of four digits was named to me, I should see so many squares, discs, lines, and dots. When the next number of four digits was named, I should see my sets of squares, discs, lines, and dots correspondingly increased. When a new number was named these sets would be again correspondingly increased. And so on, until there were several hundreds of squares, of discs, of lines, and of dots.

These (when the last number had been named) could be at once transmuted into a number, which would be the total required.

Take for instance the numbers, 7234, 9815, 9127, 4183. When the first was named the mind's eye would picture 7 squares, 2 discs, 3 lines, and 4 dots. When the second (9815) was named there would be seen 16 squares, 10 discs, 4 lines, and 9 dots. After the third (9127), there would be 25 squares, 11 discs, 6 lines, and 16 dots; after the fourth (4183), there would be 29 squares, 12 discs, 14 lines, and 19 dots. This being all, the total is at once run off from the units' place; the 19 dots give 9 for the units, one 10 to add to the 14 lines (each representing ten), making 15, so that 5 is the digit in the tens' place, while 100 is added to the 12 discs or hundreds, giving 13 or 3 in the hundreds' place, and 1,000 to add to the 29 squares or thousands, making 30, or for the total 30,359. The process has taken many words in describing, but each part of it is perfectly simple, the mental picturing of the constantly increasing numbers of squares, discs, lines, and dots being almost instantaneous (in the case, of course, of those only who possess the power of forming these mental pictures). The final process is equally simple, and would be so even if the number of squares, discs, lines, and dots were great. Thus, suppose there were 324 squares, 411 discs, 391 lines, and 433 dots. We take 3 for units, carrying 43 lines or 434 in all, whence 4 for the tens, carrying 43 discs or 444 in all, whence 4 for the hundreds, carrying 44 squares or 468 in all, whence finally 468,443 is the total required.

We can understand then how easy to Bidder must have been the summation of the fifteen products of cross-multiplication to the carried remainder-they would be added consecutively in far less time than the quickest penman could write them down. Probably they would be obtained as well as added in less time than they could be written down. Thus digit after digit of the result of what appears a tremendous sum in multiplication would be obtained with that rapidity which to many seemed almost miraculous. We must further take into account a circumstance pointed out by Mr. G. Bidder. The faculty of rapid operation,' he says, speaking of his father's wonderful feats in this respect, was no doubt congenital, but it was developed by incessant practice and by the confidence thereby acquired. I am certain,' he proceeds, that unhesitating confidence is half the battle. In mental arithmetic, it is most true that "he who hesitates is lost." When I speak of incessant practice, I do not mean deliberate drilling of set purpose; but with my father, as with myself,1 the mental handling of

1 Mr. G. Bidder's powers as a mental arithmetician would be considered astonishing if the achievements of his father and others were not known. 'I myself,' he says, can perform pretty extensive arithmetical operations mentally, but I cannot pretend to approach even distantly to the rapidity and accuracy with which my father worked. I have occasionally multiplied 15 figures by 15 in my head, but it takes me a long time, and I am liable to occasional errors. Last week, aftor speaking to Prof. Elliot, I tried the following sum to see if I could still do it:

378,201,969,513,825
199,631,057,265,413

and I got, in my head, the answer, 75,576,299,427,512,145,197,597,834,725: in which, I

numbers or playing with figures afforded a positive pleasure and constant occupation of leisure moments. Even up to the last year of his life (his age was seventy-two) my father took delight in working out long and difficult arithmetical problems.

We must always remember, in considering such feats as Bidder and other calculating boys' accomplished, that the power of mentally picturing numbers is in their case far greater than we are apt to imagine such a power can possibly be. Precisely as the feats of a Morphy seem beyond belief till actually witnessed, and even then (especially to those who know what his chess-play meant) almost miraculous, so the mnemonic powers of some arithmeticians would seem incredible if they had not been tested, and even as witnessed seem altogether marvelous. Colburn tells us that a notorious free-thinker who had seen his arithmetical achievements at the age of six, 'went home much disturbed, passed a sleepless night, and ever afterwards renounced infidel opinions." 'And this,' says the writer in the 'Spectator,' from whom I have already quoted, 'was only one illustration of the vague feeling of awe and open-mouthed wonder, which his performances excited. People came to consult him about stolen spoons; and he himself evidently thought that there was something decidedly uncanny, something supernatural, about his gift.'

But so far as actual mnemonic arithmetical power is concerned, the feats of Colburn, and even of Bidder, have been surpassed. Consider, for instance, the following instances of the strong power of abstraction possessed by Dr. Wallis:-December 22, 1669.-În a dark night in bed,' he says in a letter to his friend, Mr. Thomas Smith, B.D., Fellow of Magdalen College, 'without pen, ink or paper, or anything equivalent, I did by memory extract the square root of 30000,00000,00000,00000,00000,00000, 00000,00000, which I found to be 1,77205,08075,68077,29353, ferè, and did the next day commit it to writing.'

And again: February 18, 1670.-Johannes Georgius Pelshower (Regiomontanus Borussus) giving me a visit, and desiring an example of the like, I did that night propose to myself in the dark, without help to my memory, a number in 53 places: 24681357910121411131516182017192122242628302325272931, of which I extracted the square root in 27 places: 157103016871482805817152171 proximè; which numbers I did not commit to paper till he gave me another visit, March following, when I did from memory dictate them to him.' Mr. E. W. Craigie, commenting on these feats, says that they are not perhaps as difficult as multiplying 15 figures by 15, for while of course it is easy to remember such a number as three thousand billion trillions, being nothing but noughts, so also it may be noticed that there is a certain order in the row of 53 figures; the numbers follow each other in little sets of arithmetical progression (2, 4, 6, 8), (1, 3, 5, 7, 9), (10, 12, 14), (11, 13, 15), (16, 18, 20), and so on; not regularly, but still enough to render it

think, if you will take the trouble to work it out, you will find 4 figures out of the 29 are wrong.' I have only run through the cross-multiplication far enough to detect the first error, which is in the digit representing thousands of millions, This should be 4

not 7.

an immense assistance to a man engaged in a mental calculation. A row of 53 figures set down at hazard would have been much more difficult to remember, like Foote's famous sentence with which he puzzled the quack mnemonician; but still we must give the doctor the credit for remembering the answer.' Mr. Craigie seems to overlook the circumstance that remembering the original number, and remembering the answer, in cases of this kind, are utterly unimportant feats compared with the work of obtaining the answer. If any one will be at the pains to work out the problem of extracting the square root of any number in 53 places, he will see that it would be a very small help indeed to have the original number written down before him, if the solution was to be worked out mnemonically. Probably in both cases, Wallis took easily remembered numbers, not to help him at the time, but so that if occasion required he might be able to recall the problem months or years after he had solved it. Anyone who could work out in his mind such a problem as the second of those given above, would have no difficulty in remembering an array of two or three hundred figures set down entirely at random.

I have left small space in which to consider the singular evidence given by Prof. Elliot and Mr. G. Bidder respecting the transmission in the Bidder family of that special mental quality on which the elder Bidder's arithmetical power was based. Hereafter I may take occasion to discuss this evidence more at length, and with particular reference to its bearing on the question of hereditary genius. Let it suffice to mention here that, although Mr. G. Bidder and other members of the family have possessed in large degree the power of dealing mentally with large numbers, yet in other cases, though the same special mental quality involved has been present, the way in which that quality has shown itself has been altogether different. Thus Mr. G. Bidder states that his father's eldest brother, who was a Unitarian minister, was not remarkable as an arithmetician; but he had an extraordinary memory for Biblical texts, and could quote almost any text in the Bible, and give chapter and verse.' A granddaughter of G. P. Bidder's once said to Prof. Elliot, Isn't it strange: when I hear anything remarkable said or read to me, I think I see it in print?' Mr. G. Bidder 'can play two games of chess simultaneously,' Prof. Elliot mentions, 'without seeing the board.' 'Several of Mr. G. P. Bidder's nephews and grandchildren,' he adds, 'possess also very remarkable powers. One of his nephews at an early age showed a degree of mechanical ingenuity beyond anything I had ever seen in a boy. The summer before last, to test the calculating powers of some of his grandchildren (daughters of Mr. G. Bidder, the barrister), I gave them a question which I scarcely expected any of them to answer. I asked them, "At what point in the scale do Fahrenheit's thermometer and the Centigrade show the same number at the same temperature?" The nature of the two scales had to be explained, but after that they were left to their own resources. The next morning one of the younger ones (about ten years old) came to tell me it was at 40 degrees below zero, This was the correct answer; she had worked it out in bed.' RICHARD A. PROCTOR, in Belgravia.