instantly and quite correctly, when asked how many farthings there are in 868,424, 121/.

A correspondent X. in the Spectator,' referring to a somewhat earlier part of Bidder's career as a youthful calculator, says, 'In the autumn of the year 1814, I was reading with a private tutor, the Curate of Wellington, Somersetshire, when a Mr. Bidder called upon him to exhibit the calculating power of his little boy, then about eight years old, who could neither read nor write. On this occasion, he displayed great facility in the mental handling of numbers, multiplying readily and correctly two figures by two, but failing in attempting numbers of three figures. My tutor, a Cambridge man, Fellow of his College, strongly recommended the father not to carry his son about the country, but to have him properly trained at school. This advice was not taken, for about two years after he was brought by his father to Cambridge, and his faculty of mental calculation tested by several able mathematical men. I was present at the examination, and began it with a sum in simple addition, two rows, with twelve figures in each row. The boy gave the correct answer immediately. Various questions then, of considerable difficulty, involving large numbers, were proposed to him, all of which he answered promptly and accurately. These must have occupied more than an hour. There was then a pause. To test his memory, I then said to him, Do you remember the sum in addition I gave you?" To my great surprise, he repeated the twenty-four figures with only one or two mistakes.'* It is evident, therefore, that in the course of two years

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*This feat is remarkable, because the power of picturing numbers distinctly before the mental eye, and dealing with them as readily as though pen and paper were used, is not necessarily accompanied by the power of retaining such numbers after they are done with; on the contrary, it must be an advantage to the mental calculator to be able to forget all merely accidental groups of numbers, though of course it is equally an advantage to him to be able to retain all numbers which he may have to use again. I have very little doubt myself that the power of selecting things to be forgotten and things to be remembered is a most useful mental faculty; and that those minds work best in the long run which can completely throw off all recollection of useless matters.

his powers of memory and calculation must have been gradually developed.

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he said,

·

Bidder was unable at this time to explain the process by which he worked out long and intricate sums. He did not appear burdened by his mental calculations. As soon as a question was answered,' says X., 'he amused himself with whipping a top round the room, and when the examination was over, he said to us, "You have been trying to puzzle me, I will try to puzzle you. A man found thirteen cats in his garden. He got out his gun, fired at them, and killed seven. How many were left?" Six," was the answer. "Wrong," none were left. The rest ran away." I mention this to show that he was a cheerful and playful boy when he was about ten years old, and that his brain was not overtaxed. It would be curious to inquire whether Bidder was really the inventor of the now time-honored joke with which he puzzled his examiners. If it had been as well known in 1816 as now, he would hardly have asked a roomful of persons, even though they were college fellows, a question which some one or other of them would have been sure to have heard before. he really invented the puzzle, it was clever in so young a lad.

If

The next evidence is more precise. It is given in a letter from Mr. C. S. Osmond, and is derived from an old pamphlet of thirty-four pages, published about the year 1820. From this we learn that when Bidder was ten years old, he answered in two minutes the following question: What is the interest of 4,444/. for 4,444 days at 41 per cent. per annum? The answer is, 2,434. 16s. 54d. A few months later, when he was not yet eleven years old, he was asked, How long would a cistern 1 mile cube be filling if receiving from a river 120 gallons per minute without intermission? In two minutes he gave the correct answer: 14,300 years, 285 days, 12 hours, 46 minutes. A year later, he divided correctly, in less than a minute, 468,592,413,563 by 9,076. I have tried how long this takes me with pen and paper; and, after getting an incorrect result in one and a quarter minute, went through the sum again, with correct result, (51,629,838 and 5,875 over) in about the same time.

At twelve years of age he answered in less than a minute the question, If a distance of 9 inches is passed over in a second of time, how many inches will be passed over in 365 days, 5 hours, 48 minutes, 55 seconds? Much more surprising, however, was his success when thirteen years old, in dealing with the question, What is the cube root of 897,339,273,974,002, 153? He obtained the answer in 2 minutes, viz., 964,537. I do not believe one arithmetician in a thousand would get out this answer correctly, at a first trial, in less than a quarter of an hour. But I confess I have not tried the experiment, feeling, indeed, perfectly satisfied that I should not get the answer correctly in half a dozen trials.

No date is given to the following case: The question was put by Sir William Herschel, at Slough, near Windsor, to Master Bidder, and answered in one minute : Light travels from the sun to the earth in 8 minutes, and the sun being 98,000,000 of miles off' (of course this is quite wrong, but sixty years ago it was near enough to the accepted value), if light would take six years and four months travelling at the same rate from the nearest fixed star, how far is that star from the earth, reckoning 365 days and 6 hours to each year, and 28 days to each month?' The correct answer was quickly given to this pleasing question, viz., 40,633,740,000,000 miles.

On one occasion, we learn, the proposer of a question was not satisfied with Bidder's answer. The boy said the answer was correct, and requested the proposer to work his sum over again. During the operation Bidder said he felt certain he was right, for he had worked the question in another way; and before the proposer found that he was wrong and Bidder right, the boy told the company that he had calculated the question by a third method.

The pamphlet gives the following extract from a London paper, which, if really based on facts, proves conclusively that Bidder was a more skilful computer than Zerah Colburn :-'A few days since, a meeting took place between the Devonshire youth, George Bidder, and the American youth, Zerah Colburne (sic), before a party of gen

tlemen, to ascertain their calculating comprehensions. The Devonshire boy having answered a variety of questions in a satisfactory way, a gentleman proposed one to Zerah Colburne, viz., If the globe is 24,912 miles in circumference, and a balloon travels 3,878 feet in a minute, how long would it be in travelling round the world? After "nine minutes'" consideration, he felt himself incompetent to give the answer. The same question being given to the Devonshire boy, the answer he returned in two minutes-viz., 23 days, 13 hours, 18 minutes-was received with marks of great applause. Many other questions were proposed to the American boy, all of which he refused answering, while young Bidder replied readily to all. handsome subscription was collected for the Devonshire youth.' This account seems to me to accord very ill with what is known about Colburn's skill in mental computation. That Bidder could deal more readily with very large numbers was admitted by Colburn. But the problem which Colburn is said to have failed in solving during nine minutes is far easier than some which he is known. to have solved in a much shorter time. It should be noted that Colburn was nearly two years older than Bidder.

A

And now let us consider what we know respecting Bidder's method of computation. On this point, fortunately, the evidence is far clearer than in Colburn's case. Colburn, when asked how he obtained his results, would give very unsatisfactory answers-in one case blurting out the rude remark, God put these things into my head; I cannot put them into yours.' Bidder, on the other hand, was ready and able to explain how he worked out his results.

we

The first point we learn respecting his method seems to accord with the theory advanced by myself in 1875, but it will presently be seen that in Bidder's case that theory cannot possibly be maintained. From his earliest years, are told by his eldest son, he appears to have trained himself to deal with actual objects, instead of figures, at first by using pebbles or nuts to work out his sums. In my opinion,' proceeds Mr. G. Bidder, he had an immense power of realising the actual number.' However, in multiplying he made use of the

ordinary arithmetical process called cross multiplication, by which the product of two numbers is obtained, figure by figure, in a single line. 'He was aided, I think,' says his son, by two things first, a powerful memory of a peculiar cast, in which figures seemed to stereotype themselves without an effort; and secondly, by an almost inconceivable rapidity of operation. I speak with some confidence as to the former of these faculties, as I possess it to a considerable extent myself (though not to compare with my father). Professor Elliot says he,' meaning Mr. G. P. Bidder, saw mental pictures of figures and geometrical diagrams. I always do. If I perform a sum mentally, it always proceeds in a visible form in my mind; indeed, I can conceive no other way possible of doing mental arithmetic.' This, by the way, is a rather strange remark from one possessing so remarkable a power of conception as the younger Bidder. Assuredly another way of working sums in mental arithmetic is common enough; and even if it had not been, it might easily have been conceived. Many, probably most persons, in working sums mentally, retain in their memory the sound of each number involved, not an image of the number in a visible form. Thus, suppose the two numbers 47 and 23 are to be multiplied in the mind. The process will run, with most ordinary calculators, in a verbal manner: thus, three times seven, twenty-one, three times four, twelve and two fourteen-one four one. (These digits being repeated mentally as if emphasised, and the mental record of the sound retained to be presently used when the next line is obtained.') Ágain: twice seven, fourteen, twice four, eight and one nine-nine four. Then the addition mentally thus, one, four and four eight, nine and one ten-one, nought, eight, one, the digits of the required product. I happen to know that this is the way in which most persons would work a sum of this kind mentally, retaining each necessary digit by emphasising, so to speak, the mental utterance of the digit's name. Of course the process is altogether inferior to the visual process, so to call that in which mental pictures are formed of the digits representing a NEW SERIES.-VOL. XXX., No. 2

number. But not one person in ten has the power of forming such pictures.

Of course, one who, like Bidder, could picture at will any number, or set of numbers, and carry on arithmetical processes with such numbers as freely as though writing on paper, would have a great advantage over a computer using ink and paper. He would be saved, to begin with, all inconvenience from the quality of writing materials, necessity of taking fresh ink, and so forth. The figures would start into existence at once as obtained, instead of requiring a certain time, though short, for writing down. They would also always arrange themselves correctly. But this would be far from being all. Indeed, these advantages are the least of those which mental arithmeticians using the visual method possess over the calculator with pen and paper. The same power of picturing numbers which enables the mental worker to proceed in the confident assurance that every line of a long process of calculation will remain clearly in his mental vision to the end of that process, enables him to retain a number of results by which all ordinary processes of calculation can be greatly shortened. He may forget in a day or two the details of any given process of calculation, because he not only makes no effort to retain such details, but purposely hastens to forget them. He would, however, be careful to remember any results which might be of use to him in other calculations. The multiplication table, for instance, which with most persons ranges only to the product 12 times 12, and even then is not retained pictorially in the mind, with Bidder ranged probably to 1000 times a 1000, or even farther. This may seem utterly incredible to those unfamiliar with the wonderful tenacity and range of memory possessed by such men as Bidder the arithmetician, Morphy the chess-player, Macaulay the historian, and others, each in their own special line. There is a case in print showing that a much less expert arithmetician than Bidder possessed a much more complete array of remembered numbers than he did—the case, namely, of Alexander Gwin, a native of Derry, one of the boys employed for

12

calculation in the Ordnance Survey of Ireland, who at the age of eight years knew the logarithms of all numbers from I to 1000. He could repeat them either in regular order or otherwise. Now, every one of these logarithms (supposing Gwin learned them from tables of the usual form) contains seven digits, and there is no connection between these sets of digits by which the memory can be in any way aided. If young Gwin at eight years old could remember all these numbers, we may well believe that Bidder, who probably possessed an even more powerful memory, retained a far larger array of such numbers.

Thus we can partly understand the marvellous rapidity with which Bidder effected his computations. Professor Elliot says on this point that the extent to which Bidder's arithmetical power was carried was to him incomprehensible, as difficult to believe as a miracle. You might read over to him fifteen figures, and another line of the same number, and without seeing or writing down a single figure he would multiply the one by the other, and give the result correctly. The rapidity of his calculations was equally wonderful. Giving his evidence before a parliamentary committee rather quickly and decidedly with regard to a point of some intricacy, the counsel on the other side interrupted him rather testily by saying, "You might as well profess to tell us how many gallons of water flow through Westminster Bridge in an hour." I can tell you that, too,' was the reply, giving the number instantaneously.' however, be it remembered, proved rather how retentive Bidder's memory was than how rapidly he could compute. For either he knew or did not know the precise breadth, depth, and rapidity of the Thames at Westminster Bridge. If he did not know, he could not have made the computation. If he did know, it could only have been because he had had special occasion to inquire, and we cannot readily imagine that any occasion can have existed which would have required the very calculation which Professor Elliot supposes Bidder to have made on the spur of the moment.

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This,

Professor Elliot proceeds to remark on the power of Bidder in retaining vivid impressions of numbers, diagrams,

&c. 'If he saw or heard a number, it seemed to remain permanently photographed on his brain. In like manner, he could study a complicated diagram without seeing it when walking and apparently listening to a friend talking to him on some other subject.' Every geometrician, I imagine, can do this. At least, I know that I have often found myself better able to solve geometrical problems of difficulty when walking with a friend, and really (not apparently only) listening to his conversation, than when alone in my study with pen and paper to delineate diagrams and note down numerical or other results. diagram so thought of stands out before me, as Professor Elliot says that Bidder's mind-diagrams stood out, "with all its lines and letters. The faculty is not, I believe, at all exceptional, though of course the degree in which it was developed in Bidder's case was altogether

So.

The

All the fifteen

The process of multiplying a number of fifteen digits by another such number is one which, so far as the ordinary method is concerned, everyone can appreciate. This method is doubtless the best for most arithmeticians, simply because it is one which requires least mental effort in retaining numbers, and also because the operation is one which can be readily corrected. rows of products are present for checking after the process has once been completed on paper. It would be a more difficult process to the mental arithmetician. In fact, I can hardly believe that even Bidder could have retained a clear mental picture of the set of nearly three hundred digits which form the complete “sum." At any rate, we know that the method he adopted was one which most persons would find far more difficult, even using pen and paper, but which requires a much smaller effort of memory on the part of the mental arithmetician. The process called crossmultiplication is not usually taught in books on arithmetic. This would not be the place to describe it fully. But I may be permitted to give an illustration of the process as applied to two numbers, each of three digits only. Take for these numbers, 356 and 428. The arithmetician sets these down in the usual way, and then writes down the

product in one line, figure by figure, beginning with the units' place, so that the sum appears thus :

356

428 152368

He appears to those unacquainted with the method he uses to be multiplying at once by 428, just as one multiplies at once by 11 or 12. In reality, however, the work runs thus in his mind: Eight times six, forty-eight. (Set down eight and carry four.) Five times eight, forty; twice six, twelve, making fifty-two; and with the carried four, fifty-six. (Set down six and carry five.) Thrice eight, twenty-four, twice five, ten, making thirty-four; four times six, twenty-four, making fifty-eight; and with the carried five, sixty-three. (Set down three and carry six.) Twice three, six; and four times five, twenty, making twenty-six; and with the carried six, thirty-two. (Set down two and carry three.) Lastly, four times three, twelve; making with the carried three, fifteen-which being set down completes the product.

To make a comparison between this method and the ordinary method I have set them side by side, as actually worked out; for of course there is no essential reason why the cross-method should be carried out without keeping record of the various products employed. 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 stands thus :

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 recognize 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 mental 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 exceptional aptitude for dealing with numbers.

Of the effect of practice in some arith

*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 results will not uncom monly agree, though the incorrect result is obtained in the two several cases by different