AN EXTRAORDINARY CALCULATOR.

WE are fortunate not only in being able to correct the erroneous impression made by our article, published on September 28th, on the death of the late Mr. Bidder, as to the character of his mathematical faculty, but in being able to place before our readers a very valuable and curious account of the peculiar features of that faculty, and of its more or less here- ditary nature. In another column will be found a careful study of whatever unusual faculty has manifested itself in various members of Mr. Bidder's family, as well as of the form in which it made itself conspicuous in himself ; and we venture to think that the story is one which ought to have the deepest interest, not only for all arithmeticians and geometricians, not only especially for teachers of arithmetic and geometry, but even more for psychologists and all who care to understand how a very slight graft of imaginative power on a strong memory and a habit of rapid reasoning, will tend to trans- form,—we might almost say, to transfigure,—a merely skilful computer into a thinker who can use numbers as a sort of magic wand for the transformation of the real world. We are greatly indebted to Major-General Cunningham Robertson for the very curious information he has obtained for us from Professor Elliot and Mr. Bidder, and we have no doubt that the sub- stance of his communication is certain to be transferred from our columns to works of a more permanent character, like Dr. Carpenter's "Mental Physiology," in which we may observe that there is a trace of the same error as to the temporary character of Mr. Bidder's peculiar gift, into which we had ourselves fallen. Speaking of the still more remarkable calculating powers of Zerah Colburn, of which Dr. Carpenter gives a most curious and impressive account, he says :—" It seems to have been the case with him, as with George Bidder and other calculating boys, that with the general culture of the mind, this special power faded away."* Our readers will see that, at least as regards the

late Mr. Bidder's remarkable powers, this statement is incorrect. Professor Elliot expressly says, "I never saw any appearance of Mr. Bidder's faculty falling off."

The account given by our correspondent suggests what seems to us inferences of no slight importance as to the character and

rationale of Mr. Bidder's faculty. In the first place, it is very curious

to observe that the faculty which took so remarkable a form in Mr. Bidder, was in all probability closely allied to one which at first sight might seem to be of a very different character in his eldest brother. We are told that "his 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." We should like to know,--if there be any means of knowing,— a little more of this elder brother's gift. Did he see mentally the place, and letters, and numbering of the text, as if printed in some particular Bible ; or was it only the wording and the number of the chapter and of the verse which remained in his memory ? If the former, there would be, we think, a much closer affinity between this faculty and that of the great arithmetician and

engineer, than would superficially appear. For observe that all the evidence goes to show that one most important element, at all events, in Mr. Bidder's power lay in the curious faculty,—which we have more than once called attention to as characteristic of some of the greatest chew-players of the world,—of carrying about with him a vivid mental picture of the numbers, figures, and diagrams with which he was occupied, so that he saw, as if it were on a slate, the elements of the problem he was working. His eldest son, it seems, can play two games of chess in his head, without looking at the boards; and he himself states that in working out mental arithmetic, he always sees a complete picture of the steps in the calculation. "Professor Elliot says he (Mr. 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." That other ways are possible, however, the case of Zerah Colburn seems to show. It is in evidence that when at the age of ten, he performed in a few seconds calculations which no mathematician of the period knew how to perform without a very tedious and laborious calculation, he was entirely ignorant of the ordinary methods of working on paper even a simple sum in multiplication or division, and was totally unable to explain to the eminent Francis Baily, who examined and tested him, the method by which he arrived at his results. But, however, this is not to our present purpose. It is pretty clear, we think, that Mr. Bidder's arithmetical genius, great as it was, was much more akin to that of ordinary men than Zerah Colburn's. One great element of interest in it is precisely this,—that it seems to have been a highly exalted form of the arithmetical faculties of more ordinary men, and that one of the most effectual of the means by which it gained this exaltation of power, was the capacity for seeing, as if photographed on his retina, the exact figures, whether arithmetical or geometrical, with which he was occupied at the time. Again, it would seem that his eldest son, who pos- sesses a certain share of his father's power in this respect, him- self ascribes his own power in very great degree to this photo- graphic picture of the processes through which he goes ; and that one of Mr. Bidder's granddaughters testifies to the same curious gift. "Isn't it strange ? When I hear any- thing remarkable said or read to me, I think I see it in print." Now all this seems to us to render it most likely that the gift said to have been possessed by the elder brother, of "being able to quote almost any text in the Bible, and to give chapter and verse," was a slight variation of the same faculty,—that it really arose from a power of seeing in some visionary Bible the exact place and appearance of the text in question. Probably what he recalled was the aspect of the printed text, from which he mentally read off the words and the reference. That, however, is of course of the nature of conjecture. But this, at any rate, is clear, that ix three distinct generations, in father, son, and granddaughter, memory seems to have taken a visual form, and appears rather as vision of the appropriate symbols than as mere memory,—a form of memory which, in relation to words and numbers, is, we believe, very rare. Certainly men usually remember both words and numbers principally by their sounds, and not by their written or printed symbols, though in relation to figures and colours, the • Mental Physiology. First Edition, p. 235. We have not the Second Edition by us, and do not know therefore whether any correction of this statement wilai embodied in that edition.—En. Spectator.

visionary picture is, of course, the usual form of memory. Whether this kind of memory,—namely, the vivid imagination of the symbols by which words and numbers are suggested to the eye,—be the effect or the cause of unusual strength of memory,—and either explanation is conceivable,—it is certain that those who have it, have an enormous advantage over those who remember by association only,—who do not see any picture, in contiguous lines, of the numbers or words with which they are dealing, but have to call them up by the force of association of each with the next in succession. The difference, indeed, between learning "by heart," or as it is called "by rote," and summoning up before the imagination a page on which all the words or numbers required stand forth in black and white, as the trees of your garden or the houses within sight of your home stand out in your memory the moment you choose to summon them up, is enormous. And probably the differ-

ence is great in regard to the confidence reposed in each kind of

memory. As Mr. Bidder justly says in his letter,—" I am certain that unhesitating confidence is half the battle. In mental arithmetic

it is most true that be who hesitates is lost." Now every one knows

how much more unhesitatingly we read off from a vivid mental picture, than merely repeat from association. In the former case, whether truly or falsely, we seem to ourselves to be speaking from actual sight ; in the latter, only from habit. And though of course the picture may be an erroneous picture, the mere vivid- ness of it gives us a confidence which we seldom have when relying on the law of association alone. One of the great chess- players, who can play not merely two, but twelve games at once without seeing the board, expressly stated that he had before him a perfectly vivid picture of each board, which altered instan- taneously as each move was made, and then remained printed on his mind till another move again changed the situation. Whether, as we said, that be the cause, or only the effect of an exceedingly tenacious memory, we are not sure. But whichever be the true explanation, it gives a confidence in the correctness of the memory which nothing else probably could give, and by means of that confidence enormously strengthens the power of calcula- tion and of inference.

Another point of great interest in this communication is the evidence it bears to the attractiveness of the manipulation of

numbers and figures to the minds of Mr. Bidder and his son :— "With my father, as with myself, the mental handling of numbers or playing with figures afforded a positive pleasure and constant occupation of leisure moments." This enjoyment of the subject,

is, we have no doubt, one of the most powerful causes, rather than one of the consequences, of this unusual calculating power. It is

clear that in the Unitarian minister there was something like the same tenacity of memory ; but the mental attraction to him was towards his own theme,—the subject-matter of the Bible,—and the consequence was that his great memory displayed itself in the field to which his mental interests drew him. It was the fasci- nation of quantitative calculations for Mr. Bidder and his son which determined the application of their memory in this particu- lar direction. We should be curious to know whether in those mem- bers of the family whose interests have taken a linguistic turn, the memory for phrase and idiom appears to be as strong as it is in the mathematical members of the family for numbers and figures. We suspect it is the profound fascination of chess for those who care for it at all, which has elicited so curious a power

of playing simultaneously many games blindfold, in men who have not unfrequently never shown any very tenacious memory in any other sphere of life.

The third remark which these communications suggest to us is the extraordinary advantage of learning early to realise and ap- preciate the concrete meaning of abstract numbers. "From Mr.

Bidder's earliest years," says his 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 sum. In my opinion, he had

an immense power of realising the actual number." We recommend that remark to the London arithmeticians who recently disputed

with Mr. Sonnenschein as to the value of concrete aids to the child's imagination in realising the exact meaning of the various numeri- cal landing-places, tens, hundreds, and so forth. It is clear that Mr. Bidder's power, as an engineer at least, as distinguished from his power as a mere calculator, was due in no small degree to this power of realising the concrete meaning,—in relation to the Victoria Docks, for instance,—of the numbers at his command.

Mr. Bidder must have had all the gifts needful for planning a great military campaign,—for realising how many troops he could bring to bear in a given time on a given point, and appreciating the subsidiary operations of the commissariat needful for his purpose. It was this power, too, of realising what numbers

meant when translated into things, which must have been one of the great secrets of that keen interest in the manipulation of numbers of which his son speaks. Until you know what they mean in life, numbers can have none of the interest of life ; and, it is for want of this interest, that arithmetic so often appears distasteful and dry to men of graphic imaginative power.