which is done at once by supposing the vertical row of three dots on the right of 6, placed as a horizontal row in the corner under the two neighbouring vertical rows of three each; that is, by changing the three right hand rows from

The changes from 2 on the one hand to 1, and from 3 on the other to 4 and 5, are similarly effected. If the reader will make the actual calculation (using the word calculation in its real sense as meaning pebbling), taking 37 pebbles, dice, or other objects, and marshalling them first as in 6, and then as in 2 and 1, back again to 6, and then as 3, 4, and 5, he will see how easy the transformations are. But if they are easy when actual objects are shifted about, they are much easier, at least to any one who can picture groups of objects (dots, or the like) at will, when the mind makes all the transformations. After a little practice the changes above figured for such a number as 37 would be made in a moment, and the changes for a number of several hundreds in half a minute or so this in the case of a mind not possessing exceptional power in this way. But as a Morphy or a Blackburne can play twenty games of chess blindfold, recognising in each, with amazing rapidity, a number of lines of play on both sides for nine or ten moves in advance-which seems even to an ordinary blindfold player scarcely explicable, and to an ordinary chess-player almost miraculous-so a Colburn or a Bidder would be able to apply the marshalling system above illustrated as rapidly to a number of many millions or billions, as I, when a boy, could apply it to a number of several hundreds. Accordingly I was led to recognise in this marshalling method the explanation of Colburn's wonderful achievements in finding divisors for numbers, or recognising quickly when a number has no divisors.

For it will be seen that the groupings 1, 2, 3, 4, and 5, above, at once show that 37 has no divisors but itself and unity. (Of course we know in this case that 37 cannot be divided; and even in the case of much larger numbers we may know, without the trouble of trying the division, or marshalling the pictured number, that such numbers as 2,

4, 5, 6, 8, 10, 12, 14, 15, and others, will not divide a number-for instance, if it is an odd number no even number will divide it, and if it does not end with a 5 or a 0 no number ending in 5 will divide it. But, as already explained, the number 37 is to be regarded only as selected for the purpose of conveniently illustrating the marshalling method. A larger number would have required several pages of unsightly groups of dots.) From grouping 1 we see that division by the number 2 will leave one as a remainder, for a dot remains alone on the right. From grouping 2 we see in like manner that one will be left as a remainder after division by 3, for the group shows twelve columns of three each and one over. So grouping 3 shows nine columns of four dots, and one over; grouping 4 shows seven columns of five each, and two over; and lastly, grouping 5 shows six columns of six each, and one over. We need not go on, because it is manifest from grouping 5 that if we took columns of any greater number than six each we should have fewer than six rows of them, and we have already learned that no number less than six is an exact divisor. The marshalling of our number, then, has shown that it is a prime.

In like manner, if a number has divisors, this method at once shows what they are. Thus, suppose the number had been 36, then we should have obtained groupings 1, 2, 3, and 5, without the odd man over, while the grouping 4 would have shown only one over instead of two. Thus we should have learned that 36 is divisible by 2, 3, 4, and 6 without remainder, and by 5 with remainder one.

So this method shows at once whether a number is an exact square, and if so what its square root is. Thus, if the number had been 36, the marshalling method would give (after perhaps groupings 3 and 4 had been tried) the grouping 5, without the odd man over, and we see that this grouping is a perfect square with six dots on each side. Thus we learn that 36 is a square number, its square root being 6.

For determining whether a number is a perfect cube, the plan which would probably be used by one possessing in a marked degree the marshalling power would be that of grouping his dots into sets having not only length and breadth, as in the groupings above, but height or thickness also. But one less skilful in picturing groupings would simply marshal the number into sets of equal squares, until either he found one set in which there were as many squares as there were dots in the side of each set, or else perceived that no such arrangement was possible. Thus if the number were 27 he would come, by the marshalling method, on this arrangement

three squares, each three in the side, showing that the number is thrice three times three, or is the cube of three. If the number had been 28, say, so that it had come to be grouped mentally, thus,

it would be seen at once that the number is not a perfect cube; for clearly if we try squares fewer in the side we shall have too many, and if we try squares more in the side we shall have too few. We could have a row of seven squares of four each (two in the side) with none over; but that is not what we want. And with larger numbers the result would be equally decisive; so soon as we had a set of squares nearly equal in number to the number of dots in the side of each, with or without any over, we should be certain the number was not a perfect cube; for of squares one more in the side there would be too many, and of squares one less in the side too few. Thus take the number 421. We should presently get, on marshalling, eight squares, each seven in the side, and 29 over, which would not make such a square; but we should only have six complete squares of eight in the side, and we should have eleven complete squares of six in the side.

I do not know which of the two plans described in the preceding paragraph a skilful mental-marshallist would adopt. In my own mental marshalling I never had occasion to seek for the cube roots of numbers. I should say, however, that most probably the second would be the method adopted. For while as yet the computer had had little practice this would be the only available method; and after he had once fallen into the way of it he would not be likely, I should say, to take up the other.

So much respecting the theory I adopted in explanation of Colburn's remarkable readiness in finding divisors, detecting primes, and so forth. It still seems to me probable that he largely made use of this method of marshalling, the power of which few would conceive who had not tried it-though, of course, it only has value for those who possess the power of picturing arrays of objects in great number, and of readily marshalling such arrays in fresh order. Yet it is certain that many calculators proceed on an entirely different plan. For instance, in 1875 I had the pleasure of a long conversation with Professor Safford (of Boston, Mass.), whose skill, when young, in mental calculation had been remarkable. He told me, with regard to the determination of the divisors of large numbers, that he seemed to possess the power of recognising in a few moments what numbers were likely to divide any given large number, and then of testing the matter in the usual way, by actual division, but with great rapidity. He said that to this day he found pleasure in taking large numbers to pieces, as it were, by dividing them into factors; or else, where no such division was possible, in satisfying himself on that point. He had also come to know the properties of many large numbers in this way, remembering always the divisors of any number he had examined, or its character as a prime if it had proved to be so.

What we know about the late Mr. Bidder, who was in some respects the most remarkable of all the calculating boys, leaves no room to doubt that his processes of mental arithmetic were commonly only modifications of the usual processes,-not altogether unlike them, as the theory I formerly advanced would have implied.

The facts now to be related came out in a very interesting correspondence which recently appeared in the pages of the 'Spectator.' The correspondence was suggested by certain remarks respecting the late Mr. G. P. Bidder in a well-written article on Calculating Boys, which seemed to imply that Bidder in after-life showed no marked abilities. 'He had the good sense,' says the writer in the 'Spectator,' 'after de

lighting the " 'groundlings" by performing marvellous arithmetical

feats, to study carefully a profession. He became a civil engineer of some eminence, enjoyed the confidence and esteem of Robert Stephenson, was once President of the Institute of Civil Engineers, and drew up some tables which are of use to his professional brethren.' The writer in the 'Spectator' went on to discuss the powers shown by Colburn, Bidder, and others, referred to Colburn as admittedly a mediocrity, and then said, "The only exception to the rule that juvenile calculators prove mediocrities which occurs to us is Whately, who had undoubtedly for a short time an extraordinary aptitude for figures, akin to that of Bidder and Colburn, and who, if he had been unfortunate enough to have had a father as vain and silly as Colburn's was, might have been exhibited to admiring crowds.' Major-General Robertson sent extracts from letters by Professor Elliot and Mr. G. Bidder, eldest son of the late Mr. G. P. Bidder, in which it was clearly shown that Mr. Bidder the elder showed marked abilities through life, and possessed a remarkable capacity for taking broad and accurate views of all questions in which he was engaged. On this point (which lies somewhat outside my subject) I need not say more than that the writer in the 'Spectator,' with a frankness which more than atoned for his error, admitted that he had been mistaken. What now concerns us, is the evidence adduced respecting Bidder's calculating powers.

6

In the first place, it had been noticed in the original article, quite correctly, that there was a distinction between Bidder's powers and Colburn's. It is important to notice this. It confirms my view that they adopted different methods. 'Bidder, as Colburn admits,' says the 'Spectator,' after describing some of Colburn's feats, was even more remarkable in some ways; he could not extract roots or find factors' (the special class of feats which suggested my theory) 'with so much ease and rapidity as Colburn, but he was more at home in abstruse calculations."

Next let us consider the way in which Bidder's calculating powers were developed from his childhood, one may almost say his babyhood, onwards to a certain point when the study of other matters prevented their further development and caused them gradually to diminish.

We read that at three years of age, 'Bidder answered wonderful

questions about the nails in a horse's four shoes;' but the earliest feat of which I have been able to find exact evidence belongs to his ninth year. When only eight years old, and entirely ignorant of the theory of ciphering, he answered almost instantly and quite correctly, when asked how many farthings there are in 868,424,1217.

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 twentyfour figures with only one or two mistakes." It is evident, therefore, that in the course of two years his powers of memory and calculation must have been gradually developed.

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," he said, "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

1 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,