28 DECEMBER 1878, Page 14




SIR,-I annex some extracts from letters received from Mr. James Elliot, of Goldielands, near Hawick, lately professor of mathematics in Queen's College, Liverpool, and from the eldest son of the late Mr. G. P. Bidder, having reference to the article which appeared in the Spectator of September 28th last, headed 4. Calculating Boys." The facts stated in these extracts respecting the remarkable mental powers possessed not only by the late Mr. G. P. Bidder, but also by many of his relatives and descendants, will, no doubt, interest many readers of the Spectator.—I am, Sir, &c.,

"I was very much dissatisfied with the article in the Spectator of September 28. It seemed like an attempt to depreciate Mr. Bidder's abilities. Ignorance is not a sufficient excuse, as he was so well known in London. Besides being a 'calculating boy,' he was possessed of first-rate business ability, and of a rapid and clear insight into what would pay, especially in railway matters. He became a wealthy man, which ia a good proof of his possessing that insight. I have a right to give an opinion on this subject, having known him intimately from the timo that he and I were fellow-students in Edinburgh at the University classes of mathematics and natural philosophy until the year of his death. The Spectator might have said something more specific as to the extent to which his arithmetical power was carried. To me it was incomprehensible, as difficult to believe as a miracle. You might read over to him fifteen figures, and another lino of the same number, and without seeing or writing down a single figure, he would multiply the one by the other, and give the product correctly. The rapidity of his calculations was equally wonderful. Giving his evi- dence before a Parliamentary Committee rather quickly and decidedly, with regard to a point of some intricacy, the counsel en the other side interrupted him rather testily by saying, Yon 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. 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. The diagram stood out before him with all its lines and letters. And this remarkable faculty is possessed to a certain extent by several of his descendants. Talent in a variety of forms has always been show- ing itself among his family and relatives. His eldest son, now a thriving barrister and Queen's Counsel, distinguished himself at Cam- bridge in mathematics, being seventh wrangler of his year ; and his second son in classics at Oxford being first-class man and Fellow of his college. These two sons were both pupils of mine. The eldest (the barrister) can play two games of chess simul- taneously with as many opponents without seeing the board. He, like his father, can multiply fifteen figures by fifteen without seeing them ; but then he does it by a peculiar process, which he can explain, and is willing to explain. He avails himself of a system of mnemonics which requires a first-rate memory, and would be no help whatever to me, but a hindrance. The late Mr. G. P. Bidder, I am confident, used no such artificial process. Several of Mr. G. P. Bidder's nephews and grandchildren also possess very remarkable powers. One of his nephews at an early age showed a degree of mechanical ingenuity beyond anything I have ever seen in a boy. The summer before last, to test the calculating powers of some of his grandchildren (daughters of Mr. George 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 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 is the correct answer; she had worked it out in bed. Another grandaughter once remarked to me, Isn't it strange ; when I hear anything remarkable said or read

to me, I think I see it in print.' As regards the points to which you especially refer in the article in the Spectator, taking them seriatim. First, as far as my experience has gone in teaching mental arithmetic, I have found those boys that were cleverest otherwise, best and quickest at it. I have, however, met with one exception in the case of a ball- idiot, who was remarkable in his own country district for his powers of calculation. I got him to put down his operations in a few eases on paper, and his modes of abbreviation were ingenious. The schools in which I have seen mental arithmetic carried to the greatest perfection were the Sessional School of Edinburgh and the Central Lancastrian School in London. (2.) I suspect that rapid falling off only occurs in cases where the calculating power has fallen into neglect, or been pushed aside by other pursuits. I never saw a-ny appearance of Mr. Bidder'a faculty falling off. (3.) The word precocious implies that ' all preco- cious calculators were at their best before their minds were disciplined and stored with knowledge,' but in the schools I have mentioned, high calculating powers did not diminish, but were developed along with

other powers as education progressed. Many of our greatest men have shown remarkable precocity in certain of their faculties, but this precocity did not accompany or produce feebleness

in their other faculties. (4.) No doubt some bankers, clerks and shopmen, too, are quick at mental calculation, but the work:- of such calculations differs from that of Mr. Bidder, not only in degree, it probably differs also in kind. The skill of the one is the result of the special training of ordinary faculties ; the- skill of the other is the result of the ordinary normal development of a special faculty. The power of vivid or, as it were, photographic representations, no trace of which is to be met with in ordinarily constituted minds, may perhaps be suggested as one of the characteristic marks of the possession of this special faculty. It may be noted that bankers do not much encourage mental calculation in their clerks. They prefer having everything put down on paper. An anecdote told of Sir William Curtis illustrates this preference. Four applicants for a vacant clerkship in his bank came before him in succession. He asked each of them, 'What is tho SUM of 4 and 3?' Three of the applicants- answered at once, ' Seven ;' but the fourth -(who had probably got a hint) said, 'I will tell you in a minute, Sir!' and taking out his pocket- book, put down the numbers and their sum in figures, You are the

man for me,' said Sir William, like no mental calculations.' 4. I grant that the faculty of arithmetical calculation has not much con- nection with true aptitude for mathematics, but it is by no means inconsistent with it. 5. In some cases it may be true that the sense of number comes late to children who grow up vigorous-minded men, but this is not the general rule. I think the combination of early- developed powers of calculation with dullness of the senses and a sluggish, unobservant, unimpressionable character, must also be quite exceptional."

Extracts from a letter of Mr. G. Bidder, of Ravensbury Park, Mitcham, Surrey (eldest son of the late Mr. G. P. Bidder), dated November 24th :—

" No one who knew anything of my father would have classed him among the mediocrities, as the writer of the article in the Spectator did. He was distinguished throughout his whole life by remarkable mental powers, as well as for his capacity of taking broad and accurate views of all questions on which he was engaged, and his opinion was greatly sought after and relied on, both in commercial and engineering matters. It would take take too lone to enumerate all the undertakings initiated and carried through by him, but I may, as examples, refer to the Elec- tric Telegraph Company, of which he was one of the original founders, and amongst the first to perceive the importance of the telegraph as a means of communication. The magnificent Victoria Docks at North Woolwich, and I may say the whole of that now populous district, were the creations of his mind. He was called mad for proposing to con- struct docks away from London, in what were then marshes and water- meadows, but he had formed a just estimate of the capabilities of the position, and not only succeeded in carrying out his ideas, by the construction of the docks (a work of great engineering, skill), but had such confidence in their future that he persuaded the dock company to buy at the time sufficient land for their extension to treble their original size, and this extension is now being cassis d out. He was consulted, as one of a committee, by the Admiralty, with reference to the best types of ships-of-war, and by the Chancellor of the Exchequer upon certain very ingenious altera- tions in taxation. As to his arithmetical powers and processes, the latter are described by him in a lecture which be delivered at the Institution of Civil Engineers, in the session of 1855-56, when vice- president. From his earliest years 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 he had an immense power of realising the actual number. In multiplication his process- was simply cross multiplication, so as to get the answer figure by figure. He was aided, I think, by two things,—first, a powerful memory of a peculiar cast, in which figures seemed to stereotype themselves with- out 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 my- self (though not to compare with my father). Professor Elliot says he saw mental pictures of figures and geometrical dia- grams. I always do. If I perform a sum mentally, it always pro- ceeds in a visible form in my mind ; indeed, I can conceive no other way possible of doing mental arithmetic. The second faculty, that of rapid operation, was no doubt congenital, but developed by incessant practice and by the confidence thereby acquired. I am certain that unhesitating confidence is half the battle. In mental arithmetic, it is most true that ' he who hesitates is lost.' When I speak of inces- sant practice,' I do not mean deliberate drilling of set purpose ; but with my father, as with myself, the mental handling of numbers or- playing with figures afforded a positive pleasure and constant occupa- tion 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 diffi- cult arithmetical and geometrical problems. When at college he studied very carefully (as his note-book which I have shows) the higher mathematics, including the differential and integral calculus, but he never seemed fond of algebraic analysis. It is also worthy of record that my father had an enormous store of facts, formula), and constants relating to all manner of geometrical questions and physical subjects, which were always available for the ready solu- tion of problems either in pure mathematics, or in the applications of mathematics to mechanics, hydraulics, ifte. In my opinion, this is a kind of knowledge which is not half-appreciated. I have found con- tinually immense advantage in having formulaa and constants ready to hand. On one occasion, my father was called as a witness in an im- portant railway contest in Parliament. The opposing counsel formally took exception to his being examined, on the ground that it was well know* his powers were so unexampled, that nobody could cope with him or check him, and that to allow him to be examined would there- fore place the other side at an unfair disadvantage. I need hardly say the objection did not prevail. As to the other members of my father's family, his eldest brother (who was a Unitarian minister) was not re- markable 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. Another brother was an excellent mathematician. He was actuary of the Royal Exchange Life Assurance Office. I my- self can perform pretty extensive arithmetical operations mentally, but I cannot pretend to approach even distantly to the rapidity or accuracy with 'which my father worked. 1 have occasionally multiplied 15 figures by 15 in my head, but it takes me a long time, and I am liable twoccasional errors. Last week, after speaking to Professor Elliot, tried the following sum to see if I could still do it,-

378,201,969,518,825 199,631,057,265,413

Alia I got in my head the an swer, 75,576,299,427,512,145,197,597,834,725, in which I think, if you take the trouble to work it out, you will find four figures out of the 29 are wrong. My children show considerable, I may say certainly more than average, but not extraordinary powers of doing mental arithmetic."