Мнемоника - Статьи Институт Математики и ее Применений Март/Апрель/Май,1987, Том 23, с. 68-71

Джордж Паркер Биддер: Чудо-счетчик
ДЖОЙС ЛИНФУТ, AFIMA Колледж Люси Кавендиш, Кембридж

Англия, в которой родился Джордж Биддер в 1806, была Англией в которой уже произошла индустриальная революция. В начале девятнадцатого века было беспрецедентное развитие в расширении производства товаров всевозможных видов, которое на некоторое время даже опередило необходимости как дома, так и за границей. История этого периода конечно хорошо известна, но один из аспектов о котором было сказано мало - большой объем арифметических вычислений который и обеспечил это развитие. Это было отражено в школах и учебниках: примеры, даваемые детям для работы, требовали такие сложные вычисления, что по современным стандартам они были бы просто неприемлемы. Издание учебников по арифметике стало прибыльным делом. Книга Джона Бонникастла, опубликованная в 1795 (которая по словам Биддера была единственной книгой по арифметике которую он открывал), имела огромный успех в первой половине девятнадцатого столетия, и дожила до восемнадцатого, последнего издания в 1851. Эта книга содержала такие темы, как прочтые и сложные проценты, комиссия, брокераж и страховка, скидка и предоставление ежемесячной и ежегодной рент. Обучение состояло главным образом в точном выполнении примеров в формах которым надо было точно следовать. Обученные таким образом дети становились бухгалтерами и счетоводами, вливаясь в новую индустрию. Другой человек, предложивший свой путь в удовлетворении нужд в эффективных вычислениях (в данном случае это было для создания математических таблиц, как он предполагал) был Чарльз Баббидж, который а 1822 впервые написал президенту Королевского Общества предложение сконструировать вычислительную машину. Несомненно важно, что Чарльз был в это время близким другом Джона Гершеля, который совсем недавно принимал участие в раннем образовании Биддера ... Вычисления были "в воздухе", и когда в 1815 Биддер выдал свои выступления по устной арифметике, аттракцион был не только в том, что мальчик демонстрировал удивительные способности. Интерес, проявленный множеством известных людей, подтверждает, что это искусство имело связь с нуждами того времени.

Джордж Паркер Биддер, родившийся в Мортонхэмпстоне в Девоне в 1806, был сыном каменщика. When he was 5 or 6 years old, an older brother taught him to count; he then taught himself to multiply by arranging and counting lead pellets in rectangular arrays. Soon after this he began to earn halfpennies from the villagers by doing simple calculations in his head. His father took him to the local fairs, where his earnings increased, and they then began to go further afield. With visits to Exeter, Bristol, Oxford, Birmingham..., a lucrative business was developing, and by the age of 9, Bidder had a national reputation. At Cambridge he came to the attention of John Herschel, who, with the help of some of his friends succeeded in arranging that George should be sent to a school in London, to give him a chance of becoming an educated man. Sir William Herschel also took so much interest in him that he arranged for him to give a performance to the Queen at Windsor.

But the next year his father took him away from school, and the travelling exhibitions were resumed. A likable personality, he attracted favourable attention everywhere. After a visit to Edinburgh, a subscription was raised among members of the Royal Society and others for his further education. This eventually resulted in his entering the University there. He studied mathematics and geology, and left after 3 years, at the age of 19, for a post in the Ordnance Survey. After a year he left to become the assistant of H. R. Palmer, the Engineer of the London Dock Company. Thenceforward he had a very varied and successful career as a civil engineer; he was a friend from University days of Robert Stephenson, and during the 1830s and 1840s he made great contributions to the planning and and construction of the railways of this country. (He had a formidable reputation as an expert witness in cases brought under the Railway Acts.) By many notable engineering undertakings, abroad as well as in England, he gained both fame and fortune, and became, in 1859, the President of the Institution of Civil Engineers.

There is good reason at the present time to remember George Bidder. In the middle, as we are, of a revolution in computing methods (one might almost say a "computing mania" in some ways paralleling the "railway mania" of the nineteenth century) there is some danger of forgetting how great the calculating powers of the unassisted human brain can be. This is a capability which should not be lost sight of. To understand it is to have a better understanding of our condition.

George Bidder is the ideal example for such an exposition, because there is so much incontrovertible evidence about his accomplishments. In booklets published within a few years of his performances there exist about two hundred questions, quoted word for word, which he answered during his performances. In most cases we know the time taken by him to give the answer, we know the date and place, the names of people present. The local newspapers carried detailed accounts of his visits to their towns. There can be no doubt whatever about the authenticity of these records. The men who vouched for him included Fellows of St. John's College, Cambridge, Fellows of the Royal Society of Edinburgh, and others of similar standing.

Another factor which adds considerably to our information is the lecture which George Bidder himself gave in 1856 at the Institution of Civil Engineers, under the Chairmanship of Robert Stephenson, who was then the President. In this his mathematical development is outlined, from the straightforward multiplication sums which he could perform mentally at the age of 7 or 8, to problems involving the summation of arithmetical and other progressions (including the calculation of compound interest) which he was doing, by methods he had himself devised, when he was 13 or 14. These methods were a natural development of his earlier work. In the multiplication of numbers, for instance, he had always dealt first with the left-hand digits, so that he at once obtained an idea of the size of the answer. This became progressively more refined as he added in the partial products, each as soon as it was found, so that at any stage all he had to hold in his memory was the result he had so far obtained, and the stage of the calculation which he had reached. GPB learnt from his questioners: problems which he was given at the age of 10 had introduced the idea of an arithmetical progression, and he had been forced to invent a method of summation. What he did is worth examining in detail. At this time, GPB knew no algebra: all he had was his great familiarity with numbers, and his interest in the patterns which they form. He began with the basic series 1+2+3+4+... and found successively the sums of the first, the first two, the first three, the first four... terms, thus arriving at the sequence: 1, 3, 6, 10,... each term of which was the sum of the corresponding number of terms of the first series. If, therefore, he could find the law of formation of the terms of the sequence, he had his answer. He began by comparing the corresponding terms of the series and the sequence thus:

1 1

,

3 2

,

6 3

,

10 4

,...

and he then saw that these ratios could be expressed as

2 2

,

3 2

,

4 2

,

5 2

,...

Here was the general law he needed, and he could immediately say that, for instance, the twelfth of these ratios was 13/2, and that the sum of twelve terms of the first series was 12 times this, i.e., 13 x 12/2.

Put into words, the argument seems long, but GPB had such a clear grasp of the process that to him it was obvious. Indeed, it is doubtful if he used words at all when he was thinking about it. The speed and effortlessness with which he was able to sum these progressions was a necessary factor in enabling him to answer correctly the formidable questions which were later put to him. The example which follows comes from a pamphlet published in Exeter: GPB did this computation on October 17th, 1818, when he was 12 years 4 months old: the original wording has been copied: "A and B made the following bet for one thousand guineas, to be decided on Ripley Green in the Whitsun-week. The proposer has 10 choice cricketers in full exercise, who on this occasion are to be distinguished by the first ten letters of the alphabet: these are to run, gather up, and carry singly, 1000 eggs laid in a right line, each just a fathom behind the one before it: they are to work one at a time, in the following order: A is to fetch up the first ten eggs, B the second ten, C the third ten, and so on, until K shall have carried up the thousandth egg, at 100 eggs a man: the men are to have £100 for their three days' work, if they do it, and it is to be distributed in proportion to the ground each man shall in his course have gone over." (N.B. The old alphabet was being used, in which there was no J, and K was the tenth letter. The starting point and the 1000 egg positions were all to be in a straight line: the distance between consecutive points was 2 yards.) The question was to find each man's share, and also the total distance he had travelled.

It is, of course, a highly artificial example, but certainly a very exacting test of calculating ability. Using his knowledge of arithmetical progressions, it would be easy for GPB to say at once that in collecting his first batch of eggs, A goes 4 x (1 + 2 + 3 + . . . + 10) yards, i.e., 220 yards. In his second batch, since 100 eggs have already been collected, the first egg he must pick up is 200 yards further on than the corresponding egg of the first batch, and similarly for each of the ten eggs of the second batch. This adds an extra distance of 4000 yards to the 220 yards he travelled for his first batch.

The argument can be repeated for each of his remaining batches: this gives the total distance he travels as the sum of an A.P. of which the first term is 220 the difference is 4000, and the number of terms is 10, i.e., 182 200 yards.

Similarly, B goes 4000 yards further than A, C goes 4000 further than B, and so on. Again there is an A.P. to be summed: the total distance run by all the men is found to be 2 002 000 yards.

After this point there is still a good deal of heavy calculation. The payment per yard is £300/2002000, which means that A's payment is

£300     2 002 000

X 182 200 =

£300 x 911 10010

.

Table I  GPB's results as printed in 1820   Computer results  £ s d fracts d farthings A 27 6 8 1/2 509 0 2.5894 B 27 17 0 1/2 014 0 2.0140 C 28 10 0 1/4 439 0 1.4386 D 29 2 0 464 0 0.8631 E 29 14 0 288 0 0.2877 F 30 5 11 3/4 713 11 3.7123 G 30 17 11 3/4 137 11 3.1396 H 31 9 11 3/4 562 11 2.5614 I 32 1 11 3/4 987 11 1.9860 K 32 13 11 1/4 411 11 1.4106

This answer GPB worked out in pounds, shillings, pence and farthings, and "tracts," where a fract is defined as one thousandth part of a farthing. He then went on, still completely mentally, and without making any written record, to finish the calculation, stating the payment due to each man, and the distance he had run. (For brevity, the distances are omitted.) These results which he gave for the payments, as printed in the pamphlet recording the feat, have now been checked by the use of an electronic computer. The results are shown in Table I. In only five instances (indicated by underlining in the Table) is there a difference, in each case of a single figure. It seems most probable that these are misprints.

GPB gave his answers after 32 minutes. The time taken by computer, including the time needed to write and "de-bug" the programme, was about half-an-hour.

George Bidder's ability to calculate compound interest was astonishing: the method he devised was based on his earlier work with progressions, though the age at which he first achieved it has been, and still is, a matter for conjecture. Mitchell (1907) puts his age at 11, but the example which he quotes "What is the interest on £4444 for 4444 days at 4½ per cent. per annum?" was, as the answer shows, on simple interest.

In none of the pamphlets, which cover the period up to his exhibitions in August 1819 when he was 13, is there a single example of compound interest. That autumn he went with his father to Edinburgh, where he so impressed the "members of the University, and other literary and mercantile gentlemen of Edinburgh" that arrangements were made to pay for his education at the University, and he gave no more exhibitions. The account which was published in the Edinburgh Correspondent of his first arrival in Scotland has unfortunately proved impossible to trace, but the circumstantial evidence strongly suggests that it was in the early autumn of that year that the method for the calculation of compound interest was perfected.

It is to Robert Stephenson that we owe what account we have of the way GPB thought about this problem. In 1856 he persuaded GPB, who had been a friend of his for more than thirty years, to give a lecture on "Mental Arithmetic" to the Institution of Civil Engineers, of which Stephenson was then the President. GPB had, on his own admission, great difficulty in putting his procedure into words, and in spite of great efforts he was unable to produce the usual written communication. His first draft, which exists in his own handwriting, is an amazing production, in which every number is written out in words, but it is next to impossible to follow the argument.

In the end, the lecture was given extempore, with demonstrations. What he said was taken down and published in the Proceedings of the Society. The section on compound interest has recently been republished verbatim (Smith, 1983). Not unexpectedly, it is somewhat difficult to follow, and no attempt has been made to explain or correct it. "Ideas," GPB himself said, "rise to my mind with the speed of lightning". . . and such flashes were not easily captured. But familiarity with the methods of his earlier work makes it easier to see what he is doing: the calculation of compound interest is approached as a natural extension of his method for summing an arithmetical progression.

In his lecture he gives the example of the interest on £100 for 14 years at 5 per cent. The simple interest is £5 coming in every year for the whole period, and amounting to £70.

Each £5 as it comes in begins to produce interest of 5s a year. (For convenience, the old style of currency is kept.) The next stage is to find the total sum of all these 5s. During the whole period, the first £5, which comes in at the end of the first year, produces 13 of them, the second £5 which comes in at the end of the second year produces 12, the third 11, and so on. The 13th produces 5s at the end of the 14-year period. So the total number of 5s which have come into the account is 13 + 12 + 11 + ... + 1 and this GPB arranged as 1 + 2 + 3 + 4 + ... + 13 (First Series).

He found the sum of this A.P. by his usual method, as 14 x 13/2, giving the total contributions of the 5s as 91 x5s = £22-15s, and the answer so far as £92-15s. But the 5s themselves are producing interest at the rate of 3d a year, and the next step is to count these threepences. GPB took first that contingent of (5s)s which arise from the first £5. The first comes in after 2 years and runs for 12 years, the second comes in after 3 years and runs for 11 years and so on.

The total number of (3d)s in this contingent is therefore (12 + 11 + 10 + ...) = 13. 12/2.

The second contingent, arising from the second £5, is similar, but everything is a year later, so that the number of 3ds in this contingent is 12 x 11/2; from the third £5 there are 11 x 10/2, and so on. The last 3d comes from the 12th £5, which produces 5s in the 13th year, and this has 3d interest at the end of the 14-year period.

The number of 3ds is, therefore, 13 x 12/2 + 12 x 11/2 + ... 4 x 3/2 + 3 x 2/2 + 1 which GPB as usual reversed, and gave in the form 1 + 3 + 6 + ... + 78 (Second Series).

To find this sum, GPB followed the method he had used for A.P.s and by summing successively 1, 2, 3... terms he obtained 1 + 4 + 10 + 20 + ... (Third Series), and compared each term with the corresponding term of the second series. These ratios are

1 1

4 3

10 6

20 10

,

or

3 3

,

4 3

,

5 3

,

6 3

.

Clearly the 12th ratio is 14/3, i.e., the required sum of twelve terms of the second series is 14/3 x the 12th term of the second series. But this term is 13 x 12/2 and the number of 3ds is, therefore,

14 3

X

13 x 12 2

=

14 x 13 x 12 3 x 2 x 1

.

and the contribution from all the 3ds is 364 x 3d = £4-lls.

This method could obviously be continued, but since the contributions of successive stages diminish so rapidly, not many stages are needed.

Table II Factor  Contribution  Total so far  (1)  14 x 1/20 £70           £70 (2)   13/2 x 1/20   £22-15s     £92-15s (3) 12/3 x 1/20 £  4-11s     £97- 6s (4) 11/4x1/20         12s-6d £97-18s-6d (5) 10/5 x 1/20           1s-3d £97-19s-9d (6) 9/6x1/20               1d   £97-19s-10d

What is so delightful to the mental calculator is that at each stage the contribution can be very easily deduced from that at the one which precedes it, simply by multiplying by the appropriate fractions, in this case by 1/20 for the rate of 5 per cent., and by, successively, one of the ratios 13/2, 12/3, 11/4,... and so on. In this particular example, the work goes as shown in Table II, and GPB could complete such a calculation in about 1 minute. No wonder the "mercantile gentlemen" were impressed, and we too may be disposed to marvel.

GPB had a poor opinion of the way in which, in his day, arithmetic was taught. Practical experience in counting and reckoning, and practice in mental arithmetic, should come, he thought, before the boys learned to write down numbers. "If a boy has been taught to know numbers only by the artificial signs which represent them," he said, "I should much fear his ever being able afterwards to possess a clear conception of the value of any number or quantity." A boy was in a hopeless position if he had been taught dogmatically, and arrived mechanically at his results, without any appreciation of what the figures represented. "With the usual forms of teaching arithmetic, not one boy in a hundred can give any reason for what he does."

In the 1856 lecture, GPB added the method of calculating compound interest by the use of the binomial theorem, but he made it quite clear that this was not what he had done when he was a boy. "I have endeavoured to show," he said, "by what kind of process my mind at a very early age, and when wholly unacquainted with symbolic representation, and algebraic expedients, analysed the law connecting these series and rendered them available for computation."

My thanks are due to Mr. J. S. Linfoot, who suggested and carried out the computer check on the "cricketers" problem.

Библиография

Bidder, G. P., "On Mental Calculation," Min. Proc. ICE, 1856.

Bonnycastle, John, "A Scholar's Guide to Arithmetic," London, 1795.

Linfoot, Joyce, "The Calculating Phenomenon"; Part IV of Clark, E. F., "George Parker Bidder, The Calculating Boy," KSL Publications, Bedford, 1983.

Mitchell, F. D., Am. Psych., 1907, XVIII, 61-143.

Smith, Steven B., "The Great Mental Calculators," Appendix, 1983.

Use has also been made of an anonymous compilation, "Some Extraordinary examples in Mental Calculations by G. Bidder, a Devonshire Youth," M. Bryant and Co., Corn Street, Bristol. This is one of a group of four pamphlets, dated about 1820, which contain questions answered by GPB during his exhibitions. They are to be found in the Graves Collection in the Library of University College, London, and my thanks are due to the Librarian for making them available to me.