Об этих феноменальных личностях было много сказано и написано в совершенно не критичной манере; с одной стороны их считали сверхестественным явлением, в то время как с другой стороны ни чего ненаучного в них не находили. Тем не менее, возможно мы сможем пролить свет на нормальные человеческие процессы человеческого мозга рассмотрением подобных исключительных случаев. Первая задача данной статьи дать короткое перечисление самих этих людей, и дать впервые полную библиографию данного объекта исслодований. Далее будет предпринята попытка сделать такой психологический анализ их возможностей, который поможет понять их, и возможно даст некоторые подсказки практическим инструкторам в арифметике.
НИКОМАХОС. - Луциан говорил, что он не знает как можно лучше похвалить счетовода, чем сказать, что тот считает подобно Никомахосу, из Гераса.1 Относится ли это к вычислительным способностям Никомахоса (около 100 до н. э.), или к известному Вступлению в Арифметику, написанному им, мы остаемся в неведении. Де Морган не согласен с официальным мнением,2 Кантор поддерживает последнее.3 Буквальный перевод пассажа помещает Никомахоса несомненно среди искусных вычислителей.
АФРИКАНСКИЕ ТОРГОВЦЫ РАБАМИ. - Возможно из-за требования работать на улице или, в связи с необходимостью конкурировать с Английскими маклерами, вооруженными карандашами и бумагой, многие торговцы рабами в Африке, старой закалки, похоже всегда были готовы считать, и что важно, также, для практических целей, - момент недооцененный некоторыми последующими счетчиками. "Удивительно, с каким искусством Африканские брокеры подсчитывали изменения в Европейских ценах на рабов. Один из них имел видимо десять рабов на продажу, и для каждого из них требовалось десять различных пунктов. Он мгновенно переводил их в голове в слитки, медь, унции, согласно коэффициента обмена который преобладал в той части страны с которой торговался, и немедленно подводили баланс."4
Рассказывают, что капитаны кораблей жаловались, что становится все более и более сложно совершать выгодные сделки с такими быстрыми вычислителями. Кстати, именно африканец был первым, кто проявился в этой роли в Америке.
ТОМ ФУЛЛЕР. - Упоминания о Фуллере из первых рук следующие: Письмо, прочитанное перед Пенсильванским Обществом по Запрещению Работорговли доктором Рушем из Филадельфии, которое было опубликовано, более или менее полно, в трех местах;5 и некролог который находится в Колумбийском Centinel.6 Позднейшие документы основывались на этих двух.7
Томас Фуллер, известный как Вирджинский Счетчик, был выкраден из родной Африки в возрасте четырнадцати лет и продан на плантации. Когда ему было примерно семьдесят лет, "два джентельмена, исконные Пенсильванцы, а именно Вильям Хартсшорн и Самюэль Коатс, господа честные и порядочные, услышали, во время путешествия по местам где жил раб, о его экстраординарных способностях в арифметике, послали за ним и полностью удовлетворили своё любопытство ответами, которые он дал на следующие вопросы: Во первых, Спрошенный, сколько секунд содержится в одном году с половиной, он ответил примерно через две минуты 47 304 000. Второе: Спрошенный сколько секунд прожил человек которому 70 лет, 17 дней и 12 часов, он ответил за минуту с половиной 2 210 500 800. Один из джентельменов, который был занят с карандашом и бумагой проверки вычислений, сказал, что ответ не верный, и что сумма не так велика как он сказал - на что старик поспешно ответил: 'главный, масса, вы забыли о високосных годах'.
Добавив количество секунд в високосные года, полное количество в обоих случаях полностью совпало."8 Другой вопрос был задан и получил удовлетворительный ответ. Перед двумя другими джентельменами он дал число равное произведению девяти значного на девять.
He began his application to figures by counting ten and proceeded up to one hundred. He then proceeded to count the number of hairs in a cow's tail and the number of grains in a bushel of wheat. Warville says in 1788, "he has had no instruction of any kind, but he calculates with surprising facility."9
In 1790 he died at the age of 80 years, having never learned to read or write, in spite of his extraordinary power of calculation.10
ДЖЕДЕДИЯ БАКСТОН. - Jedediah Buxton11 was born in 1702, at Elmton, in Derbyshire, England, where he died in 1772.12 Although his father was schoolmaster of the parish and his grandfather had been the vicar, his education was by some chance so neglected that he was not able to scrawl his own name.13 All his attainments were the result of his own pure industry; the only help he had was the learning of the multiplication table in his youth; "his mind was only stored with a few constants which facilitated his calculations; such as the number of minutes in a year, and of hair's-breadths in a mile."14 He labored hard with his spade to support a family,15 but seems to have shown not even usual intelligence in regard to ordinary matters of life. The testimony as to his arithmetical powers is given by two witnesses. George Saxe says : "I proposed to him the following random question: In a body whose three sides are 23,145,789 yards, 5,642,732 yards, and 54,965 yards, how many cubical 1/8ths of an inch? After once naming the several figures distinctly, one after another, in order to assure himself of the several dimensions and fix them in his mind, without more ado he fell to work amidst more than 100 of his fellow-laborers, and after leaving him about five hours, on some necessary concerns (in which time I calculated it with my pen) at my return, he told me he was ready: Upon which, taking out my pocket-book and pencil, to note down his answer, he asked which end I would begin at, for he would direct me either way. ... I chose the regular method, ... and in a line of twenty-eight figures, he made no hesitation nor the least mistake."16 "He will stride over a piece of land or a field, and tell you the contents of it, almost as exact as if you measured it by the chain. ... He measured in this manner the whole lordship of Elmton, of some thousand acres, ... and brought the contents, not only in acres, roods and perches, but even in square inches; ... for his own amusement he reduced them to square hairs-breadths, computing (I think) 48 to each side of the inch."17 Various other problems were solved by him with like facility on later occasions, before a different witness.18
From May 17 to June 16, 1725, he was (to use his own expression) drunk with reckoning, by which a kind of stupefaction was probably meant. The cause was the effort to answer the following question: In 202,680,000,360 cubic miles how many barley-corns, vetches, peas, wheat, oats, rye, beans, lintels, and how many hairs, each an inch long, would fill that space, reckoning 48 hairs in breadth to an inch on the flat! His table of measures, which he founded on experiment, used in answering this was:
Quite curious is Buxton's notation for higher numbers. His system is: Units, thousands, millions, thousands of millions, millions of millions, thousand millions of millions, tribes, thousands of tribes, etc., to thousand millions of millions of tribes; cramps, thousands of cramps, etc., to thousand million of million of cramps; tribes of cramps, etc. to tribes of tribes of cramps.
In regard to subjects outside of arithmetic, his mind seemed to have retained fewer ideas than that of a boy ten years old. On his return from a sermon he never brought away one sentence, having been busied in dividing some time or some space into the smallest known parts. He visited London in 1754, and was tested by the Royal Society. On this visit he was taken to see King Richard III performed at Drury Lane playhouse, but his mind was employed as at church. During the dance he fixed his attention upon the number of steps; he attended to Mr. Garrick only to count the words that he uttered.20 At the conclusion of the play they asked him how he liked it. He replied "such an actor went in and out so many times and spoke so many words; another so many, etc."21 He returned to his village and died poor and ignored.
АМПЕР. - The first talent shown by Andre Marie Ampere,22 *1775, at Lyon, 1836, at Marseilles, was for arithmetic. While still a child, knowing nothing of figures, he was seen to carry on long calculations by means of pebbles. To illustrate to what an extraordinary degree the love of calculation had seized upon the child, it is related that being deprived of his pebbles during a serious illness, he supplied their places with pieces of a biscuit which had been allowed him after three days strict diet.
As soon as he could read he devoured every book that fell into his hands. His father allowed him to follow his own inclination and contented himself with furnishing him the necessary books. History, travels, poetry, romances and philosophy interested him almost equally. His principal study was the encyclopedia in alphabetical order, in twenty volumes folio, each volume separately in its proper order. This colossal work was completely and deeply engraved on his mind. "His mysterious and wonderful memory, however, astonishes me a thousand times less than that force united to flexibility which enables the mind to assimilate without confusion, after reading in alphabetical order matter so astonishingly varied."23 Half a century afterwards he would repeat with perfect accuracy long passages from the encyclopedia relating to blazonry, falconry, etc.
At the age of eleven years the child had conquered elementary mathematics and had studied the application of algebra to geometry. The parental library was not sufficient to supply him with further books, so his father took him to Lyon, where he was introduced to higher analysis. He learned of himself according to his fancy, and his thought gained in vigor and originality. Mathematics interested him above everything. At eighteen he studied the Mecanique analytique of Lagrange, nearly all of whose calculations he repeated; he said often that he knew at that time as much mathematics as he ever did.
In 1793 his father was butchered by the revolutionaries, and young Ampere was completely paralyzed by the blow. Rousseau's botanical letters and a chance glance at Horace roused him after more than a year from an almost complete idiocy; and he gave himself up with unrestrained zeal to the study of plants and the Augustan poets. At the age of twenty-one his heart suddenly opened to a new passion and then began the romantic story of his love, which is preserved in his Amorum and his letters.24 Ampere became professor of mathematics, chemistry, writer on probabilities, poet, psychologist, metaphysician, member of the Academy of Sciences of Paris, discoverer of fundamental truths of electrodynamics, and a defender of the unity of structure in organized beings.25
Just as he began by learning completely the encyclopedia of the 18th century, he remained encyclopedic all his life, and his last labors were on a plan for a new encyclopedia.
ГАУСС. - The arithmetical prodigies might be divided into two classes, the one-sided and the many-sided. The former would include those who like Buxton, Colburn and Dase were mere "reckoning-machines," the other would consist of men in whom the calculating power was only a part of gifts of mathematical talent like Safford, or even of the highest mathematical genius like Gauss.
Carl Friedrich Gauss was born in 1777, in Braunschweig. He was the offspring of a poor family that had in nowise distinguished themselves, although his mother seemed to have been of finer mental build than the paternal stock. Moreover his maternal uncle was a man of unusual talent: com-
2Смитовский Словарь Греческих и Римских Биографий. Никомахос.
3Кантор, Vorlesunyen uber Geschichte der Mathematik, Лейпциг, 1880, I, 363.
4[Т. Кларксон.] Эссе о Рабстве и Комерции Человеческим Видом, в частности Африканцами. 2-е Издание, Лондон, 1788. (Пассаж на который ссылаемся не появился в Американском издании, Филадельфия., 1788, 1787, 1804).
5Американский Музей, Том V, 62, Филадельфия, 1799.
Steadman, Narrative of a five years expedition against the Revolted Negroes of Surinam, South America, 2v. 4o, Лондон, 1796, Том II, 260. Во французском переводе, Том III, 61.
Ниидлс, Исторические Мемуары Пенсильванского Общества по Запрещению Работорговли; Филадельфия, 1848, с. 32.
6Columbian Centinel of Boston, Dec. 29, 1790, No. 31 of Vol. XIV.
7Например, Gregoire; An Enquiry concerning the Intellectual and Moral Faculties, and Literature of Negroes, followed with an Account of the Life and Works of Fifteen Negroes and Mulattoes; Перевод Д.Б.Вардена; Бруклин, 1810. (The translation is from Gregoire's original manuscript.) Brissot de Warville; New Travels in the United States of America, performed in 1788; London, 1792, p. 287; 2d Ed., London, 1794, vol. I, 243; Boston, 1797 (reprint of 1st ed.), p. 158; in the original French edition, vol. II, p. 2. Williams; History of the Negro Race in America; New York, 1883, vol. I, 399. Didot's Nouvelle biographie generale v. Fuller.
11Gentleman's Magazine, 1751, Vol. XXI, p. 61, 347; 1753, vol. XXIII, p. 557; 1754, vol. XXIV, p. 251, which are the original authorities. Chalmer's General Biogr. Dictionary, London, 1812, v. Buxton; Rose, New General Biogr. Dictionary, London, 1848, v. Buxton; Didot's Nouvelle biographic generale, v. Buxton; Michaud's Biographic universelle v. Buxton.
12The dates are given on the authority of Lyson's Magna Britannia, London, 1817, vol. V, Derbyshire, p. 157.
13"His total want of education bas been attributed to his excessive stupidity when a child, and an invincible unwillingness to learn anything." Lyson's Magma Britannia, V, 157, note.
15"A day-labourer," Lyson's Magna Britannia, loc. cit. "Either a small land-owner or a day-labourer; but probably the former," The Journey-Book of England, Derbyshire; London, 1841, p. 79.
22Bibliography (for his life), Saint-Beuve, M. Ampere, sa jeunesse, ses etudes diverses, etc., in the Revue des deux Mondes, 1837, 4 ieme serie, t. IX, p. 389. M. F. Arago, Eloged' Ampere (given in a somewhat condensed form by E. Arago, in Michaud's Biogrgraphie universelle, nouv. ed., v. Ampere), translation in Smithsonian Reports, 1872, p. 111. Didot's Nouvelle biographic generale, v. Ampere. Valson, Vie d' Ampere, Lyon, 1886.
23Arago, Eulogy on Ampere, Smithsonian Reports, 1872, p. 113; Michaud's Biogr. universelle, I, p. 597.
24Andre Ampere, Correspondence et souvenirs, Paris, 1873.
25List of Works in Michaud's Biogr. universelle, nouv. ed., I, p. 611.
