VIII. ARITHMETICAL PRODIGIES
Extremely great ability to perform feats of mental arithmetic excites popular wonder and admiration to a degree far beyond that excited by most other manifestations of mental gifts. This may be due to the fact that in calculation each individual has a rather definite standard of performance, namely his own ability to calculate. When another goes far beyond him and his friends, in so definite a performance, he can see for himself that the typical has been phenomenally exceeded. The gifted person who exceeds the typical to an equal extent in perception of the fine shades of meaning in words, or in the detection of absurdities and contradictions in demagogy, creates no sensation among his fellow townsmen; for there is no way whereby the average man can “check up” in the performances, to show himself how phenomenally he has been exceeded in capacity for them.
Bidder, the famous English calculator, is recorded in history because he could perform mental arithmetic perhaps fifty times as well as typical persons. The facts that he also became one of the most successful civil engineers of his time, and made a large fortune, are noted as of merely incidental interest, and would not have given him a place in the history of unusual persons. A man may make fifty times as much money as the average man does, by meeting with fifty times as much acumen and energy the intricate, subtle, and difficult situations offered by modern economic life. Yet he is not so very likely to be regarded as prodigiously gifted. His fellowmen can and will explain the difference between him and themselves as due to luck or circumstance. But a gift for “lightning calculation” is obviously peculiar to the person, and makes of him an object of wonder.
The same general considerations hold in the case of children. Many children of extraordinary intelligence are found, because they have attracted attention to themselves by excellence in arithmetic; and upon examination show themselves to be equally excellent at those tests which measure IQ, excellence in which is not necessarily conspicuous except to the trained psychologist.
Accounts of prodigious calculators go back to ancient Greece, in Lucian’s reference to Nikomachos of Gerase. The word “calculation” means literally “pebbling,” coming from the Latin calculi, pebbles. Records of lightning calculators have been collected by Scripture and by Mitchell.
Jedediah Buxton (b. 1702) appears to be the first calculator on record in modern accounts. He lived at Elmton, England. “He labored hard with a spade to support a family, but seems not to have shown even usual intelligence in regard to ordinary matters of life.... In regard to matters outside of arithmetic he appeared stupid.” In 1754, when he was taken to London to be tested by the Royal Society, he went to see King Richard III performed. “During the dance he fixed his attention upon the number of steps; he attended to Mr. Garrick only to count the words he uttered. At the conclusion of the play, they asked him how he liked it.... He replied that such and such an actor went in and out so many times, and spoke so many words; another so many.... He returned to his village, and died poor and ignored.” It is said that he could give an itemized account of all the free beer he had had from the age of 12 years.
Tom Fuller, “The Virginia Calculator” (b. 1710), seems to be another case of highly specialized ability. He came from Africa as a slave when about 14 years old. He is first heard of as a calculator at the age of 70 years, when it is stated that he reduced a year and a half to seconds in about two minutes, and 70 years, 17 days, 12 hours to seconds in about a minute and a half, correcting the result of his examiner, who had not taken leap years into the reckoning. He also calculated mentally the sum of a simple geometric progression, and multiplied mentally two numbers of nine figures each. He was totally illiterate.
Other prodigious calculators, who are not known to have had superior general ability, are Zerah Colburn (b. 1804), Henri Mondeux (b. 1826), Jacques Inaudi (b. 1867), and Ugo Zaneboni (b. 1867). None of these individuals achieved eminence in any other respect, but this does not necessarily prove that they were not of superior intelligence. It would have been impossible, for instance, for the slave, Tom Fuller, to achieve intellectual eminence in a profession.
None of them was studied psychologically except Inaudi, who was examined by Binet. Inaudi was an Italian by birth. In childhood he tended sheep, as did Mondeux. His passion for numbers began at the age of about 6 years. At 7 years of age he could multiply five-place numbers by five-place numbers, “in his head.” His memory span for digits given orally was 42. He must hear them, the span being considerably reduced if he only saw them. He had little education, and did not learn to read and write until he was 20 years old. He lived by public exhibitions of his power to calculate. Binet concluded that he had no particular ability except the gift for calculation, and was not generally superior.
None of these calculators showed any gift for mathematics beyond arithmetic. Many others are on record who are known to have had great all-round superiority, and mathematical genius of the highest order, as is proven by their achievements. Bidder (b. 1806), Bidder, Jr. (b. 1837), Safford (b. 1836), Gauss (b. 1777), Ampère (b. 1775), Hamilton (b. 1788), and Whatley (b. 1787), all were lightning calculators.
George Parker Bidder was the son of a stonemason, of Devonshire. His family history is on record, and is quite interesting in connection with his gifts. His eldest brother, a Unitarian minister, had an extraordinary memory for Bible texts, but took no special interest in arithmetic. Another brother was an excellent mathematician and insurance actuary. Still other members of the family were distinguished in non-mathematical pursuits. Bidder’s ability was first noticed when he was 6 years old. In 1822, at the age of 16 years, he took a prize in mathematics at the University of Edinburgh. He became a distinguished engineer, and accumulated wealth, as before stated. His son, the younger Bidder, was wrangler at Cambridge, and became barrister and Queen’s counsel. He could multiply fifteen-place numbers by fifteen-place numbers, and could play two games of chess simultaneously, blindfolded. Two of his daughters “showed more than average ability in mental arithmetic.”
Truman Henry Safford was the son of a Vermont farmer, both parents having been school teachers. His power in calculation was noticed when he was 3 years old. At about 7 years of age, he began to study algebra and geometry, and soon thereafter, astronomy. In his tenth year he published an almanac, computed entirely by himself. His interests included chemistry, botany, philosophy, geography, and history in addition to astronomy and mathematics. He took his degree at Harvard in 1854, at the age of 18 years, and became an astronomer. He was professor of astronomy in Williams College for many years, until his death, and made many important astronomical calculations and discoveries.
Carl Frederick Gauss, the great mathematician, was a lightning calculator, the marvels of his performance exceeding those of nearly all others. Gauss entered the gymnasium when he was 11 years old, and in mathematics soon surpassed his teachers. He began the study of higher analysis at 10, and at 14 could read Newton with understanding. At 24 he published Disquisitiones Arithmeticæ, which is a fundamental contribution to mathematics. He himself has related that he remembers having followed by mental arithmetic a calculation concerning the wages of his father’s workmen, and of having thus detected an error in the reckoning, at the age of 3 years. He could use from memory the first decimals of logarithms, and was especially ingenious at discovering new methods. Gauss was unquestionably a person of very extraordinary general intelligence. As a child he mastered not only mathematics, but also the classical languages with wonderful ease. It is quite possible, however, that his gift for mathematics exceeded his general capacity in other respects.
The renown of André Ampère’s achievements in science is commemorated in the ampère. As a child, he showed all-round ability, and encyclopedic interests. He learned counting at 3 or 4 years of age, by means of pebbles, “and was so fond of this diversion that he used for purposes of calculation pieces of a biscuit, given him after three days’ strict diet.” There is no question that Ampère was a child of extremely high IQ, the ability at calculation being but one manifestation of his great genius. He was a chemist, a metaphysician, and a mathematician. He became professor of mathematics, and wrote on probabilities, the unity of structure in organisms, and electrodynamics. In this last field he discovered fundamental truths, and immortalized his name. He was elected to the Academy of Sciences in Paris, and is recognized as one of the world’s great thinkers, not as a calculator merely.
Richard Whatley, Archbishop of Dublin, was a prodigious calculator as a child. From 5 to 9 years of age he astonished onlookers by his feats. He afterwards ceased to interest himself in calculation, but used his intellectual capacity for achievement in other fields.
The greatest calculator on record, according to the researches of Scripture, is Johann Dase, born in Hamburg, in 1824. He could count objects with extreme rapidity. “With a single glance he could give the number, up to 30 and thereabouts, of peas in a handful, scattered on the table”; could give the number of sheep in a herd, or books in a case so quickly that his record remains unequaled. He could carry on enormous and protracted calculations, without recording figures, but seemed not to comprehend mathematical principles. He attended school when 2 to 3 years old, and began public exhibitions at 15 years of age. From the records it is not possible to prove or disprove superior general intelligence.
There are on record but three calculators, who were personally examined by psychologists, so far as the present writer can learn. Inaudi, already mentioned, and Pericles Diamandi, a Greek grain merchant, born in 1868, were examined by Binet. Arthur Griffith, son of a stonemason, born in 1880, was examined by Lindley and Bryan, in the laboratory at the University of Indiana, in 1899.
Binet concluded that Inaudi had no unusual ability except for mental calculation, and that his auditory memory for digits was a special gift. Diamandi, on the other hand, in addition to his ability in calculation, knew five languages, was an incessant reader, and wrote both novels and poetry. He entered school at 7, and remained until he was 16, always heading his class in mathematics. His methods in calculation were visual. “He has a number-form of a common variety, running zigzag from left to right, and giving most space to the smaller numbers. This number-form he sees as localized within a peculiar grayish figure, which also serves as a framework for any particular number or other object, which he visualizes.”
Griffith had, from the age of 3, a passion for counting and made fair records in all studies. He entered school at 10, and attended school seven years. In scope and tenacity of memory, and in rapidity at calculation, he ranked with the best recorded cases, according to the investigators who examined him. Memory was described as very systematic; and rapidity was seen to depend on the great number of numerical relations committed to memory, and upon reduction in number of operations through short-cut methods.
These three examinations were all conducted more than twenty years ago, before standardized methods of measurement had been developed. It is difficult to glean from them, and from the biographical material compiled by Scripture and by Mitchell, what the truth is, as regards the extent to which this gift for calculation was special in these persons. Many of them, as we have seen, were certainly men of genius, with general capacity for selective thinking. Several others probably were not of superior general intelligence, but in no case can we be certain, on the basis of anecdotal evidence alone. Some of them were peasants or slaves, born to manual toil, in the absence of free schools, and in the presence of rigid class distinctions. It is not inconceivable that a child of IQ over 170, condemned by unavoidable environment to herd sheep or pick cotton through his youth, might find relief from the monotony of his work by calculating. As Mitchell, himself a lightning calculator, says, “Given a knowledge of how to count, and later a few definitions, and any child of average ability can go on, once his interest is accidentally aroused, and construct, unaided, practically the whole science of arithmetic, no matter how much or how little he knows of other things.” This statement is probably true, if we change one word, and substitute for “child of average ability,” “child of great ability.”
All who have examined lightning calculators, or searched their biographical records, are agreed that the secret of their power lies in highly developed mechanics. Special habits of combining and recognizing numbers are formed, which differ from ordinary calculation comparatively in somewhat the same way as the method of the child who added 7 + 5 by adding 7 + 2 + 2 + 1, the latter being analogous to the usual method.
The lightning calculator memorizes combinations far beyond those ordinarily memorized, so that he is, for instance, able to add 2581 + 1763 as quickly as an ordinary person can add 15 + 8. He learns multiplication tables up to 100 × 100, whereas we learn only through 12 × 12. He devises and uses many “short cuts,” e.g. multiplying by two easy numbers and taking the difference, instead of multiplying by an awkward number. Multiplication is probably used as the fundamental operation.
This specialization in and perfection of arithmetical connections, by a person of original aptitude for and interest in numbers, results in the prodigious calculator. As Scripture concludes, “These persons had enormous ability to learn calculation, not to calculate without learning.” The rôle played by practice is seen in the fact that if interest in counting wanes, and practice at calculation ceases, the skill acquired deteriorates through disuse. Whatley, and others, who became distracted from calculation by other interests as they grew up, lost the power they had possessed. However, by resuming practice, the skill can be regained by those who have acquired it, as is the case with skills in general.
Satisfaction in mental activity for its own sake is expressed by those calculators who have given introspections. After Safford had lost the power of lightning calculation through disuse, he continued to take pleasure in factoring large numbers, or in satisfying himself that they were prime. The younger Bidder said, “With my father, as with myself, the handling of numbers or playing with figures afforded a positive pleasure, and constant occupation of leisure moments. Even up to the last year of his life,[[16]] my father took delight in working out long and difficult arithmetical and geometrical problems.”
All who have studied material relating to prodigious calculators have especially stressed the very early age at which the gift has shown itself. This is especially true of those who achieved greatness in science, as adults. Gauss, Whatley, and Ampère were all first noted at the age of 3 years, and Safford and Bidder at the age of 6 years. It appears to the present writer to be probable that any child of IQ over 180 could be taught to be a lightning calculator. This inference comes from observing such children, as they master numbers.