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.”