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The American Journal of Psychology XVIII April 1907 – Mathematical Prodigies – Frank D Mitchell

By a "mathematical prodigy" we shall mean a person who shows unusual ability in mental arithmetic or mental algebra, especially when this ability develops at an early age, and without external aids or special tuition. We shall use the word "calculator" in the sense of "mental calculator," as a synonym for "mathematical prodigy," and shall usually mean by "calculation" "mental calculation," unless the contrary is clearly indicated by the context. A "professional calculator" will be taken to mean a mental calculator who gives public exhibitions of his talent. "Computer," however, will be restricted to mean one who calculates on paper. All problems mentioned as solved by the mathematical prodigies will be understood to be done mentally, unless otherwise indicated.

Zerah Colburn (1804-1840), the son of a Vermont farmer, was regarded as a backward child until the end of his 6th year when one day his father heard him repeating parts of the multiplication table, though the boy had had only about six weeks schooling. The father then "asked the product of 13x97 to which 1261 was instantly given in answer. He now concluded that something unusual had actually taken place; indeed he often said he should not have been more surprised, if some one bad risen up out of the earth and stood erect before him."

The elder Colburn now took Zerah about the country, giving public exhibitions of the child's powers in various cities. Colburn was thus the first professional calculator, in the sense already defined. From the list of questions answered by him at Boston, in the fall of 1810, and from the account in the body of the Memoir, it appears that even at this early date, only four months after the discovery of his talent, he was a good calculator, though of course he improved with further practice. It is clear, therefore, that his powers had been developing for some time - to judge from other cases at least six months, if not a year - before they attracted his father's attention. This may mean that he learned to count from his elder brothers and sisters, - the eldest was about seven years older than Zerah, - rather than from his own brief six weeks at school. Colburn's preference for multiplication, the extraction of roots, factoring, and the detection of primes seems to have developed early; he never became as proficient in division as Bidder, for example, and, like most of the prodigies, he used addition and subtraction only incidentally, in the service of other operations, not for their own sake. In answering catch questions and in repartee he was moderately clever.

In the spring of 1812 Zerah was taken by his father to London. Here, among other feats, he found mentally, by successive multiplication, the 16th power of 8 (=281474976710656) and the 10th powers of other 1-figure numbers, also, though with more difficulty, the 6th, 7th, and 8th powers of several 2-figure numbers. The square root of 106929 (=327) and the cube root of 268336125 (=645) were found "before the original numbers could be written down." He immediately identified 36083 as a prime number, and found "by the mere operation of his mind" the factors, 641 and 6700417, of 4294967297 (=232+1).1

While in London, Colburn learned to read and write, and later began the study of Algebra; but his education was subject to long interruptions, owing to the constant financial difficulties caused by his father's lack of business ability. After visits to Ireland and Scotland, the Colburns went, in 1814, to Paris, where Zerah spent eight month at school, studying mainly French and Latin.

Returning to England early in 1816 he entered Westminster School in September, under the patronage of the Earl of Bristol, making fair progress in the languages, and standing well in his class, in which, however, he was one of the oldest boys. He also studied six books of Euclid under a private tutor, but stowed no marked geometrical aptitude.

 In 1819 his father removed him from school, and soon after we find him, at his father's suggestion, unsuccessfully attempting the career of an actor and playwright. In 1822 he opened a small school, which ran for a year or more. His next occupation was as a computer in the service of the secretary of the Board of Longitude. Shortly after his father's death, in 1824 Zerah returned to America, and in December of 1825 joined the Methodist church, becoming a circuit preacher. After seven years of this occupation, being in need of funds to eke out his modest ministerial salary, he wrote the Memoir, carrying out a plan which his father and friends had had in view long before. In 1835 he resumed teaching, as "Professor of the Latin, Greek, French and Spanish Languages, and English Classical Literature in the seminary styled the Norwich University." He died in 1840.

From this brief account of Colburn's romantic career, it will be seem that his education, while much interrupted, was fairly good. He spent four or five years in the study of languages, for which he seems to have had a natural liking, and later was able to teach them. He began the study of algebra, but did not get beyond the elements of it; and he studied geometry, which he found easy but uninteresting, owing to the lack of any visible practical application. The literary style of his Memoir, though far from Addisonian, is always readable, the book is interesting throughout, and even the specimens of his poetry given in the appendix are not specially bad, all things considered. …..

Concerning the rapidity of Colburn's calculations not much is known. The only series of problems whose times he gives us dates from 1811, before he was 7 years old, and so is hardly typical of his performances two or three years later when he was in his prime. The times indicated are fairly short, in most cases shorter than if the work had been done on paper by a good computer. The testimony of observers as to his "extraordinary rapidity" is of little value in the absence of definite figures; especially since some of his feats, notably the extraction of square and cube roots and the finding of factors, were accomplished by the aid of extremely simple methods. Colburn's powers probably increased up to the time of his visit to Paris in 1814; but when he gave up his regular exhibitions, and became interested in other matters, he gradually lost much of his skill. There seems to be no authority, however, for the statement that after a time his powers left him entirely; in 1823, at any rate, after a considerable period of disuse, they were readily revived for purposes of written longitude computations.

Of his methods of calculation Colburn has left us a very good account; the only calculator of whom we have a fuller account is Bidder, whose methods closely resembled Colburn's. Both men, in multiplication, began at the left, instead of at the right as we usually do in written computations; and both, by the aid of certain properties of the 2-figure endings of the numbers used, were able to find with remarkable ease and rapidity the square and cube roots of exact squares and cubes and also, though less rapidly, the factors of fairly large numbers.

Colburn had two physical peculiarities that need to be mentioned. (1) He possessed an extra finger on each hand and an extra toe on each foot. This peculiarity he shared with his father and two of his brothers. (2) In his early years his calculations were accompanied by certain bodily contortions, similar to those of St. Vitus' dance. They seem to have passed away rather early; Colburn himself has no recollection of them, and mentions them simply on the authority of persons who saw him when "quite a child."

Sources:  Scripture, op. cit., p. 11; A Memoir of Zerah Colburn, written by himself, Springfield, 1833; Philosophical Magazine, XL, 1812, p. 119; XLII, 1813, p. 481; Analectic Magazine, I, 1813, p. 124; Carpenter, Mental Physiology, §205, p. 232; Cornhill Magazine, XXXII, 1875, p. 157; Belgravia, XXXVIII, 1879, p. 450; Gall, Organology, §XVIII, pp.84-7 (in On the Functions of the Brain, V, Eng. tr., Boston, 1835). Scripture gives two other references which the writer has been unable to consult: The American Almanac, 1840, p. 307, and the Medical end Philosophical Journal and Review, III, 1811, p. 21. Gall's account, however, seems to be based upon this last article.

We read in Baily's account (Analectic Magazine, I, 1813, p. 124) that "any number, consisting of 6 or 7 places of figures, being proposed, he [Colburn] will determine, with... expedition and ease, all the factors of which it is composed."