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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.

George Parker Bidder (1806-1878), "the elder Bidder," was the son of a stone-mason of Devonshire, England. The indications of hereditary influence are stronger in the Bidder family than in that of any other calculator. Bidder's eldest brother, a Unitarian minister, had an extraordinary memory for Biblical texts, but no special arithmetical gift; another brother was an excellent mathematician and an insurance actuary; a nephew early showed remarkable mechanical ability; Bidder's eldest son, George Parker Bidder. Jr. (hereafter referred to as "the younger Bidder"), inherited in considerable degree his father's gift for mental arithmetic, together with his uncle's mathematical ability, being seventh wrangler at Cambridge in 1858; and two daughters of the younger Bidder showed "more than average, but not extraordinary powers of doing mental arithmetic." Other members of the family were distinguished in non-mathematical ways.

At the age of 6 Bidder learned from an elder brother to count to 10, then to 100; this was the only formal instruction in figures he ever received. From counting by units to counting by 10's, and then by 5's, was a natural development. He then set about learning the multiplication table up to 10x10, with the aid of shot, marbles, etc., until, as he expresses it the numbers up to 100 became his friends, and he knew all their relations and acquaintances.

A year or so later his readiness in solving simple problems mentioned in his hearing attracted attention, and he acquired a considerable local reputation. Bits of mathematical information (such as that 10x100 means 1000, etc.) and halfpence contributed by his admirers conduced to the gradual development of his talent, aided by his natural keenness in reasoning about numerical relations; so that he was soon able to perform 4-, 5-, and 6-figure multiplications mentally.

Meantime he came to observe various interesting properties of numbers, - the formulas for the sums of numerous series, casting out the 9's, short cuts in multiplication, properties of squares and of 2-figure endings, and the like. As a result of this "natural" algebra and number-theory he hit upon some ingenious methods of performing complex operations; in particular, by his 11th year he was already in possession of a method by which he could solve compound interest problems mentally in an amazingly short time, in fact, almost as rapidly as a good computer using a table of logarithms. Later, after his meeting and competitive test with Zerah Colburn, in 1818, he acquired great skill in the extraction of roots and the finding of factors, by methods similar to Colburn's, but with improvements of his own.

Colburn (Memoir, p. 175), "Some time in 1818, Zerah was invited to a certain place, where he found a number of persons questioning the Devonshire boy. He [Bidder] displayed great strength and power of mind in the higher branches of arithmetic; he could answer some questions that the American would not like to undertake; but he was unable to extract the roots, and find the factors of numbers."

Bidder's reputation soon became more than local, and when about 8 years old he was exhibited in various places by his father, after the fashion so recently set by the Colburns. But Bidder's admirers, more energetic than Colburn's, actually raised a fund to pay for his education, and put him in school.

Later on, when his father resumed the profitable exhibitions, friends once more intervened, this time with permanent success. The boy was placed with a private tutor, and in 1819 attended classes in the University of Edinburgh, where he took a mathematical prize in 1822.

Leaving the university in 1824, he held positions successively in the Ordnance Survey and in an assurance office. But by the advice of his friends he later decided to devote himself to civil engineering, and ultimately became one of the most successful engineers of his time. He was connected with several engineering undertakings of the first magnitude, and as a member of the Institution of Civil Engineers took a prominent part in the controversies then before the profession. Constant use kept up his calculating powers, and in various railway and other contests before Parliamentary committees his great command of statistics and keen powers of analysis made him a formidable witness.

It would seem that Bidder's powers of mental calculation increased steadily at least up to the beginning of his university days, if not later, and thereafter remained almost undiminished to the end of his life.

Both in numerical calculations and in his study of higher mathematics he was interested in general principles, practical applications, and striking properties, rather than in intricate analysis for its own sake, or calculations with numbers chosen merely for their length, [words missing here] Edinburgh he maintained a good class standing in mathematics including differential and integral calculus, but only by hard study.

In the solution of problems where special properties or symmetries played a part he was equalled, if at all, only by such great calculator-mathematicians as Gauss and Ampere. In division his skill was considerable. In multiplication he was able, on one occasion, to handle two 12-figure numbers, but only by "a great and distressing effort"; in general, he did not cultivate his calculating power much beyond the limits of its practical usefulness to him. In his lecture "On Mental calculation," before the Institution of Civil Engineers, to which reference has already been made, Bidder has left us an excellent account of his methods of calculation.

Sources Scripture, op. cit., p. 23; Proceedings Institution of Civil Engineers, XV, 1855-6, p. 251; LVII, 1878-9, p. 294; Colburn's Memoir, p. 175; Phil. Mag., XLVII, 1816, p. 314; Spectator, LI, 1878, pp. 1634-5; LII, 1879, pp. 47. III.

Spectator (loc. cit.): "If I perform a sum mentally, it always proceeds in a visible form in my mind; indeed, I can conceive no other way possible of doing mental arithmetic"