MATHEMATICAL PRODIGIES. By FRANK D. MITCHELL.

THE AMERICAN
JOURNAL OF PSYCHOLOGY

VOL. XVIII

APRIL, 1907.

No. 1

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MATHEMATICAL PRODIGIES¹ By FRANK D. MITCHELL.

The object of the present paper is threefold:
(1) To give a summary of the mathematical prodigies² described in the literature of the subject, without, however, duplicating unnecessarily the work of previous writers.
(2) To give a brief account of the writer's own case, which is, it is believed, fairly typical, despite certain peculiar limitations to be described later, and which will shed light on certain factors in mental calculation that have not hitherto received fail recognition.
(3) To set forth a new theory of mental calculation, based upon the foregoing data, and incidentally to criticise certain other theories hitherto advanced in this field.
I.

In view of the incompleteness of existing data in most cases, and the inaccessibility of some even of the existing sources of information, a complete history of the mathematical prodigies would be out of the question. We shall, therefore, simply attempt to give a reasonably complete list of those of whom definite information is available, together with a statement of the significant facts known about them. A few names - that of Euler, for example - have been omitted on account of the absence of any satisfactory data that would shed light on the theory of mental calculation; and no attempt has been made to collect the accounts of new prodigies found every now and then in the newspapers. Such accounts are not readily accessible, and

¹From the Psychological Seminary of Cornell University.
²By 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.

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are usually so popular and unreliable that they have little scientific value.
There are several possible bases for a classification of the mathematical prodigies. We might group them chronologically, as Scripture¹ does; or by the extent of their power, as measured either by the size of the numbers they could handle or by the rapidity of their calculations; or by the degree of their mathematical ability, as shown by the character of the problems they solved and the processes they used. Or we might classify them according to memory type, as either visual or auditory calculators. No one of these classifications, adhered to consistently throughout, would quite answer the purpose here, owing to the great unevenness of the material at hand in the case of the different calculators. An arrangement has therefore been adopted which is in part chronological, but which is modified by most of these other considerations. In this way, so far as the crossing of the different principles of division permits, those men are in the main brought together who are most naturally compared, and the important points of resemblance and difference come out more conveniently than if an abstractly logical arrangement were adopted.
We begin, then, with Fuller and Buxton, who have much in common, and who are the first modern calculators about whom reliable data are available. Colburn, Mondeux, and Inaudi form the next group, followed by Zaneboni, Diamandi, and Dase. Then come the two Bidders and Safford, followed by Gauss and Ampere, and finally those who may be called "minor prodigies," whether because of limited powers of calculation or because the available information is not sufficient for a more detailed account.
Tom Fuller² (1710-1790), "the Virginia calculator," came from Africa as a slave when about 14 years old. We first hear of him as a calculator at the age of 70 or thereabouts, when, among other problems, 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 failed to take account of the leap-years.³ He also found the sum of a simple geometrical pro-

¹In his article on "Arithmetical Prodigies," in the American Journal of Psychology, IV, 1891, pp. 1-59. We shall hereafter have frequent occasion to refer to this article, the only one in English in which a comprehensive study of the subject is attempted.
²Scripture, op. cit., p. 2; Binet, Psychologie des grands calculateurs et joueurs d'echecs, 1894, p. 4; American Museum, V, 1789, p. 62. This last date is erroneously given by Scripture as 1799.
³Binet, op. cit., p. 5, notes that the harder problem was done in less time than the simpler one, and is inclined to suspect that the records are unreliable. But in the case of so slow an plodding a calculator

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gression, and multiplied mentally two numbers of 9 figures each. He was entirely illiterate.
Jedediah Buxton¹ (1702-1772) was very stupid even from boyhood. Though his father and grandfather were men of some education, he remained illiterate all his life, and was of less than average intelligence; even the statement of a mathematical problem he comprehended, we are told, "not without difficulty and time." In calculation he was, like Fuller, extremely slow; but he had a prodigious memory, and could retain long numbers for days or even months, so that be performed enormous calculations, which in some cases occupied him for weeks. On one occasion he mentally squared a number of 39 figures, in 2½ months. His methods were original, but very clumsy; to multiply by 378, in one instance, he multiplied successively by 5, 20, and 3 to get 300 times the number, then by 5 and 15 to get a second partial product, and finally by 3, to complete the operation. Thus instead of adding two zeros to multiply by 100, he multiplied first by 5 and then by 20. This fact, together with his slowness, shows pretty clearly that his methods were of counting rather than multiplication, though we are told that he had learned the multiplication table in his youth. He could give from memory an itemized account of all the free beer he had had from the age of 12 on. He was able to calculate while working or talking, and could handle two problems at once without confusion. At a sermon or play Buxton seems to have paid no attention to the speaker's meaning, but to have amused himself by counting the words spoken, or the steps taken in a dance, or by some long self-imposed calculation. He could call off a number from left to right or from right to left with equal facility, and by pacing a piece of ground could estimate its area with considerable accuracy.
Zerah Colburn² (1804-1840), the son of a Vermont farmer,
as Fuller, little importance can be attached to such discrepancies, especially since the times given are only approximate. Moreover, Fuller was at this time about 70 years old himself, and may therefore have had in his memory, already calculated, the number of seconds in 70 years. The times given seem to indicate that he used a process of modified counting, rather than multiplication in the ordinary sense. The importance of this distinction will appear later.
¹Scripture, op. cit., p. 3; Gentleman's Magazine, XXI, 1751, pp. 61, 347; XXIII, 1753, p. 557; XXIV, 1754, p. 251.
¹Also spelt Colborne. 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

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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."⁸ 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, be 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
the writer has been unable to consult: The Amerian 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.
¹Memoir, pp. 11-12. Scripture (op. cit., p. 12) is "tempted to ask for the authority on which the statements were made", and inclined not to "put too much faith in the figures", on the ground that Colburn never speaks of himself as having any extraordinary power of memory for long periods of time. But the full passage as quoted above makes it clear that the father had told the incident repeatedly to awe-stricken listeners in Zerah's hearing; moreover, the remembering of such a simple problem could hardly require "extraordinary power of memory" in a person used to mental calculation. Colburn's feats in factoring large numbers are hard to explain except by supposing that he remembered at least those numbers which he had previously examined and found prime. This would simply a rather considerable development of his memory for figures. At any rate, there is nothing improbable in his remembering the figures quoted in the text, even for some years after his calculating powers had declined.
It may be noted that later in his article Scripture's faith in Colburn's memory increases; for on page 46 he thinks we can presuppose in the case of Colburn and certain others an extended multiplication table, perhaps even to 100x100. Reasons for rejecting this supposition, in Colburn's case at any rate, will appear later.

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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

¹Memoir, pp. 37-8, quoting from a prospectus printed in London, 1813. From Colburn's own account of his methods of factoring (pp. 183-4), it appears that the only way in which he could immediately identify as prime such a number as 36083 would be by remembering the result of a previous examination of it. Scripture (op. cit., p. 14, note) says that it "requires considerable faith" to accept the statement that Colburn factored 2³²+1. But we are not told that he did it "instantly"; a friend of Morse's says simply, "almost as soon as it was put to him" (Scripture, loc. cit., quoting from a letter in S. I. Prime's Life of Samuel F. B. Morse, p. 68; the reference is undoubtedly to this problem), while Carpenter (Mental Physiology, p. 233; the writer has not been able to find Carpenter's authority for this statement) says, "after the lapse of some weeks." Even if the time was only a matter of some minutes, the feat is not incomprehensible. The smaller factor, 641, might easily have been hit upon by a lucky trial at a very early stage of the work. 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." Now 2³²+1 is only a 10-figure number, or three figures longer than those Colburn was used to handling; and the smallness of the factor 641 renders the problem much simpler than it at first appears. Since, then, the feat is entirely possible, and since it is cited by Colburn from the publicly circulated Prospectus of 1813, and is mentioned by at least one contemporary writer who was not acquainted with the Memoir, there is no reason for believing that Colburn fabricated the incident; especially since his limited mathematical knowledge would never have shown him the importance of this particular number. Had he been inventing out of whole cloth, he would have multiplied together two prime numbers chosen at random, and would probably have made the smaller one at least a 4-figure, if not a 5-figure number. On the historical reliability of the Memoir see Appendix I.

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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. It 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,¹ 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."² 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

¹I. e., in 1832 or 1833. Cf. Memoir, p. 31, "after possessing the talent twenty-two years", from August, 1810; p. 142, "nine years' residence here" in America, from June, 1824; p. 166, "twenty-two years ago", to 1810 or 1811; p. 167, "the last seven years that he has spent in the traveling connection", from December, 1825. These passages show that the Memoir was not begun, or at any rate had not reached the third chapter, before 1832, and was not completed until shortly before its publication in 1833. Scripture's statement, therefore (op. cit., p. 11, note²), that "there is no statement regarding the time at which they [the Memoir(s)] were written, or even a date to the preface; the last year mentioned in the book is 1827", is decidedly misleading. The last date printed in figures, to be sure, so that it could be identified by a cursory glance, is 1827; but the last date "mentioned" is certainly 1832, if not 1833, even granting that all the periods of time above quoted are only approximate, and cannot be taken without an allowance of half a year one way or the other for possible error.
²Scripture, op. cit., p. 16, quoting from American Almanac, 1840, p. 307.

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considered. The question of the historical reliability of the Memoir will be discussed later; for the present it will suffice to say that, on a careful reading, the book shows scarcely a trace of that self-glorification with which it has been charged by Scripture and Binet.
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 statement¹ 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,² 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 endings³ of the

¹Scripture, op. cit., p. 15.
²Bidder's account is more detailed, better written, and in more concise mathematical language than Colburn's, as a result of Bidder's superior educational advantages; it contains, furthermore, explanations of several of Bidders's feats, such as the solving of compound interest problems, which would have been hopelessly beyond Colburn's powers. At the same time Colburn's account is perfectly clear, to the non-mathematical reader perhaps even clearer than Bidder's. In this matter, as in several others, Scripture is hardly fair to Colburn; thus he speaks of Colburn's explanations as "the least intelligible of all the explanations" (p. 50). It is no reproach to Colburn that he was excelled by Bidder; but he certainly deserves credit for what he did do, and one of the things he did was to write a very good account of his methods, over twenty years before Bidder followed his example.
³By a 2-figure ending we shall mean the last two figures of any given number; thus 56 is the 2-figure ending of 3456, 01 of 2401, 07 of 7, etc. What properties of these endings were used by the mental calculators will be explained hereafter.

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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 two¹ 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."²
Henri Mondeux³ (1826-1862) was the son of a woodcutter near Tours. Sent to tend sheep at the age of 7, he amused himself by playing with pebbles, and thus learned mental arithmetic. Jacoby, a schoolmaster at Tours, hearing of him sought him out, offered to instruct him, and gave him his address in the city; but the boy's memory outside mathematics was so poor that he forgot both name and address, and found the schoolmaster only after a month's search. He received instruction in arithmetic and other subjects, and in 1840 was exhibited before the Paris Academie des Sciences. In the committee's report on him we are told that he "carries on readily in his head not only the various arithmetical operations, but also, in many cases, the numerical solution of equations; he devises processes, sometimes remarkable, for solving

¹Colburn says (Memoir, p. 72), "his father and two of his [father's] sons," while the account in the Philosophical Magazine (XLII, 1813, pp. 481-2) says Zerah and three of his brothers. It has been assumed in the text that Zerah did not count himself, and that the other writer counted him twice; this is the simplest way of reconciling the two statements. The peculiarity had been in the Colburn family, we are told, for several generations.
²Memoir, p. 173. Scripture does not refer to this second peculiarity; but since Colburn mentions another mathematical prodigy with a similar affliction, and since Safford showed a striking nervousness in his early calculations, it haa seemed worth while to mention the matter. Gall, probably quoting from the Medical and Philosophical Journal and Review article already cited, seems to refer to this nervousness when he says (op. cit., V, p. 86): "While he [Colburn] answers, it is seen, by his appearance, the state of his eyes, and the contraction of his features, how much his mind labors." Colburn was not quite 7 years old when seen by the writer of the article on which Gall's account is based. Gall himself, however, examined Colburn in Paris, probably in 1814. Cf. Memoir, pp. 76-7.
³Scripture, op. cit., p. 21; La grande Encyclopedic, art. Mondeux; Cauchy's report on Mondeux, in Comptes rendus hebdomadaires des stances de I'Academie des Sciences, XI, 1840, pp. 840, 952; reprinted in Oeuvres Completes de Cauchy, ie Serie, 1885, V, p. 493, and in Binet, op. cit., pp. 14-22. The writer has been unable to consult the other references cited by Scripture.

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a great number of different questions which are ordinarily treated by algebra, and determines in his own way the exact or approximate value of integral or fractional numbers which satisfy given conditions." More specifically, he finds powers of numbers by rules of his own discovery which are equivalent to special cases of the binomial theorem; he has worked out formulas for the summation of the squares, cubes, etc., of the natural numbers, and for arithmetical progression and other series; he solves simultaneous linear equations by a method of his own, and sometimes equations of higher degree, especially where the root is a positive integer; and he solves such problems in indeterminate analysis as finding two squares whose difference is a given number. He "knows almost by heart the squares of all whole numbers under 100." learning a number of 24 figures, divided into four 6-figure periods, requires 5 minutes. He can solve a problem while attending to other things.
Mondeux's admirers hoped that he would one day distinguish himself in a scientific career; but this was not the case. Like his successor Inaudi, whom he closely resembles in several respects, he became a professional calculator; but he had no ability outside of mathematics, and even there his powers soon reached a limit beyond which they did not increase. He died in obscurity. If we may judge by the Academy report, he was almost the equal of Bidder in his insight into mathematical relations;¹ but on the numerical side he was far excelled by Inaudi, who could, for example, memorize 24 figures in half a minute, a feat for which Mondeux required 5 minutes.
Jacques Inaudi² (b. 1867), an Italian by birth, passed his early years, like Mondeux, in tending sheep. An anecdote which Binet regards as rather doubtful indicates a possible prenatal influence in the direction of calculation; otherwise there is nothing noteworthy in his heredity. His passion for figures began about the age of 6, and at 7 he could carry on mentally multiplications of 5 figures by 5 figures. His education is very slight; he did not learn to read and write until he was 20 years old. Outside of mental calculation he has no special ability; his memory for most things except figures is rather poor, and he is often absent-minded. At last accounts he was still a professional calculator, living by public exhibitions of his talent. He visited the United States in 1901-2,

¹Just how much Mondeux owed to Jacoby's teaching is hard to say. The writer has been unable to consult Jacoby's Biographie d'Henri Mondeux or Barbier's Vie d'Henri Mondeux; Binet, however, who cites both these works, says that Jacoby's lessons were "sans grand sncces." (Op. cit., p. 14.)
²Binet, op. cit., pp. 24-109, 199-204, et passim.

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appearing in many of the larger cities, and is said to have been fairly well received by American audiences.
Telling on what day of the week a given date falls is one of his favorite problems. The reduction of years, mouths, etc., to seconds he accomplishes almost instantly, knowing by heart the number of seconds in a year, month, week, or day. He solves by arithmetic problems corresponding to algebraic equations of the first and sometimes of higher degree, also such problems as the resolution of a given 4- or 5-figure number into the sum of four squares. In these latter cases, however he proceeds for the most part simply by trial, aided, of course' by his skill in calculation and his familiarity with many squares' cubes, and the like. At his regular performances the pro-gramme includes the subtraction of one 21-figure number from another, the addition of five 6-figure numbers, the squaring of a 4-figure number, the division of one 4-figure number by another, the extraction of the cube root of a 9-figure number and the 5th root of a 12-figure number, or such similar problems as may be proposed by the audience. As each number is announced he repeats it slowly to his assistant, who writes it on the blackboard and then reads it aloud, to make sure there is no mistake. Inaudi then repeats the number once more, after which he devotes himself to the solution of the problem, meanwhile making an occasional remark to keep the audience in good humor. Throughout the exhibition he faces the audience, never once looking at the blackboard. Actually he begins his calculation as soon as the numbers are given, and carries it on during the various repetitions of the numbers by himself and his assistant, so that by the time he seems to begin the solution he may be well advanced toward the answer. In this way he appears to work much more rapidly than he really does.
Inaudi is a well-marked instance of the auditory¹ memory type. When he thinks of numbers, in calculation or otherwise, he does not see them "in his mind's eye," as arrays of dots or other small objects, or as written or printed figures; numbers are for him primarily words, which he hears as if spoken by his own voice, and during his calculations he almost always pronounces at least some of these words, either with partial distinctness or in a confused murmur. Any interference with

¹Actually it would be more correct to call his type auditory-motor, and the same is probably true of most of the other auditory calculators we shall study, since a pure or non-motor auditory individual is rare. For convenience, however, the writer has followed Binet's terminology. The meagreness of our information in most cases makes it difficult to tell just what part the motor element plays; and this is especially true when we are dealing with a limited field like calculation, where the motor element may often play a less important part than in certain other fields.

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this habitual articulation embarrasses him, and prolongs his calculation. He remembers a number very much more readily after hearing it than after seeing it; in fact, if a written number is banded to him, he usually reads it aloud, in order to learn it by sound rather than by sight. Whether visual images are entirely absent is a purely theoretical question; it is at least clear that, if present at all, they play a negligible part in his mental computations. We shall later find reason to believe that this condition is by no means so rare as has been supposed. Owing to the traditions of English and French psychology, the visual theory of mental calculation has lain ready to hand, and has in the past found much apparent confirmation. But now that an unmistakably non-visual calculator is on record, it will no longer do to beg the whole question; we must insist on considering each case upon its own merits, either settling it by definite evidence or leaving it frankly in doubt. We shall see later how much of the supposed evidence for the visual theory falls before a careful examination.
One of Inaudi's most marked characteristics is his powerful memory for figures. In one experiment he was able to repeat, after a single hearing, though with an effort, 36 figures, read off to him slowly in groups of three; but in the attempt to repeat 50 figures under the same conditions he became confused, and got only 42 of them correct. This latter number, 42, Binet therefore takes as the limit of Inaudi's power of acquisition, or "mental span," under these conditions. In an experiment made to determine in what time he could learn 100 figures read off to him in groups as often as requested, he learned the first 36 in a minute and a half, the first 57 in 4 minutes, 75 in 5½ minutes, and the whole 100 (actually there were 105) in 12 minutes. On the other hand, he can repeat in order, at any time within a day or two, all the figures used in his last performance, whether in the statement of the problems, in the answers, or in the intermediate calculations. The number of these figures at times runs as high as 300, and the total duration of the performance is usually not more than 10 or 12 minutes. Each new performance, however, blots out of his memory almost entirely the figures used in the previous one; but such constants as the number of seconds in a year, etc., as well as many powers and products, and any particular numbers or results in which he for any reason takes a special interest, remain permanently with him. These facts show how important it is to take account of the conditions of such experiments if the figures established by them are to have scientific value. In an experiment lasting the same length of time as one of his regular exhibitions, but under very different conditions, Inaudi can learn only a third the number of figures he

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remembers with ease under his usual conditions. In these public performances, however, each number in the probem as given is repeated several times (twice by Inaudi himself and once each by his assistant and the proposer of the question), and the figures of the various calculations and the result have a logical connection in the problem. Moreover, the numbers are learned in relatively short stages, separated by intervals in which they can be assimilated.¹
Concerning the rapidity of Inaudi's calculations we have fairly full information, - so much fuller, in fact, than we have for any previous calculator, that no satisfactory comparisons can be made. Since the results of Binet's experiments are readily accessible, a brief summary of them will here suffice In each experiment the subject was given a written column of numbers, each of which was to be mentally increased or diminished, multiplied or divided, by the same number; in other words, the addend, subtrahend, multiplier, or divisor was uniform for the whole given column of numbers. The results were called off down the column as fast as obtained, and the average time for each single operation thus determined. These tests were made on some of Binet's pupils, on Inaudi, and on four department store cashiers who were thoroughly practiced in addition, subtraction, and multiplication of small numbers, and could perform mentally 2-figure multiplications²², and in some cases, though with difficulty, 3-figure multiplications. The students were of course considerably slower than Inaudi and the cashiers; but the cashiers, in dealing with the smaller numbers to which they were accustomed, were fully as rapid as Inaudi, in some cases slightly more rapid. In dealing with larger numbers, however, which exceeded the limits of their customary calculations, their inferiority to Inaudi was very marked.

¹Mondeux, it will be remembered, required 5 minutes to learn 24 figures, whereas learning this number of figures is a common incident of Inaudi's exhibitions, and takes only half a minute. Here again, however, the results are not directly comparable. Mondeux learned the number in groups of 6 figures, and presumably from a paper or blackboard, while Inaudi always groups numbers in periods of three, and learns them by audition instead of vision. We shall refer later to a distinction which must be made between the direct and immediate remembering of figures which results from deliberately committing them to memory, and the very rapid and abbreviated automatic calculations which in some of the prodigies simulate direct memory. Recollection as the result of repeated calculation may form an intermediate stage in the passage of the latter into the former. These distinctions will become important in connection with the much discussed question whether, and to what extent, the mental calculators possessed extended multiplication tables.
²By a 2-figure, or n-figure, multiplication will be understood hereafter a multiplication in which each of the two numbers contains 2 (or n) figures, and the product 3 or 4 (2n-1 or 2n) figures.

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Ugo Zaneboni¹ (b. 1867), an Italian, born in the same year as his countryman Inaudi, received a fair education. His interest in numbers began at the age of 12, and when 14 he could solve² any problem his teacher proposed to him. While serving his term in the army he was for a time stationed at a railroad depot, where he amused himself by gradually committing to memory a vast body of statistics relating to timetables, distances between different cities, population, tariffs, etc. When he later took to the stage as a professional calculator, questions based on these statistics formed part of his regular programme. Among his other usual feats are the repetition, either forwards or backwards, of a memorized number of 256 figures, the squaring of numbers up to 4 figures and the cubing of numbers up to 3 figures, finding the 5th powers of 2-figure numbers, and, conversely, extracting the 5th root of any number of 10 figures or less, the cube root of any 9-figure number, and the square root of any number of 7 figures or less, whether the given number is a perfect power or not. In these problems he is aided by his knowledge of many perfect squares, cubes, etc., as well as by various properties of 2-figure endings, with which he is thoroughly familiar. He possibly has a number-form, in which the numbers from 1 to 10, from 10 to 100, and from 100 to 1000 are arranged along three horizontal lines. This number-form, however, if it really exists, plays little or no part in his actual calculations.
Pericles Diamandi³ (b. 1868), the son of a Greek grain merchant, attributes his calculating gift to his mother, who "has an excellent memory for all sorts of things." One brother and one sister, out of a family of fourteen, share his aptitude for mental arithmetic. He entered school at the age of 7, and remained there until he was 16, always standing at the head of the class in mathematics. But it was only after entering the grain business himself, in 1884, that he discovered his powers of mental calculation, which he now found very useful. He knows five languages, - English, French, German, Roumanian, and his native Greek, - and is a great reader; he has read all he can find on the subject of mental calculation; and he has written novels and poetry, concerning whose quality, however, Binet does not enlighten us. It will thus be seen that Dia-mandi's education is much better than Inaudi's, and his range

¹Rivista sperimentale di Freniatria, etc., XXIII, 1897, pp. 132-159, 407-429. A summary of these articles, in German, is found in the Zeitschrift fur Psychologie und Physiologie der Sinnesorgane, XVI, 1898, p. 314. The writer is indebted to Mrs. Rose Harrington for a translation of considerable portions of the original Italian articles.
²Mentally, it is to be presumed, though the article is not explicit on this point.
³Binet, op. cit., pp. 110-154, 98, 187 ff, passim.

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of interests correspondingly wider, but that he was far less precocious in calculation than his rival.
Diamandi is of the visual memory type. 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. He has colored audition for the names of various persons, the days of the week, etc., and if a few figures in a given number differ in color from the rest he remembers the colors without effort. If the color scheme is more complicated, however, he first memorizes the number and then learns the colors of the individual figures. He always sees numbers as written in his own handwriting, and preferably, if the numbers are large, in a rectangle as nearly square as possible, rather than in one or two long Hues. He learns spoken figures (in French) much less readily than written, since in the case of spoken figures he must not only call forth the corresponding visual images, but translate the numbers into his native Greek, in which all his calculations are carried on. Where he seeks to learn the figures very accurately, for purposes of calculation, he is only about half as fast as Inaudi;¹ but where he is concerned with speed rather than accuracy his times are much shorter. In the one case he learned 10 figures in 17 seconds; in the other, 11 figures in 3 seconds.
In calculation Diamandi is considerably slower than Inaudi, whether the numbers concerned are large or small. His time was 127 seconds for a 4-figure multiplication, whereas Inaudi could accomplish the same feat in 21 seconds. Diamandi finds the various figures of the product in order, from right to left, by cross-multiplication; thus in such an example as
46273
729
416457
92546
323911
33733017
he finds the figures of the partial products not in the horizontal lines of the ordinary method, but in vertical lines, - first

¹Here again, however, we must be careful about direct comparisons of dissimilar data, since Diamandi learned from a paper and wrote out his results, while Inaudi depended on audition and speech. Moreover, Diamandi's times were found to be subject to considerable variation from day to day.

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7, then 5, 6, then 4, 4, 1, then 6, 5, 1, etc., - and adds each column before he proceeds to find the numbers that compose tire next column. This method has the advantage that the various figures of the partial products can be forgotten almost as fast as obtained, since that figure of the total product which depends on a given column of the partial product is found and recorded as soon as the column is known, and the numbers in that column therefore play no further part in the calculation. On Diamandi's performances in other operations than multiplication Binet gives us no data.
Johann Martin Zacharias Dase¹ (1824-1861) was born in Hamburg. Concerning his heredity we have no information. He attended school at the age of 2½ years, but attributed his powers to later practice and industry rather than to his early instruction. He seems to have been little more than a human calculating machine, able to carry on enormous calculations in his head, but nearly incapable of understanding the principles of mathematics, and of very limited ability outside his chosen field. In this respect he resembled Buxton; but in the rapidity and extent of his calculations he was incomparably superior to Buxton, or indeed to any other calculator on record. He multiplied together mentally two 8-figure numbers in 54 seconds, two 20-figure numbers in 6 minutes, two 40-figure numbers in 40 minutes, and two 100-figure numbers in 8¾ hours; he could extract the square root of a 60-figure number in an "incredibly short time," and the square root of a 100-figure number in 52 minutes. All these times, with the exception of that for the 100-figure multiplication, are probably more rapid, in some cases much more rapid, than those of a good computer using paper. Buxton, it will be remembered, once succeeded in multiplying two 39-figure numbers; other calculators, however, seem to have been unable to handle multiplications much above 15 figures. But if there was any definite limit to Dase's powers, the experiments of which we have record do not show it. We shall later find reason for believ-

¹Also spelt Dahse. The full name is given on the authority of Brockhaus's Konversations-Lexikon, ed. 1898, art. Dase. Scripture, following the title-page of Dase's posthumously published Factoren-Tafeln (3 vols., 1862-5), gives the name as simply Zacharias Dase, which seems to be the way in which Dase usually wrote it. On Dase'a life and calculations see Scripture, op. cit., p. 18; Briefwechsel zwischen Gauss und Schumacher, Altona, 1861, III, p. 382; V, pp. 30, 32, 277-8, 295-8, 300-304; VI, pp. 27-8, 78, 112; Crelle's Journal (Journal f.d. reine u. angewandle Mathematik), XXVII, 1844, p. 198; Zacharias Dase, Factoren-Tafeln, Hamburg, Vol. I, 1862, Preface; Schroder, Lexikon d. hamburgischen Schriftsteller, 1851, art. Dase; Preyer, "Counting Unconsciously," Pop. Sci. Monthly, XXIX, 1886, p. 221; Brockhaus's Konversations-Lexikon, 1898, art. Dase. For other references see Scripture, loc. cit.

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ing that the 100-figure multiplication was not really a sever tax upon his powers of mental arithmetic. In short, Dase's achievements so far transcend those of any other recorded calculator that he stands in a class by himself, unapproached b any of his rivals.
At the age of 15 Dase began his public exhibitions, and continued them for a number of years. He soon numbered among his friends several eminent mathematicians, however, and their influence gradually led him more and more to devote his vast powers to the service of science.¹ Among his (non-mental) computations are included the determination of the value of p to 200 decimal places² by the formula
p
4 = tan^-1 1
4 + tan^-1 1
8 + tan^-1 1
8 ,
a labor of two months; the computation of the 7-place natural logarithms of the numbers from 1 to 1005000; and factor-tables for the 7th and 8th millions (except a small portion) and parts of the 9th and 10th millions. This last task, however, was one in which his patience aud perseverance were of more value than his sjcill in calculation, since, by methods to which Gauss was careful to call his attention, the work was made mainly mechanical. Dase had planned to carry the table through the 10th million, but death cut short his labors. The

¹Scripture's statement (op. cit., p. 19) that Colburn and Mondeux "enjoyed even greater advantages [than Dase,] yet failed to yield any results" in the service of science, is misleading. With both Mondeux and Dase the trouble seems to have been not lack of opportunity to acquire mathematical knowledge, but lack of native ability to use the opportunities they had. With Colburn, on the other hand, the trouble really was at least in part lack of opportunity; he certainly did not enjoy the opportunity to attend university lectures, nor did any eminent mathematician try "in vain for six weeks to get the first elements of mathematics into his head" (ibid., p. 18; Gauss-Schumacher Briefwechsel, III, p. 382; V, pp. 32, 295), as in the case of Dase. Moreover, Colburn's description of his methods must be reckoned as an important contribution to the science of psychology, none the less important because it is somewhat inferior to Bidder's later description. For other instances of Scripture's unfairness to Colburn, see Appendix I.
²Scripture omits to mention any specific number of decimal places, though in both the references he gives (p. 18), to Crelle's Journal and to the Gauss-Schumacher Briefivechsel, the number of decimal places is made prominent. The natural inference would be that Scripture regarded p as a commensurable number of exactly 200 decimal places; but in view of his frequent use of higher mathematics in his other published works, one hesitates to attribute to him so gross an error. Of course anybody, with a logarithm table and a little knowledge of geometry, can compute the value of p to three or four places; the record of such a computation is absolutely meaningless without specific mention of the number of figures to which the computation is carried out.

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tables were completed by another hand, and published as far as the 9th million in 1862-5.
Dase had one other notable gift, doubtless related to his calculating power: he could count objects with the greatest rapidity. With a single glance he could give the number (up to thirty or thereabouts) of peas in a handful scattered on a table; and the ease and speed with which he could count the number of sheep in a herd, of books in a case, or the like, never failed to amaze the beholder. Here, again, his powers are so far in advance of those of any other recorded person that be stands in a class by himself.
George Parker Bidder¹ (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."² Other members of the family were distinguished in non-mathematical ways.

¹Scripture, 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.
²Spectator, LI, 1878, pp. 1634-5. In this article the younger Bidder is referred to as Mr. G. Bidder; but his full name was the same as that of his father, George Parker Bidder. (Cf, JOB. Foster's Men-at-the-Bar, 2nd ed., London, 1885, and The Law List, London, for 1882.) Scripture refers to both father and son, in different places, as George Bidder, and to the son usually as George Bidder, Q. C., Mr. Bidder, Q. C., or the younger Bidder; by Bidder (unqualified) he always means the elder Bidder, except in one case (p. 28), where the context prevents any misunderstanding. After noting that the similarity of the two names has caused some confusion, he tell us (loc. cit.), somewhat dogmatically, that "the only way out of the difficulty is to distinguish the son by adding his title [Q. C.]." (Why would not the son's A. B., or A. M., or his date of birth, or the father's C. E., answer just as well?)
Despite this device for avoiding difficulty, Scripture has fallen into sad confusion in dealing with the various members of the Bidder family. On p. 28 of his article he quotes from the Spectator (loc. cit.) the sentence: "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", omitting the comma after "mentally", but

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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
correctly attributing the remark to the younger Bidder. On p. 57, however, he makes the same quotation, this time adding a superfluous "of" after "conceive" and omitting the comma as before, but now attributing the quotation simply to Bidder (unqualified), meaning the elder Bidder, as the context unmistakably shows; for a little farther on he says, "This faculty was also inherited [transmitted?], but with a very remarkable difference. The younger Bidder [italics mine] thinks of each number in its own definite place in a number-form," etc.
But a worse confusion than this is still to be noted. The Spectator correspondence above cited, printed just after the elder Bidder's death, moved another correspondent (Spectator, L,II, 1879, p. 143) to quote from Brierley's journal for Jan. 25, 1879, the case of an eighteenth century Dissenting minister, the Rev. Thomas Threlkeld, who had a memory for Biblical texts similar to that of the elder Bidder's brother. On the strength of this, Scripture tells us (p. 27): "One of his [the elder Bidder's] brothers was an excellent mathematician and an actuary of the Royal Exchange Life Assurance Office. Rev. Thomas Threlkeld, an elder brother [!], was a Unitarian minister. He was not remarkable as an arithmetician, but he possessed the Bidder memory and showed the Bidder inclination for figures, but lacked the power of rapid calculation. He could quote almost any text in the Bible, and give chapter and verse. [Here Scripture gives the correct reference for this last sentence, which is taken from the younger Bidder's letter, and refers to the brother of the elder Bidder.] He had long collected all the dates he could, not only of historical persons, but of everybody; to know when a person was born or married was a source of gratification to him." Here we are given the correct reference for this last sentence, which refers to the Rev. Thomas Threlkeld, and is from the later volume of the Spectator. Thus by a piece of carelessness, hard to excuse, Scripture has inextricably confused the brother of the elder Bidder with this Rev. Thomas Threlkeld, who, so far as we know, was related to the Bidder family only by common descent from Adam!

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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.¹ Later, after his meeting and competitive test with Zerah Colburn, in 1818, he acquired great skill in the extraction of roots and the find-ing of factors, by methods similar to Colburn's, but with improvements of his own.²
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

¹On the mathematical side, if P represents the principal, r the interest (as a fraction of the principal, not as a per cent.), and n the number of years, Bidder's method amounted to the expansion of the expression P (1+r)ⁿ, by the binomial theorem, to a sufficient number of terms to insure accuracy in the last farthing. The properties of several numerical series were skillfully utilized at different stages of the expansion. (Cf. Proc. Inst. C. E., XV, p. 267, for Bidder's own account.)
²Colburn says of this meeting (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." Thus it would seem that Bidder's mind was not strongly turned in the direction of this class of problems until after this meeting with Colburn, but that once he became interested in them he soon outstripped his rival. Strangely enough Scripture, after mentioning this passage from the Memoir in his general bibliography oa Bidder, does not cite it in his account of the meeting of Colburn and Bidder, but refers only to the one-sided account of a London paper, which represents Bidder's triumph as complete. For a further discussion of this meeting, see Appendix I.

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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 analysts made him a formidable witness.
It would seena that Bidder's powers of mental calculation increased steadily at least up to the beginning of his university days, if not later,¹ and thereafter remained almost undimin. ished 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, fa Edinburgh he maintained a good class standing in mathematics including differential and integral calculus, but only by hard study.² 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";³ in general, he

¹In the Spectator, LII, 1879, pp. 111-112, are given specimens of Bidder's feats during tha years 1816-1819, with times of solution, also the London newspaper account of his meeting with Colburn to which reference has already been made. Scripture (op. cit., p. 26) argues from this series that Bidder's powers increased between 1816 and 1819. That this was the case can hardly be doubted; but it certainly is not proved by this series of examples. Even comparative times for an expert computer solving these same problems on paper would prove nothing, since in several cases there are two or three diSerent ways of doing the work, and possible short cuts which it is impossible to say whether Bidder noticed or not. Moreover, no two of the problems are alike. Perhaps the hardest problem of the lot is the compound interest question (1816, solved in 2 minutes) which is first in the list. The cube root of the 18-figure number (1819, 2 minutes) is far easier than it looks; (or by this time, a year after his meeting with Colburn, Bidder was doubtless familiar with the application of 2-figure endings to these problems, so that he had only to find the cube root of the first 9 figures by trial and approximation to get the first three figures of the root, then add on the last two by inspection from the last 2 figures of the given number, and find the missing 4th figure of the root by casting out the 9's. The algebraic problem which was solved "instantly" in 1819 was very simple, and was undoubtedly solved ty inspection; the answer, 3, was, from the nature of the question, the most natural first trial, and hence no special credit belongs to this last feat. These considerations show how difficult it is to reach definite conclusions from particular problems of this sort unless there is at hand specific knowledge of the detailed methods and short-cuts actually used iu the examples under consideration, particularly of any special peculiarities of the given numbers whereby the solution may be facilitated.
²Proc. Inst. C. E., XV, p. 253; Spectator, LI, 1878, pp. 1634-5
³Proc. Inst. C. E., XV, p. 259. In view of this explicit statement

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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,¹ to which reference has already been made, Bidder has left us an excellent account of his methods of calculation.
George Parker Bidder, Jr.² (b. 1837), "the younger Bidder," was the eldest son of G. P. Bidder. Practically the only information we have concerning his powers of calculation consists of a few facts brought out in the Spectator correspondence already referred to. He was 7th wrangler of his year, and later a thriving barrister and Queen's Counsel. He tells us that he was unable to approach his father in extent of memory and rapidity and accuracy of calculation; we have seen, however, that the father, writing in his soth year (after which his powers can hardly have shown any considerable increase), speaks only of multiplying 12 figures by 12 figures "on one occasion", by "a great and distressing effort", whereas the son was able, in several instances, to perform 15-figure multiplications, though slowly and with occasional errors. That the younger Bidder's method of multiplication was, like Dia-maudi's, cross-multiplication, we may infer from the fact that he incorrectly attributed this method to his father. Of the son's other feats in calculation, and of the degree of his precocity in this field, we have no knowledge. He was of visual memory type, and possessed a number-form running from right to left, the numbers up to 12 being arranged in a circle as on a clock. He declared that his calculations "proceed in a visible form" in his mind, and that he "can conceive no other way possible of doing mental arithmetic," which, as Proctor points out,³ is a rather strange remark. Unlike most of the other calculators, he employed a mnemonic system instead of natural memory in remembering numbers. He could play two games of chess blindfold simultaneously.
Truman Henry Safford⁴ (1836-1901) was, like Zerah Colburn, the son of a Vermont farmer; but both his parents were
from Bidder himself, his son and Elliot seem to be wrong in attributing to him (Spectator, 1878, p. 1634) great facility in 15-figure multiplications. The son's statement that his father used cross-multiplication is like-wise at variance with the father's explicit account of his method of multiplication (Proc., XV, p. 260).
¹Proceedings, XV, pp. 251 f.
²Referred to by Scripture as George Bidder, Q. C. Scripture, op. cit., p. 28; Spectator, LI, 1878, pp. 1634-5; Galton, Inquiries into Human Faculty, pp. 133-4, and Plate I, 20, opp. p. 380.
³Belgravia, XXXVIII, 1879, p. 461.
⁴Scripture, op, cit., p. 29; Appleton's Cyclo. of Am. Siog., art. Safford; Chambers's Edinburgh Journal, N. S. VIII, 1847, p. 265; Belgravia, XXXVIII, 1879, p. 456.

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former school-teachers, and persons of some education, father had a strong interest in mathematics, and the mother we are told, was of an "exquisite nervous temperament." Youag Safford showed a remarkable all-round precocity, similar to that of Ampere. In his 3rd year "the grand bias of his mind was suspected"; later his parents "amused themselves with his power of calculating numbers"; and when he was 6 years old he was able to calculate mentally the number of barleycorns, 617,760, in 1040 rods. At the age of 7 he had "gone to the extent of the famous Zerah Colburn's powers." About this time he began to study books on algebra and geometry, and soon afterwards higher mathematics and astronomy. Wanting some logarithms, he found them himself by the formulas; and in his loth year he published an almanac computed entirely by himself. The following year he published four almanacs, one of which, computed for Cincinnati at once reached a sale of 24,000 copies. In this almanac he used a new and original rule for obtaining moon risings and settings, accompanied by a table which saved a quarter of the work of their computation. About this time he also discovered a new rule for calculating eclipses, with a saving of one-third in the labor of computing.
Such feats at once made the boy a public character, and in the same year (1846) he was examined by the Rev. H. W. Adams, a skillful mathematician. He solved a number of difficult algebraic problems, doubtless in the main by algebraic methods rather than by the trial and error method of most of the other prodigies. Problems in the mensuration of solids caused him no trouble, though in one case, where the answer was a 12-figure number, he "used a few [written] figures." He extracted the cube roots of 7-figure exact cubes "instantly," doubtless by the use of 2-figure endings. Finally, he squared 365.365.365.365.365.365, entirely in his head, in "not more than one minute,"¹ though with evident effort. A three-hour examination convinced Adams that the boy had mastered and gone beyond all his text-books.
Like Ampere, Safford had a wide range of interests, and an encyclopedic memory. Chemistry, botany, philosophy, geography, and history, as well as mathematics and astronomy,

¹All these quotations are from the Chambers's Journal article cited above. The last problem is there given as 365,365,365,365,365 x 365,365,365,365,365,365, i. e., a 15-figure number multiplied by an 18-figure number; but since the answer contains 36 figures, it is obvious that another 365 is omitted from the first number, and that the problem was the squaring of an 18-fignre number. The repetition of the same figures, however, greatly simplified the work, there being only three different partial products. Scripture carries over the typographical error without comment, evidently not noticing it.

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were included in his field of study. He took his degree at Harvard in 1854, and became an astronomer. After holding various positions he became professor of astronomy in Williams College in 1376, where he remained until his death in 1901.
Safford early outstripped Bidder in range of mental calculation, but with the aid of books, whereas Bidder's methods were entirely of his own discovery. It is to be regretted that we have not more detailed information about Safford's calculations;¹ but except for the examination whose results have been given above, all we can say is that later he acquired considerable skill in factoring large numbers, seeming to be able to recognize almost at a glance what numbers were likely to divide any given number, and remembering the divisors of any number he bad once examined.²
Andre Marie Ampere³ (1775-1836), like his successor Safford, showed all-round precocity, a wide range of interests, and

¹The Chambers's Journal article is written in rather florid style, and in a tone of admiration almost verging on awe. The Rev. H. W. Adams, who is there said to have been a skillful mathematician, was by no means as critical an examiner as might be wished. Thus while he tells us that several of the problems given were among the hardest in Davies' Algebra, he later notes that Safford already owned this work and had fully mastered it, hence had seen all these problems before. The times indicate, to be sure, that Safford calculated the answers afresh; but the test is not as satisfactory as if the problems had been entirely new to him. The times given, too, are mostly in the form "about a minute," and in definiteness leave much to be desired. The big number selected for the grand final test was about as unsuitable for the purpose as any that could well have been chosen. Not only in the recurrence of the same three partial products, but in the repetition of the same group of figures within each partial product, the problem is so artificially simple that it proves almost nothing concerning Safford's power of multiplication. The number 365, too, owing to its connection with the calendar, is especially easy to remember. Adams speaks of the "long and blind sums" which Safford remembered after a single hearing; but apart from this simple 18-figure number (which would not overtax the memory of any child who could keep in mind 365 and count six), the longest numbers in the statement of any of the problems mentioned in the article were of 7 figures. Now a normal boy of 13 can, on the average, retain 8.8 figures after a single hearing, and a boy of 11, 6.5 figures. Hence, while Safford's memory for figures was probably above the average, the fact is not satisfactorily proved by Adams' examination. (Cf. American Journal of Psychology, II, 1889, p. 607. The figures are erroneously quoted by Scripture, p. 41, as 8.6 for boys of 19 years, instead of 8.8 for boys of 13 years.) The fact that the answers to some of the problems were longer numbers is not relevant here; for, as we shall see later, there is an important distinction between figure-memory as such, and memory as it stands in the service of calculation,
²Belgravia, XXXVIII, 1879,p.456.
³Scripture, op. cit., p. 6; Arago's Eloge d'Ampere, tr. in Smithsonian Report, I, 1872, p. 111. The writer has been unable to consult the other references which Scripture cites.

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an omnivorous memory. He learned counting at the age of 3 or 4, 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. He became a noted mathematician, and was also prominent in several other directions. Of his mental calculations, however, we have no specific information; his later achievements so overshadowed his early gift that his biographers are silent about it, and his case sheds little light on the problems connected with the subject.
Carl Friedrich Gauss¹ (1777-1855) was the son of a poor family; a maternal uncle of his, however, was a man of considerable mathematical and mechanical talent. When not quite 3 years old, Gauss, according to an anecdote told by himself, followed mentally a calculation of his father's relative to the wages of some of his workmen, and detected a mistake in the amount. Entering the gymnasium at the age of 11, he mastered the classical languages with incredible rapidity. In mathematics he was not only head of the class, but soon outstripped his teachers. At the age of 10 he was ready to begin the study of higher analysis, and at 14 he could read the works of Euler, Lagrange, and Newton. He became one of the foremost mathematicians of his time. His Disquisitiones Arithmeticae, published at the age of 24, is practically the foundation of the modern theory of numbers.
Concerning Gauss' mental calculations we have for the most part only general information. His power seems to have lasted all his life, and to have exceeded that of any other calculator except Dase. He had a "peculiar sense for the quick apprehension of the most complicated relations of numbers," and "an unsurpassed memory for figures," and used from memory the first decimals of logarithms in his mental operations. He was especially fertile in inventing new artifices and methods of solution,
MINOR PRODIGIES. - In the following list are grouped a few calculators about whom too little is known for an extended account, but who present one or more points of interest.
The Daughter of the Countess of Mansfield² (b. about 1804)

¹The writer has followed Scripture's account of Gauss (op. cit., p. 7), not having access to the sources there cited.
²Gall, op. cit., V, p. 88; Colburn, Memoir, p. 174; Scripture, op. cit., p. 32. The reference to the Med. and Philos. Jl. and Rev. given by Scripture can hardly be correct, since the young lady, being about Colburn's age, was in 1811 only 6 or 7 years old, and could hardly have had an American reputation. The exact words in Scripture's text are found in Gall's Organology; the Jl, and Rev. reference probably refers to Mr. Van R., of Utica. In fact, all the notes to page 32 of Scripture's article are incorrect except a few of those to Gall, where the absence of a page reference covers up the inaccuracy. The trou-

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was seen by Spurzheim iu London at the age of 13, at which time she "extracted with great facility the square and cube roots of numbers of nine places." Whether this refers only to perfect squares and cubes cannot be decided. Colburn speaks of her simply as displaying, in 1812, at the age of 8 or there-bouts, "a certain degree of mental quickness [in calculation] uncommon in her sex and years." Except for Bidder's two granddaughters, whose powers were but little above the average, she is the only girl calculator on record.
Richard Whately¹ (1787-1863) began to calculate at the age of 5, and retained the power for about three years; he probably surpassed Colburn, but did not happen to hit on Colburn's favorite problem of extracting square and cube roots. When he went to school the power left him, and at ciphering he was always "a perfect dunce."
Mr. Van R., of Utica² like Whately, developed a gift for calculation at an early age (6 years), but lost it at the age of 8.
Dr. Ferrol³ (b. 1864) has a sister about a year his elder, who shares his gift for mental calculation. His father was an architect and a good reckoner, and his mother's mind was occupied with architectural computations at the time of the birth of these two children; whether this prenatal influence had any effect on their mental powers cannot be determined. Ferrol's gift showed itself at an early age, but as soon as he learned the elements of algebra, at the age of 10, he developed a preference for mental algebra instead of mental arithmetic. He was head of his class in mental arithmetic, but below the average in all other studies. He is a remarkably poor visualizer. His processes are "intuitive"; the answer to a problem, he tells us, comes "instantly," and is always correct. His general memory is probably about normal; his figure memory depends on mnemonics.
A blind Swiss mentioned by Johannes Huber⁴ not only solved
ble seems to be due to a transposition; note¹ should be note⁷ and all the others should be moved up a line, ² becoming ¹, etc. Colburn's account of the daughter of the Countess of Mansfield is quoted in full in Appendix I.
¹Scripture, op. cit., p. 10. The writer has been unable to consult the Life ot Whately there cited. By an inadvertance, Scripture, on p. 57, gives the age of Whately's first calculations as 3, whereas, on p. 10, the statement is "between five and six."
²Gall, op. cit,, pp. 87-8, quoting Med. and Philos. Jl, and Rev., III, N. Y., 1811. Gall mentions several other calculators, but it has not seemed worth while to enumerate them all here.
³P. J. Mobius, Die Anlage zur Mathemaiik, 1900, p. 73. The name is given simply as "Dr. Ferrol"; we are not told whether he is an M. D., or what are his initials.
⁴Das Gedachtniss, Munich, 1878, p. 43.

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the most difficult problems, but could repeat a series of 150 figures either forwards or backwards after a single hearin, or name at once the 30th or 50th figure, e. g., from either end. Before becoming blind he had been a man of very weak memory; but afterwards, busying himself with exercises in calculation, he discovered a very simple method of dealing with the largest numbers, and tried to sell his secret in England for a high price.¹
Vito Mangiamele² (b. 1827), the son of a Sicilian shepherd himself tended sheep, and when examined by the Academie des Sciences, at the age of 10, answered several questions among them the cube root of 3,796,416 (=156), which he found in half a minute. Cauchy, in his Academie report on Mondeux, already cited, complains that Mangiamele's masters have always kept secret the boy's methods of calculation; it is not clear whether this means that they knew and refused to tell, or that the boy himself was unable to enlighten them. He was quite uneducated. A brother and a sister of his were also noted calculators.
Prolongeau³ (b. about 1838), at the age of 6½, solved mentally with great facility problems relating to the ordinary operations of arithmetic, and to the solution of equations of the first degree.
Grandmange⁴ (b. about 1836). born without arms or legs, performed, mentally, very complicated calculations and solved difficult problems.
Matheu le Coq⁵ (b. about 1656), an Italian boy, "at the age of 6, without knowing how to read or write, commenced to perform all the most difficult operations of arithmetic, such as the four elementary operations, the rule of three, partnership (compagnie), square and cube root, and that, too, as soon as the question was put to him." He learned to calculate by stringing beads.
Vincenzo Succaro⁶ (b. 1822), a Sicilian, appeared in public as a calculator at the age of 6, received a good education, but showed no special mental ability outside of calculation.

¹Euler, it is well known, possessed considerable powers of mental calculation after becoming blind; but to what extent he had the power before his blindness, and just what feats he could perform, the writer has been unable to discover.
²Comptes rendus hebdomadaires des seances de l' Academie des Sciences, IV, 1837, p. 978; Riv. sper. di Fren., XXIII, 1897, p. 434.
³C. R. Acad. des Sci., XX., 1845, p. 1629.
⁴Ibid., XXXIV, 1852, p. 371.
⁵Binet, op. cit., p. 3; Riv. sper. di Fren., XXIII, 1897, p. 430.
⁶The source for the remaining calculators is the Riv. sper. di Fren., XXIII, 1897, pp. 429f. A summary of this article in German is found in the Zeits. f. Psy. u. Physiol. d. Sinnesorgane, XVI, 1898, pp. 317-8.

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Guiseppe Pugliese (b. "a little later"), also a Sicilian, took to the stage at the age of 5, and was exhibited in Italy and Germany. An attempt was made to teach him Geometry; but he was unable to deal with geometrical forms.
Luigi Pierini (b. 1878) learned late to speak and to walk, suffered from many children's diseases, and was an epileptic. He tended sheep, and thus learned to count. He developed a remarkable talent for mental arithmetic, and at an early age became a professional calculator.
II.

The writer will now give an account of his own case, which differs in three respects from those hitherto considered:
(1) The power is almost confined to dealing with the last two figures, or 2-figure endings, of the numbers used. It is readily seen that, with certain limitations in division and evolution, the last two (or n) figures of the numbers used in a given problem determine the last two (or n) figures of the answer, no matter what the preceding figures may be. Now the writer's mental calculations take the form almost exclusively of tracing the last two figures through the different operations, ignoring all the other figures. This evidently simplifies the work immensely.
(2) By a further specialization, the problems which he solves most often and most readily are of the general form of finding the last two figures of any power (or integer root) of any number.
(3) Finally, he has a strongly marked preference for working with even numbers. By a special method, to be explained later, he practically always changes odd numbers into even numbers for purposes of calculation, where only the last two figures of the answer are required; the even number thus obtained is readily converted into the desired odd number by very simple rules.
It will thus be seen that the writer's calculations are highly specialized, and in extent perhaps not comparable to those of any calculator heretofore considered. At the same time, some of these specializations are found in other calculators; and in the general features of its development the writer's case is typical of many or most of the others, and will, it is hoped, throw light on several points which have hitherto not been fully understood. While many of the details are in themselves of little importance, they will serve to illustrate the sort of numerical properties which not only facilitate mental calculation, but arouse the interest of the calculator, and hence furnish the motive for continued practice until the calculating habit becomes firmly established.

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In the matter of heredity the only circumstance that need be mentioned is that the writer's younger brother has shown rather more than average ability as a chess-player, and has, on a few occasions, played a game blindfold; but by what psvcho logical processes, or to what extent the power could be in creased by further practice, cannot be stated. Nor need the writer speak of his school and college work, except to say that while he has always been fond of mathematics, it has no better claim than two or three other subjects to be called his favorite study.
His interest in mental calculation dates from the time he learned to count, at the age of 4, or possibly 3.¹ He learned to count to 10, then to 100, then beyond, and also to count by 2's, 3's, etc. Now in these latter series 2x2, 2x2x2, 3x3, 3x3x3, etc., in short, the powers of the number by which he was counting, were natural resting-places, and awakened his interest, so that before long he began to count in the power series of different numbers (2, 4, 8, 16, 32, etc., 3, 9, 27, 81 etc.) for considerable distances. At first he simply emphasized the powers as they occurred in the complete series of multiples, but gradually he learned to omit the intermediate multiples, and simply count in the power series proper: 2, 4, 8, 16, etc., 3, 9, 27, 81, etc. But almost always, when the number exceeded 100, he emphasized the last two figures, and gradually got into the habit of ignoring all the others. Thus instead of saying 3, 9, 27, 81, 243, 729, 2187, etc., he usually counted 3, 9, 27, 81, 43, 29, 87, and in this simplified form counted along the different power series for considerable distances. Multiplication naturally grew out of this counting process; but it was really counting rather than multiplication proper, since he did not learn the multiplication table until some time later, when he went to school. Thus to find 9x7 at this time he would count 9, 18, etc., to 63; and even now, except within the limits of the multiplication table as he learned it to 12x12, his mental multiplications are abbreviated countings of this sort (skipping most of the intermediate links) rather than true multiplications. We have already seen reason to suspect that neither Buxton nor Tom Fuller really got beyond this counting process into true multiplication, i. e., with the use of a memorized multiplication table.
In the course of these calculations or countings, a number of properties gradually attracted the writer's attention; such as that every power of a number ending with 0 or 5 ends with 0

¹Unfortunately, definite dates cannot be given. The power developed very slowly, never really becoming important for any but psychological purposes, so that no one but the writer himself knew of its existence until a much later date.

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or 5, that the 4th power of any other number ends with 1 or 6, according as it is odd or even, that the 5th power ends with the same figure as the 1st, the 6th with the same figure as the 2nd, etc.; and chat if 76, or any number ending with 76, is multiplied by a multiple of 4, the last two figures of the product are the same as those of the multiplier (e. g., 76x12=912). Then he noticed that the ending¹ 76 occurs at various points in the power series of different numbers (the 5th power of 6, the 4th power of 32, the 2nd power of 24, the l0th power of 4, the 20th power of 2, etc.,), and that from these points the series of endings repeats, except that in some cases the ending of the next power will differ by 50 from that of the original number. Thus theendings of the first 20 powers of 2 are 02, 04, 08, 16, 32, 64, 28. 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76; the 21st is 52 instead of 02; but the 22nd is 04, like the 2nd, and thereafter the endings recur in regular order. Finally it turned out that the 20th power of every even number (not ending with 0) had the ending 76, and that odd numbers had a similar property, the 20th power ending being, however, 01 instead of 76, and even the 21st power being always the same as the 1st, except for multiples of 5.
After discovering these and similar properties, the writer found it a simple matter to find the last two figures of any power of any number, by counting along the proper series. The process was always, however, of the counting type already indicated. Thus to find the 8th power of 3 the process would be 3, ⁶, 9, ¹⁸, 27, ⁵⁴, 81, ⁶², 43, ⁸⁶, 29, ⁵⁸, 87, ⁷⁴, 61; i. e., he would count up to a power of 3, then by this power to the next, and so on, but passing very lightly over the intervening multiples, and in time learning to omit them altogether. In fact, before long the process came to be simply, 3, 9, 81, 61, i. e., simply squaring each number to get the next, the intermediate countings taking place so rapidly and automatically as hardly to appear in consciousness at all, except as brief "flashes." And even these "flashes" may sometimes be almost absent, so that only the 3 and the 61 stand out, the rest remaining a mere blur.
It happened that about the time he learned to count, and for perhaps two or three years thereafter, the writer was frequently ill. This, of course, left a large amount of time free for his calculating exercises, and probably had not a little to do with strengthening his bent in that direction.
We come now to the third peculiarity mentioned above: the writer's preference for even numbers. An examination of the

¹The term "ending" (unqualified) will hereafter betaken as synonymous with "2-figure ending."

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following table of the endings of certain products will form the best introduction to this subject.

07 32 57 82

23 61 36 11 86

48 36 36 36 36

73 11 36 61 86

98 36 36 86 36

It will be observed that the numbers at the left, 23, 48, 73, 98, differ in pairs by 25, and so with the numbers at the top, 07, 32, 57, 82; and that in each case there is one multiple of 4 (48, 32), one odd multiple of 2 (98, 82), one number of the form 4c+1 (73, 57), and one number of the form 4c-1 (23, 07). Now the 16 numbers in the body of the table, it will be seen, all belong to a similar series, 11, 36, 61, 86. If either of the factors is a multiple of 4, the product has the ending 36, as shown by the 2nd line and the 2nd column; if both are odd multiples of 2 (98, 82), the product again ends with 36; if one is an odd multiple of 2 (98, 82), and the other an odd number, the product has the ending 86, =36+50. Finally, if both numbers are odd, the ending of the product is 36+25, i.e., either 11 or 61: - 61 (a number of the form 4c+1) if the numbers multiplied are either both of the form 4c+1 (73x57), or both of the form 4c-1 (23x07); and 11 (a number of the form 4c-1) if one of the factors is of the form 4c+1 and the other of the form 4c-1 (73x07, 23x57). Thus by applying a few simple rules, any one of the 16 products in the table can be made to depend on the single product, 48x32, of the two multiples of 4 in the table. Hence to find the ending of the product of two odd numbers, change each into a multiple of 4 by adding or subtracting 25, multiply these multiples of 4 together, and then add or subtract 25, as the case may require, to get the answer. A similar principle obviously applies to the power series of any odd number; simply find the required power of the corresponding even number, and then either add or subtract 25.
Now these properties early attracted the writer's attention, and he soon got into the habit of transforming odd numbers into even numbers in practically all his calculations. The result was that (if we leave out of account multiples of 5, which belong to a class by themselves and are very easy to multiply) the whole of multiplication, so far as the endings were concerned, was reduced to the 200 possible products of any two of the 20 numbers 04, 08, 12, 16, 24, 28, 32, 36, etc.; whereas in order to do the same work without this transformation, the 3200 combinations of the whole eighty 2-figure end-

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ings prime to 5 would have to be considered. In finding powers, again, he had to deal with only 20 different series, each of which repeated after 20 terms or less; so that the whole problem of finding the last two figures of any power of any number was reduced to less than 400 simple cases, instead of an indefinite number of cases. He never committed these products and powers to memory; it was not necessary; with practice he was soon able to count to any desired one with great rapidity, in fact, just as rapidly, in the simpler cases, as he could have recalled the answer if it had been previously memorized.
To recapitulate: The writer's mental calculations usually deal only with the 2-figure endings of numbers, rejecting all previous figures if there are any; by far the commonest problem is to find (the ending of) some given power of a given number, or to investigate some property of some power or group of powers of one or more numbers; and problems involving odd numbers (except, of course, odd exponents) are almost always solved by changing the odd numbers into multiples of 4 (by adding or subtracting 25), and changing back to an odd number in the same way, if necessary, after the work of calculation is over. He might go on and indicate many other properties of numbers, or rather of endings, which be discovered and used in calculating; but enough has already been said to give a fair idea of the general nature of the processes employed, the gradual development of the calculating power, and the advantages of the various specializations which came to be adopted.
Of course his calculations are not absolutely confined within these limits. Besides finding endings in the power series of even numbers, he can also multiply endings very readily, and add or subtract them (by counting forwards or backwards) somewhat less rapidly, or divide them where the division is known to be exact; and he can work, though very much more slowly, with odd numbers. But even in the power series of 3, the odd series with which he has worked oftenest, it is easier in most cases to change 3 into 28; and in any other odd series he can scarcely work at all, except with the greatest effort. The even series in these other cases are so much easier and more familiar that it is practically impossible to resist the temptation to work in them, even when he tries to work laboriously in the odd ones as such.
When the calculation takes account of all figures of the result, not merely of the last two, the writer's powers of mental arithmetic are probably very little above the average, certainly not equal to those of any one who has had a moderate amount of practice in the work. Even the multiplication of two 2-fig-

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ure numbers takes him longer mentally than on paper; and with 3-figure numbers it is such an effort for him to remember the partial products that usually each one must be repeated aloud two or three times, and even then he is apt to forget the first partial product by the time he has found the third. With small 2-figure numbers, however, he finds no difficulty in multiplying (on paper, using only one figure of the multiplicand at a time,) in a single operation, especially where the number is even, e.g., 24 or 36. With 19 or 23, too, it would probably be easier for him to multiply in a single operation than in two operations in the ordinary way; but in such a case after the products exceeded 100, the multiplication would often tend to resolve itself into counting, - rapid and automatic, but counting nevertheless. Thus up to 23x5=115 he would probably count by 23 directly, or depend on his memory; but after that, to pass to 23x6=138, he would first count in the 3, then the 20, thus reaching 138 from 115 via 118 and 128.
There are two cases in which the writer can find complete products with fair readiness. The first is in squaring numbers; here, however, the process is usually neither counting nor multiplication directly, but an application of some algebraic formula. Up to perhaps 32, and in certain other cases, such as 36, 48, 54, 64, 72, 81, 96, 144 (i. e., numbers containing no other prime factors than 2 and 3), he would give the squares from memory; but usually he finds only the last two figures by memory, and gets the rest by interpolation between two known squares or by the formula for (a+b)².
The second case is where two numbers are to be multiplied, neither of which contains any prime factors except 2 and 3. Here his method is to count (multiply) by 2's or 3's to some convenient multiple of one of the numbers, then by that multiple to some other, and so on, until the required product is reached. Thus to find 48x64 be would count by 48 to 384 (=48x8), then by 384 to 1536, then to 3072 (=384x8=48x64), the required answer. To square 162, again, the stages would be 486, 1458, 2916, 8748, 26244, i. e., multiplying successively by 3, 3, 2, 3, 3.¹ In these cases much of the work would be automatic and half-unconscious. Thus up to 2916 (=162x18=54²) in the second example the numbers in full would be very familiar, and perhaps only the 58 of 1458 would

¹Buxton, it will be remembered, in multiplying 456 by 378, multi plied successively by 5, 20, and 3, to get 300x456; then multiplied 456x5 and that product by 15, and added the result to 300x45r, to get 375x456; and finally completed the operation by adding 3x456. This indicates pretty clearly that his method was like the one described above, a counting in the series of multiples of the multiplicand, rather than the ordinary method.

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be distinctly formulated; but above that he would have to formulate all the figures distinctly and take account of them in the counting process. In the first example, 48x64, perhaps only the 84 of 384 and the 36 of 1536 would be distinctly formulated until the end, when 3072 would be given as a whole. It will be seen, then, that part or all of the intermediate numbers of the calculation may remain below the level of clear consciousness, and that where the numbers are familiar, part of a number may be in clear consciousness and not the rest of the number. At the same time the whole number functions in the calculation, otherwise the correct answer would not result.
There is just one class of problems in which the writer could compete with the real mathematical prodigies, viz., finding the square and cube roots of exact squares and cubes. In fact, extracting the roots of perfect powers and testing the possible factors of given numbers are the only fields in which the properties of 2-figure endings are really useful, and even these problems, however interesting to the mathematical prodigy, are of little practical importance to the mathematician. Bidder. Colburn, and Safford made a specialty of these problems, and there is good evidence that all three solved them by the aid of the properties of 2-figure endings. A brief description of the "method of endings" will therefore not be out of place.
Given the last two figures of a number, the last two figures of its square are known; but given only the last two figures of a perfect square, the last two figures of the square root are not definitely known, although the possible values are usually only four in number. Similarly, an odd ending has only one possible cube root, but an even ending has either none, or two which differ from each other by 50. Now, suppose a given number is known or suspected to be a perfect square or cube, and its root contains only three figures. The first figure can readily be determined by inspection; and the last two figures must be one of a limited number of possible roots of the ending of the given number. It is usually easy, after a little practice, to tell almost at a glance which of the possible roots to choose in a given case. In doubtful cases (multiples of 5. e. g., where the number of possible roots is greater) such expedients as casting out the 9's, squaring or cubing one of the suspected answers or some number near it, or using the 3-figure instead of the 2-figure ending, will help to decide which is the correct root.
The application to factoring is still simpler. If the number to be factored is not already odd and prime to 3 and 5, it is easily made so by simple division. Now, in the case of an odd number prime to 5, if the last two figures of one of its

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divisors are known, and the division is exact, the last two figures of the other can have only one value; and it is easy to construct a table showing the different pairs of endings in the factors which will produce a given ending in the product. Now, suppose it is suspected that a given number is a factor. From the table, or by a computation in accordance with simple rules, which need not be considered here, find the last two figures of the other factor; if desired, the hundreds figure can also be determined by casting out the 9's. This done, it is necessary to carry the division only far enough to decide whether the required last two or last three figures can result; as soon as this is seen to be impossible, the work is abandoned, since only an exact divisor is wanted. It is thus evident that much work may be saved, especially where the numbers involved are not very large; indeed, a factor may often be rejected almost at a glance which would otherwise have to be divided through to the end.
So much for the application of 2-figure endings to evolution and to factoring. The latter problem never attracted the writer, owing to the habit he so early developed of confining his attention to the last two figures; but in any case where a given number is known to be a perfect square or cube, and its root contains not more than three figures, he finds no difficulty in discovering the root by inspection. This would apply almost equally to higher roots, except that in some cases it would be difficult to tell the root if it contained more than two figures; but in general, the higher the root the easier the problem, and square and cube roots are the only ones which often come up. It is evident, however, that skill in solving this class of problems does not imply special skill or quickness in other branches of mental arithmetic, and that a careful distinction must be made between the cases where the given number is a perfect power and those where it is not. Where the root is not an integer, the ending gives no aid in finding it; memorization of a large number of perfect squares and cubes, or some process of real calculation, must then be resorted to, instead of the simple method of guessing by inspection of the ending of the given number.
Before closing this part of the paper, the writer may say a few words about his memory type. He learned to count orally, and his calculations began at once, without further aid; he cannot remember ever counting on his fingers, using pebbles, or the like; and even when he learned to make written figures later on, they never came to be associated with his mental calculations, which remained strictly auditory (or auditory-motor) throughout. Ordinarily the motor element is almost entirely absent; when the calculations remain in the familiar fields

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already described, they are accompanied by no perceptible innervation of the muscles of speech. When he attempts un practiced feats, however, such as complete 3-figure multiplications, the tendency to pronounce some or all of the figures is marked.¹
But while the writer's type is unquestionably auditory in calculation, the presence of written figures is not a hindrance to him, as it is to Inaudi. On the contrary, if the numbers involved are at all large, - say a 9-figure number whose cube root is to be found, - the presence of the number on a sheet of paper before him is a distinct aid, saving a considerable effort of memory, and greatly facilitating such tests as casting out the 9's. Outside of calculation the writer's type is predominantly auditory; but he can use visual images at will with no special difficulty, and in geometry or similar fields uses them habitually as a matter of course. In general, then, his type is mixed, but with a slight predominance of auditory images.
It only remains to add that his calculating powers have increased, though very gradually, from the time he learned to count until the present, constantly taking advantage of the results of his mathematical studies, and at intervals following out new lines of inquiry and classes of problems based upon new properties of numbers and endings. There has been no tendency, however, to enter the broader fields of calculation cultivated by the mathematical prodigies; in the main, his calculations are confined within the limits already described, and even within these limits it often happens that of two problems which, to an ordinary calculator, would be of equal difficulty, one will be far easier for him than the other, owing to the peculiar preferences which have guided the distribution of his practice in calculation. While mental arithmetic has sever absorbed a disproportionate share of his time, there is scarcely a day in which some of the old familiar series do not at some odd moment or other run through his head, usually quite automatically. He has never had any fondness for written computation for its own sake, and is perhaps, if anything, a trifle slower at it than the average man with an equal knowledge of mathematics. He is liable to occasional errors unless he carefully tests every stage of his work.²

¹Much the same thing was true of Safford; we are told (Chambers's Journal, VIII, p. 265) that it was his custom to talk to himself when originating new rules, but, by implication, not when carrying on computations by familiar rules.
²In the foregoing account an attempt has been made for the most part to avoid technical terms that would not be. clear to the non mathematical reader. The student of the theory of numbers will readily recognize that "2-figure endings" are least positive residues

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III.

We are now ready to interpret the facts thus far set forth, and to construct from them an explanation of the mathematical prodigy.
Heredity. The table¹ gives such information as could be found concerning the heredity of the different calculators. Just what part these various circumstances actually played in the development of the different prodigies is a difficult question, which it would hardly be worth while to discuss here. This much is clear, however, that whatever the influence of heredity in some cases, it is in no sense an explanation of mental calculation, but at most a favoring circumstance. A satisfactory theory must rest on a much more definite basis than such general terms as heredity, environment, and the like can afford; it must explain the cases where hereditary influence is lacking, as well as those where such influence seems to be present. Hence we may safely leave the question of the relation of heredity to mental calculation for other investigators, and devote our attention to other questions.²
Development. - (a) Precocity. There is nothing more striking about the mathematical prodigies, nothing which has been the subject of more uncritical amazement, than their almost uniform precocity. Gauss began his calculations before he was 3 years old; the present writer, at 4; Ampere, between 3 and 5; Whately, at 5; Pugliese and Succaro, at about 5; Colburn, at 5; Safford, at 6 or earlier; Mathieu le Coq, Mr. Van R. of Utica, Bidder, Prolongeau, and Inaudi, at 6; Mondeux, at 7; the Countess of Mansfield's daughter, at 8 or earlier; Ferrol, Mangiamele, Grandmange, and Pierini, at early ages not definitely stated. Buxton's mental free beer record began from the age of 12; Zaneboni's calculations began at the same age; Dase attended school at the age of 2½, and took to the stage at 15. In short, precocity is unmistakably the rule; if we
(modulus 100), and that the writer's process of changing odd numbers into even simply changes the modulus to 25 instead of 100, using residues which ≡0 (mod 4). It would be easy to generalize many of the properties described above, and to show their application to n-figure endings and congruences in general; but such a task would carry us, far beyond the limits of the present paper.
¹See Appendix II.
²There are two points, however, on which a word may be saw. In the first place, it is a pretty safe guess that Colburn's extra fingers and toes were an accident, as far as his calculating power was concerned, and had no connection with his mental abnormalities. On the other hand, the nervousness which he showed, and which he shared with Safford and an unnamed calculator in the neighborhood of Troy, N. Y. (Memoir, p. 173), may have predisposed him to less active participation in childish games, if not to actual illness, and so have increased the time available for his mental calculations.

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count as unprecocious Zaneboni, Buxton, Dase, Diamandi (who began to calculate at 16, but had excelled in mathematics in school from the age of 7), the slave Tom Fuller, and the younger Bidder (about whom nothing definite is known in this respect), we have at the worst 6 unprecocious calculators as against 18 who were precocious.¹
To understand this precocity we must note, first of all, that arithmetic is the most independent and self-sufficient of all the sciences. Given a knowledge of how to count, and later a few definitions, as in Bidder's case, 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 be knows of other things. Addition is only a shortened form of counting. The same is true of multiplication;² the writer's own case shows that the calculator need not even sit down and teach himself the multiplication table, as Bidder did, but may multiply by simple modifications of his counting process. Involution is simply a modification of multiplication; it has already been pointed out that the powers of numbers are natural resting-places in counting along the series of multiples of the numbers. The inverse operations of division and evolution grow naturally out of the direct operations of multiplication and involution; much more easily and naturally in mental than in written arithmetic. Once these elementary operations are mastered, such processes as reduction of years to seconds, compound interest, and any

¹From this list has been omitted Huber's blind Swiss, who learned to calculate, presumably, late in life, by artificial methods, and obviously does not belong to what Binet calls the "natural family of great calculators." Binet's average of 8 years (op. cit., p. 191) for the precocious calculators is too high; it is obtained by rejecting (without sufficient ground, so far as the writer can see) the cases of Gauss (3 years old) and Whately (who, as we have seen, began to calculate at 5, not at 3 as stated inadvertently in Scripture's table), and by taking in several cases the age at which the prodigy was exhibited before the Acad. des Sciences at the age when his calculations began. But on Binet's own showing, Mondeux had calculated for three years before he was exhibited in Paris; so that it will not do to average together such dissimilar data. Where the age of exhibition is later than 7, no attempt has been made to date the beginning of the calculations; if we then average the ages of the known cases of precocity (some of which are undoubtedly too high by a year or more), we get 5 to 5½ as an average, not 8. This is much more natural if the "natural calculator" usually begins to calculate from the time he learns to count. Of the six men not known to be precocious, two (Fuller and Buxton) were densely ignorant, and two of the others belong to the visual type, which, as we shall see later, is in certain respects intermediate between the "natural" or auditory and the "artificial" type.
²In multiplication the counting is, of course, done in the series of multiples of the multiplicand, not in the series of natural numbers; cf. part II of the present paper.

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other arithmetical problems are simply a matter of understanding the meaning of the question and then applying known rules, plus a varying amount of ingenuity, to the solution. In accordance with the tendency of all mental operations, psychological shortenings of the processes involved will come with practice, and mathematical properties of the sort already described still further facilitate the work; so that in favorable cases the whole process may become in large measure automatic, and may go on while active attention is given to something else.
Moreover, the various symmetries and properties of numbers and series attract the attention of the calculator from the start, and keep up his interest until the habit of mental calculation has been firmly fixed. After that, if nothing intervenes to change that interest, there is practically no limit to which he may not attain, as the case of Dase abundantly shows.
We must note, furthermore, that practically an unlimited amount of time may be available for these calculations if the prodigy wishes so to use it. Mental arithmetic requires no instruments or apparatus, no audible practice that might disturb other members of the family, no information save such chance scraps as may be picked up almost anywhere for the asking, or absorbed, without even the trouble of asking questions, from older brothers and sisters as they discuss their school lessons. The young calculator can carry on his researches in bed, at the table, - if he allows himself to be "seen and not heard," - during the perhaps laborious process of dressing or undressing; in short, at almost any time during the twelve or fourteen hours of his waking day, except when he is engaged in conversation or active physical play.
Thus, if an interest in counting once takes hold of a child either not fond of play or not physically able to indulge in it, - and stringing beads, counting the ticks of a clock, or even a chance question like "Let's hear if you can count up to 100", may start such an interest, which will then furnish all the material for its own development, - he may go on almost indefinitely, and become a prodigy long before his parents suspect the fact. Indeed, the interest in counting may seem so natural to the child that he may never think of doubting that every one else possesses it, and months or even years may elapse before some accident reveals the direction of his interest to his astonished relatives. Several of the calculators - Mondeux, Mangiamele, Pierini, Inaudi - were shepherd-boys, an occupation which, since it requires an ability to count and affords ample leisure, is peculiarly favorable for practicing calculation; several, again, - Grandmange (born without arms or legs), Safford, Pierini, the present writer, - were sick or otherwise

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incapacitated for active play to a greater or less extent, and thus enjoyed an equally good opportunity to practice calculation. Fuller and Buxton, on the other hand, whether precocious or not, were men of such limited intelligence that they could comprehend scarcely anything, either theoretical or practical, more complex than counting; and their purely manual occupations left their minds free to carry on almost without limit their slow and laborious calculations.
These considerations put the whole matter of mathematical precocity in a new light. Instead of joining in the popular admiration and awe of these youthful calculators, - and even psychologists have not been wholly free from this uncritical attitude, - we must say that precocity in calculation is one of the most natural things in the world. If a person is to become a calculator at all, he will usually begin as soon as he learns to count, and in most cases before he learns to read or write; and his development, while it will of course be gradual, - in Bidder's case probably a year elapsed between his learning to count and the early incidents which made his gift known, - will be so greatly facilitated by the amount of time available, the intrinsic interest of calculation, and the ease with which new information can be picked up as needed, that he may become a full-fledged calculator before he is suspected of being able to count without the aid of his fingers. His preoccupation with his calculations may give rise to a false appearance of backwardness, or be may really be of very low intelligence, or he may be an all-round prodigy like Safford, Gauss, and Ampere; mental arithmetic is so completely independent and self-sufficient that it is equally compatible with average endowments or with either extreme of intelligence or stupidity.
Mathematical precocity, then, stands in a class by itself, as a natural result of the simplicity and isolation of mental arithmetic. There is nothing wonderful or incredible about it. The all-round prodigy like Ampere or Sir William Rowan Hamilton or Macaulay is possible only in a well-to-do and cultured family, where books are at hand and general conditions are favorable, and he must possess genuine mental ability. The musical prodigy, again, - Mozart is the stock instance, - must come of a musical family, hear music, and have at least some chance to practice, and hence cannot long hide his light under a bushel. But the mathematical prodigy requires neither the mental ability and cultured surroundings of the one nor the external aids of the other. He may be an all-round prodigy as well, like Gauss, Ampere, and Safford; it is not improbable that Bidder, under favorable conditions, would have developed into such an "infant phenomenon"; but he may also come of the humblest family, and be unable, even under the most

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favorable conditions, to develop average intelligence. He may proclaim himself to the world almost at once, like the all-round or the musical prodigy, or keep his gift a secret for months or even years. If we are to call him a prodigy at all, it is important to realize how widely he may differ from other prodigies, and to avoid carefully the popular confusion due to the misleading associations of the words "prodigy" and "precocious."
(b) Loss of Power. Mental calculation, then, starts from an interest in counting; at the outset it demands only that ability to count by 1's, 2's, 3's, 7's, and the like, which all of us require for such every-day purposes as keeping track of the days of the week. But if for any reason this interest in counting is lost, practice in calculation will cease, and the skill already acquired will disappear, just as the pianist's skill is lost when interest and practice cease. There are two striking instances of this among mental calculators: Whately and Mr. Van R. of Utica, both of whom began to calculate at an early age, but lost the power after two or three years. Here, again, however, there need be no mystery; the disappearance of the gift with the loss of the interest in which it originated is as natural and normal as its original appearance.
Just what caused the loss of interest is not always easy to say. In Whately's case the trouble may have been that on going to school he was taught arithmetic or "ciphering" by methods very different from his old ones, became confused, failed to establish a connection between the two, and lost his interest in calculation as a result of his distaste for "ciphering." In Colburn's case the loss of skill seems to have been much more gradual, and probably never complete. In this respect he is like the pianist who reta