Popular Science Monthly/Volume 42/November 1892/The Latest Arithmetical Prodigy
MATHEMATICIANS, doctors, and philosophers have lately enjoyed a rare opportunity to study a new calculating prodigy, a young man twenty-four years old, who performs mentally, with surprising rapidity, operations in arithmetic involving a large number of figures. We purpose, pertinently to his case, to consider the psychological aptitudes which serve as the basis of mental calculation. We shall use in our study the report of the committee of the Academy of Sciences which examined M. Inaudi, and the results of our own personal observations of his powers, by which we are convinced that he can bear comparison, for the extraordinary development of his memory, with all other known calculators. Jacques Inaudi was born at Onorato, in Piedmont, on October 13, 1867, of a family in modest circumstances. He passed his earlier years in tending sheep. At the age of six years lie was taken with a passion for figures, and began to combine numbers in his head while at watch over his flock. He did not try to give his calculations a material form by counting on his fingers, or with stones, but the whole operation was mental. He conceived numbers by the names which his elder brother had recited to him. Neither he nor his brother could read then. He learned by ear the numbers to hundreds, and exercised himself in calculating with what he knew. When he had done his best with these numbers he asked to be taught those above a hundred so that he might extend the sphere of his operations. He has no recollection of his brother teaching him the multiplication table. At seven years of age he was capable of performing in his head multiplications of five figures. In a little while he started with his brother to wander through Provence, the brother playing the organ and Jacques exhibiting a marmoset and holding out his hand. To increase his receipts he proposed to the people he met to perform mental calculations for them; at the markets he assisted the peasants in making up their accounts, and performed difficult arithmetical operations in the cafés. A manager engaged him to give representations in the cities. He came to Paris for the first time in 1880, and was presented to the Anthropological Society by Broca, who wrote a brief note on the case.
Since 1880, M. Inaudi has made great progress. First, he learned to read and write, and then the sphere of his operations widened. His education, which was slow, is still rudimentary on many points; but he has a receptive intelligence and an inquiring spirit, is pleasant and modest, converses agreeably, with good sense, and sometimes with irony; and is ready at cards and billiards. It would be wrong to regard him as a simple calculating machine.
The operations he performs are additions, subtractions, multiplications, divisions, and extractions of roots. He also resolves by arithmetic problems corresponding with equations of the first degree. These are to him exercises of mental calculation, by which we mean a calculation made in the head, without the employment of figures or writing, or any material means to assist the memory. His general process is as follows: first, when the problem is stated to him aloud, he listens attentively and repeats the data, articulating them clearly, to fix them well in his mind; if he does not comprehend the problem, he has it repeated. It may be communicated to him by writing, but he prefers to receive it by hearing; and if we insist upon his reading it, he pronounces it in a low tone. When he has fully grasped the question, he says, "I begin," and proceeds to whisper very fast, in an indistinct murmur, in which we can catch from time to time a few names of numbers. At such times nothing can move him or distract him; he performs the most complicated operations in the midst of the excitement of public representations. He can even talk while mentally working; he answers questions properly, and even keeps up a regular conversation without disturbance to his arithmetical operations. During his exercises he is sometimes seen to lift his hand to his forehead or to close his fist, or to draw imaginary lines with the forefinger of his right hand in the palm of his left hand. These are little tricks of no importance, that vary from one day to another. Finally, after an interval which is always short, he says, "I am done," gives the solution of the problem, and proves it for his own satisfaction.
The two remarkable features in M. Inaudi's mental calculations are the complexity of the problems he undertakes, and, in a less degree, the rapidity with which he finds the solution. Most of the questions that are put to him involve the use of a considerable number of figures; he can add in his head numbers composed of twelve ciphers each; he multiplies by one another numbers composed of eight or ten figures each; he tells how many seconds there are in an arbitrarily selected number of years, months, days, or hours. These operations, to be well carried on, require the subject to keep in mind the data of the problem and the partial solutions till the moment when the definitive solution is found. For so considerable a task M. Inaudi takes, they say, an extremely short time—so short as to convey the illusion of instantaneousness. It has been published on this subject that "he adds, in a few seconds, seven numbers of eight or ten figures. He completes the subtraction of two numbers of twenty-one figures in a very few minutes, and finds as rapidly the square root or the cube root of a number of from eight to twelve figures, if the number is a perfect square or cube, but needs a little more time if there is a remainder. He likewise finds, with incredible celerity, the sixth or seventh root of a number of several figures. He performs a division or a multiplication in less time than it takes to announce it." M. Inaudi found in thirteen seconds the answer to the question, How many seconds are there in eighteen years, seven months, twenty-one days, and three hours?
But while M. Inaudi calculates rapidly, he is not much more rapid than a professional calculator who is permitted to work out his problems on paper; M. Inaudi's merit is that he performs his operations in his memory.
His processes are not ours, and although he has been able to read and write for four years and is acquainted with the ordinary methods of calculation, he does not use them. M. Charcot caused him to perform at the Salpêtrière two divisions of equal difficulty, one on paper according to our method, and the other in his own way; the second required four times less time than the first. M. Inaudi, faithful to the processes of his infancy, manages them with surprising dexterity. He has perfected, developed, and enlarged them, but has not changed their nature.
The basis of his calculations is multiplication; even in dividing or extracting the square root, he multiplies. He makes a series of multiplications of approach. In a division, for example, he finds the quotients by groping; seeking and trying a number which, multiplied by the divisor, will produce the dividend.
He follows a course in multiplication which is peculiar to him. If more than one figure is included, he does not perform the process all at once, for he has no more extended multiplication table than ours; but his method consists in decomposing a complex multiplication into a series of simpler ones. If he is to multiply 325 × 638, M. Inaudi calculates thus:
In short, he makes six multiplications instead of one. He begins on the left, multiplying, therefore, the figures of the highest value. In other cases he changes the data around. Instead of multiplying by 587, he multiplies by 600 and then by 13, and subtracts the second product from the first. The observation of M. Inaudi brings a new factor to the theory of partial memories. It is usual to employ the word memory in a general sense to express the property, common to all thinking beings, of preserving and reproducing the impressions they have received; but psychological analysis and a large number of facts in mental pathology have shown that memory should not be regarded as a sole faculty, having a distinct seat; in the final analysis, memory is a group of operations. There exist, according to the report of the committee of the Academy, partial, special, local memories, each of which has its special domain, and which are so independent that one of them may be enfeebled, may disappear, or may develop to excess without the others necessarily presenting any corresponding modification. The older psychologists missed this truth. Gall was probably the first to assign its proper memory to each faculty, and founded the theory of partial memories. It is at the present time supported by multiplied facts, a large number of which have been furnished by M. Taine. He has cited, among others, the cases of those painters, designers, and statuaries who, after having carefully regarded a model, can make its portrait from memory. They supply fine examples of the development of visual memory. Then there are cases of musical memory. The subject has been revived of late years in the study of diseases of language. Cases have been cited of patients in whom the single memory of language, very limited and special, is abolished, while the other memories remain intact; there are patients who, without being paralyzed, can no longer write, but continue to speak; others lose the faculty of reading while they keep that of writing, so that they can not read the letter they have just written.
The study of arithmetical prodigies presents the same question under another aspect: no memory is destroyed in them; but one of the memories, that of figures, acquires an abnormal extension that excites enthusiasm and admiration, while the other memories, regarded as a whole, present nothing peculiar. They even sometimes continue below the common grade. Subjects of this class are real specialists who interest themselves during the whole course of their existence in but one thing—numbers. Pertinently to this point, a characteristic anecdote is related of Buxton, a celebrated calculator, who was taken to a performance by Garrick. At the conclusion of the play he was asked what he thought of the piece. He replied that a certain actor had entered and made his exit so many times, and had pronounced so many words, and so on. That was all the recollection he had of the play. The committee of the Academy has taken the measure of the different kinds of memory in M. Inaudi, and has concluded that he has not a greatly developed memory for forms, events, places, or musical airs, and I have found that his memory for colors is very weak. He gives surprising results only in numbers. This inequality in the development of memories assumes a remarkable character when we compare in him two things nearly identical, the memory for figures and that for letters. A series of letters was pronounced in his presence which he was asked to repeat exactly, and the same was done for figures. It would seem at first sight that the articulated sound of a pronounced letter would be as easy to hold in the ear as that of a figure, so that a person capable of repeating, for example, twenty-four figures, as M. Inaudi does without much effort, would have no more difficulty in repeating twenty-four letters. But this was not the case. It was found, not without surprise, that M. Inaudi could not repeat more than seven or eight letters from memory. He hesitated, lost his usual self-possession, and wanted to withdraw from the experiment; and when two lines of French were read to him, he could not repeat them exactly after a single hearing.
The recollection of the figures is a necessity for every mental calculator. It is of service to him, first in retaining the details of the problem, and then in retaining the partial solutions till the complete solution is found. The complexity of the problems which a person can hold in his head gives an idea of his memory. But there is a more direct and simpler means to measure the extent of the memory for figures, and that is to cause him to repeat a series of figures, seeking to find by trial the maximum number that he can repeat without mistake. Such trials are common in psychological laboratories. According to my personal observations, persons can repeat on an average from seven to ten figures without making a mistake, when they are pronounced with a rapidity of two per second. The division of figures into groups, the special vocal intonation, or some kind of rhythm, are artifices which may sometimes increase the number, and make the effort to repeat less painful. These results agree with those of an American psychologist, Mr. Jastrow, who mentions 8.5 as the average number found among pupils in his country.
M. Inaudi has practiced this kind of repetition for a long time. We repeat the number, dividing it into periods of three figures each, and giving the value of each period. For example, to make him repeat the number 395,820,152,873,642,586, we give it out, three hundred and ninety-five quadrillions, eight hundred and twenty trillions, one hundred and fifty-two billions, eight hundred and seventy-three millions, six hundred and forty-two thousand, five hundred and eighty-six. We are careful to dwell on the articulation of the numbers. M. Inaudi repeats, as fast as he comprehends it, each period of three figures; then, when he has taken in the complete number, he says confidently, "I know it," and repeats the whole series with great volubility.
I have witnessed his repetition in this way, without mistake, of a series of twenty-four figures. M. Charcot, in order to compare his capacity with that of Mondeux, another famous calculator, repeated with him the experiment, which had been tried with Mondeux, of telling off a number of twenty-four figures, divided into four periods, so that he might announce at will the six figures included in each of the periods. Mondeux took six minutes to reach the result; M. Inaudi only had to hear the figures given out. Thus a single hearing suffices M. Inaudi to fix in his mind a long series of figures or the statement of a complicated problem; he does not go back to repeat the numbers several times as we are obliged to do. He only asks, when the series of figures is a little long, to have it pronounced slowly. Once fixed in his memory, the number is retained with a precision and sureness which it is hard to conceive. M. Inaudi can not only repeat a number of twenty-four figures in the order in which he heard it, but in an inverse order, beginning with the units; he can repeat half the number in one direction, and the other half in the other direction; and all this without hesitation, without fatigue, and without mistakes.
Ordinary persons can recollect a number of many figures only a few seconds unless they have aids to their memory. M. Inaudi's memory retains for a very long time the numbers that have been given to him. He is in the habit of repeating at the end of a sitting all the numbers on which, he has been set to work, in the different questions put to him. This experiment, which I saw made at the Salpêtrière, gives really incredible results. A large number of problems were given to M. Inaudi during the afternoon, the data of all of which were preserved in writing, in order to verify the exactness of the repetition. On this day he repeated two hundred and forty-two problems. It is said that he repeated four hundred at a sitting given in the Sorbonne.
These numbers, however, should not be taken as the measure of M. Inaudi's memory for figures, because he did not learn them one after another, without interruption. They were contained in distinct experiments, in which the calculator burdened his memory each time with only twenty-four figures. He therefore had intervals of rest, however brief; and these rests probably facilitated the assimilation of the whole mass, which was really enormous. Usually, he told us, he did not try to retain groups of more than twenty-four figures. One day, twenty-seven were given out to him. That was the maximum number that was essayed. I proposed to him to recite twenty-six, and he was able to repeat them all exactly by employing his usual processes. The experiment tired him a little. After a short rest, I read fifty-two figures to him. In the middle of the experiment, when he had reached the twenty-sixth figure, I pronouncing them and he repeating them, he stopped. He was troubled, and expressed a fear that he would forget the whole. He then repeated rapidly from memory the figures which had just been pronounced, after which he asked me to continue. I went on then to fifty-two figures. He then tried to repeat them all. He did it, but with some transpositions and confusions, and about ten mistakes. The number fifty-two seems to constitute a limit for him.
We have now to examine a little more closely what is meant by the memory for figures; for there are an immense variety of psychological types, and the same mental operation may be comprehended and performed by two persons under absolutely different forms. There are many ways of fixing figures in the memory and calling them out again; or, in other words, several images of a different kind are employed. According to the investigations of the committee of the Academy, M. Inaudi's processes are the contrary of those which arithmetical prodigies are generally supposed to use.
These persons, according to their own testimony, are accustomed to take visual memory as the basis of their mental operations. They have an inner vision of the numbers that are pronounced, and those numbers, during the whole time of the operation, stand before their imaginations as if they were written on a tablet set before their eyes. This process of visualization was that of Mondeux and Colburn, and of all who have given clear explanations of themselves. With this, nothing is easier than to account for the faculty of mental calculation—that is, of calculating without reading or writing anything. Whenever any one has a clear and sure visual memory, he does not need to have the figures before his eyes to read them and write them out in order to be able to combine them; he can turn away his eyes from the slate, because they are written as if with chalk on the tablet which his memory presents to him. This explanation appears so satisfactory that Bidder, one of the greatest mental calculators of the century, wrote in his autobiography that he could not comprehend the possibility of mental calculation without this faculty of representing the figures to himself as if he was looking at them.
This interpretation has been confirmed by the researches of Mr. Galton. Inquiring of a large number of calculators and mathematicians of every kind and every age, he has learned that most of them have a visual image of the figures during their calculations; the natural series of figures is presented in a straight line, or follows the bendings of a curved line. With some persons the figures appear placed as if in relation to the rounds of a ladder; with others they are inclosed in squares or circles. Mr. Galton calls these images number-forms. The visual image must be very clear for it to be possible to recognize so many details. M. Taine, who has studied the phenomenon of the image with much care, has discovered a resemblance between mental calculators and checker-players who do not have to look at their boards. He explains their faculty by the clearness of their visual images. "It is evident," he says, "that every move, the figure of the whole checker-board, with the order of the different pieces, is presented to them as in an inner mirror; else they would not be able to foresee the probable consequences of the move that has been made upon them and of the one they are about to order." The direct testimony of players confirms this interpretation. "With my eyes turned to the wall," says one of them, "I see at once the whole board and all the pieces as they really stand. . . . I see the pieces exactly as the turner has made them—that is, I see the checker-board in front of my adversary, and not some other checker-board."
In the light of so many facts we are naturally led to believe that all mental calculators work by the considerable development of their visual memory. But the study of M. Inaudi shows that we can not draw a general conclusion from them, and that there are other means than mental vision that seem to have the same efficaciousness and power. M. Inaudi declares that no figure is presented to him under a visual form. When he endeavors to retain a series of twenty-four figures that have just been pronounced, or when he combines numbers to solve a problem, he does not see the figures, but hears them. "I hear the numbers," he says distinctly, "and it is my ear that retains them. I hear them sounding in my ear, just as I have pronounced them, in my own voice, and that inner hearing persists through a good part of the day." At another time he told M. Charcot: "Sight is no help to me; I do not see the figures. I will say even that I have more difficulty in recollecting the figures and the numbers when they are communicated to me in writing than when I receive them by speech. I feel cramped in the former case. I do not care about myself writing the figures. Writing does not help me recollect them. I prefer to hear them."
His words are confirmed by his actions. When the numbers are given to him written, he pronounces them aloud, putting himself in substantially the same position as if they had been communicated to him by the hearing; then, when he begins his calculation, he turns his eyes away from the written figures, the sight of which, instead of aiding his memory, is only an embarrassment to him. "How can I depend on seeing the figures," he says, "when it is only four years since I learned to read and write? Yet I calculated mentally before that time."
Our supposition that M. Inaudi relies on auditive images in his calculations is not absolutely correct. A pure auditive image is very rare. Auditive images and sensations of words are associated with the motions of the larynx and the mouth required to pronounce them; and when a person represents to himself a word under the form of a sound, he should at the same time experience special sensations in the organs of phonation, as if the word was about to be pronounced. In other words, so far as concerns language, the auditive type has the closest connections with the motor type; the two are often combined.
This probably takes place with M. Inaudi. We have seen that his lips are not wholly closed when he is at work. They move a little, and an indistinct murmur issues from them, in which we may catch from time to time a few names of figures. The whispering sometimes becomes so intense as to be heard several metres off. I have assured myself, by taking the respiratory curve of the subject, that it bears very clear marks of this phenomenon, even when we do not hear it. His sounding organs are then really active while he is calculating in his head. M. Charcot, wishing to determine the importance of these movements, and see what would happen if they were not executed, asked M. Inaudi to make a calculation with his mouth open. But this device did not wholly prevent the motions of articulation, which were still apparent. I tried to prevent M. Inaudi from articulating sounds in a low tone, and asked him to sing a vowel during his calculation; if the sound of the vowel kept pure in tone, it was certain he did not articulate the figures. The experiment caused M. Inaudi great embarrassment. He was still able to calculate in his head, but it took him four or five times as long as under the usual conditions, and he succeeded in doing it only by cheating a little—that is, he made some articulations of figures in a low voice, the production of which was at once detected on listening attentively to the sound of the sung vowel.
These experiments showed that articulation constitutes an integral part of M. Inaudi's mental calculations, as well as that every experimental artifice that interferes with articulation makes the calculation longer or modifies its accuracy. In other words, M. Inaudi uses auditive and motor images of articulation concurrently. We have no experimental means of determining which is the predominant factor. M. Inaudi thinks that the sound guides him, and that the motion of articulation intervenes only to re-enforce the auditive image. We might be liable to suppose, in view of the part that is played by the memory in mental calculation, that it is the only faculty developed in arithmetical prodigies; and some authors have fallen into this error. But it will be well to guard against such a supposition, which is contrary to the most certain and best established psychological facts. If we take any elementary act of the mind and analyze it, we shall find that it involves the concurrence of a large number of co-ordinated operations; with much stronger reason must such a concurrence be supposed necessary for acts as complex as mental calculations. We have found in our studies of M. Inaudi that a considerable number of his faculties have attained an extreme development, and they are precisely the ones that concur in operations of mental calculations. Perception, attention, and judgment, to the extent and in the shape in which they are needed in his work, have acquired the same perfection as his memory for figures.
It remains to inquire how these aptitudes for calculation have been formed. When we examine the history of these arithmetical prodigies, we are struck by the three facts of their precocity; the impulsive, in a certain sense all-possessing, character of their passion for calculation; and the generally illiterate, often miserable, medium in which they have grown up. Their stories have many traits in common. They are most frequently children of poor and ignorant parents. They are seized with the passion for calculating in their earliest years—at from five to ten years of age on the average—the age when most children are living in the illusions of plays and stories; they begin to combine numbers in their heads, apparently without any exterior provocation, and without the influence of parents or schoolmasters. As they grow up some of them become mathematicians like Gauss and Ampère, while others continue all their lives what they were in the beginning, simply specialists in figures. We do not know whether this distinction arises in the nature of things, or simply results from the chances of life. Very good minds think there is a relation between the calculating faculty and mathematical talent, and believe that, if these prodigies were intelligently given a special education, they might most of them become remarkable mathematicians. Experiment has not given a definite result on this point. M. Inaudi has determined not to go to the mathematical school, but will preserve and develop his natural gifts. Another question arises as to the influence of heredity in these cases. For a long time physicians have been accustomed, when an abnormal combination of talents appears in a particular person, to find a number of special characteristics in his family. Sometimes these have appeared through several generations, as in certain noted families of musicians and naturalists. Sometimes the peculiarity appears in the shape of eccentricity. No such peculiar family traits have been found associated with M. Inaudi, nor any special antecedents in himself. He has never been ill, and his development has been normal.
The study of M. Inaudi has been fruitful for psychology. On one side it has brought a remarkable confirmation to the theory of partial memories; and on another side it has made us familiar with a new form of mental calculation, the auditive form. It may also have taught us something else. We have found that it is possible for some faculties, like memory, to acquire an extent double and triple that of the normal. The fact permits us to descry in how large a measure the human mind is still capable of improvement.—Translated for The Popular Science Monthly from the Revue des Deux Mondes.