The Magic of NUMBERS. Robert Tocquet. 1960

The Magic of NUMBERS
Robert Tocquet. 1960 (First publicated in France in 1957 by Pierre Amiot as "2 + 2 = 4")

While the facts given in this book are sound, the author's intention is to amuse rather than instruct. For this reason "textbook" terminology has been avoided wherever possible.

Chapret 1 Yesterday and Today

LIGHTNING CALCULATORS, especially when illiterate, have drawn the attention of the public in all Ages by their extraordinary abilities. They can solve in their heads, sometimes instantaneously and without apparent effort, problems often so complicated that most of us, even mathematicians accustomed to juggle with figures, could solve them only with pencil and paper and over a much longer time, without being sure even then of succeeding. Some of them, too, when they have been set a problem, can talk freely with bystanders, discussing subjects completely foreign to the question they are dealing with, and then suddenly give the required solution, as if a cerebral mechanism had been working within them without their knowledge.

As a general rule, and this is a fact which should be emphasised immediately, the lightning calculators, apart from the faculty they have of handling figures with exceptional virtuosity, are of below average intelligence; sometimes they are even mentally retarded. Thus, Colburn was always at the bottom of his class, Buxton could not even write his name and Inaudi did not learn to read or write until he was over twenty years of age. There are certain exceptions, however, to this rule, for some have been known who have educated themselves normally and there have even been geniuses who were phenomenal calculators: Ampere, Arago, Georges Bidder, Whately and Gauss being examples.

Lightning calculators of the past

Among calculating prodigies who were otherwise backward or who had very little education, let us recall those who had the greatest renown in the past before examining present-day calculators in greater detail.

The Greek writer Julian mentions a certain Nikomachos, who lived at Gerasa in Palestine in the second century of our era, and who found solutions to difficult problems very rapidly.

Balthasar of Monconys, in an account of his third journey in Italy, records that in 1664 Mathieu le Coq, then aged eight and unable to read or write, had been performing advanced arithmetical operations, such as multiplications with five or six figures and extractions of square and cube roots, for some two years previously.

Thomas Fuller, nicknamed the Virginian Calculator, or the Negro Calculator, was almost totally ignorant. A slave in Virginia in the middle of the eighteenth century, he could neither read nor write and he died at the age of eighty without ever having learned to do so. Scripture records the following story about him in the American Journal of Psychology:

"When Fuller was about seventy years old, two gentlemen of Pennsylvania, William Hartshorne and Samuel Coates, both men worthy of confidence, heard of the calculator and had the curiosity to have him brought before them and put to him the following problems: First, how many seconds are there in a year and a half? Fuller replied in two minutes that there are 47,340,000 seconds. Secondly, how many seconds has a man lived who is aged seventy years, seventeen days and twelve hours? Fuller replied at the end of a minute and a half: 2,210,800,800. One of the gentlemen who examined him had taken the trouble to do the calculation on paper and told Fuller he was wrong and that the number of seconds was less. But Fuller pointed out promptly that this difference in the two results had to do with leap years."

Jedediah Buxton, who was born at Elmeton, near Chesterfield, lived in England from 1702 to 1762. A poor labourer, totally illiterate to the point that he was unable to scribble his own name, of below average intelligence, Buxton had the greatest difficulty in learning the multiplication table. This was in fact the only education he received. On the other hand, like most lightning calculators, he had a highly developed memory for figures. For example he knew the exact number of seconds in a day and in a year. Possessed of a positive mania for arithmetic, "he could perceive only figures, and pretexts for mental operations", writes Alfred Binet, "his mind being completely closed to anything else". When he came to London he was taken to Drury Lane Theatre to see Richard III played by Garrick. When he was asked afterwards if the performance had pleased him it appeared that he had found in it only an opportunity to perform calculations; during the dances he had fixed his attention on the number of steps executed; there were 5202; he had also counted the number of words the actors had spoken: 12,445; he had also remembered the number of words spoken by Garrick, and all this was found to be accurate. In his calculations he reduced all lengths to an eccentric standard, the thickness of a hair, which he had probably fixed arbitrarily.¹

He was not merely a mental calculator of great capacity; he had also a very accurate eye, a sort of divinatory power for areas. When he went through a field or a simple piece of land, he could, so it is said, give the area as exactly as if he had measured it with a surveying chain. In this way he determined the extent of the Elmeton estate in acres, and for his personal satisfaction, worked out the result in square inches and in squares having the thickness of a hair.

Zerah Colburn, who was born in the State of Vermont, U.S.A. in 1804, began calculating without knowing how to read or write. One day his father noticed his unusual aptitude by accident. The child was repeating aloud the products of the multiplication table without, it seems, ever having had the table at his disposal. Astonished, Mr. Colburn asked him: "How much is 13 times 97?" and the child replied at once: "1261". He was then six years old.

Mr. Colburn saw in this gift for calculation a means of earning money and had the idea of exhibiting his son. Zerah Colburn thus became the first of the professional lightning calculators. He performed at Boston, and then, at the age of ten, came to London, and to Paris where, apparently, his performances had no great success. When an attempt was made to get him to reveal the secret of his mysterious gift, he invariably replied: "It is God who has put these things into my head and I would not know how to put them into yours".

Thanks to the support and patronage of Washington Irving he was admitted as a pupil at the Lycee Napoleon in Paris, then at the College of Westminster, but it was found that apart from mental calculation his mind was closed to all normal forms of instruction. He lost his faculty to calculate, without any apparent reason, at the age of twenty. It is worthy of note that Colburn had one curious physical peculiarity; a certain sign of degeneracy: he had an extra finger on each hand and an extra toe on each foot.

Zacharias Dase, born in Germany in 1824, distinguished himself from the majority of lightning calculators by the fact that he placed his ability at the service of science. He calculated the natural logarithms of the numbers from 1 to 100,500 and the table of factors and prime numbers from the seventh to the eighth million. But he was never able to learn conventional mathematics, despite the efforts of the eminent teachers who interested themselves in him; in addition he showed no intelligence in anything which did not have to do with figures or numbers. Both his faculty for counting and his memory were prodigious: the astronomer Gauss made him multiply mentally one by the other, numbers each composed of a hundred figures. One is literally stupefied to realise pen in hand what an enormous array of figures such an operation entails. Yet Schumacher reports that Dase multiplied two numbers of 8 figures each in fifty-four seconds and two numbers of 20 figures each in six minutes. He had also a great rapidity in perception and a visual memory for estimating the number of objects contained in a group, for instance the number of books in a library.

Vito Mangiamele was a little ten-year-old Sicilian shepherd boy when he came to Paris to be examined by Arago in 1837. Though he had had no education he had worked out for himself calculating processes which have never been properly explained. At the Academie des Sciences where he was introduced he was asked, for example:

"What is the cube root of 3796416?"

At the end of a minute he replied: "156", which was correct. It took him no longer to solve the two equations:

x³ + 5x² - 42x - 40 = 0 and

x⁵ - 4x - 16799 = 0

Henri Mondeux became very famous. Born in 1826 at Neuvy-le-Roi, near Tours, he was the son of a poor woodcutter. At the age of seven, when he could neither read nor write, he used to amuse himself by doing involved calculations while guarding his sheep. Being ignorant of figures he counted with little pebbles arranged in different ways. A Tours schoolmaster, M. Jacoby, who had been told of this, tried to teach him but without any success.

Ultimately he was taken to Paris and presented to the Academic des Sciences. An examining committee was set up which included Arago, Serres, Sturm, Liouville and Gauchy.

It was speedily recognised that the child had an abnormal faculty for mental calculation, a prodigious memory for numbers, but an almost total absence of memory for the names of places or people, or of objects which did not interest him. It was observed also that while solving a problem he could engage in completely unrelated activities. Finally the committee published some of the calculating processes he had invented. These are given in the report by Gauchy from which the following passages are taken:

"In many instances Henri Mondeux invents methods, which are sometimes remarkable, for solving a number of questions which are ordinarily treated by algebra, and determines in his own fashion the exact or approximate values of whole numbers or fractions which must fulfil required conditions. When he has to multiply whole numbers by one another Henri Mondeux frequently splits these numbers into groups of two figures. He has succeeded in finding out for himself that the operation becomes simpler in cases where the factors are equal, and the rules he then uses to arrive at the product or the power required are precisely those which would yield the formula known as the binomial theorem. Guided by these rules he was able to give at the very instant he was asked for them, the squares and cubes of a multitude of numbers, for example the square of 1204 or the cube of 1006. As he knows pretty well by heart the squares of all the whole numbers below 100, the splitting of larger numbers into groups of two figures enables him to obtain their squares more easily. Thus he was able to find the square of 755 almost immediately while in the presence of the Academic.

"Henri has succeeded unaided in discovering the established method for finding the sum of an arithmetical progression. Among the rules which he has invented for solving different problems are several which can be inferred from certain algebraical formulas. The method he has worked out for calculating the sum of the cubes, and the fourth and fifth powers of natural numbers, can be cited as an example. To solve two simultaneous equations of the first degree Henri has made use of a device which is worth mention. He has sought first for the difference between the unknown quantities and to do this he has subtracted one equation from the other, after having multiplied the first by the ratio which exists between the sums formed successively by adding the one and the other to the coefficients of the two unknowns.

"Where it is a case of solving not simultaneous equations of the first degree but a single equation of a degree higher than the first, Henri habitually uses a method which we will explain by an example. We put a problem to him of which these are the terms: Find a number such that its cube, increased by 84, furnishes a sum equal to the product of this number when multiplied by 37. Henri gave as a solution the numbers 3 and 4. To obtain them he began by transforming the equation which it was required to solve by dividing the two numbers by the number sought. The question was then reduced to the following: find a number such that its square, increased by the quotient obtained on dividing 84 by this number, gives 37. By the aid of this transformation Henri Mondeux was able to recognise immediately that the number sought was inferior to the square root of 37, by consequence to 6; and very soon a few simple trials led him to the two numbers we have indicated.

"Even questions of indefinite analysis are not beyond the reach of Henri Mondeux. One of us asked him for two squares of which the difference was 133. As solution he gave immediately the system of the numbers 66 and 67. It was insisted that a simpler solution be obtained. After a moment of reflection he indicated the numbers of 6 and 13.

"This is the manner in which Henri proceeded to reach the two solutions. The difference between the squares of the numbers sought exceeds the square of their difference by a quantity equal to twice this difference multiplied by the smaller. The question put can thus be reduced to the following: subtract from the number 133 a square such that the remainder will be divisible by double the root. If a trial is made one after the other of the squares 1, 4, 9, 16, 25, 36, 49 ... it will be recognised that among these squares 1 and 49 are the only ones which satisfy the new question. By subtracting these from 133 and dividing the remainders 132 and 84 by the doubled roots, that is by 2 and 14, the quotients 66 and 6 are obtained, each of which responds to one of the solutions given by Henri Mondeux. It is perceived that in following the procedure we have described Henri did not first arrive at that of the two solutions which seems to us the simpler, but that which offers the squares whose roots are nearest to each other."

Gauchy ends his report by expressing the hope that Henri Mondeux would distinguish himself one day in a career of science, but despite this optimistic hope this lightning calculator died in obscurity.

Contemporary Lightning Calculators

Jacques Inaudi

We come now to the best known and most popular lightning calculator of our own time: Jacques Jnaudi.

Jacques Inaudi was born of a very poor family, at Onorato, in Piedmont in 1867. He was a shepherd boy, nearly six years old, when he was seized by a passion for figures. While guarding his flocks, he practised working out numbers in his head, so that by the age of seven he was already able to do mental multiplication involving five-figure numbers. And he did not know the multiplication table! After his mother died he left the neighbourhood and set out to wander with his brother, exhibiting a tame squirrel. The brother played the organ, while Jacques displayed the squirrel and collected the money.

He talked to bystanders they met about mental calculations, of which they naturally understood nothing. At the markets he helped the peasants do their accounts: "As a matter of fact", he writes, "I was very surprised that these men, who were generally fairly astute, should not have known the totals of their accounts, which I knew myself, almost instantaneously, just by listening to them; this gave me the courage one day to take part in an argument about the settlement of an account between two peasants who were about to come to blows, and whom I was able to calm down after demonstrating to them that they were both wrong; this altercation naturally attracted a crowd, astonished that a little chap like me should know how to count better than the grown-ups. Those who were good at figures put various questions to me to which I replied correctly and very easily, still remaining amazed that anyone should be ignorant of answers which seemed so natural to me. Afterwards the peasants came to seek me out whenever a difficulty arose".

Soon Inaudi made his appearance in cafes where he was noticed by a commercial traveller, a M. Dombey, who became his impresario and took him on tours in the provinces and then to Paris. There he attracted the attention of Camille Flammarion, who devoted some articles to him in various scientific journals. The well known anthropologist, Paul Broca, examined him in his turn and prepared a short report on his case. Broca noted that Jacque Inaudi's head was very large and irregularly shaped. The young prodigy afterwards gave performances in the Salle des Capucines in Paris, and later at the Robert Houdin Theatre. By this time he was aged thirteen.

It was 1892 when he returned to Paris; he had learned meanwhile to read and write and his intelligence had developed somewhat. He was now a young man of 24; short and stocky, with a strong head, eyes half closed, the facial angle very developed, almost straight. According to Binet, he had a gentle and modest disposition, spoke little and was rather reserved. His education was not yet extensive; hence subjects for conversation remained limited.

His impresario, who was then M. Thorcey, introduced him to the Academic des Sciences, which formed a committee to study the calculator. It included MM. Darboux, Poincare, Tisserant and Charcot. Alfred Binet became a member later. After numerous tests the committee gave its conclusions, via the pen of M. Darboux.

"It should be pointed out first of all", the well known mathematician wrote, "that the results of which we were the witnesses depended above all upon a prodigious memory. At the end of a session attended by our students, Inaudi repeated a series of numbers comprising more than 400 figures. At one of our meetings we gave Inaudi a number of 22 figures. He was able to repeat it eight days after, although we had not warned him that he would be asked for it again. A second point, which seems to me of the highest importance, has been neglected by the majority of those who examined him. They have analysed with great care the processes, certainly very simple in themselves, which Inaudi uses for various operations; but they have not noted one very evident fact: that these methods have all been invented by the calculator himself, and that they are wholly original. And, what is also interesting is that these rules differ from those which are taught everywhere in Europe, though some of them resemble in certain respects those which are followed for example, among the Hindus. This will be evident from the following:

Addition. - Inaudi easily adds 6 numbers of 4 or 5 figures; but he proceeds successively, adding the first two, then their sum to the next and so on. He always starts adding from the left, as the Hindus do today, instead of from the right as we do.

Substruction. - This is one of Inaudi's triumphs. He subtracts easily, one from the other, two numbers consisting of a score of figures, beginning always from the left.

Multiplication. - The methods used are all elementary, but they require the memory of an Inaudi. For example, to multiply 834 by 36 he makes the following breakdown:

800 x 30 =
800 x 6 =
30 x 30 =
4 x 30 =
Total

24000
4800
1080
2 144
30024

In all these partial multiplications, none of the factors ever has more than one significant figure. Despite this Inaudi knows and uses the property of the factor 25; he knows that to multiply by this number it is sufficient to take the quarter of the centuple. For example, for the square of 27, he will make the following breakdown:

25 x 27 =
2 x 27 =
Total

675
54
729

Sometimes he uses partial products bearing the index sign - (minus). For example, for the cube of 27, that is the product of 729 multiplied by 27, he makes this breakdown: (700 x 20; 700 x 7; 30 x 20; 30 x 7) - 27
or 730 x 27 = 19,710 - 27 = 19,683.

Division. - Here Inaudi fundamentally follows the ordinary rule, which reduces the division to a subtraction, but sometimes uses simplifications made possible by his memory, to which we must always return.

Raising to powers. - For raising numbers to powers, Inaudi knows and applies the rule relative to the square of a sum. For example, for the square of 234567 he uses the breakdown: 234000² + 2 x 234000 x 567 + 567².

Extraction of roots. - Here no rules are followed; there is only a simple trial and error. For example, to find a root which is 14,072, Inaudi would have tried 14000 and 15000, then 14600, then 14650, 14660, 14670 . . . and each time, the power of the number tried would have been subtracted from the higher number.²

"Inaudi also solves difficult questions in arithmetic and algebra to which the answer is furnished by whole numbers."

Here are some of the problems which are not mentioned in the report by M. Darboux:

1. Find the number whose square root and cube root differ by 18. Answer: 729, given in one minute fifty-seven seconds. (Revue Scientifique.)

2. Find a number of two figures such that the difference between four times the first figure and three times the second equals 7, and which, reversed, reduces the number by 18. Negative solution found at the end of two minutes.

3. Find a number of four figures of which the sum is 25, given that the sum of the figures of the hundreds and thousands is equal to the figure of the tens, that the sum of the figures of the tens and the thousands is equal to the figure of the units and such that, if the number is reversed, it increases by 8082.

Answer: Since the number increases by 8082 when it is reversed, therefore the figure of the thousands should be 1 and the figure of the units 9; I therefore take away 9, which is the figure of the units, from 25; I have 16 left for the three other figures. Next, the figure of the thousands plus that of the hundreds is equal to that of the tens; the figure of the tens must necessarily be the half of 16, that is 8. Three of the figures being known, it is sufficient to take them from 25 to have the hundreds, 7, and to find that the number required is 1,789. (Revue Scientifique.)

4. The sum of three numbers is 43 and that of their cubes 17299. Answer: 25, 11, 7.

5. Find a number of four figures of which the sum of the figures shall be 16, given that the 3rd is twice the 1st, that the 4th equals three times the 1st plus the 3rd. This number reversed increases by 3456. Answer: 1825.

6. The distance from Paris to Marseilles is 863 kilometres. A train starts from Paris at 8.15 a.m. for Marseilles at a speed of 39 kilometres an hour. Another train starts from Marseilles for Paris at 10.30 a.m. at a speed of 46 kilometres 500 metres per hour. Find at what distance from the two cities the trains will meet.

Answer: The trains will meet at 7 hr. 31 min. 13 and 4/6 sees, p.m., at 419 kilometres 451 metres and 80 centimetres from Marseilles, and at 344 kilometres 548 metres and 20 centimetres from Paris.

7. M. Laurent, examiner at the Ekole Polytechnique, having told Alfred Binet that the calculator Winckler was able to break down a number into four squares, the test was tried with Inaudi.

Alfred Binet proposed the number 13411.

In three minutes Inaudi gave the four following numbers:

115, of which the square is 13,225; 13, of which the square is 169; 4, of which the square is 16; 1, of which the square is 1. Total of the four squares: 13411.

One minute later the calculator found another solution:

113, of which the square is 12,769; 25, of which the square is 625; 4, of which the square is 16; 1, of which the square is 1. Total of the four squares: 13411.

Finally, some time afterwards (the exact time has not been specified), Inaudi produced a third solution:

113, of which the square is 12769; 23, of which the square is 529; 8, of which the square is 64; 7, of which the square is 49. Total of the four squares: 13411.

M. Lebesgue, author of Introduction a la Theorie des Nombres, has admitted that he would have needed a fortnight to arrive at a similar result.

Inaudi indeed calculated at an astonishing speed. Thus in 1924 Maurice d'Ocagne had the notion of organising a competition at the Societe des Ingenieurs Civils to match the calculator with calculating machines of the period. Inaudi beat the machine in addition, subtraction, raising to powers, extraction of roots and in most of the multiplications. It was in multiplications of five figures upward that the machine showed itself quicker than the man.³

In the other cases Inaudi had already given the answer to the problem before the machine had finished taking down the factors. Moreover, like the majority of lightning calculators, Inaudi gave the day corresponding to any date almost instantaneously, which the machine could not do.

Alfred Binet, who studied Inaudi from a psychological point of view, has shown that the calculator was essentially an "auditory" type and that his memory was very specialised. While he had the faculty of remembering hundreds of figures he was incapable of repeating more than five or six letters enunciated in a certain order: a, r, g, f, s, m, t, u, for example. He displayed the same inability to recite two lines of verse or prose. On the other hand he could maintain a conversation and answer questions wittily and to the point, while solving a complicated problem in his head.

Pericles Diamandi

The lightning calculator, Pericles Diamandi, who was a contemporary of Inaudi, does not belong to the group of calculators of less than average intelligence or of slight education. On the contrary he was a highly cultivated man, but as he was studied at length by Alfred Binet and sought to match himself against Inaudi we will mention him here.

Diamandi, born in 1868 at Pylaros in the Ionian Islands, came of a family of grain merchants. During his student period he was consistently first in mathematics. In 1884 he left school and began work in the grain business. It was at this time that he realised that he had exceptional powers of mental calculation. He cultivated these and discovered methods of simplification. At the same time he learned Rumanian, French, German, English and two other secondary languages, and wrote verses and novels. Having read by chance in a newspaper one day an account of a performance by Inaudi, he was seized with a desire to emulate him and gave performances himself of mental calculation in Greece and Rumania. He came to Paris in 1893 in order to match himself with Inaudi; but for various reasons the encounter did not take place. Diamandi soon had himself presented to the Academic des Sciences, before whom he wished to demonstrate his aptitudes as a calculator. The Academic entrusted the examination of the young man to the same committee which had tested Inaudi. An investigation was made, and at its conclusion Professor Charcot published a study of Diamandi in the Revue Philosophiqiie.

Later Alfred Binet revised and completed Charcot's work find also published the result of his own investigations in the same review. These showed that Diamandi had approximately the same capacities as Inaudi but that his memory was "visual" whereas that of Inaudi was, as we have seen, "auditory". Diamandi "saw" the figures in his head, in front of the frontal lobes, as if they had been written up on a screen. They seemed to remain fixed there until by an effort of will he caused them to be effaced.

When, instead of seeing figures, he heard them, he was obliged to operate a sort of transformation from an auditory to a visual image before starting to calculate. Diamandi therefore asked that for preference the numbers on which tests were to be based should be written down on a piece of paper or on a blackboard. For example he asked one of the assistants to write any five numbers of five figures on a blackboard: 4 9 3 5 7
8 0 2 4 6
9 5 3 1 4
2 7 6 9 5
7 6 2 3 2

He looked at them for a few seconds, then, from memory, repeated them at will from bottom to top, from top to bottom, from left to right, from right to left, diagonally and in any other desired direction. Spectators had the impression that these figures had been photographed on his memory and that he read them within himself with the same facility as if he were reading them from the blackboard.

Like Inaudi, Diamandi could perform arithmetical operations very rapidly. He was asked, for example, to multiply a number of fifteen figures by a number of four figures. He dictated the answer at the end of half a minute. He took about two minutes to extract the square root of a number of ten figures. At the end of the proceedings he repeated from memory the figures concerned in all the operations he had been carrying out and which were inscribed on the blackboard, first in their correct order, then in the reverse order, without hesitation, without error, and so fast that it was almost impossible to follow. On several occasions he had to be asked to repeat them, by so much did his inward eye outdistance the normal eye of the examiner.

Diamandi could also give instantaneously the day of the week for any given date, but whereas Inaudi performed a calculation to obtain this result, Diamandi had devised a table on which the dates forming the last hundred years were inscribed in circular form. On this table a sort of grille turned, carrying the names of the months and days of the week. To know the day of a given date he had only, by turning the grille, to bring the name of the month together with that of the year in question. The name of the day appeared automatically. But Diamandi had no need to have this apparatus before his eyes. He had engraved it on his memory, and upon that mental screen he could read as upon a real table.

Rightly or wrongly, Diamandi believed that the faces of individuals represent one of the ways in which their psychology is expressed. Thanks to his exceptional visual memory he had formed a mental collection of the faces of innumerable people whose characters he knew; thus when he found himself confronted by people unknown to him he could, he said, describe their state of mind.

In this form of exercise his successes appear to have been remarkable and many who consulted him declared themselves absolutely astounded by the disclosures he made to them about the most secret inclinations of their minds, but it is clear that in this field a check is almost impossible.

"During my ten years of travel throughout the world", he confided to Gaston Mery, director of the Echo du Merveilleux, "I had opportunities of observing faces of all categories. In this way a sort of cinematographic museum was formed in my mind, extending over an immense variety of types. When I study a face its image evokes all the corresponding images, and these stand out as from a group and appear before me; I have only to compare. I know that this wrinkle, that fold of the lips, that expression of the eyes, that shape of nose are the mark of this or that psychological disposition, the stigmata of this or that way of thinking or feeling. It is true that I had to work by trial and error for a long time, and that for certain traits of character I still do. But on balance, if in face reading I have not succeeded in attaining the absolute precision of a mathematical operation, I have achieved a relative precision, which I shall try to improve more and more by new observations, but which already seems to me to be far from negligible."

One of his sisters, Uranie, was also a lightning calculator. It was at the age of seven that she noticed her special aptitude. Her brother's first successes later encouraged her to cultivate this gift. Her visualising power extended not only to figures, which appeared to her as coloured; similarly she attributed colours to the letters of the alphabet and the names of the days.

Here are the colours she associated with figures, as well as certain letters and days of the week:

0 White (like the letter O).

1 Black (like the letter I)

2 Bright yellow (like the letter S and the word "Sunday")

3 Vermilion (like "Wednesday")

4 Very dark brown

5 Royal blue (like "Saturday")

6 Bright yellow (as for 2, but darker)

7 Very dark navy-blue

8 Grey blue

9 Sepia (like F and "Thursday").

Uranie Diamandi affirmed that she remembered most easily numbers containing figures with light and conspicuous colours framed by figures of dark or dull colours. She found that in these cases the association of colours and figures aided her memory.

"For instance", she said, "104 (black, white, brown) is easy to learn and to remember because 0, which is white, is here placed between two dark colours. In the same way 129 (black, bright yellow and sepia) is better learned and remembered because of the contrast."

The mental arithmetic carried out by Uranie Diamandi was somewhat similar to that performed by her brother. Five rows of five figures were inscribed on a blackboard. She looked at the square for about a minute then, turning her back, recited it in all directions and named any figure whose position in the square was given her. She added up the rows mentally, carried out the most varied operations upon the numbers: subtractions, multiplications, divisions, raising to powers, etc., and was moreover able to raise a figure up to the twentieth power, which corresponds to a number formed from between 7 and 20 significant figures.

Louis Fleury

Inaudi, who has been justly considered for fifty years as the giant of lightning calculators, has had rivals worthy of him in recent years: Louis Fleury, Mlle. Osaka and Maurice Dagbert.

Louis Fleury, born near Belfort on April 21, 1893, was afflicted from birth with a double ophthalmia which made him completely blind. Abandoned by his parents at the age of eighteen months, he was placed by the Public Assistance in a family of small farmers. At the age of ten he could barely walk and could neither wash nor dress himself. An effort was made to give him some education at the school for the blind at Arras; it appeared that calculation was his weak point: he learned addition and subtraction with difficulty and multiplication with even more difficulty, while the mechanism of division remained totally incomprehensible to him. In contrast to most of the great calculators, Fleury was thus a mentally retarded case where elementary arithmetic was concerned.

At fifteen, considered as ineducable, he was placed in a home for incurables.

"He had been there two months", writes Dr. Osty, "when he received a sudden and violent shock. A man of about forty, his neighbour at table, uttered a loud cry and rolled on the floor in an epileptic fit. In his night of blindness, the convulsions of the patient and the cries of those about him assumed terrifying proportions for Fleury. The emotional shock was so great that he became ill for several days. This first and violent terror long haunted his mind, like an agonising obsession.

"A mental transformation resulted. For psychologists this is perhaps the most interesting aspect of Fleury's case.

"Seeking within himself for a cure for his obsession, he had the idea of concentrating upon work which was the most absorbing for him because it was the most difficult. He set himself to do mental additions, subtractions and multiplications which up to then he had been able to do in the writing of the blind only up to a certain degree of complexity. It was miraculous! All the calculations he attempted resolved themselves with wonderful ease, rapidity and sureness. Even division, that irreducible fortress, was as easy as the other operations.

"From then on the abstract world of figures became his real inward life; his mind worked therein without effort and with delight. Mental calculation became his great distraction, a kind of sport, the intellectual sport of a man whom circumstances and blindness had condemned to live most of his time sitting down. A sport however without much real progress, for all that he undertook he achieved. His practice in calculation was not so much a forward march towards greater accuracy or facility, as the exploration of the extent of his capacities.

"And this gift for calculation, which had emerged from the depths of a psychotic crisis, brought about a general, improvement. His mind, hitherto clouded, became wholly cleared. This was manifested by a feeling of greater aptitude for learning and a desire to educate himself."

Fleury in fact asked to return to a blind school, but the Public Assistance would not agree to this. He then resolved to escape at all costs from the depressing futility of the environment in which he lived, and to this end he simulated madness. He was admitted to an Armentieres mental hospital, where it was rapidly perceived that he was not mad, but that on the contrary he possessed exceptional powers of mental calculation. In order to widen the scope of his capacities it was explained to him what the square of a number is, and he at once calculated the squares of numbers of three and four figures. Next, a square root was defined for him, but without indication of its method of extraction. Within a few days Fleury discovered a method enabling him to extract square roots for four figure numbers mentally and without mistake.

When he came of age, Fleury left the hospital at Armentieres and gave some exhibitions in France; then he went to England and to the United States, where he gave performances in schools, theatres and travelling circuses. He returned to France in 1927, and it was at this time that he was examined at the International Psychical Institute by Dr. Osty and his colleagues. The programme for the tests was the following: M. Sainte-Lague, who held a mathematical degree and was a professor at the Conservatoire National des Arts et Metiers of Paris, played what amounted to the role of examiner. Before the demonstrations he prepared a certain number of operations to be carried out, as well as their answers. The questions were put orally at a rapid rate; as soon as one answer was supplied by Fleury, another problem was put. Dr. Osty measured the time for each calculation with a chronometer.

Here are some of the questions, with the answers and the duration of the mental calculation:

Multiply 553 by 88. Answer: 48664, in 2 seconds.

Multiply 649 by 367. Answer: 238183, in 10 seconds.

Divide 5364 by 43. Answer: 124, remainder 32, in 4 seconds.

Divide 20700 by 48. Answer: 431, remainder 12, in 3 seconds.

Raise 5287 to its square. Answer: 27,952,369, in 10 seconds.

Raise 94 to the power of 4. Answer: 78,074,896, in 15 seconds.

Raise 2 to the power of 20. Answer: 1,048,576 in 20 seconds.

Raise 2 to the power of 30. Answer: 1,073,741,824, in 40 seconds.

Extract the square root of 13,250. Answer: 115, remainder 25, in 4 seconds.

Extract the square root of 222,796. Answer: 472, remainder 12, in 12 seconds.

Extract the cube root of 456,609. Answer: 77, remainder 76, in 13 seconds.

Extract the fifth root of 1,935,752,415. Answer: 72, remainder 834,783, in 3 minutes 10 seconds.

Problems of a different kind were also put to him. Thus M. Sainte-Lague supplied the total of a number raised to its cube added to another number of four figures. Fleury had to give the number which had been cubed and the number which had been added to this cube.

The number given was 707,353,209. Answer: 891 cubed and 5238, in 28 seconds.

The number given was 211,717,440. Answer: 596 cubed and 8704 in 25 seconds.

Another problem: break down 6137 into four numbers which are perfect squares. Fleury gave three answers successively.

First answer: 5,476, square of 74; 400, square of 20; 225, square of 15; 36, square of 6, in 2 minutes 10 seconds.

Second answer: 6,084, square of 78; 36, square of 6; 16, square of 4; 1, square of 1, in 10 seconds.

Third answer: 5,776, square of 76; 225, square of 15; 100, square of 10; 36, square of 6, in 1 minute 20 seconds.

Finally, for any date, past or future, by the Gregorian or the Julian calendar, Fleury gave the day of the week almost instantaneously.

While the majority of lightning calculators are "visuals", Fleury is of the "tactile" type, which is very rare. He said that he "felt the outlines of imaginary cubarithms passing beneath his fingers", that is to say the embossed counting symbols used by blind people. "When he was carrying out an operation", wrote Dr. Desruelles, who studied Fleury at the asylum at Armentieres, "his fingers moved with extreme rapidity. With the right hand he grasped the fingers of the left hand one after another; one represented hundreds, another tens, a third units. He moved his fingers feverishly over the lapel of his jacket and it was curious to watch him using these tactile images to obtain sensations corresponding to those he would have bad in touching cubarithms."

Like all lightning calculators, Fleury employed some simplifying processes, but, to be frank, these were in no way remarkable and in consequence do not deserve mention. On the other hand it is of interest to note the manner in which he proceeded to extract square or other roots, to break down a number into perfect squares and to give the day of the week corresponding to a given date. But let us emphasise at once (and this observation applies to a great number of operations performed by lightning calculators) that these processes, which are carried out practically instantaneously, are for the most part unconscious. It is thanks to analysis that it has been possible to reconstruct them. Take for example the extraction of the square root of 1526. Fleury's technique, arrived at after successive trials, is the following: the root required is greater than 30 and smaller than 40; 30 squared gives 900 - this number is too small; 33 squared gives 1089; 37 squared is equal to 1369; these two squares are both lower than 1526, and their roots will not fit; 40 squared gives 1600; this number is too high. The square root is probably 39. In fact 39 squared is equal to 1521. The correct answer is therefore 39, with a remainder of 5.

The breakdown of a number into four perfect squares is an application of the preceding process. Take, for example, the breakdown of 12315 into four perfect squares. Fleury seeks first for a perfect square fairly close to 12315. This is the case with 10000, square of 100, but 10000 is not sufficiently close to 12315; 110 squared gives 12100 and this number can serve; thus there remains 215 to be broken down into three squares. Fleury now seeks a number which, squared, gives a result near to 215, but he finds that the remainder cannot furnish two perfect squares; he therefore leaves it and makes a second attempt. He tries 105 squared, which gives 11025; there remains 1290, which must be split into three squares; he next takes 35, which he squares, giving 1225; the remainder is 65; he then sees easily that the square of 7, which is 49, and the square of 4, which is 16, together total 65. The four numbers sought are therefore: 11025, 1225, 49, 16.

Where a day of the week to correspond with a date is needed, this, according to Dr. Osty, is Fleury's method. It is necessary to remember:

(a) that for a date before 1582 (Julian Calendar) the 1st of January 1582 was a Monday;

(b) when a date later than 1582 (Gregorian Calendar) is involved, what day of the week the 1st of January of the present year is (example: January 1, 1927 was a Saturday);

(c) that the first day of the eleven other months of the year correspond to this or that day of the first fortnight of January;

(d) that the following correctives must be used: 1. forward displacement of one day for the months of leap years, less February; 2. in consideration of the displacement of one day in the leap years, to make a reduction for the Gregorian Calendar of 12 days in every twenty years or of 61 days in every hundred years, since a date must fall on the same day every seven years; 3. for the Julian Calendar reduce by 12 days for ever twenty years and by 62 days for every hundred years.

It should be noted that Fleury worked out all these ratios in his head while at the Armentieres asylum. It can easily be imagined what their discovery represented in terms of research and effort and in innumerable comparisons.

Two examples given by Dr. Osty show how Fleury used these points of reference.

Question: What was the day of the week on August 13, 1911?

Base of reference: January 1, 1927 is a Saturday.

1st stage: Determine the day of January 1, 1911. Given the 16 years of interval between 1927 and 1911, refer back twenty years, that is to 1907, and reduce by 12 days to find January 1, 1907, which gives a Tuesday. And, as each new year involves a forward displacement of one day (except for leap years: 2 days) it follows that January 1, 1908 is a Wednesday; January 1, 1909 a Friday (2 days for 1908, a leap year); January 1, 1910, a Saturday; January 1, 1911 a Sunday.

2nd stage: To know which day August 1, 1911 was. August 1 of a year falls on the same day as January 3 when the year is not a leap year, and January 4 when it is a leap year. Therefore, as January 1, 1911 is a Sunday, January 3, 1911 is a Tuesday and August 1, 1911 is also a Tuesday.

3rd stage: Determine the day of August 13, 1911. August 1, 1911, Tuesday; August 15, 1911, Tuesday; August 13, 1911, Sunday.

Now, and this is what is worthy of attention, all these points of reference and all these reasoning processes passed through Fleury's mind in less than one second, since he gave the correct answer at the end of that time.

Another question: What was the day of the week of September 19, 139?

Basis of reference: January 1, 1582 is a Monday.

1st stage: Apart from the points of reference already mentioned, Fleury knew, since he had worked it out, that, in the Julian Calendar, a date falls on the same day of the week every 28 years. This enabled him to go straight back 1400 years (1400 being divisible by 28). Hence 1582 - 1400 = the year 182. January 1, 182: Monday.

2nd stage: Find the day of the week of January 1, 180, distant from the the year 140 by twice 20 years. January 1, 182, Monday; January 1, 181, Sunday; January 1, 180, Friday (reduction of two days for leap year).

3rd stage: Find the day of the week for January 1, 140. From 180 to 140 there are twice 20 years, hence a reduction of twice 12 days, which gives January 1, 140: Thursday.

4th stage: Find the day of the week for January 1, 139. This day will be one day before that of January 1, 140, therefore: Wednesday.

5th stage: Find the day of the week for September 1, 139. September 1 of a year always falls on the same day of the week as January 6, except in leap years when it falls on the same day as the 7th. Therefore September 1 is a Monday.

6th stage: September 1, Monday; September 15, Monday; September 17, Wednesday; September 19, Friday.

Here, the answer was given after 4 seconds.

Mlle. Osaka

If Fleury was, as we have said, of the "tactile" type, the woman lightning calculator Mlle. Osaka belongs, like most of the great calculators, to the category of the "visuals".

This girl, whose borrowed Asiatic name in fact disguises a Frenchwoman born a few miles from Bagneres, was a mentally retarded child. She began walking and talking only at the age of four and a half. Because she was sickly she scarcely attended school, so that at twenty-six she could only just read and write. Her knowledge of arithmetic was confined to addition. One day she was present at a performance given, if not by a lightning calculator, at least by a calculating virtuoso, and without knowing why, she felt that she would easily be able to perform the same feats.

"Let us reflect", Dr. Osty justly notes in this connection, "upon the important psychological significance of this fact, occurring to so many people who have been ignorant of their own exceptional gifts up to the day when an accidental circumstance has arisen to stir in them the sense of their special capacities in this or that direction: artistic, literary, scientific or others. In such cases it is as though the subconscious mind is made aware of the individual's hidden resources, and under the impulse of circumstance, succeeds in forcing its way across the threshold of consciousness and imposing upon it the vital spark."

However this may be, Mlle. Osaka, impelled by this strange certainty, set herself to learn the rules of calculation of which she was ignorant, that is to say subtraction, multiplication and division, but found herself held up by this last operation, of which she was never able to understand the workings. Two facts of great importance however soon made themselves evident to her mind. She found on the one hand that she could calculate with extreme rapidity, and on the other that she could retain in her memory the numbers she had manipulated in her head. This second discovery induced her to turn her training in another direction. She stopped trying to study calculation itself and sought instead to memorise larger and larger numbers. From this moment on, her progress was extremely rapid, so that she could fulfil her secret desire to exhibit herself in public. She perfected her aptitudes, learned by heart a colossal mass of numbers which she calculated on paper: the powers of numbers of 1 and 2 figures, up to the 10th; the powers of numbers of 3 figures up to the 7th or 8th, the numbers of hours, minutes, seconds corresponding to ages, etc.

In these conditions her mental numerical luggage was literally indestructible, and Mlle. Osaka was able to give the answer immediately and without a mistake, within the framework of her knowledge, to every question on powers or roots; and it was equally possible for her to give the number of seconds lived by a person of this or that age, etc., with the same facility. When she wishes to remember numbers she sees them as if they were outside herself; when a hundred or so figures are dictated to her it seems to her as though they were written up on a blackboard and are, as she says, "more legible than the real ones". Thus, if she is asked for the sixth power of 97, she will "see" all the multiplications which she has used to calculate the powers of 97, from the second to the sixth. If a board is covered with figures and she is asked what number is inscribed on the fifth line, Mlle. Osaka, who has only heard the figures called out, sees all the numbers on the board instantly and clearly. Her capacity for mental retention is so great that she can repeat them either in the normal order or backwards, with the geatest of ease. The following experiments carried out by Dr. Osty at the International Psychical Institute give an idea of her extraordinary mnemonic capacities.

Dr. Osty asked for the square of 97, then for the 10th power of the same number, both of which the calculator gave instantly. After that he asked for the sixth root of 402,420,747,482,776,576, then for the square root of the same number and again these were given at once and correctly. This done, he wrote down, absolutely at random, a succession of a hundred figures and then read them out at a speed of approximately one figure per second.

"When I had finished this test", writes Dr. Osty, "Mlle. Osaka repeated the hundred figures in the order in which they had been spoken. About forty-five minutes afterwards, and when we had been talking about a number of things, I said to Mlle. Osaka, who had no reason to expect the question:

'Could you repeat to me the hundred figures which I dictated to you nearly an hour ago?'

'Very easily', she replied.

'Would it be possible to do this backwards?'

'I will try'.

And she succeeded."

In the course of a demonstration conducted with a small audience, twenty people each wrote a number on a piece of paper, each paper bearing a control number from 1 to 20. The papers were mixed up, drawn at random, and the numbers on them read aloud, then written as they were called on a blackboard containing twenty sections numbered from 1 to 20. They varied from millions to nonillions.

Mlle. Osaka, who was standing facing the public, heard them but did not see them. She turned her back to the board which, in fact, could have been removed.

In order to complicate the test Mlle. Osaka asked that a few problems should be put to her before she repeated the numbers on the board. One of the audience asked her for the square and then for the 10th power of 27, another for the 10th power of 55, then to go through all its powers in descending order. It was child's play for this girl to answer these questions.

Then Mlle. Osaka was asked to repeat the twenty numbers written on the blackboard in the order from 1 to 20, which she had not heard since the order of inscription had been determined by chance. This she did immediately with extraordinary rapidity and without any mistake.

"This was followed", writes Dr. Osty, "by requests for the powers of 3 figure numbers. Someone asked for the 2nd, the 3rd, the 4th, then the 5th powers of 223. These were given exactly and without delay.

"To those who had supplied the date of their birth, Mlle. Osaka at once gave the number of days, hours, minutes and seconds they had lived, leap years being taken into account. One person proposed the mental multiplication of 624,987 by 2,358. Mlle. Osaka did this at slow speed, figure by figure. Forty-eight figures were thus produced in seven minutes without apparent effort and without error. Mlle. Osaka was then asked if she could repeat the numbers which had been inscribed on the blackboard, starting at the end. She did this immediately, figure by figure, from section 20 to section 5, then by groups of three figures for the remaining sections. She was then asked to state again the numbers of section 6, section 13, etc. Another person asked her to repeat the number in section 7 backwards. The correct answers were immediately given them. At this moment Dr. Montier, who had earlier asked for the multiplication of 624987 by 2358, asked Mlle. Osaka if she could again give the result of this multiplication. The 48 figures were at once enumerated at very high speed. M. Morice, an architect, asked, as an addition to the blackboard test, that this should be turned face to the wall and that Mlle. Osaka should again repeat the numbers. To amuse herself, the girl repeated them all as fast as she could utter them."

These feats appear really prodigious when it is remembered that their execution is based upon a colossal memory for numbers. To remember thousands of numbers, each formed from 15, 20, 30 and up to 40 figures, to bring them out instantly and exactly from the depths of the subconscious, is a terrifying operation, which as I believe, and shall emphasise later, borders on the paranormal.

And here is another strange aspect of the extraordinary memory of Mlle. Osaka, an aspect which we have already encountered with other lightning calculators. Apart from numbers, she has difficulty in remembering current facts and things she learned at school. She has never been able to learn the correct order of the letters of the alphabet.

Maurice Dagbert

M. Maurice Dagbert, the lightning calculator who came into prominence at the conjurors' congress held in Paris in 1947, and who later displayed his full capacities at the Lausanne Congress in 1948, has certainly not the gigantic memory of Mlle. Osaka; nevertheless his powers of memory are exceptional. Moreover his capacities as a mental calculator are such that they seem at least to equal Inaudi's. Among other feats performed before the Academic des Sciences, he extracted a fifth root (answer: 243) in 14 seconds; a seventh root (answer: 125) in 15 seconds; a cube root (answer: 78,517) in 2 minutes 15 seconds; a fifth root (answer: 2189) in 2 minutes 3 seconds; and raised 827 to its cube in 55 seconds.

Here is the report on M. Dagbert by MM. Gaston Fayet, Jean Chazy and Joseph Peres, taken from volume 220 of the Comptes rendus hebdomadaires des seances de l'Academie des Sciences:

"At the request of the permanent secretaries we have examined the 'mental calculator' Maurice Dagbert, who wishes to be presented to the Academic. The results of this examination have been conclusive and appear to us to be worth publishing in the Comptes rendus.

"M. Dagbert's powers of calculation seem to be comparable with those of Jacques Inaudi, presented to the Academie by Darboux in 1892. Like Inaudi, M. Dagbert is served by an exceptional memory. He has informed us that he works out figures with the aid of extremely vivid images which he obtains by shutting his eyes or by staring at a white object (the ceiling of the hall in which he is operating, for example). He sees the figures appear, as they are announced to him, as though he had written them himself on a blackboard.⁴

"In the course of the examination, which lasted 2 hours 30 minutes, M. Dagbert had occasion to carry out varied calculations (perpetual calendar for Gregorian or Julian dates, multiplications, powers and extraction of roots). The details have been recorded in a report on the meeting which will be preserved in the Academic archives.

"M. Dagbert has had only a primary education and his knowledge of mathematics and elementary algebra are approximately nil. His taste for calculation was very precocious and a visit he made to Inaudi when he was 14 led him to make a personal effort which evidently has been very fruitful. He has told us that he has discovered for himself the rules he uses in his calculation, rules which for him are purely empirical and of which he does not explain the reasoning. He has given us the example, a particularly simple one, of the rule he employs to evaluate the cube of a number of two figures: he utilises two key numbers, determined by the figure of the units u, which he knows by heart but whose origin eludes him. It is recognised at once that the first key number x is the figure of the units, the second y the number of the tens in 3u². Dagbert's rule thus appears as the result of the development of the binomial (10d + u)³, the calculation being directed to obtain the various figures of the result successively: u³ gives the figure of the units and the remainders to be carried; xd, to which the amount carried forward is added, gives the figure of the tens and fresh remainders; (3u + dy)d, by adding the remainders, the figure of the hundreds; finally, the carrying forward accomplished, d³ is added, giving the thousands of the answer.

"It is not surprising in these conditions that M. Dagbert, whose speed and powers of calculation are remarkable in the fields he knows so well, should be put off the track by quite simple questions (like those put to Inaudi at the time of his presentation in 1892) but which necessitate some algebraical transformations.

"After the demonstration was over, M. Dagbert was presented to the members of the Academic and carried out before them some of the very complicated mental calculations of which he is capable."

In his exercises in public the arithmetical operations he performs mentally overlap, so that the cascades of figures which pour out over the audience, almost without a break, form a strange medley. First one person is invited to give his age, then five numbers of two figures are proposed by the public. A little while afterwards the calculator gives the third power of the first number, the fourth power of the second number and the fifth power of the third; he then stops to indicate to a spectator that he has lived for so many hours, minutes and seconds, and shows, by a calculation on the blackboard, that he has taken leap years into account. He rounds off by supplying the sixth and seventh powers of the last numbers, these two answers, be it noted, having 11 and 13 figures respectively.

More difficult feats are then proposed: raising 3 figure numbers to their cubes, then the extraction of roots. Someone for instance gives a 15 figure number, another one of 19 figures and finally dates on either the Gregorian or the Julian calendar; all these are given simultaneously. Instantly the artist indicates the day of the week to which these correspond, then announces the cube root of the first number and, partially, the fifth root of the second. He replies to further requests for dates. Finally he gives the complete cube root of the second number. Similar operations follow with the greatest rapidity, interspersed by replies concerning the dates of Easter, Ascension, Whitsun and the phases of the moon.

Finally he ends the performance by repeating all the numbers that have been announced during the session, that is to say, some hundred and fifty figures.

It should be added that in the course of his exhibitions M. Dagbert often plays brilliant pieces on the violin while he is solving complicated calculations in his head. Thus while he has been playing a fantasy from Il Trovatore with remarkable skill I have seen him carry out the extraction of twenty cubic roots of 3 figures and the multiplication of a 5 figure number by another number of 6 figures. The total operation lasted seven minutes. As he put down his instrument the calculator gave the twenty one solutions straight off, absolutely without error. At no time did he make use of pencil or paper, even to note the terms of the problems.

The Psychology of Lightning Calculators

After this brief study of the principal lightning calculators, let us consider in what conditions their aptitude makes its appearance and by what mechanism it is developed.

First of all it seems that the gift of calculation is not inherited. The only two exceptions known are those of Bidder and Diamandi. The first passed on his gifts to his children and grandchildren, while Diamandi had one brother and one sister who possessed aptitudes for mental calculation similar to his.

Among all of them the gift has appeared spontaneously, without any outside stimulus. Indeed, many of the lightning calculators were born of poor and even very poor parents who paid little attention to the teaching or education of their children. It should be added, as we have already pointed out, that several calculators were at first considered to be backward children. At the age of 17, the Belgian calculator Oscar Verhaeghe expressed himself like a baby of 2.⁵

It has been mentioned earlier that Zerah Colburn displayed a sign of degeneracy: an extra finger or toe to each limb. Another lightning calculator, whom we have not mentioned so far, was Prolongeau, who was born without arms or legs. Mondeux was an hysteric.

To sum up, neither the external environment in which the lightning calculators developed, nor their general intelligence, at least in the case of most of them, gives us the explanation of their faculty: they have had neither the influence of a teacher nor an example; they have not been drawn into their way of life by guidance or by a normal education; their level of intelligence has often been very much inferior to their extraordinary arithmetical capacities. As Alfred Binet has rightly said: "In the birth of their faculty, there is something which resembles a sort of spontaneous generation".

Another trait which characterises the lightning calculator is precocity. In the cases of Gauss and Whateley the faculty made itself manifest at the age of three years. It is reported that Gauss's father was in the habit of paying his workmen at the end of the week and that he added to their wages a bonus for any extra hours they had done, based on the total earned for each day. One day, when Gauss senior had just finished his calculations and was preparing the money, the child, who was then barely three and who had been following his father's operations without being noticed, cried out:

"Father! Father! You've done it wrong. This is the amount." The sum was done again carefully and to everybody's astonishment it was discovered that the right amount was indeed that given by the little boy. In the same fashion Ampere brought off quite lengthy mental operations at the age of four, when he knew neither letters nor figures. He used nothing but a few pebbles or beans. A story told by Arago is enough to demonstrate the point to which love of calculation had taken possession of the child. Maternal tenderness having deprived the young Ampere of his beloved beans during an illness, he replaced them with fragments of a biscuit which he had been given after a three days' fast

Ampere, as we know, became one of the greatest French physicists, but, strange to say, in proportion as his knowledge of classical mathematics and science developed, so he progressively lost his aptitude for mental calculation.

It was the same with Whateley, who notes: "My ability to calculate displayed one peculiarity. It showed itself at the age of four and lasted three years. I could do the most complicated additions in my head, and a good deal more rapidly than others who did them on paper, and no one ever found any mistakes in them. But by the time I began going to school, my ability to calculate had vanished and since then I have always been very weak in mathematics".

Safford, according to M. Scripture, could do multiplications in his head with answers running up to 36 figures at the age of five. As he had remarkable aptitude for classical mathematics, he became a professor of astronomy, but entirely lost his faculty for lightning calculation. On the other hand the Swiss, Leonard Euler, who began calculating at the age of five and who, according to his biographer Lacroix, "deserves to occupy in mathematics the place which Voltaire holds in literature", retained his prodigious facility for mental calculation until extreme old age. Possessed of an encyclopaedic mind and gifted with a colossal memory, he not only solved the most complex problems of analysis or geometry in his head, but also knew by heart the Aeneid of Virgil, his favourite author, and had a profound knowledge of physics, chemistry, zoology, botany, geology and medicine as well. He was also well versed in history and the Greek and Latin languages.

In 1937, an English Sunday newspaper published a Reuter report from Vienna that Meho Focie, aged 5, the son of a bootmaker, had amazed mathematics teachers in Zagreb. He had not learned to read or write, but he could solve any complicated problem in his head. He was able to multiply or divide six figure numbers and find their square roots without a mistake in a few seconds. A mathematician, watching him at play, asked him: "I am fifty-one today; how many days have passed since my birth?" Without stopping his game the child gave the exact answer, taking leap years into account.

Colburn, Prolongeau and Inaudi began calculating at the age of six, and Bidder, Mondeux and Mangiamele, the less precocious examples of the series, at the age of eleven. Alfred Binet has suggested eight as the average age for the first manifestations of the gift of lightning calculation. However that may be, what should be especially noted on this subject of precocity is the number of them who began to calculate before knowing how to write figures. Thus they worked out numbers in their heads without, understanding the meaning of a written figure. This is a rather mysterious fact, difficult for an adult accustomed to calculate with these precise and convenient symbols to understand.

An exceptional memory for numbers, the ability to keep them stored in the consciousness in a manner bordering on the supernatural also characterises the lightning calculators. It is in both these characteristics specifically that they surpass ordinary mortals, and in the second that they resemble hypnotic or psychic subjects in whom sensation can take the form of hallucination.

The memory of the lightning calculator is generally a visual one, but it may also be auditory, tactile or motory.

Mondeux and Colburn saw figures form before their eyes, as if traced by an invisible pen. Diamandi perceived them before his frontal lobes. Mlle. Osaka saw them, when they were dictated to her, as if written "in white upon a blackboard". It was the same for Dagbert. Dismer, the shepherd who lived not far from Stuttgart, and Pierre Annich, the herdsman from the Tyrol, who were also lightning calculators, were also "visuals".

Inaudi, on the other hand, was primarily an "auditory".

"I hear a voice which calculates", he often said.

This voice, however, did not prevent him from following a conversation, or from simultaneously carrying out easier calculations than the main problem put to him, or from playing the flute. The mysterious voice continued its soliloquy and at the end of a certain time supplied Inaudi with the answer to an extraordinarily complicated calculation.

Fleury, as we have said earlier, was a "tactile".

Finally it seems that the majority of the calculators have also brought their "motor" memories into play, either by involuntarily executing certain movements with their hands or by silently articulating the figures. It was thus that Jean Hutzinger, a lightning calculator who had his hour of fame, constantly moved his lips while calculating. To sum up, it is very probable that a particular form of memory predominates in this or that case, but that all the memory forms play their part to a greater or lesser degree.

The Hidden Powers of the Mind

To explain the gift of mental calculation the classical psychologists invoke certain quite ordinary psychic qualities carried to a high degree. Their view is clearly expressed by the Swedish neurologist Dr. Jakobson in Acta Medica Scandinava.

"The lightning calculators", he says in substance, "are subjects gifted with a highly developed visual memory and an excellent memory of association, who are apparently unaffected by mental fatigue and are capable of concentrating their attention upon complex operations very rapidly and for a prolonged time. Impelled from a very early age by the development of a spontaneous faculty for mental calculation, they acquire progressively an automatic memory for arithmetical operations which is independent of all mathematical training properly so called. In this way they can greatly extend, by simple memory, the numbers of the multiplication tables of which they know the products by heart. Such subjects have no more need to reflect, that is to say engage even in a mental operation, in order to multiply two numbers of three or even four figures, than we have to multiply the ten first numbers in Pythagoras' table. Starting from this data they can easily reduce the most complex multiplications to a small number of operations of which they add up the products mentally. To do this they literally see the figures as "written in the air". Divisions are done in the same way, simplified by a method consisting in extracting from the dividend the squares, known by heart, of the round numbers into which the divisor can be resolved.

"Every arithmetical operation can be broken down in this way into a certain number of 'prefabricated' operations, of which the results ure known by heart and which it is then only necessary to add up with the remainders. In essence, therefore, it is more a question of a very special form of memory, which has been developed by training, than of a genuine aptitude for calculation."

In reality this explanation is clearly inadequate. It explains neither their precocity nor their innate gift, nor the fact that it can show itself in mentally backward persons, nor the often prodigious character of the arithmetical memory of many of the calculators. Moreover it docs not sufficiently emphasise the extent to which they perform their feats automatically.

When Inaudi hears a voice which calculates within him while he continues to converse or is consciously doing quite different calculations, when Dagbert plays a brilliant piece on the violin and during this time solves twenty-one complicated problems in his head, do we not see here appearing the "unknown guest" of Maeterlinck, emerging from the normal personality, superimposing itself upon it and proclaiming its independent existence? Certain calculators have moreover actually felt this "guest" living within them and have distinctly recognised it.

"From my childhood", writes Ferreol, "I calculated in an entirely intuitive manner, to such an extent that I frequently had the impression of having lived before. If I was set a difficult problem the answer sprang directly from my mind without my knowing at first how I had obtained it; I then looked for the method by starting from the result. This intuitive power of comprehension, which never made mistakes, developed in close parallel with the exercises which it was called upon to do. I still often have the impression that someone is beside me who whispers to me the answer I wish for, the paths I am seeking, and generally these are paths which no one, or almost no one, has trodden before me and which I should not yet have found if I had set out to lookforthcm." (Reportedby Moebius, Uber die Anlage zur Mathematik.}

With this calculator the separation between conscious thought and the subconscious or psychic faculties seems to have been complete, the conscious control being only an accessory condition.

It was the same with Bidder:

"He possessed", writes M. V. Pole, "an almost miraculous faculty for finding, more or less intuitively, factors which when multiplied together gave this or that large number. Thus, given the number 17861, he could see at once that this resulted from the multiplication of 337 by 53 ... He said that he would not be able to explain how he did it; with him it was almost a natural instinct."

Much the same can be said of Verhaeghe, who is quite unable to explain the mechanics of the complicated mathematical operations he carries out with such extraordinary rapidity and accuracy. When he is asked how he sets about it, he replies: "I do not know. It comes to me like that".

Elsewhere, as with Buxton, Inaudi and Dagbert, there existed or exists an instant-by-instant collaboration between the conscious and the unconscious, a contact between the two levels of the mind, and it is certain that among calculators the artifices which enable operations to be simplified and which are apparently discovered by the conscious mind, are in fact, automatically introduced by the unconscious.

This is clearly implied in a statement by the "amateur" calculator M. Paul-Aumont Lidoreau, who uses a certain number of established tables in the extraction of cube roots.⁶

"I do my calculations entirely in my head", he said, "without any cerebral fatigue. I am aided in this in an incomprehensible manner by my subconscious mind, and it is this, I suppose, which docs the most important work. Thus, to extract the cube root of a 15 figure number, I have to perform an average of 12 to 15 operations in 20 seconds. Several of these calculations do themselves simultaneously in my mind without my knowing exactly how."

Added to these are other facts of a slightly different kind which demonstrate that in our psychic personality the unconscious is an active region, capable of creation. "It is not I who think", confided Lamartine, "it is my ideas which think for me." Similarly Alfred de Musset said: "You don't work, you listen; it is as if a stranger spoke in your car". Schopenhauer has also defined the role of the unconscious: "My philosophic postulates have all produced themselves without my intervention, at moments when my will has been asleep and my mind not engaged ... Thus my personal self has been a stranger to my work".

This subconscious activity also shows itself in the scientific field. "Instead of forcing myself to understand a proposition on the spot", Arago notes, "1 admit provisionally that it is true; and the next day I am astonished to comprehend perfectly what seemed wholly obscure the night before."

Answering a question by Paul Valery, Professor Langevin made a similar observation: "Each time", he said, "one thinks with intensity and has thus to some extent prepared the work of the subconscious, this latter continues of its own accord and sometimes gives \varning when it has finished. I have very clear recollections of an inner shock informing me at a given moment that a question has been solved and that there is nothing left to do but to consciously express the result".

Joliot-Curie too stated that he had had "sudden illuminations" supplying him with the best means of producing and observing a phenomenon, with the immediate sensation that the method thus suggested was unique and that any other would be less simple. "This sensation", he says, "I can recall in at least two circumstances, of which one was the proof of the explosion of the uranium atom."

On his side Professor Jacques Hadamard reveals: "I have found a long-sought solution at the precise moment of a sudden awakening, caused by the motion of a car, and this solution has revealed itself to me in a direction quite different from that in which I had sought it until then".

Leaping over a mass of intermediate reasoning and outstripping years and even centuries of research, the unconscious is sometimes capable of bringing surprising truths to the notice of the conscious.

At the death of Pierre Fermat, one of the greatest mathematicians of the seventeenth century, there was found in his home a copy of the works of the Greek mathematician Diophantes, bearing in the margin the following annotation in Format's handwriting: "I have demonstrated... (he states a concept which it seems needless to quote here) but I cannot write down the demonstration, as the margin is inadequate".

"Now", notes Jacques Hadamard, "this demonstration for which the margin seemed to him too narrow, has been sought for in vain for three centuries past. Essential progress has been made along the lines of his theorem: it has already been demonstrated in limited cases. But for this it has been necessary to bring to bear a whole arsenal of algebraical theories scaffolded one upon another, and of which none was known in Permat's time, of which none was even imagined at that time, and no allusion by him indicates that he even suspected their existence."

¹ A few years ago Dr. Ginestous drought to the Bordeaux Anatomical Society a young man who in a sense behaved in somewhat the same way as Buxton. He continuously counted in his head all the letters contained in the sentences he read, wrote, spoke, heard or thought. Whether he was reading a newspaper, writing something down, or chatting with a friend or a stranger, he counted everything without this fantastic labour occasioning him the slightest fatigue, causing him any physiological difficulty or hindering him in his profession. From his childhood he counted all the letters in his lesson hooks and those in books he looked through by chance, and in the lines of the Aeneid or the Iliad which he had to learn by heart. This strange mental activity ceased at night and he did not dream. But it recommenced on waking. When he was not being spoken to or was not reading he made up sentences which he ceaselessly counted.

This subject displayed one other peculiarity. He had the phenomenon of coloured vision. The months and the days of the week presented themselves to him in a colour lighter or darker according to the passage of time towards the last month of the year or the last day of the week. January and Monday appeared to him as perfectly white; the other months and the other days of the week were slightly tinged with grey, which darkened gradually to attain complete blackness in December and on Sunday.

² In I948, when he was eighty-one year of age, Inaudi discovered a general law which he expressed in these terms:

"To find, for instance, the sum of the 25 first cubes, I multiply 25 by 26, obtaining 650, divide the result by 2, which gives 325, and I raise it to its square, which gives the number required: 105625." Thus empirically Inaudi found the formula which gives the sum of the cubes of the n first whole numbers and which is expressed:

S = (

n(n + 1)
2

)²

Henri Mondeux had also discovered this process.

³ Electronic "brains" and modern calculating machines make it possible to carry out, at the speed of light and in infallible fashion, the most complex calculations which would otherwise occupy teams of specialists for weeks or even whole years. These machines can give the product of two numbers often figures in less than 50 millionths of a second.

These have sometimes been described as "thinking machines". In reality this is true only within the limits set to the invention by man and in relation to the final result which he has assigned to it. It is certainly possible to make these machines carry out reasoning processes, such as syllogisms, on condition, of course, that the appropriate symbols are chosen and the right circuits established. But between this and genuine "thinking" there is a gulf which it is impossible to bridge.

⁴ M. Dagbert can, with absolute accuracy, represent the figures to himself as if they were written in white on a blackboard. His mental pictures are less clear if the figures are red on a blue background, and even less clear with yellow figures on a green bark ground.

⁵ Oscar Verhaeghe, horn on April 16, 1926 at Bousval (Belgium), in a family of modest civil servants, belongs to the group of calculators whose intelligence is well below the normal. The raising to various powers of numbers formed from the same figures is one of his specialities. Thus, 888,888,888,888,888 is raised to its square in 40 seconds and 9,999,999 to its fifth power in 60 seconds, the results involving 35 figures. Oscar Verhaeghe has been put through a certain number of tests by various learned groups and by the eminent mathematician Kraichit, of Brussels University.

⁶ M. Paul-Aumont Lidoreau was initiated into mental calculation from a very early age. When he was at school, he was already able to extract the roots of numbers of from 9 to 15 figures in his head by methods of his own which he later perfected.

Like all the virtuoso calculators he performs the perpetual calendar operation, and can calculate the age of a person in days, hours, minutes and seconds very quickly. But his speciality is the extraction of cube roots not ending in 5. In this class he has even beaten the record of Inaudi (which was 2 minutes 15 seconds for the extraction of a cube root of a number of 15 figures) by giving at the end of 20 seconds the cube root of a perfect cube of 15 figures and in 46 seconds that of a perfect cube of 18 figures.

In the course of a demonstration organised at the International Psychical Institute, M. Lidoreau extracted the cube roots of the following numbers in his head in a few seconds and without a mistake:

37,246,609

599,930,290,504

924,579,746,488

13,055,567,849,956,664

In contrast to the majority of lightning calculators, M. Lidoreau does not possess an exceptional memory for numbers.