SIX FEMALE MATHEMATICIANS

BY JULIAN L. COOLIDGE

1. HYPATIA, 375-415

WHOEVER has made the least study of the history of mathe­matics must have been struck by the fact that both in ancient times and the more recent periods the names quoted are almost

always those of members of the male sex. At once the answer comes, "Women do not so easily take to mathematical reasoning or mathe­matical speculation, the female mind is not naturally mathematical." Frankly I do not know whether this is true. I do not naturally accept without proof such wide conclusions, yet it is an undeniable fact that the number of women who have attained real eminence in the mathe­matical field is lamentably small. I will speak of six who have at­tained some real distinction, a small number, all things considered.

Everyone who has heard of the history of mathematics has heard of Hypatia, daughter of Theon of Alexandria. The books or articles dealing with her are, for our purposes, many, but unsatisfactory. Some are devoted to denouncing the narrowness or bigotry of the early fathers of the Church, others take the name of Hypatia as an excuse for pro­claiming the rights of women to an equal share with men in the recog­nition of a fit place in the intellectual history of the race. Both are, doubtless worthy causes, but they tell us little about Hypatia the mathematician, There are some highly documented monographs whiclr deal with her place in relation to the historians of her epoch, but are equally lacking in precision as to her mathematical status; the reason is quite simple. We have not today a single authentic work which comes from her hand. About Hypatia the martyr we have much information, about Hypatia the scientist we have only the broadest hints.

It seems to be well authenticated that she wrote a treatise on the conics of Apollonius. There is no mention of this in the collection of her contemporary Pappus, perhaps his work stimulated hers, but she is merely mentioned as a follower of Apollonius, who flourished five hun­dred years earlier. She is also said to have written a commentary on some work of Diophantus; the story that she also published some sort of an astronomical table seems less plausible. This is all we can defi-

20

JULIAN L. COOLIDGE 21

nitely say about her writings; on the other hand, there can be no doubt that she was highly regarded by her contemporaries. She re­ceived the best of training, for her father Theon was connected with the Museum, the intellectual center of Alexandria, and a learned mathe­matician himself; he must have given his talented daughter the best education in his power. She was the outstanding leader of the N eo­Platonic School of philosophers and took an active part in the last attempt to oppose the Christian religion and lead mankind back to the ancient rights of heathen worship. She would have lived and died a harmless philosopher if she had not been caught up in a desperate quarrel between two of the leading men of her time, Orestes the Gover­nor of Egypt, and Cyril the bishop. Orestes seems have been easy­going by nature, and well disposed towards Hypatia. According to Kingsley he made love to her and more or less wound her around his finger; Cyril was a cruel and ambitious man, bound to keep the prefect below him. He interpreted his position to mean that it was his duty to take every means to put down the high-minded, inoffensive leader of a non-Christian cult. Hypatia was the victim of this struggle for mastery, she was killed and mutilated by a body of Christian fanatics who did implicitly what they were told by Bishop Cyril.

A dreadful story is this. The fact that she was very beautiful doubt­less added something to the great popularity of her lectures, but she was regarded as the unquestioned leader of her school of philosophy. It is true that she was professedly opposed to Christianity, but she had Christian followers, most notable of whom was Bishop Synesius. We really know very little about Hypatia as a contributor to mathe­matical science, but we may surely put her down as the first mathe­matical martyr.

2. MARIA GAETANA AGNESI, 1718-1799

This extraordinary Italian seems to have been born with a quite unusual intellectual capacity in different directions, and might have made her mark in various ways, had she not been so devoted to reli­gious exercises and a life of unselfish service. She spoke French at the age of five, at nine she published a Latin defense of liberal studies as a proper exercise for members of her sex, at eleven she was familiar with Greek, German, and Spanish, and presently added Hebrew to this list. Her Propositiones philosophicae appeared when she was twenty. She turned seriously to mathematics at the age of nineteen, her principal teachers being her father and the Olivetan Father Rampinelli; later she became herself associated with this order.

Agnesi's first mathematical effort seems to have been connected with

22 SIX FEMALE MATHEMATICIANS

L'Hospital's Trait!! des sections coniques, but her great year was 1748 when she was elected to the Bologna Academy of Science, and pub­lished her I nstituzioni analitiche ad uso della gioventu italiana. This is a really extraordinary piece of exposition. She felt that mathe­matics was advancing so fast that it was hard for a beginner to estab­lish contact with the most recent progress, and she wished to publish a single work that would place the reader abreast of the most novel ad­vances of the day. The book was translated into both French and English, the latter being accomplished by James Colson, Lncasian Professor in Cambridge, the first translator of Newton into his mother tongue. John Hellins, who brought out Colson's translation, wrote, "He (Colson) found Agnesi's work to be so excellent that he was at the pains of learning the Italian language at an advanced age for the sole purpose of translating that work into English; that the British Youth might have the benefit of it, as well as the Youth of Italy." As for the French translation, or rather the opinion of it as a worthy subject for translating, we read in the Registres de l'Academie, 1749, "L'ordre, Ia clarete, Ia precision regnent dans toutes les parties de cet ouvrage. On n'a point vu paraitre dans aucune langue des Institutions d'analyse qui puissent mener aussi vite, ni conduire aussi loin ceux qui voudraient penetrer dans les Sciences Analytiques. N ous le regardons comme le Traite le plus complet, et le mieux fait qu'on ait en ce genre."

I think that this praise is well bestowed, and the book is very clearly and systematically written. The first section handles the Analysis of Finite quantities and deals with the construction of loci, including the conic sections, and then passes to simple problems of maxima and minima, tangents and inflections, but there are so far no questions of infinitesimals. In Book II we run across infinitely small quantities. These are defined as quantities that are so small that compared with the independent variable the proportion is less than any assigned quan­tity. When an infinitesimal is added to or subtracted from the vari­able itself, the difference is not noticeable. Infinitesimals are called Differences or Fluxions. It is astonishing that differences which are variables tending to zero, and fluxions which are finite rates of change should be treated as essentially the same thing.

Agnesi lived in a time when the final word as to the real meaning of the calculus was still to be written. I do not know what letters she actually used, Colson uses Newton's notation writing x for the fluxion of x, but he treats it as if it were Leibniz difference dx. There is a long and elaborate discussion of maxima and minima, flexure and evolutes, proceeding like Huygens from the evolute to the involute.

JULIAN L. COOLIDGE 23

Agnesi's third book is given to the integral calculus. The subject was still in its early stages. She gives some specific rules for integra­tion, and a section on the expression of a function as a power series; naturally there is no discussion of the extent of convergence. Book IV which is given to the "Inverse method of tangents" discusses some very simple differential equations. Whoever has heard of Agnesi has heard of the curve to which she gave the name of the Witch. It has the Cartesian equation y2x + a2(x - a) = 0. She starts with the geometrical fact that if corresponding points on the curve and a cer­tain semicircle have equal abscissas, the square of the abscissa is to the square of the radius of the semicircle in the same ratio in which the ab­scissa: would divide the diameter of the semicircle. We, I say, asso­ciate this curve with Agnesi, and she gave it its name; it was first studied by Fermat.

It is often said of Agnesi that she was professor of mathematics at the University of Bologna, and, as a matter of fact, she was nominated to that station by Pope Benedict XIV in 1750. As a matter of fact, surprising to relate, she never really occupied the chair, never really taught at Bologna. She had· no worldly ambition, no desire to shine except in religious work. When she was 44 years of age the University of Turin asked her considered opinion of some recent scientific work, including most original articles by the youthful mathematician La­grange on the calculus of Variations. Thesehad been highly appre­ciated by Euler. Agnesi replied that such matters no longer occupied her thoughts. I cannot doubt that her natural aptitude for mathe­matics was altogether extraordinary, she is even said. to have worked out difficult mathematical formulae in her sleep. But the constant urge to give all of her talents to religious work very early deprived the world of the fruits of her unusual mathematical capacity.

3. EMILIE, MARQUISE DU CHATELET, 1706--1749

It would be indeed difficult to find a more complete contrast to Maria Gaetana Agnesi than her contemporary the Marquise du Chatelet. The extremely religious Italian was ahnost dragged by force from a life of religion to one devoted for a short time to the study of mathematical science, who returned as soon as possible to her first love, while her pendant was brought up in a society which had no moral priociples at all, who was mistress of many men, especially Voltaire, and who died consumed by an almost insane passion for the little-moved Marquis de Saint Lambert. Her biographer' says that she was certainly more in­teresting as a woman than as a mathematician. I cannot dispute that,

1 Hamel, An Eighteenth Century Marquise, London, 1910.

24 SIX FEMALE MATHEMATICIANS

it is a matter of taste, but I incline to believe that her talent for mathe­matics was really remarkable, and for that reason she deserves our notice.

Of the Marquise du Chatelet's love life I will make no further men­tion, her interest in mathematics and ambition to do something with it seems to have been genuine. Her closest mathematical friends were Koenig, John Bernoulli, Maupertuis, and especially Clairaut. She was not as gifted for languages as was Agnesi, but she learned English from Voltaire in order to master the works of Newton and subsequently, with the same teacher, read the works of Tasso and Ariosto in Italian. The fact that she was so anxious to study Newton is curious for she originally took the other side in the Newton-Leibniz controversy. To back up her conviction she wrote her Institutions de Physique, but then ran to the other extreme with the result noted. Her great work' must have occupied a good deal of time and energy. The editors cast an interesting light on the accuracy and originality of this work by writ­ing, "L'illustre interprete, plus jalouse de saisir !'esprit de !'Auteur que ses paroles, n'a pas craint en quelques endroits d'ajouter au de trans­poser quelques idees pour donner au sens plus de clarete. En conse­quence on trouvera souvent Newton plus intelligible dans cette traduc­tion que dans I' original. A 1' egard de Ia confiance que le Public doit a voir dans cette traduction, il suffit de dire qu' elle a ete fait par feu Madame Ia Marquise et qu'elle a ete revue par M. Clairaut.... La seconde partie de l'ouvrage est un commentaire des principes relatifs au Systeme du Monde."

In a disgustingly laudatory historical introduction to this volume by Voltaire we read, "On a vu deux prodiges, l'un que Newton ait fait cet ouvrage, !'autre qu'une femme l'ait traduit et l'ait eclair€." The puz­zle is, how far did Emilie really understand what she wrote, how far did she merely follow Clairaut? Voltaire says in a historical preface: "A I' egard du Commentaire algebrique c' est un ouvrage au dessus de Ia traduction. Madame du Chatelet travailla sur les idees de M. Clair­aut, elle fit taus les calculs elle m~me, et quand elle avait acheve un chapitre, Clairaut !'examinait et le corrigeait. Ce n'est pas tout ... M. Clairaut faisait encore revoir par un tiers les calculs quand ils etait mis au net."

I judge from all this that the work, in so far as it was a translation of Newton, was originally her composition, but that she did not care to run the risk of making mistakes, and so had Clairaut check it up, the commentaire algebrique was presumably entirely Clairaut's work, but

2 Principes mat!Wmatiques de Philosophic naturelle par feu la Marquise du Chatelet, Paris, 1769.

JULIAN L. COOLIDGE 25

Voltaire had no particular interest in eulogizing this scientist. We are told that they sometimes became so absorbed in their combined labors, that Voltaire was really upset by the loss of so much attention from his mistress.

Another indication of the Marquise's knowledge of mathematics is found in a letter she wrote to a friend at the age of 28. "I feel how much I should lose if I did not profit by the kindness you have shown in deigning to condescend to help my weakness, and to teach such sub­lime truths in an almost jesting manner . . . I confess to you that I understand nothing of Guisnee alone." This is perhaps a document that goes against my thesis. One is compelled to feel that the main idea was to get hold of the particular gentleman no matter by what means, there is little in Guisnee' s Traite de l' application de l' algebre a la geometrie that could offer real difficulty to one who had mastered Newton's Principia.

Madame du Chatelet jllade one other excursion into scientific writing in competing for a prize offered by the Academy of Science for an article on fire. It did not receive the prize, but was printed at the Academy's expense.

I leave the Marquise with a disappointing feeling that perhaps I have misjudged her. She wrote nothing original of a mathematical nature, but, with the possible exception of Hypatia, no woman before her time did anything of the sort. She may have been a complete failure, using the work of abler men, notably Clairaut, but one can put up a fair claim for a more charitable view of this extraordinary woman.

4. MARY SOMERVILLE, 1780-1872

This extraordinarily long-lived woman, whom her biographer in the Dictionary of National Biography, Ellen Mary Clarke, describes as "the most remarkable woman of her generation," certainly deserves a place in any list of female contributors to mathematics, even though, like others who preceded her, she has little to show in the way of abso­lutely original contributions to the Queen of Sciences. One has at once, however, the conviction that she was a real lover of learning, and that advances in the most intricate scientific theories appealed to her for their own sakes. Age was no hindrance to her. At 90 she was study­ing Peirce's Linear Associative Algebra, at the time of her death, two years later, she was interested in Higher Algebra and the Calculus of Quaternions. She wrote of herself, "Mathematics was the natural bent of my mind."

26 SIX FEMALE MATHEMATICIANS

Mrs. Somerville's long life was singularly even and untroubled. Her first marriage to Samuel Grieg was terminated by death after three years, her second marriage to William Somerville lasted forty years. They lived at first in the ~arious places required by his service as a Naval surgeon, her later years were spent largely in Italy.

Mrs. Somerville's circle of acquaintances, or better friends, was extraordinarily large. No leading British scientists were omitted from that number. She was especially intimate with the Herschells, father and son; her French friends included Arago, Poisson, Lacroix, Laplace, besides many others not especially scientific. The list of learned so­cieties to which she belonged was truly astonishing.

The most famous of Mrs. Somerville's mathematical writings was her translation of Laplace's Mecanique Celeste which she translated under the title of Mechanism of the Heavens. Laplace is said to have­observed that she was the single woman. who understood his work. Poisson did not specify the sex but said there were not twenty people in France who could read what she had written. After the death of Nathaniel Bowditch she received a letter from his son asking her to write a review of Bowditch's Commentary on Laplace published in four volumes. She replied "I refused to undertake so formidable a work, fearing that I should not do justice to the memory of so great a man." One wonders whether this was Laplace or Bowditch. As a matter of fact, she was at the moment pretty busy on her own account. In 1834 she wrote her Connexion of the Physical Sciences which was trans­lated into German and Italian and ran to its seventh edition, her Physical Geography had to be content with six. Her Form and Rotation of the Earth dealt with the difficult question of the shape of rotating bodies, and in 1869 she wrote on Molecular and Microscopic Science. She never seems to have shied away from a topic because of its diffi­culty, or because the weariness of old age had reduced her capacity for work. She once said of herself, "She wrote because she had to;" what a wonderful reason for working so mightily for the intellectual progress of mankind.

5. SOPHIE GERMAIN, 1776-1831

The French scientist whose name stands above was nearly a contem­porary of Mary Somerville, but had a very different story. The con­trast was not, perhaps, as violent as that between Maria Agnesi and Emilie du Chatelet, but still placed them in quite different classes, at least socially. Sophie had first to overcome the doubts of her family, and whereas the Englishwoman's reputation rested in a large part on her translation of Laplace, her French contemporary went boldly, per-

JULIAN L. COOLIDGE 27

haps too boldly, into original work. In fact, if we except Hypatia, abont whom we really know very little, Mlle. Germain was the first of her sex to undertake original problems, even if her solutions were, in places, of doubtful rigor.

Sophie's progress in mathematics was stimulated by the fate of Archimedes, killed by a Carthaginian soldier while occupied with geom­etry. Her progress was begun because she followed the new revolu­tionary practice of handing in her compositions to her teachers. In this case the teacher was Lagrange, and he was struck by her work, al­though he only knew her by the nom de plume of M. LeBlanc. In 1808 there appeared in Paris an Italian named Chladni who conducted experiments in the vibrations of elastic membranes which tended to establish that there was a definite underlying mathematical law. Part of the technique consisted in dusting on some fine powder, and noting the figures formed by the nodal points. The theory for the cor­responding problem in one dimension had already been worked out, but the corresponding theory for two dimensions seemed too difficult to be attractive. Laplace a.ssumed that it would yield only to some new mathematical method, and when the Academie des Sciences offered a prize for the best essay on the mathematical theory of elastic surfaces with a comparisom with experimental data, the most famous mathe­maticians of France left the subject severely alone. Not so Sophie Germain. In 1811 she sent in an anonymous solution of the problem. The commission charged with studying her communication contained famous names, Malus, and the four great L's, Laplace, Laguerre, Lacroix, and Legendre. They seem to have done precious little about it, and the last named wrote to her that he would rather accord to her the rights of priority than to quarrel with her on a subject she had evi­dently studied deeply. The one certain thing is that she did not receive the prize. The fact was that her knowledge of mathematics was not up to the task. Lagrange wrote that her method of passing from a line to a surface did not seem to him correct. Her biographer, 3 Stupuy, says, "Le vrai, c' est que Sophie Germain, travailant pour ainsi dire par instinct, et sans avoir fait un cours regulier d'analyse, n'avait pas resolu !a question; mais son mernoire, dont !a sagacite fut remarquee, ouvrit si bien !a voie, que Lagrange en tira 1' equation exact."

A second competition on the same subject came in 1813 with the same result, but although there still remained difficulties, the commission voted to accord to her an "Honorable mention." The third and last time came in 1816. Sophie wrote under her own name and was ac-

3 Stupuy, Sophie Germain, Paris, 1896.

28 SIX FEMALE MATHEMATICIANS

corded the prize even though Fourier had doubts as to whether the correctness of the equation was completely demonstrated (reference 3, p. 30). One thing is certain, as a result Sophie had rendered herself famous in France, the fact that a woman had done all this was a stimu­lus to N a tiona! pride. She was called the Hypatia of the nineteenth century. She seems to have devoted at least three considerable mono­graphs to elastic surfaces, "Recherches sur Ia Theorie des Surfaces e!astiques," of 1824, "Remarques sur Ia nature, les barnes et l'etendu de Ia question des surfaces,'' of 1826, and ''Memoire sur l'epaisseur dans la Theorie des Surfaces Elastiques," 1824. It would take too long to analyze these further. My general impression is that the mathe­matical technique is not very remarkable, and that in places she makes assumptions which are distinctly open to question. But what a re­markable young woman!

Among the people with whom Sophie corresponded was Legendre, and that brings out her interest in the Theory of Numbers. On page 17 of his monograph, "Sur le Theoreme de Fermat," he mentions with high praise the proof by Mlle. Germain of the theorem that Fer­mat's equation x• + y• = z" is not soluble for x, y, z; n not divisible by an odd prime where n < 100. I do not know the date of this article, but Legendre refers to Sophie as the winner of the Academie prize for the vibrations of ellastic lamina.

Sophie's interest in the theory of numbers appears in another con­nection. In 1806 the French had occupied the city of Brunswick. This was, at the moment, the place of residence of the mathematician Gauss, and Sophie, fearing that some mischance might come to the great man, wrote in distress to her family friend, General Pernety, who was at the moment besieging Breslau. She pleaded her cause so well that the general sent an emissary to find out how Gauss was getting along. This officer found out that the German was doing well but denied all knowledge of Mlle. Germain, the fact being that he had only had contact with M. LeBlanc. The difficulty was finally cleared up to everyone's satisfaction, and they exchanged several letters. I have the impression that he recognized her ability, but he does not seem to have taken an interest in the remark in one of her letters that she had generalized one of his formulae.

I cannot leave Mlle. Germain without referring to her article "Sur la courbure des Surfaces," which appeared in Grelle, Vol. 7, for 1831, presumably after the author's death from tuberculosis. This is de­voted to mean curvature, which she defines as the sum of the reciprocals of the radii of principal curvature. (Some writers describe this as twice the mean curvature.) She is familiar with the work of Meunier,

JULIAN L. COOLIDGE 29

and refers to Gauss; she did not grasp how much more important was the Gaussian curvature than what she treats in her rather elementary article. The high reputation she enjoyed among French mathemati­cians is based on more serious grounds.

6. SONYA KOVALEVSKY, 1850-1891

I have noted before that the female mathematicians I have men­tioned have shown their interest more in the works of others they translated rather than by their own individual compositions. Such was not the case with Sophie Germain, perhaps the first on record. Still less was it so in the case of the other Sophie whose name stands above.

Nearly everything connected with S6nya was unusual. There are a variety of ways of spelling her name, I follow that used in the book by Anna Carlotta, Duchess of Cajanello. 4 There seems a great doubt about the year of her birth, 1850 to 1853. She was a highly talented and completely undisciplined young woman, given to extreme affection and overwhelming jealousy and a requirement for a return that no human being could maintain indefinitely. There was a cult among the members of the young Russian circle which she joined to contract more or less Platonic marriages, in order to be free to go wherever desired, with or without the attendance of the partner. S6nya desired to take advantage of this convenient arrangement. When permission was re­fused by her more conventional parents, she sneaked out unknown to them to the rooms of her willing accomplice, then sent word where she was. The old couple accepted the inevitable, announced that she was engaged to Vladimir Kovalevsky, and presently accepted the marriage. The young people stayed for a while in St. Petersburg, then went to Heidelberg where S6nya continued her favorite study of mathematics more or less for two years; the partners drifting apart in the process. Her biographer, the Duchess Cajanello, remarks, 5 "She required too much freedom from those she loved and who loved her, and thought to gain by force and what they would have given her spontaneously had it been demanded.... Her own individuality was far too pro­nounced for her to live in harmony with others. Kovalevsky was also, in his way, restless by nature." They drifted further and further apart. In 1870 she went to Berlin to study with Weierstrass.

She took her Ph.D. in 1874, strangely enough in Gottingen. For the next nine years she seems to have occupied herself in various unscien-

4 Anna Carlotta, Duchess of Cajanello, Biography of SOnya KovaUvski, translated from the Swedish by A.M. Clive Bayley, New York, 1895.

' Op. cit., p. 177.

30 SIX FEMALE MATHEMATICIANS

tific pursuits, but iu 1883, owing to the activity of Mittag Leffier, she was appointed Privat Docent in Stockholm. She was duly raised to a professorship and she held that office until her death in 1891.

S6nya's peak moment came in 1888 when she received from the French Acadetnie the famous "Prix Bordin" and the amount was raised from fr. 3000 to fr. 5000 owing to the unusual merit of the work. But unalloyed happiness was not for S6nya. We are told,' "In this way both the happiness of her love and the triumph of her ambition were spoiled. Separately they would have given her great joy. Her tragic destiny gave her all she desired in life, but under such circum­stances that, as she herself complained, the sweetness was turned to gall. . . . She expected from her lover such absolute devotion and self -abnegation as must have surpassed the powers of all but a very few exception,al men. On the other hand she could not decide to cut her life in two at one blow, surrender her work and become merely a wife. On the rocks of the impossibility of reconciling such different claims, their love suffered its final shipwreck." We are never told who this man was, we are left with the pang that such great genius was fated to come to final shipwreck by the absolute lack of balance and self -control of her character.

Let us finally look at what she wrote. She was essentially the pupil of Weierstrass. Mittag Leffier, in speaking of her, wrote (op. cit., p. 318), "She came to us from the centre of modern science full of faith and enthusiasm for the ideas of her great master of Berlin, the venerable old man who has outlived his favorite pupil. Her works, all of which belong to the same order of ideas, have shown by new discoveries the power of Weierstrass's system." Her doctor's dissertation which was published in Grelle, Vol. 80, was called "Zur Theorie der partiellen differentialgleichungen" and deals with a very general system of such equations of the first order, in any number of variables. Weier­strass had given an analogous structure for total equations, S6nya's contribution consists in extending this to partial equations. In Vol. IV of the Acta Mathematica we have from her an article whose title is self -explanatory, "Ueber die Reduction einer bestimmten Klasse Abelscher Integrale 3ten Ranges." She next wanders into astronomy, showing how Weierstrass enables us to improve on Laplace in "Zusatz und Bemerkungen zu Laplace's Untersuchungen," Astronomische Nachrichten 2643 of 1885. In Acta Mathematica of 1890, we have "Sur une propriete d'un systeme d'equations," based on an earlier study of hers, and a short article in Acta, 1891, "Sur Ia Theorie de M. Bruns." S6nya's most famous article, for which she received the

6 Op. cit., p. 268.

JULIAN L. COOLIDGE 31

Prix Bordin in 1888, was entitled "Sur un cas particulier du probleme de la rotation d'un corps au tour d'un point fixe" published in full in the 1W'emoires presentes a l'Academie des Sciences (Memoires ertrangers), Vol. 31, 1884. Till then only two cases had been found which gave the complete solution of the differential equations involved; S6nya found a totally new case where the complete solution is thoroughly worked out, the underlying basis depending on the ideas of Weierstrass and ultraelliptic integrals.

HARVARD UNIVERSITY

CURIOSA 254. A Study in Multiplication. Denoting by l,. a number with the digit 1 repeated n

times, we find for values of n > 10:

Similarly

l' ' l' " l' 4

11' 1112

11114

1 2 1 1 2 3 2 1

1234321

~~ 123454321 ~ 1 2 3 4 5 6 5 4 3 2 1 l~ 1234567654321 z: 123456787654321 z: 12345678987654321

The symmetric arrangement of the digits seems to disappear when n becomes greater than 9. Thus,

1,: ~ 1 2 3 4 5 6 7 8 9 0 0 9 8 7 6 5 4 3 2 1,

a number formed by two blocks of digits. The right-side block, 0987654321, is made up of all the 10 digits written in reverse order.

We shall denote it by RI. In the left-side block, which will be later referred to as L, the digit 8 is missing, all others appear in their natural order. For n = 11, 12,,, 18 we have

L 1 2 R 1 L1232R1

Ll23432R1 L12345432R1

' ' . . . . . '

1,: ~ L 1 2 3 4 5 6 7 8 7 6 5 4 3 2 R 1 l,l ~ L 1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 R 1

Comparing l~ with l1 ~, l~ with l1 ~, in general!? with l£+ 9 , we find that in every case lf+9 is formed by inserting between the constants L and R the digits of I? except the unit digit 1. Denoting this truncated value of li by Tk we can express the relationship between li+9 and lk by the fonnula

lf+9 = L Tk R 1

where L is the constant 1234567890 and R = 098765432. With the help of the constants L and R the formula can be generalized to numbers > 18.

Thus

lfs+~< = L L Tk R R 1 or symbolically L2 Tk R2 1.

In generall:+u. = D Tk R1 1.

D. R. KAPREKAR, Demlo Numbers (1948), p. 18

DEVLAL!, INDIA