"All abstract speculations, all knowledge which is dry, however useful it may be, must be abandoned to the laborious and solid mind of man.... For this reason women will never learn geometry."

In these words Immanuel Kant, more than a century ago, gave expression to an opinion that had obtained since the earliest times respecting the incapacity of the female mind for abstract science, and notably for mathematics. Women, it was averred, could readily assimilate what is concrete, but, like children, they have a natural repugnance for everything which is abstract. They are competent to discuss details and to deal with particulars, but become hopelessly lost when they attempt to generalize or deal with universals.

De Lamennais shares Kant's opinion concerning woman's intellectual inferiority and does not hesitate to express himself on the subject in the most unequivocal manner. "I have never," he writes, "met a woman who was competent to follow a course of reasoning the half of a quarter of an hour—un demi quart d'heure. She has qualities which are wanting in us, qualities of a particular, inexpressible charm; but, in the matter of reason, logic, the power to connect ideas, to enchain principles of knowledge and perceive their relationships, woman, even the most highly gifted, rarely attains to the height of a man of mediocre capacity."

But it is not only in the past that such views found acceptance. They prevail even to-day to almost the same extent as during the ages of long ago. How far they have any foundation in fact can best be determined by a brief survey of what woman has achieved in the domain of mathematics.

AthenÆus, a Greek writer who flourished about A.D. 200, tells us in his DeipnosophistÆ of several Greek women who excelled in mathematics, as well as philosophy, but details are wanting as to their attainments in this branch of knowledge. If, however, we may judge from the number of women—particularly among the hetÆrÆ—who became eminent in the various schools of philosophy, especially during the pre-Christian era, we must conclude that many of them were well versed in geometry and astronomy as well as in the general science of numbers. Menagius declares that he found no fewer than sixty-five women philosophers mentioned in the writings of the ancients[108]; and, judging from what we know of the character of the studies pursued in certain of the philosophical schools, especially those of Plato[109] and Pythagoras, and the enthusiasm which women manifested in every department of knowledge, there can be no doubt that they achieved the same measure of success in mathematics as in philosophy and literature.[110]

The first woman mathematician, regarding whose attainments we have any positive knowledge, is the celebrated Hypatia, a Neo-platonic philosopher, whose unhappy fate at the hands of an Alexandrian mob in the early part of the fifth century has given rise to many legends and romances which have contributed not a little toward obscuring the real facts of her extraordinary career. She was the daughter of Theon, who was distinguished as a mathematician and astronomer and as a professor in the school of Alexandria, which was then probably the greatest seat of learning in the world. Born about the year 375 A. D., she at an early age evinced the possession of those talents that were subsequently to render her so illustrious. So great indeed was her genius and so rapid was her progress in this branch of knowledge under the tuition of her father that she soon completely eclipsed her master in his chosen specialty.

There is reason to believe—although the fact is not definitely established—that she studied for a while in Athens in the school of philosophy conducted by Plutarch the Younger and his daughter Asclepigenia. After her return from Athens, Hypatia was invited by the magistrates of Alexandria to teach mathematics and philosophy. Here in brief time her lecture room was filled by eager and enthusiastic students from all parts of the civilized world. She was also gifted with a high order of eloquence and with a voice so marvelous that it was declared to be "divine."

Regarding her much vaunted beauty, nothing certain is known, as antiquity has bequeathed to us no medal or statue by which we could form an estimate of her physical grace. But, be this as it may, it is certain that she commanded the admiration and respect of all for her great learning, and that she bore the mantle of science and philosophy with so great modesty and self-confidence that she won all hearts. A letter addressed to "The Muse," or to "The Philosopher"—?? F???s?f?—was sure to be delivered to her at once. Small wonder, then, to find a Greek poet inditing to her an epigram containing the following sentiment:

"When I see thee and hear thy word I thee adore; it is the ethereal constellation of the Virgin, which I contemplate, for to the heavens thy whole life is devoted, O august Hypatia, ideal of eloquence and wisdom's immaculate star."[111]

But it was as a mathematician that Hypatia most excelled. She taught not only geometry and astronomy, but also the new science of algebra, which had but a short time before been introduced by Diophantus. And, singular to relate, no further progress was made in the mathematical sciences, as taught by Hypatia, until the time of Newton, Leibnitz and Descartes,—more than twelve centuries later.

Hypatia was the author of three works on mathematics, all of which have been lost, or destroyed by the ravages of time. One of these was a commentary on the Arithmetica of Diophantus. The original treatise—or rather the part which has come down to us—was found about the middle of the fifteenth century in the Vatican Library, whither it had probably been brought after Constantinople had fallen into the possession of the Turks. This valuable work, as annotated by the great French mathematicians Bachet and Fermat, gives us a good idea of the extent of Hypatia's attainments as a mathematician.

Another of Hypatia's works was a treatise on the Conic Sections by Apollonius of Perga—surnamed "The Great Geometer." Next to Archimedes, he was the most distinguished of the Greek geometricians; and the last four books of his conics constitute the chief portions of the higher geometry of the ancients. Moreover, they offer some elegant geometrical solutions of problems which, with all the resources of our modern analytical method, are not without difficulty. The greater part of this precious work has been preserved and has engaged the attention of several of the most illustrious of modern mathematicians—among them Borelli, Viviani, Fermat, Barrow and others. The famous English astronomer, Halley, regarded this production of Apollonius of such importance that he learned Arabic for the express purpose of translating it from the version that had been made into this language.

A woman who could achieve distinction by her commentaries on such works as the Arithmetica of Diophantus, of the Conic Sections of Apollonius, and occupy an honored place among such mathematicians as Fermat, Borelli, and Halley, must have had a genius for mathematics, and we can well believe that the glowing tributes paid by her contemporaries to her extraordinary powers of intellect were fully deserved. If, with Pascal, we see in mathematics "the highest exercise of the intelligence," and agree with him in placing geometers in the first rank of intellectual princes—princes de l'esprit—we must admit that Hypatia was indeed exceptionally dowered by Him whom Plato calls "The Great Geometer."

There is still a third work of this ill-fated woman that deserves notice—namely, her Astronomical Canon, which dealt with the movements of the heavenly bodies. It is the general opinion that this was but a commentary on the tables of Ptolemy, in which event it is still possible that it may be found incorporated in the work of her father, Theon, on the same subject.

In addition to her works on astronomy and mathematics, Hypatia is credited with several inventions of importance, some of which are still in daily use. Among these are an apparatus for distilling water, another for measuring the level of water, and a third an instrument for determining the specific gravity of liquids—what we should now call an areometer. Besides these apparatus, she was likewise the inventor of an astrolabe and a planisphere.

One of her most distinguished pupils was the eminent Neo-platonist philosopher, Synesius, who became the Bishop of Ptolemais in the Pentapolis of Libya. His letters constitute our chief source of information respecting this remarkable woman. Seven of them are addressed to her, and in four others he makes mention of her. In one of them he writes: "We have seen and we have heard her who presides at the sacred mysteries of philosophy." In another he apostrophizes her as "My benefactress, my teacher,—magistra—my sister, my mother."

In science Hypatia was among the women of antiquity what Sappho was in poetry and what Aspasia was in philosophy and eloquence—the chiefest glory of her sex. In profundity of knowledge and variety of attainments she had few peers among her contemporaries, and she is entitled to a conspicuous place among such luminaries of science as Ptolemy, Euclid, Apollonius, Diophantus and Hipparchus.[112]

It is a matter of regret to the admirers of this favored daughter of the Muses that she is absent from Raphael's School of Athens; but, had her achievements been as well known and appreciated in his day as they are now, we can readily believe that the incomparable artist would have found a place for her in this masterpiece with the matchless form and features of his beloved Fornarina.

After the death of Hypatia the science of mathematics remained stationary for many long centuries. Outside of certain Moors in Spain, the only mathematicians of note in Europe, until the Renaissance, were Gerbert, afterward Pope Silvester II, and Leonardo da Pisa. The first woman to attract special attention for her knowledge of mathematics was Heloise, the noted pupil of Abelard. According to Franciscus Ambrosius, who edited the works of Abelard and Heloise in 1616, the famous prioress of The Paraclete was a prodigy of learning, for besides having a knowledge of Latin, Greek and Hebrew, which was something extremely rare in her time, she was also well versed in philosophy, theology and mathematics, and inferior in these branches only to Abelard himself, who was probably the most eminent scholar of his age.[113]

Many Italian women, as we have seen in a preceding chapter, were noted for their proficiency in the various branches of mathematics. Some of the most distinguished of them flourished during the seventeenth and eighteenth centuries. Among these were Elena Cornaro Piscopia, celebrated as a linguist as well as a mathematician; Maria Angela Ardinghelli, translator of the Vegetable Statics of Stephen Hales; Cristina Roccati, who taught physics for twenty-seven years in the Scientific Institute of Rovigo, and Clelia Borromeo, fondly called by her countrymen gloria Genuensium—the glory of the Genoese. In addition to a special talent for languages, she possessed so great a capacity for mathematics and mechanics that no problem in these sciences seemed to be beyond her comprehension.[114] Then there was also Diamante Medaglia, a mathematician of note, who wrote a special dissertation on the importance of mathematics in the curriculum of studies for women, Alle matematiche, alle matematiche prestino l'opera loro le donne, onde non cadano in crassi paralogismi—"To mathematics, to mathematics," she cries, "let women devote attention for mental discipline."[115]

The most illustrious, by far, of the women mathematicians of Italy was Maria Gaetana Agnesi, who was born in Milan in 1718 and died there at the age of eighty-one. At an early age she exhibited rare intelligence and soon distinguished herself by her extraordinary talent for languages. At the age of five she spoke French with ease and correctness, while only six years later she was able to translate Greek into Latin at sight and to speak the former as fluently as her own Italian. At the early age of nine she startled the learned men and women of her native city by discoursing for an hour in Latin on the rights of women to the study of science. This discourse—Oratio—was not, as usually stated, her own composition, but a translation from the Italian of a discourse written by her teacher of Latin. That a child of nine years should speak in the language of Cicero for a full hour before a learned assembly and without once losing the thread of her discourse was, indeed, a wonderful performance, and we are not surprised to learn that she was regarded by her countrymen as an infant prodigy.[116]

In addition to Italian, French, Latin and Greek, she was acquainted with German, Spanish and Hebrew. For this reason she was, like Elena Cornaro Piscopia, the famous "Venetian Minerva," called Oracolo Settilingue—Oracle of Seven Languages.[117]

But it was in the higher mathematics that Maria Gaetana was to win her chief title to fame in the world of learning. So successful had she been in her prosecution of this branch of science that she was, at the early age of twenty, able to enter upon her monumental work—Le Instituzioni Analitiche—a treatise in two large quarto volumes on the differential and integral calculus. To this difficult task she devoted ten years of arduous and uninterrupted labor. And if we may credit her biographer, she consecrated the nights as well as the days to her herculean undertaking. For frequently, after working in vain on a difficult problem during the day, she was known to bound from her bed during the night while sound asleep and, like a somnambulist, make her way through a long suite of rooms to her study, where she wrote out the solution of the problem and then returned to her bed. The following morning, on returning to her desk, she found, to her great surprise, that while asleep she had fully solved the problem which had been the subject of her meditations during the day and of her dreams during the night. Could the psychiatrist who so loves to deal with obscure mental phenomena find a more interesting case to engage his attention or one more worthy of the most careful investigation?

Finally Maria Gaetana's opus majus was completed and given to the public. It would be impossible to describe the sensation it produced in the learned world. Everybody talked about it; everybody admired the profound learning of the author, and acclaimed her: "Il portento del sesso, unico al Mondo"—the portent of her sex, unique in the world. By a single effort of her genius she had completely demolished that fabric of false reasoning which had so long been appealed to as proof positive of woman's intellectual inferiority, especially in the domain of abstract science. Maria Gaetana's victory was complete, and her victory was likewise a victory for her sex. She had demonstrated once for all, and beyond a quirk or quibble, that women could attain to the highest eminence in mathematics as well as in literature, that supreme excellence in any department of knowledge was not a question of sex but a question of education and opportunity, and that in things of the mind there was essentially no difference between the male and the female intellect.

The world saw in Agnesi a worthy accession to that noble band of gifted women who count among their number a Sappho, a Corinna, an Aspasia, a Hypatia, a Paula, a Hroswitha, a Dacier, an Isabella Rosales who, in the sixteenth century, successfully defended the most difficult theological theses in the presence of Paul III and the entire college of cardinals. And so delighted were the women—especially those in Italy—with the signal triumph of their eminent sister that they defied the traducers of their sex—muliebris sapientiÆ infensissimis hostibus—to continue any longer their unreasonable campaign against the rights of women which were based on the intellectual equality of the two sexes.

So highly did the French Academy of Science value Agnesi's achievement that she would at once have been made a member of this learned body had it not been against the constitutions to admit a woman to membership. M. Motigny, one of the committee appointed by the Academy to report on the work, in his letter to the author, among other things, writes: "Permit me, Mademoiselle, to unite my personal homage to the plaudits of the entire Academy. I have the pleasure of making known to my country an extremely useful work which has long been desired, and which has hitherto"—both in France and in England—"existed only in outline. I do not know any work of this kind which is clearer, more methodic or more comprehensive than your Analytical Institutions. There is none in any language which can guide more surely, lead more quickly, and conduct further those who wish to advance in the mathematical sciences. I admire particularly the art with which you bring under uniform methods the divers conclusions scattered among the works of geometers and reached by methods entirely different."

As an indication of the exceptional merit of Agnesi's work, even long after its publication in 1748, it suffices to state that the second volume of the Instituzioni Analitiche was translated into French in 1775 by Antelmy and annotated by the AbbÉ Bossuet, a member of the French Academy and a collaborator of D'Alembert on the mathematical part of the famous EncyclopÉdie.

A still greater proof of the estimation in which Agnesi's work was held by men of science is the fact that it was translated in its entirety into English by the Rev. John Colton, Lucasian Professor of Mathematics in the University of Cambridge, and published in 1801, fifty-two years after it had appeared in Italian. His impression of the methods followed by the Milanese savante was so favorable that, in the words of a contemporary writer, it "gave rise to his very spirited resolution of learning a new language at an advanced period of life, that he might make himself perfect master of them."[118]

Gratifying, however, as were the tributes of admiration and appreciation which came to Agnesi from all quarters, from learned societies, from eminent mathematicians, from sovereigns—the Empress Maria Theresa sent her a splendid diamond ring and a precious crystal casket bejeweled with diamonds—that which touched her most deeply was, undoubtedly, the recognition which she received from the great MÆcenas of his age, Pope Benedict XIV. As Cardinal Lambertini and Archbishop of Bologna, he had taken a conspicuous part in the honors showered on Laura Bassi when she received her doctorate, and was specially delighted when she was made professor of physics in his favored university. Being himself familiar with the higher mathematics, he recognized at once the exceptional merit of Maria Gaetana's work and showed his appreciation of it not only by letters and presents, but also by having her, motu proprio, appointed by the Bolognese senate as professor of higher mathematics in the University of Bologna.

In advising her of this appointment, he writes her that he had in view the honor of the University in which he had always taken a special interest, and that the appointment carried with it no obligation of thanks on her part but rather on his—che porta seco ch'ella non deve ringraziar Noi, ma che Noi dobbiamo ringraziar lei. The interest that this wise and broad-minded pontiff exhibited in the advancement of learned women and the rewards he was ever ready to accord to their achievements in science and literature—especially in the cases of Laura Bassi and Maria Gaetana Agnesi—is in keeping with the policy pursued by his predecessors, and accounts in great measure for that large number of learned women in Italy who, since the opening of the first universities, have been the glory of their sex and country.

But ardent as was the desire of the Supreme Pontiff to have Agnesi occupy the chair of mathematics, and numerous as were the appeals of her friends and the members of the university faculty to have her accept the appointment that carried with it such signal honor, she could never be induced to leave her beloved Milan. For, after completing her masterpiece, she resolved to retire from the world and devote the rest of her life to the care of the poor, the sick and the helpless in her native city. She did not, however, as is so frequently asserted, enter the convent and become a nun.[119] During many years after her retirement from the world, she lived in her own home, a part of which she had converted into a hospital. During the last fifteen years of her life she had charge of the Pio Albergo Trivulzio—a large institution founded by Prince Trivulzio for the aged poor who were without home or assistance.

She had devoted ten years of the flower of her life to the writing of her Instituzioni Analitiche—prepared primarily for the benefit of one of her brothers who had a taste for mathematics—and, after it was finished, she entered upon that long career of heroic charity which was terminated only at her death at the advanced age of eighty-one.

One loves to speculate regarding Maria Gaetana's possible achievements if she had continued during the rest of her life that science in which, during a few short years, she had won such distinction. She had made her own the discoveries of Newton, Leibnitz, Roberval, Fermat, Descartes, Riccati, Euler, the brothers Bernouilli, and had mastered the entire science of mathematics then known. Her pinions were trimmed for essaying loftier flights than any hitherto attempted, and her intellect was prepared, as one of her scientific friends expressed it, "for fixing the limits of the infinite." But while the world of science was still sounding her praises and predicting for her still greater triumphs in the field of analysis, it learned with surprise and sorrow that she had bid adieu to those studies in which she had achieved such extraordinary success, and had consecrated her life to the service of the poor and the afflicted. She disappeared completely from those literary and scientific reunions where she had so long been the most conspicuous figure, and was thenceforth known only as the ministering angel of the suffering and the abandoned. For half a century hers was a life of the most heroic charity and self-abnegation. Very readily, therefore, we can understand why a recent representative of the scientific world should desire to see her name placed on the calendar of saints.[120]

Had Agnesi devoted her entire life to science instead of abandoning it just when she was prepared to do her best work, she might to-day be ranked among such supreme mathematicians as Lagrange, Monge, Laplace and the Bernouillis, all of whom were her contemporaries. Even as it was, she has been placed beside Cardan, Leibnitz and Euler for her remarkable powers of analysis of infinitesimals, while the best proof of the literary value of her Instituzioni Analitiche is the fact that it has been selected by the famous society Della Crusca as a testo di lingua—a work considered as a classic of its kind and used in the preparation of the great authoritative dictionary of the Italian language.

But by consecrating herself to charity she probably accomplished far more for humanity and for the well-being of her sex than if she had elected to continue her work in the higher mathematics. There had been many learned women in Italy before her time and many since; many who were distinguished as Hellenists, as Latinists, as polyglots, as mathematicians—women like the Roccati, the Borghini, the Brassi, the Ardinghelli, the Barbapiccola, the Caminer Turra, the Tambroni; but Maria Gaetana Agnesi surpasses them all, not only in knowledge, but as a potent influence for the diffusion of culture and the spirit of brotherhood, for the expansion of benevolence and charity, and, above all, for the elevation of woman. She was also, as her latest and best biographer beautifully expresses it, "an inspired condottiera who, in the field of civility, anticipated the conquests of these latter days." She was, indeed, as her epitaph informs us, pietate, doctrina, beneficentia insignis, and as such she will live in the memory of our race as long as men shall admire genius and love virtue.

In the year following the publication of Agnesi's Instituzioni Analitiche was recorded the premature and tragic death of the distinguished French mathematician, the Marquise Émilie du ChÂtelet. She has been described as a "thinker and scientist, prÉcieuse and pedant, but not the less a coquette—in short, a woman of contradictions."[121] To most readers she is better known by reason of her liaison with Voltaire, of whom she is regarded as a mere satellite, than for her work in science. But she was far more than a satellite that shone by the light received from the sage of Ferney. For there can be no doubt that she was a highly gifted woman who, besides having a thorough knowledge of several languages, including Latin, possessed a special talent for mathematics. It was said of her that "she read Virgil, Pope and algebra as others read novels," and that she was able "to multiply nine figures by nine others in her head." No less an authority than the illustrious AmpÈre declared her to be "a genius in geometry."

Among her teachers in mathematics were Clairaut, Koenig, Maupertuis, PÈre Jaquier and Jean Bernouilli, the immediate predecessors of such distinguished mathematicians as Monge, Lagrange, d'Alembert and Laplace. At her Chateau of Cirey, where she and Voltaire spent many years together, she was visited by learned men from various parts of Europe. Among these was the Italian scholar, Francisco Algarotti, who was the author of a work entitled Newtonism for Women. And as Mme. du ChÂtelet was an ardent admirer of Newton, the author of the Principia soon became a strong bond of union between her and the brilliant Italian. She called the savants who frequented her chÂteau at Cirey the Émiliens and purposed writing memoirs to be entitled Emiliana—a design, however, which she was never able to execute.

The first work of importance from the pen of the Marquise was entitled Institutions de Physique. In it she gave an exposition of the philosophy of Leibnitz and dissertations on space, time and force. In the discussion of the last topic she seems to have anticipated some of the later conclusions of science respecting the nature of energy.

Her most noted achievement, however, was her translation of Newton's Principia, the first translation into French of this epoch-making work. To translate this masterpiece from its original Latin, it was necessary that the Marquise, in order to make it intelligible to others, should have a thorough understanding of it herself. To the translation she added a commentary, which shows that Mme. du ChÂtelet had a mathematical mind of undoubted power. She labored assiduously on this great undertaking for many years and completed it only shortly before her death; but it was not published until ten years after her demise.

In his Élogie Historique on the Marquise's translation of the Principia, Voltaire, in his usual flamboyant style, declares "Two wonders have been performed: one that Newton was able to write this work, the other that a woman could translate and explain it." In an effort to express in a single sentence all his admiration for his talented friend he does not hesitate to state: "Never was woman so learned as she, and never did anyone less deserve that people should say of her, 'She is a learned woman.'" Again he refers to her with characteristic Frenchiness as "a woman who has translated and explained Newton, in one word a very great man—en un mot un trÈs grand homme."[122]

But, although the extent of her attainments and her ability as a mathematician were unquestionable, she fell far short of her great contemporary, Gaetana Agnesi, both in the depth and breadth of her scholarship and in her power of infinitesimal analysis. As to her moral character, she was infinitely inferior to the saintly savante of Milan. She was by inclination and profession an Epicurean and an avowed sensualist. In her little treatise, RÉflexions sur le Bonheur—Reflections on Happiness—she unblushingly asserts "that we have nothing to do in this world except procure for ourselves agreeable sensations." Considering her profligate life, bordering at times on utter abandon, we are not surprised that one of her countrymen has characterized her as "Femme sans foi, sans moeurs, sans pudeur,"—a woman without faith, without morals, without shame.[123]

Anna Barbara Reinhardt of Winterthur in Switzerland was another woman of exceptional mathematical talent. She is remarkable for having extended and improved the solution of a difficult problem that specially engaged the attention of Maupertuis. According to so competent an authority as Jean Bernouilli, she was the superior, as a mathematician, of the Marquise de ChÂtelet.

Of a more original and profound mathematical mind was Sophie Germain, a countrywoman of the Marquise du ChÂtelet. Hers was the glory of being one of the founders of mathematical physics. A pupil of Lagrange and a co-worker with Biot, Legendre, Poisson and Lagrange, she has justly been called by De Prony "the Hypatia of the nineteenth century."

Her success, however, was not achieved without overcoming many and great difficulties. In the first place, she had to overcome the opposition of her family, who were decidedly averse to her studying mathematics. "Of what use," they asked, "was geometry to a girl?" But in trying to extinguish her ardor for mathematics they but augmented it. Alone and unaided she read every work on mathematics she could find. The study of this science had such a fascination for her that it became a passion. It occupied her mind day and night. Finally her parents, becoming alarmed about her health and resolved to force her to take the necessary repose, left her bedroom without fire or light, and even removed from it her clothing after she had gone to bed. She feigned to be resigned; but when all were asleep, she arose and, wrapping herself in quilts and blankets, she devoted herself to her favorite studies, even when the cold was so intense that the ink was frozen in her ink-horn. Not infrequently she was found in the morning chilled through, having been so engrossed in her studies that she was not aware of her condition. Before such a determined will, so extraordinary for one of her age, the family of the young Sophie had the wisdom to permit her to dispose of her time and genius according to her own pleasure. And they did well. Like the great geometer of Syracuse, Archimedes, who had ever been her inspiration in the study of mathematics, she would have died rather than abandon a problem which, for the time being, engaged her attention.

She first attracted the attention of savants by her mathematical theory of Chladni's figures. By the order of Napoleon, the Academy of Science had offered a prize for the one who would "Give the mathematical theory of the vibration of elastic surfaces and compare it with the results of experiment." Lagrange declared the problem insoluble without a new system of analysis, which was yet to be invented. The consequence was that no one attempted its solution except one who, until then, was almost unknown in the mathematical world; and this one was Sophie Germain.

Great was the surprise of the savants of Europe when they learned that the winner of the grand prix of the Academy was a woman. She became at once the recipient of congratulations from the most noted mathematicians of the world. This eventually brought her into scientific relations with such eminent men as Delambre, Fourier, Cauchy, AmpÈre, Navier, Gauss[124] and others already mentioned.

It was in 1816, after eight years of work on the problem, that her last memoir on vibrating surfaces was crowned in a public sÉance of the Institut de France. After this event Mlle. Germain was treated as an equal by the great mathematicians of France. She shared their labors and was invited to attend the sessions of the Institut, which was the highest honor that this famous body had ever conferred on a woman.

The noted mathematician, M. Navier, was so impressed with the extraordinary powers of analysis evinced by one of Mlle. Germain's memoirs on vibrating surfaces that he did not hesitate to declare that "it is a work which few men are able to read and which only one woman was able to write."

Biot, in the Journal de Savants, March, 1817, writes that Mlle. Germain is probably the one of her sex who has most deeply penetrated the science of mathematics, not excepting Mme. du ChÂtelet, for here there was no Clairaut.[125]

Like Maria Gaetana Agnesi, Mlle. Germain was endowed with a profoundly philosophical mind as well as with a remarkable talent for mathematics. This is attested by her interesting work entitled ConsidÉrations GÉnÉrales sur l'État des Sciences et des Lettres aux DiffÉrentes Époques de Leur Culture. All things considered, she was probably the most profoundly intellectual woman that France has yet produced. And yet, strange as it may seem, when the state official came to make out the death certificate of this eminent associate and co-worker of the most illustrious members of the French Academy of Sciences he designated her as a rentiÈre—annuitant—not as a mathÉmaticienne. Nor is this all. When the Eiffel tower was erected, in which the engineers were obliged to give special attention to the elasticity of the materials used, there were inscribed on this lofty structure the names of seventy-two savants. But one will not find in this list the name of that daughter of genius, whose researches contributed so much toward establishing the theory of the elasticity of metals,—Sophie Germain. Was she excluded from this list for the same reason that Agnesi was ineligible to membership in the French Academy—because she was a woman? It would seem so. If such, indeed, was the case, more is the shame for those who were responsible for such ingratitude toward one who had deserved so well of science, and who by her achievements had won an enviable place in the hall of fame.[126]

Four years after the birth of Sophie Germain was born in Jedburgh, Scotland, one whom an English writer has declared was "the most remarkable scientific woman our country has produced." She was the daughter of a naval officer, Sir William Fairfax; but is best known as Mary Somerville. Her life has been well described as an "unobtrusive record of what can be done by the steady culture of good natural powers and the pursuit of a high standard of excellence in order to win for a woman a distinguished place in the sphere naturally reserved for men, without parting with any of those characteristics of mind, or character, or demeanor which have ever been taken to form the grace and the glory of womanhood."[127]

The surroundings of her youth were not conducive to scientific pursuits. On the contrary, they were entirely unfavorable to her manifest inclinations in that direction. Having scarcely any of the advantages of a school education, she was obliged to depend almost entirely on her own unaided efforts for the knowledge she actually acquired. She, like Sophie Germain, was essentially a self-made woman; and her success was achieved only after long labor and suffering and in spite of the persistent opposition of family and friends.

When she was about fifteen years old, the future Mrs. Somerville received her first introduction to mathematics; and then, strange to say, it was through a fashion magazine. At the end of a page of this magazine, "I read," writes Mrs. Somerville, "what appeared to me to be simply an arithmetical question; but in turning the page I was surprised to see strange-looking lines mixed with letters, chiefly X's and Y's, and asked 'What is that?'" She was told it was a kind of arithmetic, called algebra.

Her interest was at once aroused; and she resolved forthwith to seek information regarding the curious lines and letters which had so excited her curiosity. "Unfortunately," she tells us, "none of our acquaintances or relatives knew anything of science or natural history; nor, had they done so, should I have had courage to ask of them a question, for I should have been laughed at."

Finally she was able to secure a copy of a work on algebra and a Euclid. Although without a teacher she immediately applied herself to master the contents of these two works, but she had to do so by stealth in bed after she had retired for the night. When her father learned of what was going on, he said to the girl's mother, "Peg, we must put a stop to this, or we shall have Mary in a straightjacket one of these days." The mother, who had no more sympathy with her daughter's scientific pursuits than had the father, and, fully convinced, like the great majority of her sex, that woman's duties should be confined to the affairs of the household, strove to divert her daughter's mind from her "unladylike" pursuits. But her efforts were ineffectual. The young woman, in spite of all obstacles and opposition, contrived to continue her cherished studies; and, through her uncle, the Rev. Dr. Somerville, afterward her father-in-law, she was able to become proficient in both Latin and Greek. When she was thirty-three years of age she became the happy possessor of a small library of mathematical works. "I had now," she writes, "the means, and pursued my studies with increased assiduity; concealment was no longer necessary, nor was it attempted. I was considered eccentric and foolish, and my conduct was highly disapproved of by many, especially by some members of my own family."[128]

In March, 1827, Mrs. Somerville received a letter from Lord Brougham, who had heard of her remarkable acquirements, begging her to prepare for English readers a popular exposition of Laplace's great work—MÉcanique CÉleste. She was overwhelmed with astonishment at this request, for her modesty made her diffident of her powers; and she felt that her self-acquired knowledge of science was so far inferior to that of university men that it would be sheer presumption for her to undertake the task proposed to her. She was, however, finally persuaded to make the attempt, with the proviso that her manuscript should be consigned to the flames unless it fulfilled the expectations of those who urged its production.

In less than a year her work, to which she gave the name of The Mechanism of the Heavens, was ready for the press. But it was far more than a translation and epitome, as originally intended by its projector, Lord Brougham; for, in addition to the views of Laplace, it contained the independent opinions of the translator respecting the propositions of the illustrious French savant. No sooner was the work published than Mrs. Somerville found herself famous. She had, as Sir John Herschel expressed it, "written for posterity," and her book placed her at once among the leading scientific writers and thinkers of the day. She was elected an honorary member of the Royal Astronomical Society at the same time as Caroline Herschel, they being the first two women thus honored. Her bust, by Chantry, was placed in the great hall of the Royal Society, and she was made a member of many other scientific societies in Europe and America. In recognition of her services to science she was granted by the government a pension of £200 a year—a sum which was shortly afterward increased to £300. In addition to all this, Mrs. Somerville had the satisfaction of learning that her work was so highly esteemed by Dr. Whewell, the great master of Trinity, that it was, chiefly on his recommendation, introduced as a textbook in the University of Cambridge and prescribed as "an essential work to those students who aspire to the highest places in the examinations." What Mme. du ChÂtelet had done for Newton, Mrs. Somerville did for Laplace.

Among other books from the pen of this highly gifted woman is her Connection of the Physical Sciences and a work entitled Physical Geography, which, together with the Mechanism of the Heavens, was the object of the "profound admiration" of Humboldt. Then there is a number of very abstruse monographs on mathematical subjects, one of which is a treatise of two hundred and forty-six pages On Curves and Surfaces of Higher Orders, which, she tells us, she "wrote con amore to fill up her morning hours while spending the winter in Southern Italy."

Her last work was a treatise On Molecular and Microscopic Science embodying the most recondite investigations on the subject. This book, begun after she had passed her eightieth birthday, occupied her for many years and was not ready for publication until she was close upon her ninetieth year. Her last occupations, continued until the day of her death at the advanced age of ninety-two, were the reading of a book on Quaternions and the review and completion of a volume On the Theory of Differences.

Like her illustrious friend, the great Humboldt, Mary Somerville was possessed of extraordinary physical vigor, and, like him, she retained her mental powers unimpaired until the last. And like her great rival in mathematics, Maria Gaetana Agnesi, she was always "beautifully womanly." Her scientific and literary occupations did not cause her to neglect the duties of her household or to disregard "the graceful and artistic accomplishments of an elegant woman of the world." Her daughter Martha writes of her: "It would be almost incredible were I to describe how much my mother contrived to do in the course of the day. When my sister and I were small children, although busily engaged in writing for the press, she used to teach us for three hours in the morning, besides managing her house carefully, reading the newspapers—for she was always a keen and, I must add, a liberal politician—and the most important new books on all subjects, grave and gay. In addition to this, she freely visited and received her friends.... Gay and cheerful company was a pleasant relaxation after a hard day's work."[129]

The life of Mary Somerville, like that of Gaetana Agnesi, proves that the pursuit of science is not, as so often asserted, incompatible with domestic and social duties. It also disposes of the fallacy, so generally entertained, that intellectual labor is detrimental to the health of women and antagonistic to longevity. The truth is that it is yet to be demonstrated that intellectual work, even of the severest kind, is, per se, more deleterious to women than to those of the stronger sex.

Scarcely less remarkable as a mathematician was Mrs. Somerville's distinguished contemporary, Janet Taylor, who was known as the "Mrs. Somerville of the Marine World." She was the author of numerous works on navigation and nautical astronomy which in their day were highly prized by seafaring men. In recognition of her valuable services to the marine world she was placed on the civil list of the British government.

As an eminent mathematician as well as a "representative of the highest intellectual accomplishments to which women have attained," SÓnya KovalÉvsky will ever occupy an honored place among the votaries of science. In many respects this richly endowed daughter of Holy Russia was par excellence the woman of genius of the latter half of the nineteenth century.

She was born in Moscow in 1850, but although her career was brief it was one of meteoric splendor. At an early age she exhibited an unusual talent for mathematics and an unquenchable thirst for knowledge. Not being able to obtain in her own country the educational advantages she desired, she resolved at the age of eighteen to go to Germany with a view of pursuing her studies there under more favorable auspices.

She first matriculated in the University of Heidelberg, where she spent two years in studying mathematics under the most eminent professors of that famous old institution. Thence she went to Berlin. She could not enter the University there, as its doors were closed to female students; but she was fortunate enough to prevail on the illustrious Professor Weierstrass, regarded by many as the father of mathematical analysis, to give her private lessons. He soon discovered to his astonishment that this child-woman had "the gift of intuitive genius to a degree he had seldom found among even his older and more developed students." Under this eminent mathematician SÓnya spent about three years, at the end of which period she was able to present to the University of GÖttingen three theses which she had written under the direction of her professor. The merit of her work and the testimonials which she was able to present from Weierstrass, Kirchhoff and others were of such supreme excellence that she was exempted from an oral examination and was enabled, by a very special privilege, to receive her doctorate without appearing in person.

Not long after receiving her doctor's degree—one of the first to be granted to a woman by a German university—she was offered the chair of higher mathematics in the University of Stockholm. She was the first woman in Europe, outside of Italy, to be thus honored. But her appointment had to be made in the face of great opposition. No other university, it was urged by the conservatives, had yet offered a professor's chair to a woman. Strindberg, one of the leaders of modern Swedish literature, wrote an article in which he proved, "as decidedly as that two and two make four, what a monstrosity is a woman who is a professor of mathematics, and how unnecessary, injurious and out of place she is."[130]

The fame that came to SÓnya through her achievements in the German and Swedish universities was immensely enhanced when, on Christmas eve, 1888, "at a solemn session of the French Academy of Sciences, she received in person the Prix Bordin—the greatest scientific honor which any woman had ever gained; one of the greatest honors, indeed, to which any one can aspire."

She became at once the heroine of the hour and was thenceforth "a European celebrity with a place in history." She was fÊted by men of science whithersoever she went and hailed by the women of the world as the glory of her sex and as the most brilliant type of intellectual womanhood.

Mme. KovalÉvsky's printed mathematical works embrace only a few memoirs including those which she presented for her doctorate and for the Prix Bordin. But brief as they are, all of these memoirs are regarded by mathematicians as being of special value. This is particularly true of the memoirs, which secured for her the Prix Bordin; for it contains the solution of a problem that long had baffled the genius of the greatest mathematicians.

The prize had been opened to the competition of the mathematicians of the world, and the astonishment of the committee of the French Academy was beyond expression when it was found that the successful contestant was a woman.[131]

Everyone admired her varied and profound knowledge, but, above all, her amazing powers of analysis. A German mathematician, Kronecker, did not hesitate to declare that "the history of mathematics will speak of her as one of the rarest investigators."[132]

Shortly before her premature death, she had planned a great work on mathematics. All who are interested in the intellectual capacities and achievements of woman must regret that she was unable to complete what would undoubtedly have been the noblest monument of woman's scientific genius. She was then in the prime of life and perfectly equipped for the work she had in mind. Considering the extraordinary receptive and productive power of this richly dowered woman, there can be little doubt, had she lived a few years longer, that she would have produced a work that would have caused her to be ranked among the greatest mathematicians of the nineteenth century.

It is pleasant to record that this woman of masculine mind, masculine energy and masculine genius, far from being mannish or unwomanly, was, on the contrary, a woman of a truly feminine heart; and that, although a giantess in intellectual attainments, she was in grace and charm and delicacy of sentiment one of the noblest types of beautiful womanhood. She could with the greatest ease turn from a lecture on Abel's Functions or a research on Saturn's rings to the writing of verse in French or of a novel in Russian or to collaborating with her friend, the Duchess of Cajanello, on a drama in Swedish, or to making a lace collar for her little daughter, Fouzi, to whom she was most tenderly attached.[133]

Little more than a quarter of a century has elapsed since Strindberg, expressing the sentiment of the great majority of the men of his time, declared that a woman professor of mathematics is a monstrosity. But during this short period what a change has been effected in the attitude of the world toward women who devote themselves to the study and the teaching of science! Women mathematicians are found to-day in all civilized countries, and no sane person now considers it any more "unwomanly" or more "monstrous" for them to study or teach mathematics than for them to teach music or needlework. Yet more. They are now frequent contributors to mathematical magazines and to the official bulletins of learned societies, and not infrequently they are on the editorial staffs of publications devoted exclusively to mathematics. They are also found as computers in some of the largest astronomical observatories, where the speed and accuracy of their work have evoked the most favorable comment.

Of women in America, who have distinguished themselves by their work in the higher mathematics, it suffices to mention the name of Miss Charlotte Angas Scott, recently deceased, who was for years professor of mathematics in the College of Bryn Mawr. Her writings on various problems of the higher mathematics show that she faithfully followed in the footsteps of her illustrious predecessors,—Hypatia, Agnesi, du ChÂtelet, Germain, Somerville and KovalÉvsky.

[108] "Ipse mulieres Philosophas in libris Veterum sexaginta quinque reperi," Historia Mulierum Philosopharum, p. 3, Amstelodami, 1692.