Woman in Science — Women in Mathematics, Part 1

CHAPTER III WOMEN IN MATHEMATICS

"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.

Athenaeus, a Greek writer who flourished about A. D.

200, tells us in his _Deipnosophistae_ 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 hetaerae--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"--[Greek: Te Philosopho]--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 sapientiae 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 Abbe Bossuet, a member of the French Academy and a collaborator of D'Alembert on the mathematical part of the famous _Encyclopedie_.

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 Maecenas 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.