Maria Gaetana Agnesi
María Gaetana Agnesi (Milan, May 16, 1718 - Milan, January 9, 1799) was an Italian philosopher, mathematician, linguist, philanthropist, writer and theologian.
Biography
Agnesi was born in Milan on May 16, 1718, the first daughter of Pietro Agnesi and Anna Brivio. She is said to be the eldest of 21 siblings, born to the three wives her father had. She is considered a child prodigy.
Agnesi's occupation is not clear, although her long-supposed relationship with the academic world of the University of Bologna tends to be dismissed, and she is considered solely a wealthy businesswoman; less is she even known of her mother. It is certain that they were rich, perhaps because of the silk business, and above all enlightened, and that they took great care in the education of María with means that -for those who could afford them- were common at the time: tutors and private teachers, and meetings of intellectuals called in the living room of the family home, in which philosophical issues were discussed, especially natural philosophy: issues close to what we now call physics.
Although these events displeased María Gaetana, with her withdrawn and solitary character, respect for her father and his precocious discursive faculties, which stood out in these groups, together with the musicals of her sister María Teresa, made the sessions in the salon of the Agnesi became famous, with a point of legend, for the gift of languages and the discretion of the eldest daughter, with the musical accompaniment of the youngest.
Agnesi is credited with having mastered Italian, Latin, Greek, Hebrew, French, Spanish and German before she was thirteen. Precocity and polyglotty are favorite targets of myth, so depending on the source consulted, the reference age can go down to 5 years, and the number and name of the languages can fluctuate.
The example of her first philosophical essay is revealing about how much truth there is and how much the narrators emphasize the precocity of María Gaetana: some want her to write it in Latin at the age of 9, and the subject is also dealt with by claiming the right to higher education for women. The story deserves skepticism, but it has surprising realities behind it, and of course an ornamental omission: it seems that the essay existed, its topic was indeed the defense of female academic training, and the age of the girl was really 9 years old but it was about from a translation exercise proposed by one of his tutors, who had written the original text in Italian himself or taken from another source. María Gaetana translated it into Latin, supervised or not by her tutor, memorized it and exposed it publicly.
The descriptions of these performances by young María that some witnesses have left us suggest an exhibitionist effort on the part of her father, bordering on circus spectacle. The 20-year-old girl who spoke in perfect Latin ("pure, easy and precise", "like that of an angel", witnesses say) about the origin of springs and rivers, the tides, the theories of Ptolemy and Newton or the function of the liver, and that she had been offering similar functions since she was 5 or 9 years old, she had to feel fed up with her role. A visitor puts in her mouth: & # 34; I'm sorry I subjected you to this; I know that for every interested listener I have twenty bored to death".
The training given to María Gaetana in her childhood and youth always had religion side by side with science: the tutors, preceptors and intellectuals in the hall of the family palace frequently wear the habit. Such is the case with the Jesuit and geometer Giovanni Saccheri, with the monk and mathematician Ramiro Rampinelli, or with the also Jesuit and mathematician Vincenzo Ricatti (son of Jacopo Francesco Ricatti, famous for the equation that bears her name). It was the environment that some author has baptized as "the Catholic Enlightenment".
It is not surprising then that María Gaetana, withdrawn and lonely at heart, very religious, and with a scientific vocation, aspires to leave the world and enter a convent, as her sister Giuseppa Teresa had already done. Some say that an adolescent illness reinforced those convictions and that desire.
The death of her mother during the delivery of her eighth child gives rise to a pact between María Gaetana and her father. In exchange for not taking the habit, continuing to live at home, and taking care of him and his siblings, he asks his father to be able to go to mass whenever he wants, dress simply and humbly, and not have to attend dances. and parties". Her father will marry two more times, his second wife dying after giving birth to two children, followed by eleven from the third. María Gaetana is attributed the role of mother to her twenty siblings, the corresponding burden, and also the pain of losing them; most did not survive infancy, and only four are said to have exceeded thirty years of age.
In 1738 Pietro Agnesi was able to publish a profound book by his 20-year-old daughter, Propositiones Philosóphicae, which summarized the defense of 191 philosophical theses debated or proposed in those social gatherings that Maria Gaetana hated.
Math Journey
Studies and Legacy in Mathematics: The "Instituzioni#34;
From the age of 20, Agnesi abandons all social activities and concentrates on the study of mathematics and religion; Her withdrawal would not have been greater had she taken the robes. The great influence that the mathematical monk Ramiro Rampinelli, who had taught mathematics in Rome and Bologna, had on her formation, emphasizes that scientific-monastic environment that presided over the life of Italian mathematics. Rampinelli gave Agnesi contact with the Ricatti, who also had a great influence on her; We know that Vincenzo offered to read the final version of the Instituzioni at the suggestion of his father, and also that he contributed his own material, which Maria Gaetana waited for to start printing the book.
In 1748, Agnesi's most famous work, Instituzioni analítiche ad uso della gioventú italiana, was published in Milan, which she had to pay for and publish herself. Surprisingly, the printing press is in the Agnesi mansion, and Maria Gaetana herself directs the work. The first volume is dedicated to finite magnitudes, while the second deals with the analysis of infinitesimals.
The work quickly gained notoriety among mathematicians of the time. The Instituzioni clearly expose the concepts through the judicious use of multiple examples, and have the virtue of harmonizing the works, hitherto dispersed, of many mathematicians, homogenizing them into a single and coherent set. Remember that Newton's fluxions and Leibnitz differentials were still being talked about, and that the creation of the symbols we use today in calculus, due mainly to Leibnitz and Euler, was very recent. The 1,000 pages of text and 50 pages of illustrations are nevertheless very familiar to the modern reader, reflecting Agnesi's greatest merit: creating the first complete text on Calculus, from algebra to differential equations. Furthermore, surpassing previous attempts, particularly that of L'Hopital in his book Analyse des infiniment petits.
The treatment of maximums and minimums has been highlighted in the book, and it is credited with having been the first textbook that treated differential calculus and integral calculus together, also explaining their nature as inverse problems, an idea that in 1748 it wasn't old or obvious. The clarity, order, precision, and successful use of examples have been repeatedly praised. Of course, the pioneering character of the work also implied some shortcomings: the trigonometric functions had little presence (the French edition added material to correct this), and the power series were not treated, among other gaps.
Between 1750 and 1752, it is recorded that she was a professor of mathematics at the University of Bologna, probably on an honorary basis. For the next forty-seven years she dedicated her life and estate to charity and the care of the poor, either as a resident, as a nun of the congregation, or more likely as both. This sense of vocation she sustained until her death in the same hospice that she had directed.
In 1775 the Royal Academy of Sciences published the French edition in Paris, and in 1801, two years after Maria's death, the English one was published, translated by John Colson, from Cambridge (he had to translate it much earlier, because died 1760).
Agnesi also wrote a commentary on the Traite analytique des sections coniques, by the Marquis de L'Hôpital, which unfortunately was never published, despite the fact that those who had the opportunity to see the manuscript considered it of great importance.
In 1786 María was related to some important works by Isaac Newton, in one of them on natural mathematical principles Newton described the force that makes all bodies fall as equal to the force that allows the moon and the planets to stay in the orbit
Agnesi curve
Among the lucky examples in the book there is one, at the end of the first volume, that got María Gaetana Agnesi a place in the name indexes of textbooks, and in manuals of formulas and mathematical tables, and that has made her made famous to a greater extent than all his other merits: The Curve of Agnesi.
This is a curve that Fermat had studied in 1630, and for which Guido Grandi, in 1703, had given a construction method.
The naming of this curve as: "witch", is a translation error; only English and languages that have copied the name from English use that term. Guido Grandi called, in 1718, the curve versoria in Latin, and versiera in Italian. It is a naval term, which identifies the rope or rope that turns the sail. María Gaetana Agnesi in turn wrote la versiera, adding the feminine article. John Colson, a Cambridge translator with little knowledge of Italian, calls the curve witch ('bruja'), because it "confused" versiera with avversiera (which in Italian means 'devil' or 'demonia'). The dependence that the Spanish language had on the English language ended up bewitching her also in Spanish. Other languages speak of loci (Latin for geometric 'places', curves) of Agnesi.
The curve is asymptotic to the X axis, to the right and to the left, and is therefore only represented in a neighborhood of the origin, in which it reaches a maximum just as it crosses the and. This mountainous environment, and the height of the maximum, are determined by a single parameter a, which is precisely the height of the maximum point reached at x = 0, that is, the point (0, a) is always on the curve and is also its maximum value.
The construction method is simple; to get any point on the curve:
- Bring a circle, center at point (0, a/2)
- From the origin, (0, 0), bring straight crosses with the straight y=a (OA record in the figure, in which a=10)
- Point P of the witch will be the one where the BP (horizontal lines passing through the cut between OA and the circumference) and AP (vertical that passes through the cut between OA and the straight y=a).
The set of lines OA of the plane determines the set of points on the Agnesi curve.
With a bit of geometry (only angle equality, triangle similarity, and the Pythagorean theorem are required, plus very little algebra) it is shown that Agnesi's witch equation is:
And the parametric equations are:
Agnesi does not present parametric equations, despite the fact that the treatment would have been easier, through:
Main contributions
For the history of mathematics, Agnesi is important for its influence in the popularization of calculus. She is also one of the most cited characters in reflections on the historical role of women in mathematics: it is enough to consider that the Instituzioni analítiche are, according to some, the oldest surviving mathematical work authored by women.
Her name is sometimes in the index of analytical geometry and calculus books, always associated with the curve misnamed: "witch", Agnesi Curve. The two nouns are uncertain: Agnesi did not discover that curve, nor did she intend it, and the name & # 34;witch & # 34; surely chance provided it with a bad translation into English, which also appeared in Spanish.
Many of his works were translated into English and French, the Instituzioni had a great impact on teaching, as they harmonized, in a single discourse, dispersed and heterogeneous materials from previous mathematicians, showing for the first time a logical and didactic sequence from algebra to differential equations. His unpublished works are kept in the Biblioteca Ambrosiana in Milan, which occupy twenty-five volumes.
Religious career
In 1750 Maria's father became seriously ill, and she was appointed by Pope Benedict XIV to the chair of mathematics and natural philosophy at the University of Bologna. Between 1750 and 1752 she held the chair, surely in an honorary way.
Pietro Agnesi dies in 1752, and from that moment Maria gives herself to the study of Theology, apparently especially Patristics, dedicates her fortune to works of charity, ending in misery, has exercised since 1771 by designation of the archbishop Tozzobonelli as director of the Trivulzio Hospice in Milan, concentrated on caring for the needy and sick, especially older women, and died herself in the institution she directed, on January 9, 1799.
Uncertainties and legend accompany her until death: some suggest that if she died in the Trivulzio Hospice it is because her donations had plunged her into poverty, and she was now just another destitute resident. Others maintain that she had finally fulfilled her wishes, and she was an Augustinian nun (or & # 34; blue nun & # 34;, because of the color of the habit) of the hospice.