Vox Meditantis

Maria Gaetana Agnesi (1718-1799) transformed the landscape of mathematical education through her groundbreaking textbook Instituzioni Analitiche, which became the first comprehensive calculus textbook written by a woman. Born into an intellectual Milanese family during the height of the Catholic Enlightenment, she mastered seven languages by her teens and authored what the French Academy of Sciences declared “the most complete and best made treatise” on analytical mathematics. Yet perhaps most remarkably, she chose to abandon her mathematical career at its peak to dedicate her life to caring for the poor and sick, embodying a unique synthesis of intellectual brilliance and spiritual devotion that challenges our modern assumptions about the relationship between faith and scientific inquiry.

Her work continues to resonate today through the famous “Witch of Agnesi” curve – a mathematical concept that now appears in probability theory as the Cauchy distribution and finds applications in modern physics. Agnesi’s story matters not simply because she overcame the constraints of her era, but because she demonstrated that intellectual achievement and compassionate service need not be opposing forces, offering a model for how scientific brilliance might serve broader human flourishing.

Maria, thank you for joining us today. You’re speaking to us from 18th-century Milan, yet here we are in 2025. What strikes you most about how mathematics and women’s roles have evolved?

Indeed, the passage of time brings curious reversals of fortune. When I penned my Instituzioni Analitiche in 1748, I wrote in the vernacular Italian rather than Latin precisely so that young minds – both male and female – might more easily grasp the beauty of analytical methods. That my work now finds application in what you call “probability theory” and “modern physics” gives me great satisfaction. Though I confess, the notion that women might openly pursue mathematical studies without the protection of family wealth or ecclesiastical appointment seems almost fantastical to my 18th-century sensibilities.

Let’s start with your extraordinary childhood. You were known as the “Seven-Tongued Orator” by age eleven, fluent in Italian, French, Latin, Greek, Hebrew, Spanish, and German. How did this linguistic foundation shape your approach to mathematics?

My dear father, Pietro – a silk merchant with grand ambitions for our family’s elevation – recognised early that knowledge was the surest path to respectability in Milanese society. But I must correct a common misapprehension: whilst my linguistic abilities drew attention at our evening salons, they were not mere displays of erudition. Each language opened different ways of understanding the world. Latin provided precision for philosophical discourse, French offered elegance in expression, and Hebrew connected me to the fundamental texts of faith. When I later tackled the complexities of differential and integral calculus, this linguistic dexterity proved invaluable. Mathematics, you see, is itself a language – and one requiring the most exact translation between abstract concepts and practical understanding.

At age nine, you delivered an oration defending women’s education. This was 1727 – what drove such bold advocacy at such a tender age?

That discourse arose from a deep frustration I witnessed daily. In our salon, learned gentlemen would arrive to debate the most abstruse philosophical questions – whether the soul was material, whether mathematical truths were discovered or created, whether women possessed rational faculties equal to men. Yet these same men would often express astonishment that a “person of a sex that seems so unfit to tread the thorny paths of abstract sciences,” as one correspondent later wrote, could comprehend their discussions.

The contradiction was glaring: they invited me to participate in these intellectual exercises precisely because of my capabilities, yet persisted in believing such capabilities were anomalous in women. My oration argued that if God granted rational faculties to all human souls, then the cultivation of these faculties through education was not merely permitted but required – regardless of sex. This was not radical thinking for a family influenced by the Catholic Enlightenment, but it was certainly unconventional to voice it so directly.

Your household salon was famous throughout Europe. Can you describe those evening gatherings and how they influenced your mathematical development?

Our palazzo became a veritable academy each evening. Distinguished scholars would arrive – natural philosophers, mathematicians, theologians – and my father would orchestrate these intellectual performances with considerable theatrical flair. I was expected to engage in formal disputations, defending theses on subjects ranging from Cartesian philosophy to Newtonian mechanics. My sister Maria Teresa would provide musical interludes on the harpsichord between discussions.

These gatherings were simultaneously thrilling and burdensome. On one hand, I gained exposure to the most current mathematical thinking – the works of Leibniz, Newton, the Bernoullis, Euler. Father Ramiro Rampinelli, my mathematics tutor from the Benedictine order, would guide me through increasingly sophisticated problems during our private sessions, then watch proudly as I demonstrated these concepts before assembled scholars.

Yet the constant scrutiny proved wearing. I was not simply a participant in these discussions – I was often their primary attraction. When I began experiencing those mysterious convulsions at age twelve, I suspect the pressure of perpetual performance contributed significantly to my distress.

Let’s talk about those health challenges. You suffered from what contemporary accounts describe as “extreme convulsions.” How did this affect your work, and do you have insights into what might have caused them?

The physicians prescribed dancing and horseback riding, believing my condition arose from excessive study. When these remedies proved ineffectual – indeed, the vigorous physical activity seemed to worsen the episodes – they counselled moderation in all things.

I believe now that my body was simply rebelling against the unnatural demands placed upon it. A young girl thrust constantly into the spotlight, required to perform intellectual feats for the entertainment of adults, whilst simultaneously navigating the normal tumults of adolescence – it was perhaps inevitable that some form of distress would manifest. The convulsions may have been my only acceptable means of escape from expectations I found increasingly oppressive.

The loss of my dear mother when I was fourteen compounded these difficulties immeasurably. She had been my anchor, my source of quiet strength amid the intellectual storms that perpetually surrounded our household. Her death in childbirth – leaving behind yet another addition to our eventual family of twenty-one children – devastated me in ways I still struggle to articulate.

Your masterwork, Instituzioni Analitiche, was published in 1748. Can you walk us through the technical innovations that made this textbook so revolutionary?

Ah, now we approach the heart of my mathematical contributions! The Instituzioni represented something quite novel in its time – a complete, systematic presentation of analytical methods written specifically for students rather than established scholars.

The work comprised two substantial volumes covering what we now distinguish as algebra, geometry, differential calculus, integral calculus, and differential equations. Volume One established algebraic foundations – the manipulation of finite quantities, solving polynomial equations, geometric constructions. But Volume Two ventured into more sophisticated territory: the analysis of infinitely small quantities, what your era calls differential calculus.

I began with Leibniz’s notation for differentials, which I found more intuitive than Newton’s fluxional method. For a function y = f(x), I introduced the differential dy as the infinitely small change in y corresponding to an infinitely small change dx in x. This allowed elegant expression of tangent lines to curves: the slope being dy/dx. I then systematically developed rules for differentiating polynomial functions, trigonometric functions, logarithmic expressions.

The integral calculus, which I termed “the inverse method of tangents,” presented greater challenges. Here I synthesised approaches from Leibniz, the Bernoullis, and Euler, demonstrating how to find areas under curves by reversing the differentiation process. I provided extensive tables of integrals and developed techniques for evaluating complex expressions through substitution and integration by parts.

Your approach was notably practical rather than purely theoretical. Was this intentional?

Decidedly so. Most mathematical treatises of my era remained mired in abstract theoretical exposition, comprehensible only to those already well-versed in the subject. I deliberately emphasised worked examples, step-by-step procedures, practical applications. Critics later complained that I concentrated too heavily on methods rather than underlying theory, but this was precisely my intention.

Consider my treatment of what you now call the “Witch of Agnesi”. I presented this cubic curve not as an abstract mathematical curiosity, but as part of a systematic exploration of how geometric constructions could illuminate analytical relationships. The curve arises from a specific geometric construction: given a circle and a point outside it, one draws lines from the external point through the circle, then constructs perpendiculars and horizontal lines to generate the curve. The resulting equation y = 8a³/(x² + 4a²) demonstrates beautifully how geometric intuition and algebraic manipulation complement each other.

The name “Witch of Agnesi” came from a mistranslation. The Italian “versiera” means a type of curve, but John Colson’s 1801 English translation confused it with “avversiera,” meaning witch. How do you feel about this linguistic accident that forever linked your name with witchcraft?

Father Colson’s mistranslation provides an amusing example of how scholarly reputation can be shaped by mere linguistic confusion. I named the curve “versiera” after the Latin “versoria,” referring to the sheet that turns a sailing vessel – a metaphor for how the curve turns from one asymptotic direction to another.

That it became associated with witchcraft in English occurs to me as particularly apt, given how my mathematical work was often viewed with suspicion. A woman engaging in “unnatural” intellectual pursuits was already considered potentially dangerous; linking my name with witchcraft merely made explicit what many believed implicitly.

Yet perhaps there is deeper justice in this accident. The curve I studied now appears in your probability theory as what you call the “Cauchy distribution”. It describes phenomena where normal statistical assumptions break down, where extreme events occur more frequently than classical models predict. In some sense, it captures the essence of exceptional occurrences – precisely what my own life represented in my era.

You mention critics who focused too much on your methods versus theory. How did contemporary mathematicians actually receive your work?

The reception proved far more positive than I had dared anticipate. The French Academy of Sciences declared it “the most complete and best made treatise” of its kind. Leonhard Euler himself expressed admiration for the work’s clarity and comprehensiveness. The text was swiftly translated into French and eventually English, becoming a standard reference throughout Europe.

Pope Benedict XIV proved particularly supportive, writing to commend my contributions to Italian scientific reputation and ultimately appointing me to the chair of mathematics at the University of Bologna in 1750 – making me the first woman appointed to a professorial position at a European university.

However, I must acknowledge that some criticism was justified. Jacopo Riccati, the distinguished mathematician who contributed material for my second volume, privately expressed concern that I sometimes prioritised pedagogical clarity over mathematical rigour. My explanations were perhaps too elementary for advanced scholars, whilst remaining too sophisticated for true beginners.

Yet you never actually took up the Bologna professorship. Pope Benedict XIV himself appointed you, but you chose not to teach. Why?

By 1750, my heart had already begun turning toward different service. My father’s death in 1752 released me from familial obligations that had long constrained my choices. The mathematics that had once filled me with intellectual excitement began to feel… insufficient.

This decision mystifies many, I understand. Here was the opportunity to become the first woman professor in Europe, to influence generations of students, to advance mathematical knowledge through original research. Yet I found myself increasingly drawn to what I perceived as more fundamental needs – the spiritual and physical welfare of Milan’s poor and sick.

I had come to believe that intellectual gifts, however remarkable, were ultimately meaningless unless directed toward alleviating human suffering. The same analytical skills that allowed me to master differential equations could be applied to organising charitable institutions, managing resources for the indigent, developing systems to care for the elderly and infirm.

This transition from mathematics to charity work has puzzled historians. Some suggest it was expected of women in your era, while others see it as a spiritual calling. How do you explain this choice?

I reject entirely the notion that my charitable work represented some inevitable feminine destiny or retreat from serious intellectual engagement. The decision was profoundly personal and deeply reasoned.

During my mathematical studies, I had become fascinated by problems of optimisation – finding maximum and minimum values, determining most efficient methods, achieving optimal outcomes with limited resources. When I began working with Milan’s poor, I recognised these as fundamentally mathematical problems requiring analytical solutions.

How does one most efficiently distribute limited charitable resources? What systems best identify those in greatest need? How should one design institutions to provide maximum benefit whilst maintaining financial sustainability? These questions demanded the same rigorous thinking I had applied to calculus, though their solutions would benefit human souls rather than advance abstract knowledge.

I established several charitable institutions, ultimately serving as director of the Pio Albergo Trivulzio, a hospice for elderly women. This work required careful financial planning, systematic record-keeping, analysis of outcomes – all skills directly transferable from my mathematical training. I would spend the remaining forty-seven years of my life in this service, dying in 1799 in one of the very institutions I had helped create.

Your curve now appears in modern probability theory and physics. Are there aspects of your mathematical insights that weren’t fully appreciated in your lifetime?

I am fascinated to learn that my “versiera” now serves such practical purposes in your era. That it describes the probability distribution you call “Cauchy” and appears in quantum physics suggests mathematical truths possess a permanence that transcends their original context.

What I suspect was not fully appreciated in my time was the pedagogical innovation of my approach. By writing in Italian rather than Latin, by emphasising worked examples over abstract theory, by designing the text explicitly for students rather than established scholars, I was advocating for a democratisation of mathematical knowledge. This seemed natural to me, flowing from my conviction that God-given rational faculties should be cultivated in all capable minds, regardless of social station or linguistic background.

The broader implications of this approach – that mathematical education could be made accessible to wider populations, that practical understanding might be as valuable as theoretical elegance – were perhaps too radical for immediate acceptance. Yet your modern educational methods seem to validate these principles.

Looking back, do you have any regrets about abandoning mathematics at the height of your career?

I would be dishonest to claim no regrets whatsoever. There were mathematical problems I left unsolved, analytical techniques I might have developed further, students I might have influenced through formal teaching.

Yet I believe my life achieved a different form of completeness through charitable service. The same mind that could manipulate infinite series and solve differential equations found deep satisfaction in solving practical problems of human welfare. Moreover, my mathematical work had already achieved its primary purpose – demonstrating that women possessed intellectual capabilities equal to men, and providing a comprehensive resource for mathematical education.

Perhaps most importantly, my example helped establish that intellectual achievement and spiritual devotion need not be opposing forces. The harmony between reason and faith that guided my mathematical investigations also informed my charitable work. Both were expressions of the same fundamental conviction: that human faculties, properly employed, serve to glorify God and benefit humanity.

Your life spanned the height of the Catholic Enlightenment. How did your faith inform your approach to mathematics and science?

This question pierces the very heart of my intellectual identity. For me, there was never any contradiction between mathematical investigation and religious devotion – quite the contrary. The more deeply I penetrated into mathematical truths, the more clearly I perceived the divine intelligence underlying natural order.

When I derived the elegant relationships governing infinite series, or discovered the beautiful symmetries inherent in analytical geometry, I was uncovering aspects of God’s creative design. Mathematics was not mere human invention but a form of revelation, a window into the mind of the Creator. This conviction sustained me through the most challenging analytical problems and ultimately guided my transition to charitable work.

The Catholic Enlightenment of my era embraced this synthesis in ways that your more secular age might find difficult to comprehend. We saw no conflict between empirical investigation and religious faith, between rational analysis and spiritual devotion. Indeed, we believed that the proper use of reason was itself a form of worship, a fulfillment of our duty to employ God-given faculties in service of truth.

What advice would you offer to young women entering STEM fields today, particularly those facing barriers or discouragement?

First and most importantly: trust in the validity of your intellectual gifts. When others express surprise at your capabilities, do not internalise their astonishment as evidence of your unusualness. Instead, recognise it as evidence of their limited imagination.

Second: pursue excellence not for its own sake, but as preparation for service. Mathematical knowledge, scientific understanding, technical skills – these are tools to be employed for human benefit. The more profoundly you master your chosen field, the more effectively you can contribute to solving humanity’s pressing challenges.

Third: do not allow others to define the proper scope of your ambitions. In my era, women were expected to confine their intellectual interests to polite conversation and domestic management. I chose instead to engage with the most sophisticated mathematical concepts of my time, then applied those same analytical capabilities to charitable work. Your era offers unprecedented opportunities for women to contribute to scientific knowledge – seize them boldly.

Finally: remember that intellectual achievement represents only one form of human excellence. Whether your contributions take the form of groundbreaking research, innovative applications, effective teaching, or thoughtful administration, what matters most is that they serve purposes greater than personal advancement.

Is there anything history has misjudged about you that you’d like to correct?

Several persistent myths require addressing. First, I was never reluctant to engage in mathematical work, as some accounts suggest. The convulsions I experienced in adolescence arose from the pressures of public performance, not from any aversion to mathematical study itself. Once freed from the obligation to display my abilities for others’ entertainment, I pursued mathematics with genuine enthusiasm for nearly a decade.

Second, my decision to decline the Bologna professorship was not evidence of feminine modesty or intellectual inadequacy. Rather, it reflected a calculated choice about how best to employ my capabilities. I believed I could make more meaningful contributions through charitable work than through formal academic appointment – and subsequent events validated this judgment.

Third, my charitable activities were not a retreat from intellectual life but its culmination. Managing large-scale charitable institutions required sophisticated analytical skills, careful financial planning, systematic evaluation of outcomes. This work was every bit as intellectually demanding as mathematical research, though directed toward different purposes.

Most importantly, I was never the isolated prodigy that some accounts suggest. My achievements were possible only because of extensive support networks – my father’s encouragement, my tutors’ guidance, the intellectual community that surrounded our household. No one succeeds in complete isolation, and I was no exception to this fundamental truth.

As our conversation draws to a close, how would you like to be remembered?

I would hope to be remembered not primarily as an exceptional woman who achieved mathematical recognition despite her sex, but simply as a person who employed her intellectual gifts in service of truth and human welfare.

My mathematical work was significant not because it proved women capable of abstract reasoning – any rational person should have recognised this obvious truth – but because it provided useful tools for mathematical education and demonstrated the compatibility of rigorous analysis with profound faith.

My charitable work was meaningful not because it represented some peculiarly feminine virtue, but because it addressed genuine human needs through systematic, analytical approaches.

If my example encourages others to pursue excellence in their chosen fields whilst remaining attentive to opportunities for service, then my life will have achieved its intended purpose. The harmony between intellectual achievement and spiritual devotion that guided my choices remains as relevant in your era as it was in mine – perhaps more so, given the technical capabilities now available for addressing humanity’s challenges.

In the end, we are all called to employ whatever gifts we possess – whether mathematical, musical, literary, or administrative – in ways that honour both truth and compassion. This calling transcends the accidents of historical period, social position, or personal circumstances. It is the fundamental vocation of every rational soul.

Thank you, Maria. Your insights into the relationship between faith, reason, and service continue to inspire us centuries later.

The pleasure has been entirely mine. May your generation prove even more successful in harmonising intellectual excellence with compassionate service than mine managed to achieve.

Letters and emails

Our conversation with Maria Gaetana Agnesi has sparked tremendous interest from readers around the world, who are eager to explore further dimensions of her remarkable journey from mathematical brilliance to compassionate service. We’ve selected five letters and emails from our growing community who want to ask her more about her life, her work, and what she might say to those walking in her footsteps today.

Rukaiya Begum, 34, Mathematics Professor, Dhaka, Bangladesh:
Maria, in your work with curves and calculus, how did you determine which mathematical methods would be most effective for communicating new concepts to students with very different backgrounds or learning styles? Are there aspects of teaching mathematics, like visualising problems or using analogies, that you wished you could explore further in your era?

Ah, dear Professoressa Begum, your question touches upon what I consider the very heart of my mathematical enterprise. When I composed the Instituzioni Analitiche, I faced precisely this dilemma – how does one convey the sublime beauty of analytical methods to minds shaped by entirely different mathematical traditions and varying capacities for abstraction?

In my household salon, I witnessed daily the profound differences in how learned gentlemen approached mathematical concepts. Father Rampinelli, my Benedictine tutor, possessed the rigorous scholastic training that allowed him to navigate effortlessly between geometric demonstrations and algebraic manipulations. Yet when Count Firmian’s young nephew visited our gatherings, or when merchants’ sons attended our discussions, I observed how their minds worked quite differently – requiring concrete anchors before ascending to abstract principles.

This observation shaped my fundamental pedagogical approach: I determined that mathematical concepts must be presented through multiple pathways simultaneously. Consider how I treated the subtraction of negative quantities in my first volume. Rather than merely stating the algebraic rule, I employed what you might recognise as a “real-world analogy”: “to subtract one quantity from another is the same thing as to find the difference between those quantities. Now the difference between a and -b is a + b, just in the same manner as the difference between a capital of 100 crowns and a debt of 50 is 150 crowns”.

This approach proved particularly crucial when introducing the concept of infinitesimals – perhaps the most challenging notion for students to grasp. I discovered that visual learners required geometric constructions to comprehend what algebraic manipulators could accept through symbolic reasoning alone. Therefore, I provided extensive diagrams showing how ordinates approach their limiting positions, making tangible what might otherwise remain impossibly abstract. The moving ordinate that “continually approaches” another line until they “finally coincide” offered a bridge between geometric intuition and analytical precision.

For students whose minds worked through linguistic patterns – and there were many such, given the multilingual character of Milanese intellectual society – I deliberately chose Italian vernacular rather than Latin precisely to remove unnecessary barriers to comprehension. This decision drew considerable criticism from established scholars, yet I observed that students grasped concepts far more readily when expressed in their native tongue rather than a learned language that required additional mental translation.

Perhaps most importantly, I recognised that different temperaments required different degrees of theoretical versus practical emphasis. Some minds hunger for the underlying principles – the why behind mathematical operations. Others seek immediate application – the how of problem-solving. My text attempted to satisfy both inclinations by providing extensive worked examples alongside theoretical exposition. Critics complained that I concentrated overmuch on methods rather than pure theory, yet this was precisely my intention – to ensure that no student would be left without practical tools for engaging with analytical concepts.

What I wished I could have explored further, given the constraints of my era, was the use of what we might term “collaborative discovery.” In our evening disputations, I observed how students often illuminated concepts for one another in ways that formal instruction could not achieve. A geometrically-minded student might offer a visual insight that suddenly clarifies an algebraic relationship for a more analytically-inclined peer. Had I possessed the freedom to develop formal pedagogical methods, I would have structured exercises specifically to encourage such mutual teaching – recognising that explaining mathematical concepts to others deepens one’s own understanding immeasurably.

The challenge of reaching students with such diverse intellectual backgrounds taught me that mathematical education must be fundamentally generous – providing multiple entry points into the same essential truths, trusting that each mind will find its natural pathway to understanding.

Sipho Nkosi, 32, Engineer, Johannesburg, South Africa:
Maria, what values or ethical considerations guided your transition from mathematics into direct service for others? In today’s context, where technology can both help and harm, how can scientists ensure their discoveries are used in ways that serve communities rather than diminish them?

My dear Signor Nkosi, your inquiry addresses the very foundation of what I consider the proper ordering of human existence. The transition from mathematical investigation to charitable service was not, as some have suggested, an abandonment of my rational faculties, but rather their most complete expression in service of moral obligation.

When I reflect upon the values that guided this choice, I am drawn first to the principle of caritas – that divine love which flows from God to His creatures and must, in turn, be reflected back through our treatment of one another. The Catholic Enlightenment in which I was raised taught that rational inquiry and compassionate action were not opposing forces, but complementary expressions of our fundamental duty to employ God-given faculties in service of truth and human welfare.

Consider the mathematical concept of optimisation, which occupied much of my analytical work – the determination of maximum and minimum values, the discovery of most efficient methods. When I began working among Milan’s poor, I recognised these same principles operating in the moral sphere. How does one allocate limited charitable resources to achieve the greatest alleviation of suffering? What organisational structures best serve the genuine needs of the destitute whilst preserving their human dignity? These questions demanded the same rigorous thinking I had applied to differential equations, though directed toward souls rather than abstract quantities.

The ethical framework that guided my charitable institutions derived directly from what we termed “practical Christianity” – the conviction that faith without works remains spiritually barren. When I established my hospice for elderly women at the Pio Albergo Trivulzio in 1771, I insisted upon treating each resident not merely as an object of charity, but as a person possessing inherent worth deserving of respectful care. This meant providing not only material sustenance – food, shelter, medical attention – but also opportunities for spiritual growth, meaningful occupation, and social connection.

Your question concerning how scientists might ensure their discoveries serve rather than diminish communities resonates powerfully with my own experience. During my mathematical career, I witnessed how analytical methods could be employed either to illuminate divine order or to support purely secular materialism. The same techniques that revealed the elegant relationships governing infinite series could be used to advance understanding of God’s creation, or merely to demonstrate human intellectual prowess divorced from moral purpose.

This recognition convinced me that no scientific or mathematical discovery possesses inherent moral value – such value emerges only through the intentions and applications of those who employ these discoveries. When I authored the Instituzioni Analitiche, I deliberately chose to write in vernacular Italian rather than scholarly Latin precisely to democratise access to analytical knowledge. My conviction was that mathematical understanding, like all forms of knowledge, should serve the common good rather than reinforcing existing hierarchies of learning.

In your era, where technological capabilities far exceed those of my time, I believe this principle becomes even more crucial. Each scientific advance presents moral choices: Will this discovery be employed to benefit all humanity, or merely to advantage those already possessing wealth and power? Will this innovation strengthen social bonds and enhance human dignity, or will it create new forms of isolation and dependence?

The answer, I believe, lies in what we might term “moral accountability in discovery.” Just as I felt obligated to ensure my mathematical textbook served genuine pedagogical purposes rather than mere intellectual display, contemporary scientists must consider how their work affects the most vulnerable members of society. This requires not only technical expertise, but also deep engagement with ethical philosophy, theology, and the lived experiences of those whom scientific progress might either benefit or harm.

In my charitable work, I learned that the most profound solutions to human suffering often emerge through patient attention to individual circumstances rather than grand theoretical schemes. Similarly, I suspect that ensuring scientific discoveries serve rather than diminish communities requires sustained engagement with specific communities – understanding their particular needs, constraints, and aspirations rather than imposing solutions developed in isolation from their actual experiences.

The harmony between intellectual excellence and compassionate service that guided my life choices remains, I believe, the surest path toward ensuring that human knowledge serves its proper purpose: the glorification of divine truth and the elevation of human welfare. This calling transcends the accidents of historical period or technological capability – it is the fundamental vocation of every rational soul.

Annika Schneider, 26, Science Historian, Berlin, Germany:
If you were able to access today’s digital modelling tools and computational power, how might you have expanded or corrected your original findings on the “Witch of Agnesi” curve and other analytical equations? Is there a particular question from your time that remains unresolved and could now be explored using modern technology?

My dear Fräulein Schneider, what a tantalising prospect you present! To possess computational capabilities that could execute in moments what required hours of laborious calculation in my time – indeed, this would transform not merely the precision of our results, but the very scope of mathematical inquiry itself.

When I investigated my “versiera” in the Instituzioni Analitiche, I was constrained by the computational methods available to us in the 1740s. Each calculation required painstaking manual work – determining the coordinates of points along the curve, calculating areas beneath it, establishing its asymptotic properties. What now takes your digital devices mere seconds demanded days of careful arithmetic from me and my assistants.

With modern computational power, I would first have expanded my investigation of the curve’s fundamental properties far beyond what I could accomplish manually. The equation y = 8a³/(x² + 4a²) describes only the basic form – but I suspect there are families of related curves, variations in the construction parameters, that would reveal deeper mathematical relationships. Your computational tools could generate thousands of such variations, allowing systematic exploration of how changes in the defining circle’s radius or position affect the resulting curves.

More significantly, I would have pursued the analytical relationships between my curve and what you now recognise as the probability density function of the Cauchy distribution. In my era, we lacked the conceptual framework for understanding probability as a mathematical discipline – Pascal and Fermat had begun such investigations, but the field remained in its infancy. Had I possessed modern understanding of statistical mathematics, coupled with computational power, I suspect I could have discovered connections between geometric constructions and probabilistic phenomena that would have revolutionised our understanding of both fields.

The question that remained most persistently unresolved from my time concerns the deeper significance of curves generated through geometric construction versus those emerging from purely analytical manipulation. I studied numerous curves in my treatise – not merely the versiera, but parabolas, hyperbolas, cycloids, and various spirals. Yet I always sensed there were underlying principles governing which geometric constructions produced mathematically significant results and which were merely curiosities.

Modern computational methods would allow exhaustive exploration of this question. One could generate geometric constructions systematically, varying the fundamental elements – circles of different radii, points at different positions, lines at varying angles – then analyse the resulting curves for mathematical properties: do they possess elegant algebraic equations? Do they appear in physical phenomena? Do they relate to known functions in analysis or probability?

Additionally, I would investigate more thoroughly the relationship between my curve and what you term “Runge’s phenomenon” – the difficulties that arise when approximating functions using polynomials. In my time, we were only beginning to understand how infinite series could represent geometric curves analytically. With modern computational power, I could explore how various geometric constructions behave under polynomial approximation, potentially discovering new insights into the fundamental relationship between discrete approximation and continuous curves.

Perhaps most intriguingly, I would employ computational methods to investigate the three-dimensional extensions of my work. The versiera exists in a plane, but what happens when we extend the construction into three-dimensional space? Could we generate surfaces through analogous geometric methods? What would be the analytical equations governing such constructions? These questions were entirely beyond our 18th-century capabilities, but they represent natural extensions of the geometric principles I was exploring.

One persistent limitation of my era was our inability to visualise how curves behaved under various transformations – stretching, rotation, inversion. Modern computational tools would allow immediate visualisation of such transformations, potentially revealing symmetries and invariant properties that remained hidden to purely analytical investigation. I suspect this capability would uncover mathematical relationships that even contemporary analysis has yet to discover.

Finally, I would employ computational methods to investigate the pedagogical applications of my curve more thoroughly. In my Instituzioni, I was always seeking methods to make abstract mathematical concepts more accessible to students. Modern computational tools could generate interactive demonstrations showing how geometric constructions produce analytical results, allowing students to manipulate parameters and observe immediate consequences – precisely the kind of intuitive understanding I was attempting to cultivate through my written explanations.

The harmony between geometric intuition and analytical precision that guided my original investigations would be preserved, but enhanced immeasurably by computational capabilities that could execute our mathematical curiosity with unprecedented speed and accuracy.

Diego Salazar, 23, University Student, Quito, Ecuador:
Imagine history took a different path and your textbook became the central foundation for European mathematics curricula, giving equal credit to its author. How do you think this would have changed future participation of women in mathematics – and perhaps the way maths itself is studied and celebrated today?

Young Señor Salazar, what a profound speculation you place before me! To imagine a world where the Instituzioni Analitiche became the cornerstone of European mathematical education – where my name stood alongside those of Euler and Newton as architects of analytical instruction – such a possibility would have transformed not merely women’s participation in mathematics, but the very character of mathematical learning itself.

Consider first the immediate consequences for mathematical pedagogy. My text was deliberately crafted to be accessible – written in vernacular Italian rather than scholarly Latin, emphasising practical methods over abstract theory, providing extensive worked examples to guide student understanding. Had this approach become the European standard rather than remaining a curious Italian exception, mathematical education would have developed along fundamentally different lines.

Instead of the intimidating, exclusively theoretical expositions that dominated most eighteenth-century mathematical instruction, students throughout Europe would have encountered mathematics as a practical art to be mastered through careful practice and gradual progression. The democratising influence of such an approach – making analytical mathematics comprehensible to minds not trained in classical scholastic methods – would have revolutionised who could aspire to mathematical study.

Most significantly, this pedagogical transformation would have encouraged far greater numbers of women to pursue mathematical learning. The barriers that prevented women from mathematical education were not merely formal – exclusion from universities, prohibition from scholarly societies – but also cultural and psychological. Mathematical texts written in Latin, emphasising theoretical abstraction over practical application, demanded the kind of classical education available only to wealthy males.

Had my vernacular, practical approach become standard, mathematical knowledge would have seemed far less foreign to women educated in modern languages and practical subjects. The Instituzioni demonstrated that one could master the most sophisticated analytical methods without first spending years studying Latin rhetoric and scholastic philosophy. This would have emboldened countless women to attempt mathematical study who were deterred by the apparent necessity of classical preparation.

Furthermore, the widespread adoption of my pedagogical methods would have created institutional pressures for including women in mathematical education. Universities employing textbooks designed for accessibility rather than exclusivity would have found it increasingly difficult to justify barring capable female students. Once mathematical instruction became practical rather than purely theoretical, the artificial distinction between “masculine” abstract reasoning and “feminine” practical application would have collapsed.

I believe this would have produced a cascade of changes throughout European intellectual culture. Had women begun studying mathematics in substantial numbers during the 1750s rather than the 1850s, we might have witnessed female contributors to the mathematical advances of the later eighteenth century. Women might have participated in the development of probability theory, the refinement of analytical mechanics, the early investigations of mathematical physics that characterised the Enlightenment’s mathematical flowering.

Such early female participation would have normalised women’s mathematical capabilities in ways that could have prevented the nineteenth-century notion that mathematical talent was inherently masculine. Instead of viewing female mathematicians as remarkable exceptions requiring special explanation, European intellectual culture might have simply accepted women’s mathematical participation as natural and unremarkable.

Furthermore, the broader accessibility of my pedagogical approach would have expanded mathematical literacy among all social classes, not merely women. Merchants, craftsmen, engineers – practical people requiring mathematical skills for their occupations – would have found analytical methods far more approachable. This could have accelerated the practical applications of mathematics that ultimately drove the Industrial Revolution’s technological innovations.

Yet I must acknowledge potential limitations of such a transformation. My emphasis on practical methods over theoretical foundations, whilst pedagogically beneficial, might have hindered the development of pure mathematical research. The profound theoretical advances achieved by Euler, Lagrange, and their successors required precisely the kind of abstract thinking that my accessible approach deliberately de-emphasised.

Perhaps most intriguingly, widespread early adoption of my methods might have produced a fundamentally different mathematical culture – one more oriented toward practical application, collaborative problem-solving, and democratic participation, but potentially less capable of the daring theoretical leaps that characterised eighteenth-century mathematical genius. The accessibility that would have encouraged female participation might simultaneously have constrained the development of mathematics as a purely intellectual discipline.

Nevertheless, I believe the benefits of such democratisation would have far outweighed these limitations. A mathematical culture that included women from its formative stages would have been richer, more diverse, more attuned to the practical needs of society – and ultimately more capable of serving humanity’s genuine welfare, which was always the proper purpose of mathematical knowledge.

Kaylee Robinson, 41, STEM Advocate, Toronto, Canada:
Many innovators grapple with moments of doubt or exhaustion. Looking back, how did you replenish your sense of purpose after setbacks or resistance – especially when balancing scientific inquiry, faith, and the expectations placed on women of your time? Are there daily practices or philosophies you found sustaining?

Dear Signora Robinson, your question touches upon the deepest currents of my existence – those daily struggles that test not merely one’s intellectual capabilities, but the very foundations of one’s faith and purpose. The challenges I faced were manifold: the physical torment of those mysterious convulsions, the spiritual burden of constant public scrutiny, and the perpetual tension between my desire for contemplative retreat and the demands of family obligation.

When those dreadful seizures began in my adolescence – what the physicians termed “chorea” or St. Vitus’s dance – they recommended dancing and horseback riding as remedies. Yet I discovered that such vigorous activities often worsened my condition rather than alleviating it. The true source of my affliction, I came to understand, was not physical but spiritual: my soul was crying out against the unnatural demands placed upon it through those endless evening performances in our palazzo.

The practice that proved most sustaining during these dark periods was what I can only describe as contemplatio – the discipline of turning one’s mind steadfastly toward divine truth, regardless of external circumstances. Each morning, before the household stirred, I would spend time in what the French philosopher Nicolas Malebranche called “attention” – that state wherein, as he wrote, “attention is the natural prayer of the soul”. During these quiet hours, I would examine both mathematical problems and passages from Scripture, finding in each a different pathway toward the same ultimate Truth.

This was not the sentimental piety that many imagine, but rather a rigorous intellectual and spiritual discipline. I discovered that mathematical study itself could become a form of prayer – that working through analytical problems with complete attention was a way of participating in divine Reason. When I encountered a particularly challenging theorem or struggled to express a complex relationship in clear language for my Instituzioni, I would approach it as an act of devotion, trusting that clarity of thought was itself a form of service to God.

During the most severe episodes of my convulsions, when my body would rebel entirely against my control, I learned to anchor my consciousness in these practices of attention. Rather than fighting against the physical distress, I would turn my mind toward mathematical concepts that remained constant – the properties of curves, the relationships between variables, the elegant symmetries underlying analytical operations. These truths became my refuge when bodily experience proved unreliable.

The second sustaining practice was what I termed “holy reading” – not merely studying theological texts, but approaching all learning with reverence. Whether I was working through Euler’s methods or reading the Church Fathers, I maintained the same attitude of humble receptivity. This prevented my studies from becoming sources of pride or intellectual vanity, which I recognised as particular temptations for those blessed with scholarly capabilities.

Perhaps most crucially, I learned to view each setback not as evidence of personal failure, but as an opportunity for spiritual growth. When critics dismissed my mathematical work as too elementary, or when the convulsions prevented me from completing planned studies, I tried to receive these disappointments as invitations to greater humility and deeper trust in divine providence. This required constant vigilance against the natural human tendency toward resentment and self-pity.

The expectations placed upon women of my era created particular spiritual challenges. I was simultaneously expected to display my intellectual gifts for others’ entertainment whilst maintaining the modesty and submissiveness considered appropriate to my sex. This contradiction could have destroyed me had I not discovered ways to transform these external pressures into internal disciplines. Rather than performing my knowledge for applause, I learned to offer it as a form of service – whether to the students who would benefit from my textbook or to the scholars who sought clarification of mathematical concepts.

Daily, I maintained what the mystics call “the practice of the presence of God” – attempting to remain conscious of divine Reality underlying all natural phenomena. When I studied the properties of infinite series, I contemplated the infinity of divine attributes. When I investigated the behaviour of curves approaching their asymptotes, I meditated upon the human soul’s approach toward perfect union with its Creator. This transformed even the most abstract mathematical work into spiritual exercise.

The transition from mathematical study to charitable work was not, as many supposed, an abandonment of intellectual life, but its natural completion. The same analytical capabilities that allowed me to organise complex mathematical relationships proved invaluable in designing efficient charitable institutions. The same attention to detail required for mathematical exposition served me well in attending to the particular needs of each person under my care.

Through all these experiences, I learned that true resilience emerges not from individual strength – which inevitably fails under sufficient pressure – but from conscious participation in a Reality greater than oneself. Whether that participation takes the form of mathematical investigation, charitable service, or contemplative prayer, it provides an anchor that remains constant amid all temporal changes. This insight sustained me through decades of public scrutiny, personal illness, and eventually the peaceful obscurity of my later charitable work.

The harmony between intellectual excellence and spiritual devotion that guided my choices remains, I believe, available to any soul willing to approach both learning and service as forms of worship rather than means of self-advancement.

Reflection

As our conversation with Maria Gaetana Agnesi draws to a close, we’re reminded that she passed away on 9th January 1799, at the remarkable age of 80 – having lived through nearly the entire century she helped define. Her longevity itself becomes meaningful when we consider how she used those eight decades to bridge worlds that history has often painted as incompatible: intellectual brilliance and spiritual devotion, analytical precision and compassionate service.

Throughout our discussion, Agnesi challenged common historical narratives about her life. Far from being the reluctant mathematician suggested by some accounts, she revealed herself as deeply passionate about making mathematical knowledge accessible – a pedagogical revolutionary whose choice to write in Italian rather than Latin was itself a radical democratic act. Her transition from mathematics to charity work emerges not as feminine capitulation to social expectations, but as the natural evolution of an analytical mind committed to solving humanity’s most pressing problems through rigorous methodology.

What emerges most powerfully is how Agnesi’s story illuminates the complex mechanisms by which women’s contributions have been systematically obscured. Her textbook’s influence spread throughout Europe, yet its pedagogical innovations were often attributed to the male educators who adopted her methods. This pattern of erasure – where women’s intellectual contributions become absorbed into the broader masculine narrative of scientific progress – remains painfully familiar to contemporary researchers studying gender bias in STEM fields.

The afterlife of Agnesi’s work reveals both loss and rediscovery. Her famous curve, misnamed the “Witch of Agnesi” through mistranslation, now appears throughout modern mathematics and physics as the Cauchy distribution, finding applications in probability theory and quantum mechanics. Contemporary mathematicians like those publishing in recent journals continue to explore the curve’s properties, often unaware they’re building upon the geometric insights of an 18th-century Italian woman. This disconnect between mathematical knowledge and its historical origins reflects broader patterns in how scientific education often divorces concepts from their human creators.

Perhaps most remarkably, Agnesi’s synthesis of intellectual excellence with ethical responsibility feels startlingly contemporary. In an era when artificial intelligence and biotechnology raise profound questions about the moral obligations of scientific discovery, her insistence that knowledge must serve human flourishing rather than mere intellectual advancement offers a template for ethical STEM practice. Her conviction that mathematical education should be democratically accessible, not an elite preserve, anticipates current efforts to broaden participation in STEM fields.

The gaps in historical understanding remain significant – we know little about the private dynamics of her mathematical thinking, the specific pedagogical innovations that made her textbook so effective, or the detailed operations of her charitable institutions. Yet these silences themselves speak to broader patterns of how women’s intellectual contributions have been preserved or lost.

What emerges most clearly is a figure whose radical vision of integrated knowledge – where mathematical precision served spiritual purpose, where analytical capabilities addressed social needs – offers a model for contemporary scientists grappling with questions of purpose and responsibility. In Maria Gaetana Agnesi, we find not simply a pioneering woman mathematician, but a prophet of science as moral practice, whose voice across the centuries reminds us that the highest achievements of human intellect must ultimately serve the deepest needs of human flourishing.

Who have we missed?

This series is all about recovering the voices history left behind – and I’d love your help finding the next one. If there’s a woman in STEM you think deserves to be interviewed in this way – whether a forgotten inventor, unsung technician, or overlooked researcher – please share her story.

Email me at voxmeditantis@gmail.com or leave a comment below with your suggestion – even just a name is a great start. Let’s keep uncovering the women who shaped science and innovation, one conversation at a time.

Editorial Note: This interview represents a dramatised reconstruction based on extensive historical research and documented sources about Maria Gaetana Agnesi‘s life, work, and era. While grounded in factual evidence from her writings, contemporary accounts, and scholarly analysis, the conversational format and specific responses are imaginative interpretations designed to bring her voice and perspectives to life for modern readers. Historical figures’ exact thoughts and speech patterns cannot be definitively known, and some aspects of Agnesi’s inner life remain matters of scholarly debate. This creative approach aims to illuminate her remarkable contributions whilst acknowledging the inherent limitations of historical reconstruction.

Bob Lynn | © 2025 Vox Meditantis. All rights reserved. | 🌐 Translate