A protagonist of the 18th-century mathematics: Maria Gaetana Agnesi

Abstract

We revisit the life and work of Maria Gaetana Agnesi, Milanese mathematician, whose 300th birthday is being celebrated this year. In particular, we describe the main features of her books Propositiones philosophicae and Instituzioni analitiche ad uso della gioventù italiana. A third appendix is devoted to an algebraic curve that has taken its name from Agnesi: the versiera, or witch of Agnesi.

Appendices

Appendix 1: Propositiones Philosophicae

The research themes of my philosophy which have until now come to light in single separate parts, now come before you gathered together and expounded in order, Most Excellent Lord [1, p. 39].

Thus begins the dedication with which Maria Gaetana Agnesi opened her 1738 Propositiones Philosophicae.

The “Most Excellent Lord” was Count Carlo Belloni, the generous and selfless scientific adviser who had been supporting her for some time. The dedication continues with the acknowledgement of the scientific and human merits of her family friend and confidant, and ends by formulating the best auspices, exhorting the count, together with his promising offspring, to continue and further fuel the literary and scientific culture to which he had already given so much.

Agnesi’s book was the fulfillment of her education. It consists of 191 theses, divided into 29 chapters, in some cases preceded by laws or physical principles: short treatises—some just few lines, others longer—about the most relevant topics of the scientific and philosophical culture of the time. In her family, her father had long opened his Milanese salon to local scholars and renowned foreign visitors who admired Maria Gaetana’s erudition and were cheered by her younger sister Maria Teresa’s music: here took place the “conversations” that first formed the core of the book and were then revised.⁹

Young Agnesi wrote the Propositiones Philosophicae entirely in Latin, then still the language of scholars, of which, despite her young age, she had been a master for years. Following the teaching of her mentors, whose critical method tended to discourage a priori assumptions, her work was inspired by Edmond Pourchot’s Institutiones Philosophicae, a university text of the early eighteenth century inspired in its turn by modern philosophers, especially Descartes. While at times disagreeing with it, especially about physics, Agnesi retained its principles and its emphasis on clarity.¹⁰

The dedication to Count Belloni is followed by a preliminary discussion about the history of philosophy (six short “Prolegomena”) from which emerges the way Agnesi positions her work. On the one hand, in support of those who read and learn—and perhaps even of those who write, like herself—she expresses the conviction that learning should be extended to women (Proposition III). On the other hand, as regards the nature of human knowledge, she opines that mathematics is the most certain way to reach and contemplate truth (Proposition VI): otherwise we remain in the field of opinions, which does not permit any definitive interpretation, but only “plausibility” or “likelihood” (Proposition XIV).

This form of knowledge and contemplation of truth, in line with an apologetic perspective, reflects young Agnesi’s mystical vein: a belief in the fundamental harmony between faith and reason that would lead her ever closer to the spiritual life and works of mercy to which she later dedicated all her time and belongings.

As for the content of the Propositiones, it is worthwile to quote the thorough but concise description given by a careful biographer. After the Prolegomena, we find:

the theses about logic, ontology, pneumatology. They are followed by the theses about physics; they deal with the bodies and the motions in general, and then the laws of mechanics; gravity and the motion of bodies, and then bodies falling freely or on inclined planes, pendulums, projectiles; they concern static, hydrostatic, hydrodynamics, hydraulics, the equilibrium of solids immersed in fluids, the collision of elastic and inelastic bodies, and sensory principles and qualities of bodies, under which title several topics in physics and also in chemistry are included. Finally there are theses about so-called particular physics; they cover the universe in general and the world system, celestial bodies and the theory of central motions, the atmosphere and the weather motions, the earth and the mountains, the seas and the springs, the minerals and the plants, the animals and the animal part of man [9].

As we can see, it is an encyclopaedic system founded primarily on topics that require a vast erudite background: no philosophical argument nor scientific debate is excluded, from the faculties of the human mind (Propositions XX–XXX) to the notion of universal gravitation (Propositions LX–LXVII), to the applications of the laws of motion to ballistics and geostatics (Propositions LXVIII and LXIX), from the origins of fresh water springs (Propositions CXLI-CXLV) to animal reproduction (Propositions CLXV–CLXVI). And, alongside the most recent theories, she does not fail to infuse a genuine religious spirit, according to the dogma of the time. For instance, in Proposition LXV she writes:

gravity must certainly not be traced to another cause than continuous and perennial divine action; nor is there any reason why we should hesitate to take refuge in God, when the issue of the universal laws of Nature is raised.

Again, consider her belief that “the true age of the world must be derived from the Holy Scriptures” (Proposition CVII).

At the same time, she took a stand on the issues most debated by the scientific thinking of the time: she took sides, for instance, against Descartes’s theory of vortices (Proposition CXVI), only to go back to it, in fact, when assessing the shape of the Earth after astronomer Cassini’s model (Proposition CXXXV, but she later corrected herself).¹¹

The Propositiones Philosophicae represent the scientific philosophy prevalent in Europe in the eighteenth century: an appreciation of ideas derived from Descartes, reviewed and assessed in the light of Newton’s mechanistic theories. They do not contain original ideas, which would be impossible for someone who merely set out “conversations” that took place in her home. The discussions are expounded in academic terms around the theses that were proposed by casual visitors, and the author does not seem to be particularly interested in any particular school of thought, with the possible exception of Newton’s natural philosophy, according to which the probable causes of the celestial motions are explained “with a very beautiful and very simple theory” (Proposition CXVII).

The usual philosophical themes, of a metaphysical character, provided the opportunity to introduce the physical and mathematical aspects. She was apparently already preparing for the epistemological and cultural leap of the Instituzioni analitiche, which she performed over the next ten years.

Appendix 2: Instituzioni analitiche

The Instituzioni analitiche ad uso della gioventù italiana were published in 1748. The work is formally dedicated to Empress Maria Theresa of Austria, with a boldness that came from both of them being women:

It seems to me that in this age, which in all ages to come will, luminous and great one, bear YOUR name, all Women should serve the glory of their sex, and each, as far as she can, should contribute to the Splendour in which YOU envelop it.

The sovereign, for her part, thanked her with valuable gifts. However, as we can see from the book’s title, the aim was mainly educational. The introduction “To the reader” abandons the lofty style of the dedication and takes on the tone of someone who has something almost urgent to explain: it is the need to bring together in a single text all the variety of results that are dispersed in journals, since

As much as the need for it [mathematical analysis] is evident, so that young people ardently desire to acquire it, yet equally great are the difficulties that one encounters, since it is a well-known and undoubted fact that not every city, at least in our Italy, has people who can, or will, teach it, and not everyone is able to leave their homeland to seek teachers.

With this, Agnesi also explains why she did not want to translate her text in the erudite Latin language, as was the custom for scientific works. She began it in Italian—she says—for her own enjoyment and to instruct some of her numerous brothers “since I had in mind, more than anything else, the clarity necessary and possible”.

However, apart from the clarity that pervades the whole work, none of her readers and critics took her at her word: the mention of the brothers refers to all Italians.¹²

The printing took place in her own palace, where the printer Giuseppe Richini transferred both the workers and the presses, and the author was thus able to attend daily to the progress of her work as it was being typeset: the craftsmen would later be publicly acknowledged for the skill shown in this difficult task, which required several years.¹³ At the end of each volume appear the graphs and images mentioned in the text (311 diagrams for a total of 59 pages), made by the famous artist Marc’Antonio Dal Re who engraved them on copperplate.

The first volume is entirely occupied by Book I, entitled Dell’Analisi delle Quantità finite (About the analysis of finite quantities) or Algebra cartesiana (Cartesian algebra), which is divided into six chapters. After an initial, short account of analytic geometry, there are rules and examples for symbolic computation, use of signs, reduction, powers, radicals, etc. Chapter I concretely teaches how to operate on rational and irrational expressions, exponentials, and so on, with progressive difficulty and suitable examples. The following chapters of the first volume cover algebraic equations, graphical constructions of solutions, geometrical problems and graphs of algebraic curves of various orders. Next to each formula the graphical—or “synthetic”, as Agnesi calls it—construction that explains it is always given. For instance, the text explains that the two equal values, with opposite signs, that are obtained by extracting a square root refer to constructions that take place in two different half-planes, separated by a given line.

As regards the curves, Agnesi realises that their study cannot be complete:

It is possible in two different ways to construct loci, that is, to describe curves expressed by equations that exceed the second degree, if however either one or the other can be said to describe rather than just adumbrate, and give some idea of these curves [2, vol. I, p. 351].

First, she determines the equation on the basis of the given problem, looking for some special points or characters, such as the asymptotes, the intersections with the axes or with other lines, including the imaginary intersections, and so on. She then studies the curve using several geometric constructions involving lines and conics, or known curves of higher degree.

The methods are demonstrated on suitable examples, one of which, making its appearance in chapter V (vol. I, p. 380), is the famous versiera (see Appendix 3), the rational cubic curve to which much of Maria Gaetana Agnesi’s fame is due.

In each case, the text lacks any reference to physical reasons that give meaning and justify the origin of the studied curves: this can be explained with the awareness that this would require expounding other principles and methods too, while the actual purpose is to teach analysis and its applications to geometry.¹⁴

Books II, III and IV are contained in the second volume, and consist in a total of 13 chapters. Book II (Del Calcolo Differenziale, On differential calculus) is aimed at finding maxima and minima, tangents and osculating circles at the points of an arbitrary curve, topics already covered in geometric terms in the first volume, but only for algebraic curves.

What was Agnesi’s position on the notion of a differential?

That these differential quantities are not vain imaginings, as well as being evident from the ancients’ method of inscribed and circumscribed polygons, can be clearly seen simply by envisaging that the ordinate MN ... will continuously approach BC, until they coincide; now it is clear that before these two lines coincide, they will have a distance between them and a difference that is indeterminable, that is, less than any given quantity; let them be in this position BC and FE; thus, BF and CD will be quantities smaller than any given one, and therefore indeterminable ones, that is, differences, or fluxions [2, vol. II, p. 434–5] (Fig. 3)

Book III (Del Calcolo Integrale, About integral calculus) is about rectifying curves, quadrating surfaces and cubing volumes, also by means of integration of series and complicated substitutions. Finally, Book IV (Del Metodo Inverso delle Tangenti, On the inverse method of tangents) deals with ordinary differential equations, essentially up to the second order, studied with various substitutions and by separation of variables. The choice of Leibniz’s notation over that of her beloved Newton suggests the influence of her Italian masters.

Fig. 3

Graph of the differential of a curve, drawn by Maria Gaetana Agnesi, from Instituzioni analitiche ad uso della gioventù italiana, vol. II

At the end, for higher-order equations, she mentions “artifices” and methods correctly attributed to Vincenzo Riccati and his teacher Gabriele Manfredi. However, at this point, Agnesi says,

finding myself already at the end of the printing of this work of mine, I am no longer in time to avail myself of other very learned Dissertations ... Thus it will suffice to have pointed them out to the reader, so that he may profit from them [2, vol. II, p. 1018].

She concludes by exhorting her readers to make use of the special cases and methods “used by famous mathematicians” in journals, to “avoid” higher order derivatives, “in order to acquire that cleverness and dexterity, which is necessary”. The end.

In short, she seems to say: the variety of mathematics is richer than is shown here and we must continue to study; we need a patient observation of particular cases from which to draw information on solving specific problems.

The style is neat, the difficulty is progressive. At each step we see the care taken to avoid ambiguities and intricate procedures, to save the reader trouble and to lead him without doubt or uncertainty. Of course, there remain some mistakes that were usual at that time, for instance involving the fact of believing, and calling, “true” the positive roots of an equation and “false” the negative ones. Thus, in the use of signs, literal coefficients are to be understood as always positive, making it necessary to consider as many as four different types of equations of second degree in one unknown. This fact implies the need to consider negative coordinates separately: for example, the equation y = mx represents a radius which extends in the first quadrant and, to complete the line, one must move to the third quadrant and take the equation y = − mx, also taking into account that x is negative. This, however, is a small thing. On the other hand, there are references to imaginary solutions, in which an even number must appear if the coefficients of the equation are real, and pairwise “of opposite sign”, meaning that their sum is no longer imaginary. Morevoer, she recognises that a real equation of odd degree always has a real solution: not much more could be said at that time.

The work received praises from all over Europe and the Instituzioni were greatly successful, as evidenced by the number of reviews that appeared within a short time in scientific journals and other periodicals. At the Academy of Paris, the book was considered the best work of its kind ever published; an anonymous reviewer, comparing it to what he called the sad and dismal Treatise on Fluxions by MacLaurin (1742), went on to consider it the most comprehensive text ever written about differential calculus. The accuracy of language and clarity in the exposition were especially praised.

The second volume was translated into French as early as 1775 under the title Traités élémentaires de calcul différentiel et de calcul integral, and later, in 1801, the work was published in English under the title Analytical Institutions [4] in the translation by John Colson, Lucasian professor at Cambridge and hence the successor to Newton’s chair.

The publication of the Instituzioni was responsible for the expansion and development of studies in analysis in Italy. In conclusion, what place is to be assigned to Agnesi among other mathematicians? Some, to remain within her time, place her alongside Riccati, Grandi or Fagnano, among the greatest Italian mathematicians of the eighteenth century. Others deem it impossible to find a comparison between her analytical intelligence, which feeds on known facts, sorts them and explains them, and the synthetic intelligence of those who do research and discover new results before communicating them. In any case, there is something that is hers alone: her method to simplify the study of geometry and algebra, to communicate, not only in Italy, the most recent results in differential calculus and to successfully integrate them into the mathematical fabric of her time.

Appendix 3: Which witch?

The problem III at p. 380 of the first volume of the Instituzioni analitiche reads as follows:

Given a semicircle ADC of diameter AC, find a point M external to it, such that, having drawn [a line segment] MB normal to the diameter AC that cuts the circle in D, we have [AB : BD = AC : BM],¹⁵ and since there are infinitely many points M that satisfy the problem, find their locus.

The locus is a curve, known as “versiera”. With respect to suitable coordinates, Agnesi determines its equation:

$${y=\frac{{a\sqrt {a - x} }}{{\sqrt x }}},$$

(here, the parameter a is the radius of the circle used for construction).

This curve had already been considered by Fermat (1601–1665), who studied its squaring around 1630, and by Guido Grandi (1671–1742) (and according to some, by Leibniz and Huygens too). In particular, Grandi, in Quadratura circuli et hyperbolae (1703), gave the curve the name of scala (scale), as it is useful to measure the gradation in intensity of light sources, and described it by means of the trigonometric function “versine” (or “versed sine”).¹⁶ Later, the curve assumed for the first time the name “versiera” in Grandi’s Note al trattato del Galileo del moto naturalmente accelerato (1718), referring to the rope that makes it possible to rotate the sails during a veer (from the Latin word vertere) and alluding to the name curva con senoverso (curve with versine). By mistake, the English translator of the Instituzioni, the Lucasian Professor John Colson (1680–1760), understood the term “versiera” as an abbreviation for “avversiera” (female foe), a euphemism with which the wife of the devil was often referred to, and gave the curve the name “witch of Agnesi”, by which it is currently known in the English-speaking world.

Taking a circle of radius a as in Fig. 4, take A as the origin of the coordinates, so that the generic point M of the locus we are constructing has coordinates x = AK and y = AB. From the condition of the problem AB:BD = AC:BM, we obtain xy = AC · BD, and from the similarity of the right triangles ABD and DBC we obtain ${BD=\sqrt {y\left( {{\text{2}}a - y} \right)} }$. From these relationships, squaring and simplifying, we obtain the curve in its best-known form:

$${y=\frac{{{\text{8}}{a^{\text{3}}}}}{{{x^{\text{2}}}+{\text{4}}{a^{\text{2}}}}}}.$$

Fig. 4

(courtesy of the authors)

Graphical representation of the construction of the “versiera”

It is clear that the versiera can also be obtained by considering the intersections D and H of the straight lines of the sheaf through A, respectively, with the circumference and the tangent line at C, the point diametrically opposite to A: it is the locus of the points that have the abscissa of H and the ordinate of D. From here we can easily derive the parametric equations:

$${\left\{ {\begin{gathered} x=\frac{{{\text{2}}a}}{m} \\ y=\frac{{{\text{2}}a{m^2}}}{{{\text{1}}+{m^2}}} \\ \end{gathered} } \right.},$$

where the parameter m denotes the angular coefficient of the line AH. It is therefore a rational cubic curve. It is also seen that AB, the ordinate of M, is the versine of the angle α = BÔD, and then the curve also assumes the form y = a · (1 − cos α), as a function of α.

As regards its properties, we have that the x-axis is an asymptote with an inflection point, and it is easy to compute the area between the curve and its asymptote:

$${S=2\int\limits_{0}^{{+\infty }} {\frac{{8{a^3}}}{{4{a^2}+{x^2}}}dx} =4a\int\limits_{0}^{{+\infty }} {\frac{1}{{1+\frac{{{x^2}}}{{4{a^2}}}}}} dx=8{a^2}\int\limits_{0}^{{+\infty }} {\frac{1}{{1+{t^2}}}} dt=8{a^2}\mathop {\lim }\limits_{{t \to +\infty }} \arctan t=4\pi {a^2}.}$$

Thus, the area of this region is equal to four times the area of the base circle.

The special case with a = 1/2 provides, up to a constant 1/π, the density of probability known as Cauchy distribution:

$${p\left( x \right)=\frac{{\text{1}}}{{\pi \left( {{\text{1}}+{x^2}} \right)}}}.$$

Translated from the Italian by Daniele A. Gewurz.