The Garden of Archimedes  A museum for mathematics

Calculus in Italy

The arrival of calculus in Italy

Leibniz went to Italy in March 1689, charged with the study of the genealogy of the House of Este in relation to that of Braunschweig-Lüneburg. He stayed there until the following March, spending six months in Rome and visiting Venice, Ferrara, Bologna, Naples and Florence. During his trip, he also established contact with mathematicians and scholars to diffuse his new calculus in Italy. In 1692, the Giornale de' letterati di Modena published a solution by Leibniz to the problem of finding the equilibrium configuration of a heavy rope. The solution is prefaced by an introduction, in which the new calculus is described.
In the Italian scientific milieu, which was linked to the geometric tradition of the Ceva brothers and of Vicenzo Viviani, leibnizian calculus only started to make headway at the beginning of the eighteenth century. French journals like the Acta Eruditorum are not always easy to find. Moreover, there is a lack of study and research clubs comparable to, for example, that of Paris, where Johan Bernoulli had had the chance to meet L'Hopital.
The situation changed in the first decade of the new century. In 1707, Jacob Hermann reached Padua to hold the chair of Mathematics. Hermann had studied in Basel, where Jacob and, later, Johan Bernoulli had taught. He remained in Padua until 1713, creating a dense network of contacts and becoming the reference point for Italian mathematicians who wanted to confront the new analytical methods. His successor was Nicolaus I Bernoulli (1687-1759), while other members of the Bernoulli family, Nicolaus II (1695-1726) and Daniel (1700-1782), spent a long time in Venice. In the meanwhile, the Manfredi brothers in Bologna brought into being a nucleus of scholars.
1710 saw the birth of the Giornale de' letterati di Italia (Journal of Italian Scholars), which was to publish some of the works and of the scientific debates that developed in later years.

|    Guido Grandi    |    Iacopo Riccati    |    Giulio Carlo de' Toschi Fagnano    |    Maria Gaetana Agnesi    |   

Contributions by Guido Grandi.

Among the earliest works published by Italian mathematicians, in which the new differential calculus appears, we find those by Guido Grandi.
In 1703, he published the Quadratura circuli et hyperbolae. Here one cannot find any particularly original contributions, but an autonomous reelaboration and resystematisation of results already demonstrated by others. The first part of the work deals with the quadrature of the circle, which is obtained through the formula $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...$; the second part is, instead, dedicated to the hyperbola, and the quadrature of a region is here reconducted to the series $\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 4}+\frac{1}{3 \cdot 8}+\frac{1}{4 \cdot 16}+\frac{1}{5 \cdot 32}+...$ .
About differential methods, in his preface Grandi writes:

But I have here and there also inserted the dx, dy typical of differential calculus, and their way of being differentiated and added. Thus had I been able to introduce them in my previous pamphlets too! But then, the secrets of that method had been inaccessible to me, while now, their usefulness and fecundity having been proven, why not insert them among the other methods I am familiar with? Also, the significance of the symbols is very clear, because it only signifies an infinitely small difference between the same x and y, and you will easily find the very rules of calculus if you observe and peruse this tract carefully - unless you want to recourse to the illustrious L'Hospital who explains them in a more complete way in his tract of the infinitely small.

Actually, the use of differential methods remains quite sporadic. An occasion to expose some of its rudiments is given by a letter received from Gabriele Manfredi on the quadrature of the hyperbola, which is inserted in the second part of the book. To clarify the procedure exposed by Manfredi, Grandi in fact adds a series of notes in which he explains the steps and describes the rules of the new calculus.

* Page V.1 in the exhibition

Guido Grandi
Quadratura circuli et hyperbolae

(*) In fact the "difference" of any power of the unknown x is the same power multiplied by its exponent and "differentiated" of one of its dimensions, leaving unvaried the constants by which it is multiplied, constants, actually, for which the "difference" is nil, as it is demonstrated in the Tractatu de infinitis infinitorum et infinite parvorum in the scholium to prop. V, and therefore differentiating this series [ $abxx/2+abx^3/3+abx^4 /4+ abx^5 /5+abx^6 /6$ etc. ] one obtains the preceding one $abxdx+abxxdx+abx^3dx$ etc.

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Contributions to differential equations: Iacopo Riccati.

With the development of infinitesimal calculus, many remarkable successes were reached in problems linked to the solutions of differential equations. A general resolution method, through the developments into power-series, is used by Newton in his Treatise of the method of fluxions and infinite series for the equations that we may indicate as $y'=f(x,y)$. The solution obtained is not, however, found in an explicit form, but through a series whose coefficients can be calculated in sequence.
On the continent instead, lacking a general method, scholars studied specific classes of equations with the goal of reducing their solutions to quadratures, that is, to the calculation of some integrals.
Firstly, there came the solution to the separable variable equations, and there a series of studies whose goal was to reconduct to these, through the appropriate transformations, more and more classes of equations. A typical example is that of homogeneous equations, that is those in which the second member $f(x,y)$ is a homogenous function of zero degree, that is a function only of the ratio $y/x$. In this case, positing $y=wx$, one obtains for $w$ a separable variable equation.
In the study of these equations that allow to solve wider and wider classes of differential equations, the main players are Bernoulli, Euler, the Italian Gabriele Manfredi and Iacopo Riccati, and, later, Lagrance and Clairaut.
In particular, the general procedure for homogeneous equations is described by Manfredi in his De constructione aequationum differentialium primi gradus (1707), which was much admired by Leibniz and by other European mathematicians.
Starting from Manfredi's result, Riccati studied several generalisations and determined several cases of equations that can be reduced to the previous form with appropriate substitutions. Considering, for example, the equation which, in terms of equality among differentials is written as $x^mdx+x^ny^rdy=y^sdy$, he finds some values of the exponents that make the equation solvable. Other contributions to this were given later by D. Bernoulli and Euler.

* Page V.2 of the exhibition

Iacopo Riccati
Della separazione delle indeterminate nelle equazioni differenziali e d'altri gradi ulteriori

Case I
Let us propose in general the three-member equation $x^mdx+x^ny^rdy=y^sdy$.
If $m=n+r=s$ then we are in the case solved by Mr. Manfredi, but supposing there is not among the sums of the exponents the necessary equality, let us at least seek in which cases, with some work, one can transform the proposed formula in an equivalent one in which the prescribed condition is true. So if we cannot in general separate the variables, we will determine infinite cases, in which separation can be successfully applied.

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Rectifications: the lemniscata and Fagnano's contributions.

In relation to the problem of determining the shape assumed by a thin rod under the action of specific forces, Jakob Bernoulli introduced the curve called lemniscata, whose equation in Cartesian coordinates is $(x^2+y^2)^2=a^2 (x^2-y^2)$. In the rod problem, Bernoulli finds an irrational integral of which he is unable to find the explicit expression in terms of elementary transcendent functions. This integral has the same form of

$$\int \frac{a\,dz}{\sqrt{a^4-z^4}}$$

which expresses the length of the lemniscata arc, and that, similarly, Bernoulli claims is not expressible in terms of elementary functions. Irrational integrals of a similar kind appear also in attempts to rectify the ellipse, the length of whose arc is very important in astronomy, and are therefore called elliptic.
The first studies of elliptic integrals are not so much attempts to calculate them, but rather to reduce the more complex ones to those that are involved in the rectification of the ellipse or of the hyperbola, which also belongs to this class; or to finding sums or differences of arcs that are always explicitly expressible.
In this context, Fagnano demonstrates for example that the difference of any two elliptic arcs is algebraically expressible.
Starting in 1714, Fagnano also tackles the rectification of the lemniscata through ellipse and hyperbola arcs, and finds several algebraic relations that his arcs satisfy. He also establishes the method to determine the points which divide in a set number of parts the arc or the area of a quadrant of the curve. His research later attracted the attention of Euler, who took them up and enlarged them in various directions.

* Page V.3 of the exhibition

Giulio Carlo de' Toschi Fagnano
Produzioni matematiche

Suppositions known to those who understand infinitesimal calculus.
Let us take the lemniscata $CQACFC$(figure 24), whose semiaxis $CA=a$; we know that taking in the centre C the origin of the abscissa $x$ and calling $y$ the ordinates that are normal to the axis, the nature of the lemniscata is expressed by this equation $x^2+y^2=a\sqrt{x^2-y^2}$. We also know that if we call $z$the indetermined cord $CQ=\sqrt{x^2+y^2}$ one obtains the direct arc $\int \frac{a\,dz}{\sqrt{a^4-z^4}}$ and the inverse arc $QA=\mbox{arc.}CA - \mbox{arc.}CQ =\int \frac{\sqrt{a^4+z^4}\,dz}{\sqrt{a^4-z^4}}$ . [...]

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The Istituzioni analitiche by Maria Gaetana Agnesi.

In the introduction to her Istituzioni analitiche, Agnesi writes:

There is nobody who, being informed about Mathematical things, ignores how necessary is, especially today, to study analysis, and how much progress has been made in this, and is still made, and can be hoped for in the time to be; but I do not want nor can dwell here in praising this science, which does not need it, all the less from me. But as much as it is clear how necessary it is, so that the Young eagerly desire to acquire it, so much are great the difficulties that are met in it, being it known and doubtless that not every city, at least in our Italy, has people that can or want to teach it; and not everybody can leave our Country to find its masters.

The two volumes are thus written in order to collect and order with clarity and simplicity, leaving out all that is unnecessary, and without omitting anything that can be useful or necessary. The first is called On the analysis of finite quantities, and it contains elements of algebra and analytical geometry, with the study of several curves among which the witch, to which the name of Agnesi is still linked. The second covers calculus and it is divided into three books: On differential calculus, On integral calculus, On the inverse method of tangents.

* Page V.4 of the exhibition

Maria Gaetana Agnesi Istituzioni analitiche ad uso della gioventù italiana Taking semicircle ADC with diameter AC, we seek outside of it point M, such that drawing MB, normal to the diameter AC, which will cut through the circle in D, there will be AB:BD=AC:AM, and since there are infinite points M that satisfy the problem, I ask for its locus. Agnesi determines the equation $xy^2=a^2 (a-x)$ satisfied by the points of the curve, and proceeds with its study.

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