**Gianfrancesco Malfatti**name is often written as

**Gian Francesco Malfatti**. His parents were Giovanni Battista Malfatti and his wife Giuseppa. He began his education at the Jesuit College in Verona before continuing his studies at Bologna. He studied under Laura Bassi, Vincenzo Riccati, Francesco Maria Zanotti (1692-1777) and Gabriele Manfredi at the College of San Francesco Saverio in Bologna. Bassi, who taught physics, and Zanotti, who taught both philosophy and physics, were both strong supporters of Newtonian physics and gave lectures designed to introduce their students to that approach.

Leaving Bologna, in 1754 Malfatti went to Ferrara where he was appointed as curator of the extensive library of the Marquis Cristiano Bevilacqua which had officially opened in the previous year. This appointment suited Malfatti well since the Marquis had set up a well equipped physics laboratory in his house where he held frequent meetings for the local scholars. This enabled Malfatti to continue to undertake research both on physics and on mathematics and to teach these two topics. He was interested in the solution of algebraic equations and he wrote two articles on this topic which he sent to his former professor Vincenzo Riccati. These were published as *De natura radicum in aequationibus quarti gradus* (1758) and *Epistola altera ad ... Vincentium Riccatum* (1759). What Malfatti wanted to achieve was to solve algebraically equations of degree greater than four. It had been over 200 years since Cardan had published his great work *Ars Magna* which had given methods, devised by del Ferro, Tartaglia and Ferrari, to solve all cubic and quartic equations by radicals and Malfatti set about the task of extending the methods to equations of the fifth degree. Continuing to work on this problem he published a major memoir in 1770, namely *De aequationibus quadrato-cubicis disquisitio analytica.* In this paper, from a general equation of degree five, Malfatti produced a resolvent of degree six, now sometimes called "the resolvent of Malfatti". In general, of course, an equation of degree six seemed worse than the original equation of degree five but in fact this led to certain equations of degree five being able to be solved by radicals. This was a step forward and, nearly 100 years later, Francesco Brioschi was able to use Malfatti's resolvent to show that every equation of degree five can be solved using transcendental functions.

The University of Ferrara was erected by Alberto V who reigned over Ferrara from 1388 to 1393. The University was obtained by Pope Boniface IX as a concession in 1391 and this is usually taken as its founding date. Copernicus studied there in 1503 but the buildings which house the University today date from the end of the 16th-century. The University of Ferrara was re-established in its 16th-century buildings in 1771 and, the president of the University, Giovanni Maria Riminaldi (1718-1789), knowing that by this time Malfatti was gaining a reputation as a leading mathematician, wanted to appointed him to the chair of mathematics. This, however, presented a problem since Malfatti, although he had been educated by the Jesuits, was not himself a member of the Order and it was expected that professors would be Jesuits. Despite the difficulties, Malfatti's scholarship won the day and he was appointed to the chair.

The Jesuit Order was at this time at the centre of a major controversy due, in part, to their wish to accept pagan ideas into Christianity. The Order had been made illegal in France in 1764, then Spain and the Kingdom of the Two Sicilies acted against them in 1767 and finally pope Clement XIV dissolved the Jesuits in 1773. Although this did not affect Malfatti directly, it did have a major impact on Alessandro Zorzi (1747-1779) who was given employment by the Marquis Bevilacqua as a tutor for his grandchildren. In this way Malfatti got to know Zorzi and became interested in the project that Zorzi was planning to undertake. This was to produce an Italian encyclopaedia, *Nuova Enciclopedia italiana, *to be an Italian equivalent of Diderot and d'Alembert's *Encyclopédie*. Soon Malfatti agreed to write articles for the encyclopaedia and Zorzi enlisted the help of other scholars such as Gerolamo Tiraboschi (1731-1794), Larraro Spallanzani (1729-1799), Gregorio Fontana (1735-1803), Paolo Frisi, Antonio Mario Lorgna, Giuseppe Louis Lagrange (1736-1813), Giuseppe Toaldo (1719-1797), Giambattista Beccaria (1716-1781), Leopoldo Marco Antonio Caldani (1725-1813), Saverio Bettinelli (1718-1808) and several others. Tiraboschi was a historian of Italian literature and professor at the University of Milan before becoming a librarian to the duke of Modena. Spallanzani served as professor of logic, metaphysics, and Greek, then as professor of physics at the University of Modena before gaining worldwide recognition as a physiologist while holding a chair at the University of Pavia. Fontana was a mathematician who succeeded Boscovich in the chair of mathematics at the University of Pavia. Giuseppe Lagrange was a mathematician and astronomer who was director of mathematics at the Prussian Academy of Sciences in Berlin. Toaldo was a physicist who held the chair of astronomy at the University of Padua. Beccaria held the chair of experimental physics at Turin. Caldani was professor of anatomy in Padua. Bettinelli was a poet and literary critic who had been professor at Modena until the suppression of the Jesuits after which he retired to Mantua. This project had great promise but never came to anything since Zorzi died before it had progressed very far and nobody took over the reins.

Dirk Struik, describing the papers in [5], writes that Malfatti:-

Let us give more details on the founding of the... was a quiet, scholarly man who spent most of his life as a librarian and professor in Ferrara. He was one of the founders of the "Società Italiana delle Scienze"(1782)and was active in academic reform, especially in the Napoleonic period.

*Società Italiana delle Scienze*, the Society of the XL, which has become the National Academy of Sciences of Italy

*.*In addition to Malfatti, four men played major roles in founding the Society. These were Antonio Mario Lorgna, Carlo Barletti (1735-1800) the professor of physics at Pavia University, the mathematician Ruggero Boscovich, and Lazzaro Spallanzani who we mentioned above. These five were friends and they discussed setting up a Society over many years. On 28 June 1766 Lorgna wrote to Malfatti saying that it was important for Italy to have a scientific journal. Over the following ten years discussions went on and by 1776 Lorgna and Malfatti were discussing Lorgna's "idea of forming an Academy for all Italian scholars". Malfatti thought that "the idea of forming an Academy for all the Italian Letterati is noble and glorious" but was worried that they would not get deep enough contributions from mathematicians and other scientists. Barletti was also worried that they might not get the highest quality research for its journal

*Memorie della Società italiana*, but Lorgna, who was confident that the project would succeed, pushed their doubts aside. Another problem was over the decision to publish articles in Italian in the

*Memorie della Società italiana*. Malfatti strongly supported writing papers in Italian rather than Latin and from 1779 he wrote all his papers in Italian. Several thought that this would limit the readership of the journal to Italy but they pressed ahead with the proposal and the Society, together with its publication, played a small part in the unification of Italy. Malfatti took his obligations to the Society very seriously, and membership spurred him on to undertake intensive research. He submitted fourteen articles to the journal of the Society between 1782 and 1807.

One of the topics that Malfatti worked on was the logarithmic curve. Florian Cajori writes [9]:-

As we have seen above, Malfatti wrote an important work on equations of the fifth degree. In 1799 Paolo Ruffini published a paper on the theory of equations with his claim that quintics could not be solved by radicals as the title shows:In1795... Malfatti ... discussed at length the question whether the logarithmic curve has one or two branches. The equation ydx = dy has an infinite number of integrals even under the restriction that x =0when y =1. For any constant n, e{^{}nx}= y{2^{n}gives rise to the differential equation ydx = dy. Before integrating one must agree upon the number of values y shall have. If it is to have one value, we get e^{x}= y; if two values, we get e^{}x}= y2^{}or(e)(^{x}- ye)^{x}+ y=0. That e^{x}= y has only one branch is argued from geometric and analytic considerations.

*General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible.*Melvin Kiernan writes [19]:-

Malfatti replied to a 1802 proof by Ruffini by publishingAs representative of the "old school"[Malfatti]had long believed that the general quintic equation could be solved algebraically, and had once published a supposed proof of this result. His objections were not so much to individual mathematical points in Ruffini's paper as they were an attack on the general concept and methodology. It is easy today to underestimate the enormity of the difficulty faced by these mathematicians. The mathematics of their day was largely computational, and they were unprepared to accept, and unable to understand, the new approach through theoretical and existential arguments. Thus the most frequent objection to the works of the new mathematicians was "vagueness." ... Ruffini's exchanges with Malfatti ... gave rise to six published versions of Ruffini's proof, the last appearing in1813.

*Dubbi proposti al socio Paolo Ruffini sulla sua dimostrazione dell'impossibilità di risolvere le equazioni superiori al quarto grado*(1804). However, Ruffini was not the only mathematician with whom Malfatti had a dispute. Another was Pietro Paoli, also a founding member of the

*Società Italiana delle Scienze,*and they carried out their dispute in the Society's journal. The dispute concerned the problem of determining the pressures on the corners of a solid figure resulting from its weight.

In 1802 Malfatti considered the problem of describing in a triangle three circles that are mutually tangent, each of which touches two sides of the triangle, the so-called Malfatti problem. His solution, involving heavy calculations and complicated formulas, was published in a paper of 1803 on *Un problema stereotomica*. However, the situation was more complicated as Howard Eves explains in [11]:-

Michael Goldberg, in 1967, showed in [17] that the Malfatti circles are never the solution to the original Malfatti problem. Jacob Bernoulli had solved the Malfatti problem for an isosceles triangle while, after Malfatti, the problem was also solved by an elegant geometric solution by Jacob Steiner in 1826 and Clebsch solved it using elliptic functions. Let us give a generalisation of the Malfatti problem. A configuration of circles in a triangle is called a 'greedy configuration' if the circles are chosen one after the other in such a way that each circle is the largest that does not intersect any previous circle [24]:-In1803Malfatti proposed the problem of cutting three circular cylindrical holes out of a solid right triangular prism in such a way that the cylinders and the prism have the same altitude and that the combined volumes of the cylinders be a maximum. Malfatti intuitively reduced this problem to another, now commonly referred to as the "Malfatti problem": To inscribe within a triangle three circles each of which be tangent to the other two and to two sides of the triangle. This reduced problem has enjoyed a long history and has been solved and generalised in many ways. Malfatti considered this reduced problem as equivalent to the original problem. That such is not the case was indicated in1929, when Lob and Richmond pointed out that in the case of the equilateral triangle the inscribed circle and two of the smaller circles that can be fitted into the corners have a combined area which is greater than that given by the Malfatti arrangement ...

Malfatti's interests extended beyond the results we have mentioned above for his papers dealt with many subjects from probability to mechanics. In fact he published twenty-two memories on mathematical analysis, geometry, the theory of algebraic equations, mathematical physics, the theory of difference equations, combinatorics and probability. There are several papers in [5] which describe Malfatti's work. These include:H Mellisen has conjectured that for any arrangement of n non-overlapping circles in a triangle, the greedy arrangement has the largest total area. In1994, V A Zalgaller and G A Los verified this conjecture for n =3, thus settling Malfatti's original problem.

*Problems and methods of mathematical analysis in the work of Gianfrancesco Malfatti*,

*Contributions of Gianfrancesco Malfatti to combinatorial analysis and to the theory of finite difference equations*,

*The work of Malfatti in the realm of mechanics*,

*The geometrical research of Gianfrancesco Malfatti*,

*Gianfrancesco Malfatti and the theory of algebraic equations*, and

*Gianfrancesco Malfatti and the support problem*.

We should note one final interesting result which he obtained. In 1781 he showed in the paper *Della curva cassiniana* that the lemniscate had the property that [1]:-

Let us return to make final comments on Malfatti's career. He lived through the military and political events that shook Italy in the late eighteenth century and the beginning of the next. Like almost all scientists of his time, he participated in reforming ideas with enthusiasm and actively collaborated with the republican government. In December 1796 Napoleon Bonaparte created the Cispadane Republic by merging of the duchies of Reggio and Modena and the legate states of Bologna and Ferrara. Malfatti participated in the Cispadane Republic as a member of its Education Committee in 1796 and, in the following year, he served on the committee drafting proposals to reform schools in Ferrara. On 23 May 1799 combined Austrian and Russian forces entered Ferrara and Malfatti was removed from his chair of mathematics. The Austrians dismantled the educational reforms which had been introduced, returning to the ways before the Republic was formed. However, in 1801 French forces drove out the Austrians and retook the city. Malfatti was reinstated to his chair of mathematics but he retired shortly after this, having gained thirty years of service at the University. Since the early 1790s he had suffered from respiratory ailments and eye strain. Reduced to almost complete blindness, he regained his sight after cataract surgery and was able to continue his research up until shortly before his death. After his death in Ferrara he was buried in the chapel of St Maurelio in Ferrara cathedral.... a mass point moving on it under gravity goes along any arc of the curve in the same time as it traverses the subtending arc.

**Article by:** *J J O'Connor* and *E F Robertson*

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