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Person: Malfatti, Gian Francesco

Gianfrancesco Malfatti was an Italian mathematician who worked on geometry, probability and mechanics and made contributions to the problem of solving polynomial equations.

Mathematical Profile (Excerpt):

• 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.
• 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 Ⓣ(The great art) 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.
• Ⓣ(Analytical analysis of square-cubic equations) In this paper, from a general equation of degree five, Malfatti produced a resolvent of degree six, now sometimes called "the resolvent of Malfatti".
• 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 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.
• In this way Malfatti got to know Zorzi and became interested in the project that Zorzi was planning to undertake.
• 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.
• In addition to Malfatti, four men played major roles in founding the Society.
• 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.
• Malfatti strongly supported writing papers in Italian rather than Latin and from 1779 he wrote all his papers in Italian.
• Malfatti took his obligations to the Society very seriously, and membership spurred him on to undertake intensive research.
• One of the topics that Malfatti worked on was the logarithmic curve.
• As we have seen above, Malfatti wrote an important work on equations of the fifth degree.
• Ruffini's exchanges with Malfatti ...
• Malfatti replied to a 1802 proof by Ruffini by publishing Dubbi proposti al socio Paolo Ruffini sulla sua dimostrazione dell'impossibilità di risolvere le equazioni superiori al quarto grado Ⓣ(Doubts proposed to Paolo Ruffini on his demonstration of the impossibility of solving equations above the fourth degree) (1804).
• However, Ruffini was not the only mathematician with whom Malfatti had a dispute.
• 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.
• 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.
• Malfatti considered this reduced problem as equivalent to the original problem.
• That such is not the case was indicated in 1929, 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 ...
• 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.
• Malfatti's interests extended beyond the results we have mentioned above for his papers dealt with many subjects from probability to mechanics.
• These include: 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.
• Let us return to make final comments on Malfatti's career.
• 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.
• Malfatti was reinstated to his chair of mathematics but he retired shortly after this, having gained thirty years of service at the University.

Born 26 September 1731, Ala, Trento (now Italy). Died 9 October 1807, Ferrara (now Italy).

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References

Adapted from other CC BY-SA 4.0 Sources:

1. O’Connor, John J; Robertson, Edmund F: MacTutor History of Mathematics Archive