# Francesco Gerbaldi

### Quick Info

Born
29 July 1858
La Spezia (now Italy)
Died
29 June 1934
Pavia, Italy

Summary
Francesco Gerbaldi was an Italian geometer best known for his construction of the so-called six mutually apolar conics.

### Biography

Francesco Gerbaldi attended the University of Turin, graduating in 1879. After the award of his degree, he became an assistant to Enrico D'Ovidio who occupied the Chair of Algebra and Analytic Geometry at the University of Turin. In 1881 Gerbaldi published La superficie di Steiner studiata sulla sua rappresentazione analitica mediante le forme ternarie quadratiche which contained his work on conic sections, projective geometry and projective planes. In 1882 he published Sui gruppi di sei coniche in involuzione in which he explains his construction of the so-called six mutually apolar conics. The paper contains the famous 'Theorem of Gerbaldi':
There exists a set of six mutually apolar linearly independent nondegenerate ternary quadratic forms.

Gerbaldi left Turin to spend time at the University of Pavia, and he also went to Germany to learn about the latest mathematical advances. He was appointed as an assistant at the University of Rome and, while he was there, he entered a competition for the Chair of Analytic and Projective Geometry at the University of Palermo. Filling chairs by competition was the standard method in Italy at this time. He was appointed to Palermo in 1890.

In 1884, Giovanni Guccia had founded the Circolo Matematico di Palermo. Guccia provided:-
... the meeting place [for the Society], a library and all necessary funds. His generous offer was favourably received, and on 2 March 1884 the society's provisional statutes were signed by twenty-seven members. The goal was to stimulate the study of higher mathematics by means of original communications presented by the members of the society on the different branches of analysis and geometry, as well as on rational mechanics, mathematical physics, geodesy, and astronomy.
The publication for the new society was the Rendiconti del Circolo Matematico di Palermo. The Society soon lifted the status of mathematics at Palermo to a higher level and it was at this moment that Gerbaldi took up his chair. He did much to continue building the mathematics department at Palermo to greater heights. He led seminars and presented his students with the latest mathematical developments. Laura Martini writes [6]:-
Francesco Gerbaldi in Palermo presented for almost two decades beginning in the 1890s, the most recent progress in mathematics in his courses and seminars. In particular, he introduced two talented students to the study of group theory; Giuseppe Bagnera and Michele Cipolla became two of the most remarkable Italian algebraists of their time.
Among his publications while at Palermo we mention a four part paper Sul gruppo semplice di 360 collineazione piane which continued his investigations of the six mutually apolar conics had he constructed in 1882. The first part appeared in 1898, the second in 1899, the third in 1900 and the fourth in 1902. Marat Gizatullin writes [5]:-
The conics were used for a description of Valentiner's subgroup $G_{360}$ of $Aut(P_{2})$. As an abstract group, $G_{360}$ is isomorphic to the alternating group $A_{6}$ of even permutations of six symbols. Gerbaldi demonstrated that one can take the conics as the symbols, that is Valentiner's group interchanges the conics .... Moreover, the conics were used for a construction of a point-line configuration studied by Gerbaldi. The set of intersection points of distinct Gerbaldi's conics consists of 60 points. From pure geometric consideration, without use of Gerbaldi's conics, the configuration was constructed by W Burnside, whose attention was concentrated on 45 points of the dual plane. We will call it as the Gerbaldi-Burnside configuration.
Gerbaldi attended the first International Congresses of Mathematicians held in Zürich in 1897. He lectured on Sul gruppo semplice di 360 collineazione piane being one of only two participants to lecture in Italian: the other was Giuseppe Peano. Vito Volterra chaired the session at which Gerbaldi spoke.

Renato Calapso, the son of Gerbaldi's assistant Pasquale Calapso, describes Gerbaldi's personality in vivid terms in [2]:-
Short and bitter, like February, dark face, closed and sullen, always unhappy, Gerbaldi appeared to me "like a dark night." Lonely, childless and without a family, he seemed barren of feeling and not to suffer. Then I realized that this was not so. My father, who was his assistant for eight long years, was terrified of him, because, in truth, he was a kind of torturer. As a boy, when I saw him from afar, I changed direction. Yet the mathematics that he knew and which he tirelessly taught at Palermo, with zeal and with passion, was broad and bright as the sky. He brought to Sicily four disciples, almost the same age, who achieved great fame: Giuseppe Bagnera, Pasquale Calapso, Michele de Franchis and Michele Cipolla. Gerbaldi finally departed from Palermo for Pavia in the autumn of 1908. I with my father, only the two of us, accompanied him to the steamer. We proceeded quietly and silently in the starry night until we reached the slight lapping of waves and the acrid smell of algae. The seething dark sea with the coast in the distance twinkled with the lights the city. Gerbaldi embraced my father for the last time, and murmured, "My life is over." He cried. Then I took his hand and I kissed him, repeatedly bathing his tears. He said, "My life is over." He left. We slowly returned home, almost staggering, dumb and silent, behind the shadows of the night. Our friend still lived a quarter of a century, but did nothing: his fate was to be a great teacher, but only in the land of Sicily.
Gerbaldi's departure from Palermo for Pavia in 1908 is vividly described in the above quote. He remained as professor at Pavia until he retired due to ill health in 1931. While in Pavia, he collaborated with Gino Loria to edit a volume of papers to honour Enrico D'Ovidio on his retirement. The Preface to the volume Scritti Matematici Offerti Ad Enrico D'Ovidio In Occasione Del Suo LXXV Genetliaco, 11 Agosto 1918 was written by the editors and says as much about the feelings of the authors as it does about D'Ovidio:-
On the approach of the day on which an inflexible law would retire Senator Enrico D'Ovidio from the university chair, there arose in the minds of many students whom he has had in his long and glorious career as a teacher, the pleasant idea of choosing this occasion - which coincides with his 75th birthday - to manifest to him their sentiments of unalterable affection and, at the same time, to present to him their sincere good wishes ad multos annos. ... And we are certain that to the loved teacher our publication will be doubly gratifying in as much as it serves also to show how Italy, in the tragic hours in which we live - not less than in the more grave and decisive periods of her earlier struggles for redemption - has not ceased to feed the sacred flame of science.
The longest paper in the volume is by Gerbaldi himself. This is the paper Le frazione continue di Halphen and it was one of a series of three papers which Gerbaldi wrote on the continued fractions of George-Henri Halphen.

### References (show)

1. A Brigaglia, Francesco Gerbaldi, Dizionario biografico degli Italiani 53 (Istituto della Enciclopedia Italiana, Rome, 1999), 381-383.
2. R Calapso, Matematici di Sicilia, in Atti del quarto congresso dell'U.M.I., Taormina 25-31 October 1951 (Cremonese Rome, 1953), 276-277.
3. R Calapso, Francesco Gerbaldi, Rend. Semin. Messina 2 (1957), 109-110.
4. S Cinquini, The golden decade of Pavian mathematics (1880-90) and its reprise at the beginning of the next century (Italian), Faiths and cultures in the Padua area in the late nineteenth and the early twentieth century (Italian) (Pavia, 1991), Ann. Storia Pavese 1995 (22-23) (1995), 439-458.
5. M Gizatullin, Bialgebra and Geometry of Plane Quartics, Asian J. Math. 5 (3) (2001), 387-432.
6. L Martini, Algebraic research schools in Italy at the turn of the twentieth century: the cases of Rome, Palermo, and Pisa, Historia Mathematica 31 (3) (2004), 296-309.
7. F G Tricomi, Matematici italiani del primo secolo dello stato unitario, Mem. Accad. Sci. Torino: Cl. Sci. Fis., Mat. Natur. (4) 1 (1962), 1-120.