**Mineo Chini**was the son of Biagio Chini and Vittoria Baldi. He was born in Massa in central Italy which had been ruled by the Duke of Modena from 1829 to 1859 after which it became part of the Kingdom of Sardinia. In 1861 the Kingdom of Sardinia changed its name to the Kingdom of Italy. When Mineo Chini was born in the Kingdom of Italy, therefore, it was only five years old.

Mineo Chini studied at the University of Pisa and, in 1888, he was awarded his laurea. He began publishing papers before the award of his laurea with *Una proprietaà della lemniscata di Bernoulli* Ⓣ appearing in 1887. His research was on differential geometry and, based on his thesis, he published *Sulle superficie a curvatura media costante* Ⓣ (1889), *Sopra una classe di superficie* Ⓣ (1889) and *Sopra alcune deformazioni delle superficie rigate* Ⓣ (1890).

For a list containing most of Chini's mathematical publications, see THIS LINK.

After graduating, Chini taught in secondary schools. He was an extraordinary professor of infinitesimal analysis at the R. Accademia di Torino from 1889 to 1893. He also gave lectures on infinitesimal calculus at the University of Padua. In 1890 he was appointed to a temporary position at the Turin Military Academy. He held this post until 1896. At the Turin Military Academy the professor of projective geometry was Mario Pieri and also teaching there was Giuseppe Peano who, in addition, was teaching at the University of Turin. Both Pieri and Peano were important influences on Chini's development as a mathematician. In 1891 Peano founded *Rivista di matematica* Ⓣ, a journal devoted mainly to logic and the foundations of mathematics, and he encouraged Chini to publish there which he did in 1893. From around 1892, Peano embarked on a new and extremely ambitious project, namely the *Formulario Mathematico* Ⓣ. He explained in the March 1892 part of *Rivista di matematica* Ⓣ his thinking:-

This project was not received with enthusiasm by many mathematicians but Peano encouraged those around him to contribute and Chini was one of the contributors. Peano also presented several of Chini's papers to the Turin Academy of Sciences and four such papers by Chini appeared inOf the greatest usefulness would be the publication of collections of all the theorems now known that refer to given branches of the mathematical sciences ... Such a collection, which would be long and difficult in ordinary language, is made noticeably easier by using the notation of mathematical logic ...

*Atti della R. Accademia di Scienze Torino*between 1895 and 1899.

In 1896 Chini won a competition for a professorship at the Technical Institute in Caserta, near Naples. In 1897 he entered a competition for the chair of infinitesimal calculus at the University of Modena. A committee, with Ulisse Dini as president, considered Chini's application along with that of seven other candidates, namely Italo Zignago, Giulio Vivanti, Onorato Nicoletti, Rodolfo Bettazzi, Domenico Amanzio, Orazio Tedone and Giuseppe Lauricella. A report of the findings of the committee appeared in [3] and we give a version of their evaluation of Chini:-

The committee ranked Vivanti first followed by Nicoletti and Lauricella. However, Vivanti did not take up the position and, in January 1898, Nicoletti was appointed to the chair of infinitesimal calculus at the University of Modena.Dr Mineo Chini graduated with a Iaurea in Pisa in1888. He was an extraordinary professor of infinitesimal analysis in the R. Accademia di Torino from1889to1893; and in this position he proved to have exceptional teaching qualities, as evidenced by the attestation of the principal of the Academy, and of the professor in charge of the course. He obtained his 'libera docenza'(similar to the habilitation in that it is the 'right to teach')in infinitesimal calculus in Padua, and now teaches in technical institutes. He presents six original memoirs and a book, all strictly related to the subject of the competition. Three of these works are on differential geometry. The author investigates the properties of the surface with a constant mean curvature and of the surfaces applicable onto a rotating surface. The theorems obtained on the deformations of the ruled surfaces are elegant. In another short note the author indicates a procedure by which he obtains sequences of differential polynomials appearing in the theory of partial differential equations of the second order, with equal invariants, which have already been considered by Darboux. In a later work the author continues these researches. In the last of his memoirs, the author studies what conditions a partial differential equation of the2nd order must satisfy so that, with convenient change of variable, it can be reduced to contain only the second mixed derivative, and one of the first derivatives. The book "Exercises of infinitesimal calculus" is a work with an eminently practical aim. The exercises are well chosen, often brilliant and original or performed with complete knowledge of current scientific rigour. However, these works, while proving the talent of the candidate, are not particularly important; and it would be necessary to have more scientific activity.

Chini taught courses at the University of Pavia from 1898, for example: *Geometria infinitesimale delle linee nello spazio e sopra una superficie* Ⓣ in the academic year 1899-1900; *Teoria della equazione di Laplace* Ⓣ in the academic year 1901-1902; and *Corso speciale* Ⓣ in the academic year 1902-03.

In [5] details are given of how his career progressed:-

The School of Architecture at the University of Florence offered a five year course. It contained two years of mathematics courses in which the students were taught concepts of Algebra and Analytical Geometry, both useful for them for courses in the following years on Rational Mechanics and the Science of Construction. Chini underlined how these courses should not have the generality of those given in similar two-year courses in the School of Engineering. After teaching there for five years Chini proposed replacing the course Analysis II, taught in the second year, with the most complex course of infinitesimal analysis, so as to discourage those external students who enrolled directly into the second or third year, intending to avoid the basic course Analysis I. In December 1927 a third year of the mathematical courses was approved by the Ministry of Public Education, following pressure of the Fascist Architecture Trade Union. The influence of Fascist bodies on teaching became increasingly strong in the following years but, from around 1930, Chini's health began to deteriorate and he struggled to keep up the high standards which had been the mark of his career [5]:-... in1910he was nominated Central Inspector of Secondary Schools for the Ministry of Public Education, and was then head of the Technical Institute "G Galilei" in Florence. In1919-20he moved to the University of Florence where he taught mathematical analysis in the School of Architecture. He developed an interest in both scientific research and in pedagogy: he wrote around forty scientific papers mostly focusing on differential geometry and differential equations and also wrote various books on the problems related to teaching in secondary schools and high schools, which received noteworthy praise for their simplicity and clarity.

One of the most widely used of the books Chini wrote wasHe persevered in his work as researcher and teacher until the very end: even if worn down in the last three years of his life by a serious disease, he always desired to be with his young students at the School of Architecture. He died in Florence on the11th of November1933. Throughout his life he had been awarded many honours: the Commendation of the Italian Crown(commenda della Corona d'Italia)and the Cross of Mauritian knight(Croce di cavaliere mauriziano).

*Corso speciale di Matematiche con numero se applicazioni ad uso principalmente dei Chimici e dei Naturalisti*Ⓣ published in 1904 based on courses Chini had taught at Pavia. The review [1] states:-

Rinaldo Cervellati writes about the same book in [2]:-This small volume contains the topics of the Special Course of Mathematics which was created at the University of Pavia for students in chemistry and natural sciences. It consists of four parts. In the first part, entitled Complements of Algebra, the following topics are presented: Progressions, Logarithms, Combinatorial Analysis, Binomials, Determinants, and Systems of Linear Equations. The second part is devoted to the elements of two-dimensional and three-dimensional analytic geometry; then come, in the last two, the elements of Differential and Integral Calculus. In each of these parts the author has confined himself to the essential notions, and endeavoured to accompany them with examples which are such as to interest chemists and students of the natural sciences. Note that a chapter in the third part is dedicated to the theory of errors.

The same book is also described in [4]; we follow [5]:-In the preface the author aims to "satisfy the need for students of Chemistry and Natural Sciences, who want to modernise themselves, to acquire the knowledge of at least the mathematical theories developed in this book." He also claims to have tried to present the material "in the simplest and least arid form possible and to have endeavoured to illustrate," where he could, "the various theories through appropriate examples, taken from Physics, Chemistry ... from Mechanics and Thermodynamics." With this Chini wants to immediately give an idea of how the results of mathematics can be used outside the abstract field. He is aware that chemists, naturalists and even social scientists are users of mathematics ... Mathematical concepts and theorems are clearly presented without too many demonstrations but with many numerical examples, followed by applications to problems in Physics and Chemistry.

The following paragraphs follow [5], which in turn is a translation of [4]. A few details that have been given above are repeated here.In1904, the 'Corso speciale di matematiche con numerose applicazioni'Ⓣ, written primarily for students of chemistry and natural sciences, was published in Livorno. In the preface, the author criticises the slowness of ministers that had led to the course of mathematics for chemists and natural scientists, which had recently been established at the university, to not yet have a definite structure. With clarity, Chini was able to successfully present the concepts of sequences, logarithms, combinatorics, determinants, systems of linear equations and their resolution, elements of analytic geometry and of differential and integral calculus while trying to give an idea, through examples taken from physics and chemistry, of how the theory, still under development, could be well utilised outside the abstract field of mathematics. Seven editions of the work were published, each one slightly more detailed than the previous one and elaborated on by Chini himself who, using his didactic experience, was trying to give students all those mathematical tools needed in a degree course. In the last edition, which appeared in Livorno in1923, there is an appendix with fundamental notions of mechanics and thermodynamics, both essential to the study of chemical physics.

In 1921 the book *Lezioni sull'integrazione di equazioni differenziali in aggiunta al Corso speciale di matematiche * Ⓣ was published in Livorno. Chini treated, in a clear and ordered way, differential equations of the first and second order, integration by series of differential equations and partial derivatives of the first and second order. Again, he was identifying the deficiencies of the course in the modest weekly timetable, in the lack of parallel courses of mathematics and in the very abstract study of integration of differential equations.

In the 1919-20 academic year Chini gave an open course on this topic, consisting of a series of seminars at the Institute of Further Studies in Florence and the result of the experiment was pleasing because there was a high attendance and interest from the students; he then decided to publish the lectures that have been mentioned above. At the same time he held the chair of mathematical analysis at the School of Architecture in Florence.

Chini noted that, since the aim of these schools (there were five in the whole of Italy: in Venice, Turin, Florence, Naples and Rome) was to award a degree in architecture, it was necessary for students to have basic notions of algebra, analytic geometry and infinitesimal calculus in order to be able to attend the more advanced courses of rational mechanics and the science of constructions. In the *Lezioni di analisi matematica ad uso delle scuole superiori di architettura* Ⓣ (Livorno 1932) Chini presented the subject trying not to add too much weight to the course as "these notions should not have the extension and generality" of those that would be normally taught to first and second year engineers and mathematical-physics students. Chini's scientific output was mostly in the field of differential geometry. In November of 1890, in the *Atti dell'Accademia delle scienze di Torino*, XXVI Ⓣ (1890), p. 20-34, there appeared one of his notes, entitled *Sopra alcune deformazioni delle superfici rigate* Ⓣ. Chini examined one of the writings of Eugenio Beltrami, *Sulla flessione delle superfici rigate* Ⓣ, in which the author studied the deformation of such surfaces; Chini was able to reduce to the minimum the number of possible elements that identifies the shape of a ruled surface, he researched the formulae - rather simple in this case - that gave all the bump-shaped surfaces and applied these formulae to treat some problems of the same type, but less simple, than those tackled by Beltrami in his essay. In the field of differential geometry Chini worked mostly on surfaces of medium-constant curvature, on *W*-applicable surfaces and surfaces of rotation.

Chini also enjoyed himself in the writing stories, among which is the sketch *Dopo undici anni* Ⓣ, where he narrates a love story between a young mathematician from Palermo and a cousin from Massa.

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

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