**Duilio Gigli**was the son of Torquato Gigli and Anna Belli. Torquato Gigli (1845-1936), born in Sansepolcro, studied chemical science at Pisa and lectured at the University as well as becoming head of the municipal chemical laboratory for many years. He published many scholarly articles contributing to

*Chemiker Zeitung*Ⓣ for over 25 years. He was a lecturer in pharmaceutical and toxicological chemistry at the University of Pavia. He had a great love of music and poetry as well as being a staunch Italian patriot. In 1867 he was with Garibaldi and in 1915 he volunteered to fight in the Great War.

After his studies at a classical school in Pavia, Duilio Gigli studied mathematics at the Scuola normale superiore in Pisa, where he was taught by Luigi Bianchi. His thesis was the source of the material which he published in his first paper *Sulle superficie elicoidali e rigate dello spazio ellittico* Ⓣ in which he presented an extension to the theorems due to Eugenio Beltrami on surfaces in elliptical space. The 7-page paper appeared in the *Rendiconti of the Reale Istituto Lombardo di Scienze e Lettere* Ⓣ in 1900. Gerhard Hessenberg (1874-1925) writes in a review:-

Following his doctorate Gigli spent another year undertaking further studies at Pisa; then he was appointed as an assistant at the University of Pisa. He publishedAn elegant calculation proves the essential analytic correspondence of the deformation of the ruled surfaces for the elliptical(positively curved non-Euclidean)and the Euclidean space. Specifically, a ruled surface can be bent in such a way that a geodesic line becomes straight; the control surface formed by the binormals of a curve of constant torsion can be unwound onto a common helical surface.

*Sulle somme di n addendi diversi presi fra i numeri*1, 2, ...,

*m*Ⓣ in the

*Rendiconti del Circolo Matematico di Palermo*Ⓣ in 1902. Edmund Landau writes in a review of this paper:-

For a list of Gigli's publications, see THIS LINK.How many of the combinations of n integers from1to m have the same sum s, and how many have a sum ≤ L? With the help of a generating function, the author solves these problems in a simple way.

What follows is based fairly closely on a translation of [5] but we have added extra material.

From 1910 Gigli taught at the Foscolo High School in Pavia, then at Portoferraio, Forl“, Sondrio and finally at Sassari in Sardinia before returning again to the Foscolo di Pavia (where he was also vice-president). The High School was named for Ugo Foscolo (1778-1827) a famous Italian writer and poet; it specialised in a classical education.

Thus he divided his activity between that of research and of teaching which absorbed most of his energy. In fact, he had three promotions for distinguished merit in teaching and the inclusion into the role of honour for his commitment to teaching. He qualified as a lecturer in algebraic analysis when awarded his libero docente (the Italian equivalent, which no longer exists, of the habilitation). He was for many years a lecturer at the University of Pavia, holding teaching positions in algebra and analytical geometry, in infinitesimal analysis, and in analytic geometry.

He was an editor of the *Enciclopedia delle matematiche elementari* Ⓣ along with his two colleagues Luigi Berzolari and Giulio Vivanti. For this work he wrote three essays of great historical interest on the foundations of arithmetic and on the concept of number: *Aritmetica generale* Ⓣ (in the first part of the first volume published by Ulrico Hoepli in Milan in 1929, pages 81-212), *Aritmetica pratica* Ⓣ, in collaboration with Ettore Bortolotti (in the first part of the first volume published by Ulrico Hoepli in Milan in 1929, pages 213-268), *Teoria della misura* Ⓣ, in collaboration with Luigi Brusotti (in the first part of the second volume published by Ulrico Hoepli in Milan in 1936, pages 119-174). The topics covered range from natural numbers to absolute numbers, relative numbers and complex numbers, all also all being analysed from a logical point of view, to the problem of numbering using instruments and of numbering systems, to the theory of commensurable and incommensurable quantities with consideration for relations to infinitesimal analysis. Regarding the concept of number, Gigli referred, among the many references, directly to the arithmetic of Giuseppe Peano, and tried to use his investigations into the fundamentals in his teaching. G A Miller writes in [6]:-

In a review of volume 3 of this work, written after the death of Gigli, Raymond Archibald writes [1]:-The preparation of an Italian encyclopaedia of elementary mathematics was approved by the Society 'Mathesis' as early as1909, and the project was then viewed with favour also in other countries so that it seemed desirable to publish a German edition contemporaneously with the original. Active preparation was begun at once and in1916the proposed publication was announced in my 'Historical Introduction to Mathematical Literature', as well as in other places. Various obstacles, including the World War, delayed the publication so that the first volume, which is devoted to analysis, was only recently completed after being brought up to date by the authors of the articles contained therein. The remaining two volumes are to treat geometry and applied mathematics respectively. The latter volume is expected to include also a discussion of the teaching and the history of mathematics.

For more detailed reviews of the parts of the Encyclopaedia for which Gigli was an editor, see THIS LINK.... in1930, under the editorship of Luigi Berzolari(1863-1949), Giulio Vivanti(1859-1949), and Duilio Gigli(1878-1933), appeared the first part of the first volume of 'Enciclopedia dette matematiche elementari'; the second part was published in1932, both parts(each with its own author index)making a volume of nearly1100[pages. This volume, devoted to arithmetic, computation, algebra, and analysis,was]by9authors, ... . The general editorial plan of this volume was also carried out in the second. Only the more fundamental theorems were proved, but there is a constant wealth of bibliographic references where further information may be gleaned.

In the *Lezioni di aritmetica e di algebra elementare* Ⓣ (three volumes published by Mattei et Co., Pavia in 1914, for use in upper secondary schools), he wrote in the Preface to volume 1 that he had "endeavoured to make the numbers still be numbers of things or ratios of magnitudes to magnitudes" and of having "wanted that the relations of equality and inequality of a certain operation to be seen to be done with collections of things or between magnitudes, and the sum of numbers remained the image of a certain operation that can be done with collections of things or with quantities." Reviewing this work, Lucien Baatard writes [2]:-

He was a contributor to theIn the first part of this book, published in June1914, the author discusses the first five operations of arithmetic, progressions, decimal notation, proportions, and periodic decimal fractions. Wanting to avoid the pitfall of presenting arithmetic as a set of characteristics, he bases his deductions on propositions concerning collections of objects; the study of proportions is preceded by copious considerations of continuous quantities; the number0is the object of particular care. While paying homage to Professor Gigli's thought effort, we will admit that conciseness is not always the prime quality of his explanations and demonstrations; this may be due to the point of view he wanted to take. His work will be read with interest by mathematicians or teachers already familiar with general arithmetic.

*Questioni riguardanti le matematiche elementari*Ⓣ, edited by Federigo Enriques, for which he wrote the article

*Dei numeri complessi a due e più unità*Ⓣ. This was published by N Zanichelli in Bologna in 1912 and comprised of 147 pages. Here, after a historical investigation into the origins of the theory of imaginary numbers and an analysis of the fundamental theorem of algebra, he gave an original and general presentation of the theory of numbers in

*n*units. He also collaborated in the drafting of some entries for the

*Enciclopedia Italiana*.

Among his publications - of a scientific, didactic and historical-critical nature - the following are still remembered: *Sulle somme di n addendi diversi presi fra i numeri *1, 2,* ..., m* Ⓣ, in *Reale Rendiconti del Circolo Matematico di Palermo* Ⓣ (1902), in which he analysed and solved the problem of how many of the combinations considered will give the same sum, and how many will give a sum not exceeding a certain fixed number; *La matematica nei licei* Ⓣ, in *Rivista d'Italia* (1905), in which he discussed the programme of reforms and the cultural relevance of mathematical education alongside the humanistic one; *Dei numeri trascendenti* Ⓣ (1923), in which, through a historical-critical survey, he provided a rigorous treatment of the concept of transcendent number.

For a list of Gigli's publications, see THIS LINK.

Of further interest are the *Riflessioni sui principii dell'aritmetica* Ⓣ, in the *Yearbook* of the Reale Liceo di Pavia (1925-26), where he considered and discussed the ideas of Euclid, René Descartes, Alessandro Padoa, Gottlob Frege, A N Whitehead and Bertrand Russell. In *Definizioni in matematica* Ⓣ, in the *Yearbook* of the Reale Liceo di Pavia (1926-27), the logical-epistemological status of definitions in mathematics was finely analysed.

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