Publications of Gino Fano

We list below all the publications by Gino Fano that we have been able to find. We have taken most of the data from MathSciNet, Google Scholar, Biblioteca Digitale Italiana di Matematica, and the paper A Collino, A Conte and A Verra, 'On the life and scientific work of Gino Fano', ICCM Not. 2 (1) (2014), 43-57. The year of publication is always a problem since journals often publish volumes with a date different from the year in which the volume is published. The years of publication given below are subject to possible errors because of this. In many cases we have added a comment, usually Fano's introduction or abstract.

Gino Fano's publications.

1890

Felix Klein and Gino Fano, Considerazioni comparative intorno a ricerche geometriche recenti, Annali di Matematica Pura ed Applicata (2) 17 (1890), 307-343.
Comment
This is Gino Fano's translation of Felix Klein's Erlangen Programme.

1892
Gino Fano, Sui postulati fondamentali della geometria in uno spazio lineare ad un numero qualunque di dimensioni, Giornale di matematiche 30 (1892), 106-132.
Comment
The paper begins: The geometry of a multidimensional space, which has long been a part of the branches of mathematics, has, as is well known, been one of the subjects of study and research that has been of greatest interest, especially in Italy, for some years now. However, although it has given rise to various research projects, and in these numerous and important results have also been achieved, nevertheless up to the last few days (at least as far as I know) no real definition has been given of a linear variety or, briefly, of a space with any number $r$ of dimensions (in the most general sense of this expression). Only examples are given of these linear varieties, asserting that such are the involutions of points on a straight line, linear systems of plane curves and surfaces, or, more generally, of any connected sets, etc. when one considers as the respective element the group of points, the curve, the surface, the connected set. If we want to do some research in which, precisely in order to have greater generality, it is necessary or at least preferable to leave the nature of the element itself completely undetermined, all the authors (unless I am mistaken) agree in assuming as a starting point the representability of the elements (points) of a hyperspace by means of coordinates.

Fano's paper contains the famous quotation:
As a basis for our study we assume an arbitrary collection of entities of an arbitrary nature, entities which, for brevity, we shall call points, but this is quite independent of their nature.
1893
Gino Fano, Studio di alcuni sistemi di rette considerati come superficie dello spazio a cinque dimensioni, Annali di Matematica Pura ed Applicata 21 (1893), 141-192.
Comment
The paper begins: It is known that for every result relating to entities contained in a non-degenerate quadric of five-dimensional space one can have an application, and often a notable and interesting one, by imagining that the element or point of this quadric (the nature of which, in research, is usually left undetermined) is precisely the straight line of ordinary space $(S_{3})$ (and that quadric is therefore the set of these straight lines.
Gino Fano, Sopra le curve di dato ordine e dei massimi generi in uno spazio qualunque, Memorie della Reale Accademia della Scienze di Torino (2) 41 (1893), 335-382.
Comment
This Memoir is taken from the Doctoral Degree Dissertation presented by Gino Fano to the Faculty of Science of the University of Turin in June 1892. The Memoir begins as follows: At the competition opened by the Berlin Academy of Sciences for the awarding of the third Steiner Prize (on a theme relating to the theory of algebraic skew curves), as is known, two famous Memoirs were presented: one by Halphen, the other by Noether: both were very valuable, and in fact they shared the prize. And among the results contained in these Memoirs, the theorem that skew curves of a given order and maximum genus are all contained in a quadric is certainly very important. This proposition was then extended by Mr Castelnuovo to the curves of a linear space of any number r of dimensions, and in place of the quadric in this more general case there appears the rational normal ruled surface of order $r - 1$ (or also, for $r = 5$ , the homaloid surface $F_{2}^{4}$ of Veronese). With this extension, the determination of the various curves of maximum genus $(\pi)$ of any space $S_{r}$ (and of order > $2r$ ) can be considered exhausted; precisely because these curves are contained in very simple (rational) surfaces with well-known properties, on which it will always be easy to construct them. The question that now presents itself instead as - I will say - subsequent, and which also seems worthy of being studied, is that of carrying out a similar research also for the curves of genus $\pi - 1, \pi - 2, ...$ determining if and when these too can be on the ruled $R^{r-1}$ (or, for $r = 5$ , on Veronese's $F_{2}^{4}$ ); or, when they cannot be, in which other surfaces (possibly simple) they are contained. And this research constitutes precisely the main object of this work. Even before I began to deal with it, Mr Castelnuovo had told me that he believed that curves of genus $\pi - 1$ must necessarily lie - at least from a certain order onwards - on a surface with elliptic or rational sections. The proposition actually holds, and we will see later listed the various cases that these curves can present. A similar study will also be done for curves of genus $\pi - 2$ ; more briefly however, because many of their properties can then be established easily and with reasoning completely identical to that already used for curves of genus $\pi - 1$ . And it would perhaps be interesting to try to extend these same results also to curves of genus $\pi - 3, \pi - 4, ...$ . and, in general, $\pi - k$ ; but I do not intend to deal with this for now (as I also say at the end of §8).

1894
Gino Fano, Sulle congruenze di rette del terzo ordine prive di linea singolare, Atti della Reale Accademia della Scienze di Torino 29 (1894), 474-493.
Comment
The paper begins: Research on (algebraic) congruences of lines has so far been limited almost exclusively to those of first and second order, which however were studied extensively in the various works that from the classic Memoir of Kummer to the recent treatise by Mr Sturm contributed to enrich the literature of the Geometry of the line. Very little, however, has been done on congruences of order higher than two, and in particular already on those of third order, of which only some particular cases have been considered by Roccella and Hirst, and others by Segre and Castelnuovo.
Gino Fano, Sull'insegnamento della matematica nelle Università tedesche e in particolare nell'Università di Gottinga, Rivista di Matematica 4 (1894), 170-188.
Comment
Fano was particularly impressed with the Mathematics Library at Göttingen. He writes: Those who are enrolled in the Seminar can also make use, if they wish, of the reading room (Mathematisches Lesezimmer) and its library. The aim of this institution is to make available to students above all those books and journals that it is most frequently necessary to consult; and precisely in order to not betray this purpose and to constantly place everything at the disposal of all, it is strictly forbidden to issue books and journals on loan. Those who wish to take some volume home can contact the General Library (Universitätsbibliothek) ... Among the new collections, a major role is played by the journals, especially German and French (of Italian periodicals, unfortunately, not a single one), for a total value of 300 marks a year.

1895
Gino Fano, Uno sguardo alla storia della matematica, Atti e Memorie Reale Accademia Virgiliana (1895), 3-34.
Gino Fano, Contributo alla teoria dei numeri algebrici, osservazioni varie e parte IX del Formulario, Rivista di Matematica 5 (1895), 1-10.
Gino Fano, Sui postulati fondamentali della Geometria projettiva I, Rendiconti del Circolo Matematico di Palermo 9 (1895), 79-82.
Comment
This is a letter from Gino Fano to Federigo Enriques. Enriques replies with: F Enriques, Sui postulati fondamentali della Geometria projettiva II, Rendiconti del Circolo Matematico di Palermo 9 (1895), 82-84.
Gino Fano, Sui postulati fondamentali della Geometria projettiva III, Rendiconti del Circolo Matematico di Palermo 9 (1895), 84-85.
Comment
This is a letter from Gino Fano to Federigo Enriques adding an observation to I and replying to II.
Gino Fano, Sulle superficie algebriche con infinite trasformazioni projettive in sé stesse, Rendiconti Accademia Nazionale Lincei (V) 41 (1895), 149-156.
Comment
He explains: In this Note I propose to study the algebraic surfaces of any space, which admit a continuous group (one or more times infinite) of projective transformations, extending the various considerations also to the case of homographies with multiple united points, and thus completing the very important results already obtained (for ordinary space) by Mr Enriques.
Gino Fano, Sulle equazioni differenziali lineari del $4^{o}$ ordine, che definiscono curve contenute in superficie algebriche, Rendiconti Accademia Nazionale Lincei 41 (1895), 232-239.
Gino Fano, Sopra alcune considerazioni geometriche che si collegano alla teoria delle equazioni differenziali lineari, Rendiconti Accademia Nazionale Lincei (V) 4 (1895), 18-25.
Comment
Fano writes: The aim of this Note is to make a first contribution to a theory, which I might call geometric, of linear differential equations; that is, to show how very simple geometric considerations can lead to interesting results for such an important branch of modern analysis. The results I am now obtaining are not, at least in large part, new; but the novelty of the method may perhaps encourage someone to continue these researches with me.
Gino Fano, Sopra certe curve razionali in uno spazio qualunque, e sopra certe equazioni differenziali lineari, che con queste curve si possono rappresentare, Rendiconti Accademia Nazionale Lincei (V) 41 (1895) 51-57.
Comment
Continues the work of the previous paper.
Gino Fano, Ancora sulle equazioni differenziali lineari del $4^{o}$ ordine, che definiscono curve contenute in superficie algebriche, Rendiconti Accademia Nazionale Lincei (V) 41 (1895), 292-300.
Gino Fano, Sulle equazioni differenziali lineari di ordine qualunque che definiscono curve contenute in superficie algebriche, Rendiconti Accademia Nazionale Lincei 41 (1895), 322-330.
Gino Fano, Sulle varietà algebriche dello spazio a quattro dimensioni con un gruppo continuo integrabile di trasformazioni proiettive in sé, Atti Istituto Veneto di Scienze, Lettere ed Arti 7 (1895), 1069-1103.

1896
Gino Fano, Über endliche Gruppen linearer Transformationen einer Veraenderliche, Monatshefte für Mathematik und Physik 7 (1) (896), 297-320.
Comment
The paper begins as follows: As is well known, the general binary forms up to the fourth degree give us examples of forms with a certain (finite or infinite) group of linear transformations within themselves; however, they are not the only forms that allow such a group. On the other hand, it is also clear that a binary form with more than two different root points can only allow a finite number of linear transformations; their roots can only be exchanged in a finite number of ways, and by any permutation of these, a linear (projective) transformation of the presented form into itself, insofar as such a transformation exists at all, is uniquely determined. If we therefore ignore those (very simple) forms that have only one or two different roots, we will obtain all binary forms with linear transformations within themselves by assuming any number of root points and applying to them the operations of any finite group of linear transformations of one (or two homogeneous) variables. Those forms that arise from a single point by applying the respective transformations will be of particular interest; Such forms will allow a finite transitive group. The remaining forms will be composed as products of the latter.
Gino Fano, Aggiunta alla Nota: Sulle congruenze di rette del terzo ordine prive di linea singolare, Atti Reale Accademia delle Scienze di Torino 31 (1896), 708-715.
Gino Fano, Sulle varietà algebriche con un gruppo continuo non integrabile di trasformazioni proiettive in sé, Memorie Reale Accademia delle Scienze di Torino 46 (1896), 187-218.
Comment
The paper begins: The aim of this paper is to make a new contribution to the theory of algebraic varieties of any number of dimensions and belonging to any space, which admit a continuous group of projective transformations in themselves.
Gino Fano, Sui gruppi continui di trasformazioni cremoniane del piano e sopra certi gruppi di trasformazioni projettive, Rendiconti del Circolo Matematico di Palermo 10 (1896), 16-29.
Gino Fano, Sulle superficie algebriche con un gruppo continuo transitivo di trasformazioni proiettive in sé, Rendiconti del Circolo Matematico di Palermo 10 (1896), 1-15.
Gino Fano, Lezioni di geometria della retta (L. Laudi, Rome, 1896).
Comment
This book is a lithographic printing of a hand written text.

1897
Gino Fano, Über Gruppen, insbesondere continuierliche Gruppen von Cremona-Transformationen, der Ebene und des Raumes, Monatshefte für Mathematik und Physik 9 (1897), 17-29.
Federigo Enriques and Gino Fano, Sui gruppi continui di trasformazioni cremoniane dello spazio, Annali di Matematica Pura ed Applicata 26 (1897), 59-98.
Comment
The paper begins: The subject of this paper is the classification of continuous groups of birational (or Cremonian) transformations of space, that is, their reduction to determined types, by means of birational transformations.
Among these groups, four categories immediately appear, as a natural extension of the Cremonian groups typical of the plane: the projective groups, the conformal groups (groups of transformations that change a sphere into a sphere) and the groups (which can be called) generalised Jonquières, that is, those that possess an invariant star of straight lines or an invariant bundle of planes.

The study of the groups of these four categories has already been partly carried out (for the projective and conformal groups), and partly (for the generalized Jonquières groups) could easily be carried out, reducing oneself to already known cases, that is, making use of an appropriate composition of the binary groups and the Cremonian groups of two-dimensional varieties.
Gino Fano, Un teorema sulle superficie algebriche con infinite trasformazioni proiettive in sé, Rendiconti del Circolo Matematico di Palermo 11 (1897), 241-246.

1898
Gino Fano, I gruppi di Jonquières generalizzati, Memorie Reale Accademia delle Scienze di Torino 48 (1898), 221-278.
Comment
The paper begins: The classification of continuous groups of Cremonian transformations of space was begun by Mr Enriques and myself in a joint Memoir, in which we demonstrated that these groups can all be birationally reduced to groups of one of the following categories:
a) projective groups;
b) groups of conformal transformations (i.e. which change spheres into spheres);
c) groups that we have called form in themselves a bundle of planes, or a star of straight lines;
d) two simple, transitive, well-determined $∞^{3}$ groups of transformations of the 3rd or respectively of the 7th order.
Gino Fano, Lezioni di geometria non euclidea (Rome, 1898).
Comment
This work is produced with lithographic printing.
Gino Fano, Le trasformazioni infinitesime dei gruppi cremoniani tipici dello spazio, Rendiconti Accademia Nazionale Lincei 71 (1898), 332-340.
Comment
The paper begins: In my Note "On some continuous imprimitive groups of point transformations of space" I have shown how some of the typical groups I encountered in the classification of continuous imprimitive groups of Cremonian transformations of space also presented themselves to Mr Lie in his research on the imprimitive groups of point transformations; and for these groups I have transcribed there the symbols of the infinitesimal transformations, determined by Mr Lie himself. But also for the other typical Cremonian groups one can easily find the symbols of the generating infinitesimal transformations, deducing them, in most cases, from the finite equations of the groups themselves. And this is precisely what I intend to do in the present Note.
Gino Fano, I gruppi continui primitivi di trasformazioni cremoniane nello spazio, Atti Reale Accademia delle Scienze di Torino 33 (1898), 480-504.
Gino Fano, Sopra alcuni gruppi continui imprimitivi di trasformazioni puntuali dello spazio, Rendiconti Accademia Nazionale Lincei 71 (1898), 302-308.
Gino Fano, Über Gruppen, inbesondere kontinuierliche Gruppen von Cremona-Transformationen, der Ebene und des Raumes, in Verhandlungen des ersten interna- tionalen Mathematiker Kongresses, Zurich 1897 (Teubner, Leipzig, 1898), 251-255.
Comment
The paper begins as follows:After a brief historical overview of the development of the theory of birational or so-called Cremona transformations of the plane and space, Mr Autonne's results on finite groups of quadratic and cubic transformations were presented, as were those of Messrs Seligmann Kantor and Anders Wiman on the reduction of finite groups of birational plane transformations to specific types.

When moving on to continuous groups, reference was first made to Mr Noether's communication to the German Mathematical Society in 1896; further, regarding the reduction of continuous groups of birational plane transformations to specific types, reference was made to Mr Enriques' results.
Gino Fano, Über Gruppen, insbesondere continuierliche Gruppen von Cremona-Transformationen der Ebene und des Raumes, Monatshefte für Mathematik und Physik 9 (1) (1898), 17-29.

1899
Gino Fano, Sulle equazioni differenziali lineari che appartengono alla stessa specie delle loro aggiunte, Atti Reale Accademia delle Scienze di Torino 34 (1899), 388-409.
Gino Fano Sulle equazioni differenziali lineari del V ordine le cui curve sono contenute in varietà algebriche, Rendiconti Reale Istituto Lombardo. Accademia di Scienze e Lettere 32 (1899), 843-866.
Gino Fano, Sulle equazioni differenziali lineari del V ordine e del VI ordine, le cui curve integrali sono contenute in una quadrica, Atti Reale Accademia delle Scienze di Torino 34 (1899), 415-445.
Gino Fano, Review: Einfuehrung in die Grundlagen del Geometrie, by W Killing, Bollettino di Bibliografia e Storia delle Scienze Matematiche 2 (1899), 14-21.
Comment
This is a book review.
Gino Fano, Osservazioni sopra alcune equazioni differenziali lineari, Rendiconti Accademia Nazionale Lincei 81 (1899), 285-29.
Gino Fano, Un teorema sulle varietá algebriche a tre dimensioni con infinite trasformazioni proiettive in sé, Rendiconti Accademia Nazionale Lincei 81 (1899), 562-565.
Comment
The paper begins: It is known that every algebraic curve, which admits a continuous group $∞^{1}$ of projective transformations in itself, is rational. And so is every algebraic surface, which admits a continuous transitive group (and therefore at least $∞^{2}$ ) of projective transformations.
In this Note I propose to demonstrate that also for three-dimensional algebraic varieties the proposition analogous to the previous ones holds; namely that "Every three-dimensional algebraic variety is rational, which admits a continuous transitive group (and therefore at least $∞^{3}$ ) of projective transformations.

1900
Gino Fano, Über lineare homogene Differentialgleichungen mit algebraischen Relationen zwischen den Fundamentalloesungen, Mathematische Annalen 53 (4) (1901), 493-590.

1901
Gino Fano, Nuove ricerche sulle congruenze di rette del $3^{o}$ ordine prive di linea singolare, Atti Reale Accademia delle Scienze di Torino 51 (1901), 1-79.
Comment
The paper begins: In this Memoir I propose to fully expose my research on the congruences of lines of the 3rd order, which partly date back to 1894, and of which I have already published a brief summary at that time. Having now resumed the study of this subject, I have also been able to do some new research, and simplify some parts of the original treatment.
Gino Fano, Sopra alcune particolari congruenze di rette del $3^{o}$ ordine, Atti Reale Accademia delle Scienze di Torino 36 (1901), 366-379.
Comment
The paper begins: In this Note I intend to point out the existence of some particular congruences of lines of the 3rd order (without singular lines), which seem to me to deserve special attention due to the greater number of singular points and planes that they possess in comparison with the more general congruences having the same characteristics.
Gino Fano, Sui modi di calcolare la torsione di una linea geodetica sopra una superficie qualunque, Atti Atti della Reale Accademia Peloritana 16 (1901), 198-199.
Gino Fano, Le congruenze di rette del $3^{o}$ ordine composte di tangenti principali di una superficie, Atti Reale Accademia delle Scienze di Torino 37 (1901), 501-519.
Comment
The paper begins: In my Memoir: "New research on the congruences of lines of the 3rd order without singular lines" I have considered only those congruences whose rays have the two foci in general distinct, and are therefore double tangents (properly speaking) of the relative focal surface. When instead on each ray the two foci coincide, the lines of the congruence (always supposed to be without singular lines) are principal tangents (or tripute tangents, or asymptotic tangents) of the surface locus of the foci themselves. In the present 'Note' the congruences of lines of the 3rd order of this latter type are determined.

1903
Gino Fano, Lezioni di geometria descrittiva (Turin, 1903).
(This work is produced with lithographic printing).

1904
Gino Fano, Sul sistema $∞^{2}$ di rette contenuto in una varietà cubica generale dello spazio a quattro dimensioni, Atti Reale Accademia delle Scienze di Torino 39 (1904), 778-792.
Comment
The paper begins: In every cubic variety of the four-dimensional space $(S_{4})$ there is contained a double infinity of lines, which, if the proposed variety has no double points, is certainly irreducible and also devoid of double elements. In the present Note I propose to determine some characteristics of this system $∞^{2}$ of lines, considered as a doubly infinite algebraic entity; and, among other things, I will demonstrate that its canonical system is composed of its oo ruled intersections with the linear complexes of the same space $S_{4}$ .
Gino Fano, Ricerche sulla varietà cubica generale dello spazio a quattro dimensioni e sopra i suoi spazi pluritangenti, Annali di Matematica Pura ed Applicata 10 (1904), 251-285.
Comment
The paper begins: In this paper, some questions of a projective nature concerning the general cubic variety (without double points) of four-dimensional space are treated. Among other things, the spaces that are tangent to this variety in 2, 3 or 4 different points are completely determined, as well as the properties and characters of some entities to which those spaces give rise.
Gino Fano, Sopra una varietà cubica particolare dello spazio a quattro dimensioni, Rendiconti del Reale Istituto Lombardo di Scienze e Lettere (Milano) 37 (1904), 554-566.
Gino Fano, Sulle superficie algebriche contenute in una varietà cubica dello spazio a quattro dimensioni, Atti Reale Accad. Sci. Torino 39 (1904), 597-613.
Comment
The paper begins: Quadrics of four-dimensional space which have no double points, that is, are not cones, enjoy the remarkable property (which Mr Klein established as early as 1872) of not containing any other algebraic surfaces than those which are their complete intersections with further three-dimensional algebraic varieties of the same space. In this Note it is shown that the same property also applies to cubic varieties of the space $S_{4}$ which have no double points, and to some cubic varieties with double points, all of which are determined. For cubic varieties which do not contain planes it is also shown how all the algebraic surfaces contained therein can be constructed, whether or not these are their complete intersections with other varieties.

1905
Gino Fano, Sul sistema $∞^{3}$ di rette contenuto in una quadrica dello spazio a quattro dimensioni, Giornale di Matematiche 43 (1905), 1-5.
Gino Fano, Un pó di matematica per i non matematici. Geometria descrittiva. Calcolo infinitesimale, Rivista d'Italia 8 (1905), 1-12.

1906
Gino Fano, Sopra alcune superficie del 4-ordine rappresentabili sul piano doppio, Rendiconti del Reale Istituto Lombardo di Scienze e Lettere (Milano) 39 (1906) 1071-1086.

1907
Gino Fano, Geometria proiettiva (edited by D Pastore and E Ponzano) (Turin, 1907).
(This work is produced with lithographic printing.
Gino Fano, Gegensatz von synthetischer und analytischer Geometrie in seiner historischen Entwicklung im XIX Jahrhundert, in Enzykl. Math. Wiss. (3) 1 (1907) 221-288.
Gino Fano, Kontinuierliche geometriche Gruppen. Die Gruppentheorie als geometrisches Einteilungsprinzip, in Enzykl. Math. Wiss. (3) 1 (1907) 289-388.

1908
Gino Fano, La geometria non euclidea, Scientia 26 (1908), 257-282.
Comment
This is a generally understandable overview of the history of non-Euclidean geometry.
Gino Fano, Sopra alcune varietà algebriche a tre dimensioni aventi tutti i generi nulli, Atti Reale Accademia delle Scienze di Torino 43 (1908), 973-984.

1909
Gino Fano, Sulle varietà algebriche che sono intersezioni complete di più forme, Atti Reale Accademia delle Scienze di Torino 44 (1909), 633-648.
Gino Fano, Lezioni di geometria descrittiva date nel R Politecnico di Torino (G B Paravia and C, Turin, 1909).

1910
Gino Fano, Sui fondamenti della geometria, Bollettino della Mathesis 2 (1910) 119-127.
Comment
Gino Fano writes: The questions relating to the foundations of geometry, even if they do not form the subject of special teaching in Middle Schools, cannot fail to interest every teacher, since knowledge of them must inform and regulate their teaching work. And this work is of the highest importance for scientific culture and for the future of the country: because through Middle Schools pass (with rare exceptions) all those who will one day occupy the highest positions and offices; and the majority of these, dedicating themselves immediately afterwards to special studies, will retain as the first basis and guiding line of scientific thought, recognisable at times in all subsequent work, precisely that which they will have drawn from Middle School (unless this basis is not able to form itself, but this also by way of exception). Even more so today, despite the eminent value and usefulness of classical teaching, the centre of gravity of Middle School tends to shift from the literary field towards the scientific field, the teaching of mathematics, which constitutes the skeleton of the entire scientific part, cannot help but assume ever greater importance.
Gino Fano, Superficie algebriche di genere zero e bigenere uno, e loro casi particolari, Rendiconti del Circolo Matematico di Palermo 29 (1910) 98-118.
Gino Fano, A proposito dell'apparecchio elicoidale per volte oblique, Rendiconti del Reale Istituto Lombardo di Scienze e Lettere (Milano) 43 (1910) 177-179.
Gino Fano, Scuola operaia serale femminile - Relazione 1909-1910, Unione femminile Nazionale. Sezione di Torino (Tip. Ditta G. Derossi, Turin, 1910).

1911
Gino Fano, Matematica esatta e matematica approssimata, Bollettino della Mathesis 3 (1911), 106-126.
Comment
Gino Fano begins as follows: It has long been rightly claimed that mathematics, with its methods, has given other sciences, principally astronomy, geodesy, mechanics, physics, a powerful instrument of research and progress; and other sciences still, e.g. chemistry, the natural sciences, sociology, actuarial sciences, as the prevalence of quantitative problems over purely qualitative questions becomes accentuated in them - that is, the search, in their phenomena, for the numerical element and the law that governs it - they too tend to appropriate those same methods and to make the most of them. Infinitesimal calculus, while it permitted, in the eighteenth century, the first truly grandiose applications, had also raised ever greater hopes.

1914
Gino Fano, La filosofia contro la scienza, Lettera ad Achille Loria, Nuova Antologia di lettere, scienze e arti (1914).
Comment
Gino Fano responses to the arguments which Prof Loria developed in the journal "Nuova Antologia di lettere, scienze e arti" relating to the critical attitude of some philosophical positions towards science. Fano writes: I would like you to allow me to add a few words to the criticisms and anti-criticisms reported in your interesting article. Some of those judgments leave room for more or less broad interpretations, which can also distort the concept. When, from a certain number of individual observed facts, one arrives at a "scientific fact", one abstracts not only from the subjective conditions under which each of the first ones has revealed itself to our senses, but also from other circumstances that, however objective, the scientist's criterion has judged accidental. The scientific fact therefore has the character of a simplified, schematic fact, a type of a series of possible facts. A theory associates several scientific facts; it subordinates them to a number, as limited as possible, of primitive concepts; it investigates which other scientific facts descend from the first ones as consequences - this prediction being the primary purpose of every theory - and, to confirm such predictions, it seeks whether it is possible to observe, or experimentally provoke, course facts that agree more or less with them. But all scientific knowledge is, by its very nature, schematic, simplified, approximate knowledge with respect to reality.

1915
Gino Fano, Osservazioni su alcune varietà non razionali aventi tutti i generi nulli, Atti Reale Accademia delle Scienze di Torino 50 (1915), 1067-1072.
Gino Fano, Sulle varietà algebriche a tre dimensioni a superficie-sezioni razionali, Annali di Matematica Pura ed Applicata 24 (1915), 49-88.
Gino Fano, Osservazioni sopra il sistema aggiunto puro di un sistema lineare di curve piane, Rendiconti del Circolo Matematico di Palermo 40 (1915), 29-32.
Gino Fano, Sui fondamenti della geometria, Rivista di Filosofia (1915).
Gino Fano, Il confine del Trentino e le trattative dello scorso aprile con la monarchia Austro-Ungarica (Armani & Stein, Rome, 1915), 1-10.
Comment
Lecture held at the Turin Cultural Society on 11 June 1915 by Prof Gino Fano the Royal University of Turin.

In the World War I (1914-18), an interventionist from the very beginning, Gino Fano took off the clothes in which he normally lectured and put on his military uniform. In this propaganda speech he said he wanted to bring his contribution to the "spiritual assistance of the nation; ... that is, to maintain the public spirit in full and continuous agreement with the supreme directives" of the government. Here is the beginning of his lecture:

This lecture, held at the Turin Cultural Society, part of the series of lectures it has organised, also makes its contribution to what has already been called the Spiritual Assistance of the Nation; that is, to work to maintain the public spirit in full and continuous agreement with the supreme directives of this historic moment. To this end, it is important that many facts and problems, of various natures, be well clarified before the conscience of the country, so that the profound reasons, ideal and political, that pushed Italy towards the current path appear ever better in their true light of justice and civilisation.

In relation to such a vast and complex undertaking, my task today is limited and very modest. In undertaking it, I am supported by the thought that this national work of assistance and propaganda can only be the result of a very large number of small efforts, all equally inspired by love for the country and by the awareness of a common, very high purpose. In the negotiations with the Austro-Hungarian Monarchy, which took place in the past months and more particularly last April, proposals and counterproposals were put forward with regard to Trentino for three different border lines, all intermediate between the political border established in 1866 and the geographical border traced by the Alpine chain. Although these negotiations refer to a time now surpassed, it will perhaps not be useless to examine and illustrate those different proposals and the probable reasons for the individual requests and offers: in this way the profoundly different spirit with which the negotiations were conducted by the two parties will become clear; and while the conviction of the essential necessity of the war undertaken will acquire ever more solid and secure roots, greater knowledge will also spread of one of the regions that none of us doubt will shortly integrate and seal the national unity.

His Excellency Salandra [Antonio Salandra, the prime minister of Italy] said well in his Capitoline Oration, recalled here the other evening by the dazzling and highly patriotic words of an illustrious colleague of mine, that the objectives of Italian politics in the past negotiations could be summarised as follows:

1st. The defence of Italianness, our first and greatest duty;
2nd. A secure military border, which replaces the one imposed on us in 1866, for which all the doors of Italy are open to our adversary;
3rd. A strategic position in the Adriatic less insecure and less unfortunate than the present one.

Today's theme will lead me to speak of the first two of these only. And, with regard to the first of the two, it is first necessary to establish how far the Italian populations extend, if and where it is possible to draw a line which corresponds with sufficient approximation to the linguistic and ethnographic border.

The linguistic border.

The wide basin of the Adige, with the surrounding region, has for centuries been the scene of a formidable clash between the Latin populations emanating from Rome and the Germanic populations tending towards the south. The balance between these two opposing currents was established with a dividing line, in a general west-east direction, and in its almost totality quite clear, such that all the populations of the region south of this line remained Latin, and are therefore Italian today, while those of the region north of the line are overwhelmingly German. This line is made up of two important mountain ranges, for a good stretch over 2000 m high, one of which, to the west, forms the watershed between the Noce (val di Sole, val di Non) and the Adige (val Venosta), and the other, to the east, between the Avisio, a tributary of the Adige, and subsequently the Cordevole and Boite, tributaries of the Piave, on one side, and the Adige itself with the Isargo and the Rienza on the other. The first of these watersheds separates itself from the current political border at Mount Cevedale; the second at its easternmost point (peaks of Lavaredo) also reaches the political border, which shortly after also becomes a geographical border. In between the two is the Adige corridor ...

1918
Gino Fano, Sulle varietà algebriche a tre dimensioni a superficie-sezioni razionali, in Scritti matematici offerti ad E D'Ovidio (Turin, 1918), 342-363.

1919
Gino Fano, L'opera del Comitato regionale di mobilitazione industriale per il Piemonte: settembre 1915-marzo 1919 (Tip. Giani, Turin, 1919).
Comment
Gino Fano dedicated four years, from September 1915 to March 1919, to the direction of the regional committee for industrial mobilisation for Piedmont. An excellent expert in recent and contemporary political history, he gave various propaganda speeches, including a lecture at the Turin Cultural Society entitled The border of Trentino and the negotiations of last April with the Austro-Hungarian monarchy (June 1915). This 127 page book is the report of the work of the Comitato regionale di mobilitazione industriale per il Piemonte during the four years 1915-1919 during which time it was led by Fano.

1920
Gino Fano, Superficie del IV ordine con gruppi infiniti discontinui di trasformazioni birazionali. Nota I, Rendiconti Accademia Nazionale Lincei 291 (1920), 408-415.
Gino Fano, Superficie del IV ordine con gruppi infiniti discontinui di trasformazioni birazionali Nota II, Rendiconti Accademia Nazionale Lincei 291 (1920), 485-491.
Gino Fano, Superficie del IV ordine con gruppi infiniti discontinui di trasformazioni birazionali Nota III, Rendiconti Accademia Nazionale Lincei 292 (1920), 113-118.
Gino Fano, Superficie del IV ordine con gruppi infiniti discontinui di trasformazioni birazionali Nota IV, Rendiconti Accademia Nazionale Lincei 292 (1920), 175-182.
Gino Fano, Superficie del IV ordine con gruppi infiniti discontinui di trasformazioni birazionali Nota V, Rendiconti Accademia Nazionale Lincei 292 (1920), 231-236.
Gino Fano, A proposito di un articolo del giornale "La Sera", Bollettino della Mathesis (1920), 128-131.

1921
Gino Fano, Le Scuole di Magistero, Relazione al Congresso della Società Italiana Mathesis, Periodico di Matematiche 2 (1922), 102-110.
Gino Fano, Aufgaben aus der darstellenden Geometrie : für studierende der technischen hochschulen (Speidel & Wurzel, 1921).

1922
Gino Fano, Sur la congruence des normales à une quadrique, Comptes Rendus de l'Académie des Sciences 176 (1922), 1866-1868.
Comment
The paper gives a synthetic-geometric proof of the theorem discovered by d'Ocagne about the normals of a central surface of degree two along its lines of curvature, and additional remarks thereto, which also extend to the case of the paraboloid.

1923
Gino Fano, A Series of Special Lectures on Italian Geometry. 'Italian Geometry', 'Intuition in mathematics' and 'All geometry is theory of relativity' (W B Walker, 1923).
Comment
In 1923 Fano was invited to hold a series of lectures on "Italian geometry" at the University College of Wales, in Aberystwyth. As well as that series, he gave two lectures intended for a wider audience for which he chose a theme that is directly linked to the Erlangen Program. In 'All Geometry is theory of Relativity', and, 'Intuition in Mathematics', he developed ideas which it is very easy to see have points of view with a clear Kleinian flavour.

In Intuition in mathematics he proposes to examine the question of whether the work of the mathematician is merely logical work. He adopts a historical approach in order to show that a rigorous treatment can only be applied to well-defined concepts and therefore to materials that our mind has already processed. He cites various examples starting from Euclid's "Elements" which with their deductive logical arrangement appear as the final result of a long period of discovery and elaboration, down to the critical studies of the nineteenth century, which led to the creation of mathematical logic and to its symbolism. He then focuses on another trend that emerged at the same time in which a "philosophic and intuitive spirit" prevails, mentioning the most representative figures, Riemann and Klein. It is clear, however, that among mathematical inventors and mathematical demonstrators, whom he defines as skilled technicians, his preferences go to the former because in his opinion it is intuition, "a pioneer of progress," that opens the way to logical developments.

The lecture, All geometry is theory of relativity, starts from two questions: What is a geometric figure? What is a geometric property of this figure? To answer the first one, Fano observes that the geometric objects (a point, a straight line, etc.) that make up a figure are abstractions obtained from reality. To answer the second one, he introduces the concept of transformation group and cites Klein's Erlangen program, which classifies geometries according to the invariant properties for a particular group of transformations. Hence, the meaning of the title is clear: a geometry is something related to a group of transformations. To illustrate this statement, he presents examples for the various branches of geometry - elementary, projective, and topology - and also cites non-Euclidean geometries, and to make himself clear he resorts to commonly used objects like rubber bands, ribbons, and pancakes.
Gino Fano, Vedute matematiche su fenomeni e leggi naturali, in Annuario Reale Università di Torino 1922-23 (Schioppo Torino, 1923), 15-45.
Comment
Speech read at the Royal University of Turin at the inauguration of the academic year 1922-23 on 6 November 1922 by Prof Gino Fano, full professor of projective and descriptive geometry with drawing. The speech begins:

Excellencies, Ladies, Gentlemen,
The eternal debate on the importance to be attributed to scientific studies in comparison to classical studies was championed one day in France, around 1840, by a distinguished astronomer and physicist, Francesco Arago, and the illustrious author of the 'Méditations poétiques', Alfonso Lamartine. Arago believed he had nailed his adversary to the wall, shouting: "It is not with your beautiful words that sugar is made from beetroot; nor with Alexandrine verses that you will extract soda from sea salt!" And he added: "It is science that has destroyed prejudices forever!" Lamartine did not deny that science was useful; but: "If the human race," he said, "were condemned to lose entirely one of these two orders of truth, or all mathematical truths, or all moral truths, it should not hesitate to sacrifice mathematical truths; because, if all mathematical truths were lost, industry, the material world, would undoubtedly suffer immense damage; but if man were to lose even one of the moral truths acquired through literary studies, this would mark his death and that of all humanity". And he concluded by praising languages "which you call dead, and I call immortal".

In this speech Fano reveals his vision of science and its role not only in education but also in the development of society. His vision is not utilitarian nor even technical, but, he says, "science implies above all theoretical research, disinterested, which cannot be bound by practical concerns and utilitarian considerations; investigation of nature, discovery of its most general laws, of the links, often hidden and intimate, between different orders of phenomena ... And serious deep scientific research requires a whole complex of spiritual gifts, which are consolidated and above all refined only with progressive mental training; the ability to analyse facts minutely, from various points of view; the ability to grasp analogies, and to raise them to ingenious syntheses: a shrewd combination of observation and reasoning; all integrated by a broad, multifaceted culture, and therefore incompatible with an excessively precocious technical specialisation. - Ancient Greek philosophy, the initiator of the intellectual life of the world, can largely contribute to the formation and improvement of this scientific spirit, which offers us examples of profound analysis and criticism, speculations that sometimes, in a primordial form, admirably approach modern views; expression of an intense, continuous, tenacious effort, aimed at grasping the reality of nature, possibly coordinating the notions acquired by the senses ..."

Fano then reviews the progress made by science over the centuries and its ability to represent reality. He also demonstrates full knowledge and understanding of the latest developments (atomic theory of matter, radioactivity, relativity and the first approaches to quantum mechanics), focusing in particular on statistical physics in the description of thermodynamics, closely linked to the mathematics of probability calculus.

1924
Gino Fano, Sulle forme binarie per le quali una delle spinte su sé stesse sia identicamente nulla, Giornale di Matematiche 62 (1924), 91-98.
Gino Fano, I gruppi di trasformazioni nella geometria, Scientia (18) 36 (1924), 145-154.
Comment
The paper begins: What answers would the kind readers of "Scentia" give to the questions: What is a geometric figure? What is a geometric property of this figure? The answer to the first question does not seem difficult; indeed, we all have an idea of it, more or less determined; it is only necessary to make it concrete in precise terms. From the whole complex of objects and phenomena that fall, in any way, under our senses, we, by abstraction, by means of comparisons between some of them, intended to highlight what is in common in them, by elaboration of our mind, derive, as is known, concepts; and among the simplest concepts - simplest, because they require only observations relative to space, and not, for example, also of time, energy, or other elements - there are the geometric concepts: point, line, plane. A geometric figure, in the most general sense, is any set of these elements, in a finite or infinite number; for example, triangles, quadrangles, and other plane polygons studied by elementary geometry are figures of points (vertices) and straight lines (sides); the circumference is a figure of infinite points; etc.)

Less simple is the answer to the other question: What is a geometric property of a figure? The best answer (which I hope will be clarified by the present article) starts from the concept of group of transformations (or operations); a fundamental concept that, for about a century now, has been acquiring, in mathematics, ever greater importance, as an informing and coordinating principle of old and new theories. And this is what, for geometry and its division into various branches or branches, F Klein brought to light, with a brilliant vision, 50 years ago.
Gino Fano, L'analysis situs I, Lo studio intuitivo del continuo, Scientia (18) 36 (1924), 217-230.
Comment
The paper begins: After having given a hint, in a previous article, of some geometric groups and the corresponding geometries, I propose to now discuss a further geometry, simple in its first elements, and important in itself and for its relations with other theories. Let us consider a very thin thread (corresponding, therefore, with great approximation, to the intuitive concept of a line) and elastic, which can be curved, lengthened, shortened at will; and this, in each single segment, independently of the other segments. With respect to these deformations of the line, the distance of two points measured along the line itself is certainly not invariant, nor is the ratio of two similar distances; nor is the concept of a straight line. But if a point, on the thread, is found between two other determined points, even after the thread has been deformed in any way, it will continue to be found between them; the property of a point of being intermediate between two others, on the line, and, more generally, the natural order, or rather the two natural orders opposed to each other according to which the points on the line are followed, have an invariant character with respect to those deformations.
Gino Fano, L'analysis situs II, L'indirizzo combinatorio, Scientia (18) 36 (1924), 289-300.
Comment
The paper begins: The questions concerning the topological characteristics of a surface, mentioned in the previous article, correspond to a first stage of the Analysis Situs, which could be described as an elementary and intuitive phase, since the postulates that form the basis of the theory were not always well established, much less all explicitly stated; and even in the demonstrations recourse was made several times to intuitive considerations. The very concepts of the "continuum", of continuous deformation, of connection, had not yet been subjected to rigorous criticism. Indeed, it has recently been said that it was a kind of experimental science, based on paper, rubber and scissors, which nevertheless rendered useful services to mathematics. Later, an attempt was made to give the Analysis Situs precise foundations, from which the entire theory could be derived as a pure logical consequence.
Gino Fano and Corrado Segre, Cenno necronologico, Annuario Reale Università di Torino 12 (1924), 219-228.

1925
Gino Fano, Sulle superficie dello spazio $S_{3}$ a sezioni piane collineari, Rendiconti Accademia dei Lincei (6) 1 (1925), 473-477.
Gino Fano, Lehrbuch der allgemeinen arithmetik (Speidel & Wurzel, 1925).
Gino Fano, Lezioni di Geometria Descrittiva (Politecnico di Torino, Turin, 1925).
Comment
This is produced with lithographic printing.

1926
Gino Fano, Sulle superficie di uno spazio qualunque a sezioni piane collineari, Mem. Accademia dei Lincei (6) 2 (1926), 115-129.
Gino Fano, Sulle superficie di uno spazio qualunque a sezioni omografiche, Bollettino dell'Unione Matematica Italiana 5 (1926), 164-167.
Comment
The paper begins: In a work that will be included in the Memoirs of the Royal Academy of the Lincei, I have determined all the surfaces of any space, having the 2 by 2 homographic curve-sections; a problem that Prof Fubini and I had already solved last year, limited to the surfaces of 3-dimensional space. Here is a brief outline of the procedure used and the result obtained.
Gino Fano, La varietà delle forme binarie del $7^{o}$ ordine a sesta spinta identicamente nulla, Rendiconti Accademia Nazionale Lincei 4 (1926), 161-166.
Gino Fano, Onoranze a Corrado Segre, discorso commemorativo, Suppl. Rendiconti del Circolo Matematico di Palermo XV (1926), 3-11.

1927
Gino Fano, Intenti, carattere e valore formativo della matematica, in Alere Flammam 7 (Turin, 1927), 1-26.
Comment
This lecture by Fano was at the war school on 15 March 1924.
Gino Fano, Les cycles de la géométrie non euclidienne au point de vue projectif, in In memoriam N I Lobatschevskii 2 (Glavnauka, Kazan, 1927), 17-24.
Comment
This paper is in a collection of memoirs presented by scientists from various countries to the Kazan Physico-Mathematical Society on the occasion of the celebration of the centenary of the discovery of non-Euclidean geometry by N I Lobachevsky, held 12-24 February 1926. The paper begins: On the occasion of the centenary of the day when the famous Russian geometer N I Lobachevsky expounded at the Physiko-Mathematical Faculty of Kazan University the first principles of his new Geometry, I take the liberty of presenting some considerations concerning the different "cycles" (circles; limit curves, or horicycles; hypercycles, or lines of equal distance from a straight line) that are encountered in this Geometry, and which from the projective point of view are all, as is now well known, conics. The analogy between these different kinds of lines does not yet appear completely if we remain within the field of elementary metric geometry; while by quite simple considerations it is possible to show that they can all be generated by projective sheaves of straight lines. This is, in my opinion, a modest contribution that projective geometry can well make to Lobachevsky's Geometry, without changing anything in the general lines of its development.

1928
Gino Fano, Un esempio di trasformazione birazionale cubica inerente a un complesso lineare, Rendiconti Accademia Nazionale Lincei 9 (1928), 16-19.
Comment
The paper begins: In this Note I give an example of a birational transformation of space inherent in a linear complex. The transformation is very well known; but this particular property of it has not yet been, as far as I know, explicitly noted.
Gino Fano, Trasformazioni di contatto birazionali del piano, Rendiconti Accademia Nazionale Lincei 8 (1928), 445-451.
Comment
The paper begins: Contact transformations, in the plane, in space, and also with several independent variables, constitute a brilliant creation of S Lie, who studied them almost exclusively from the differential point of view, that is, on the hypothesis that the functions that appear there are only subject to having regular behaviour in the neighbourhood of a generic point of the region considered. They can also be studied in the algebraic or birational field; but they have not been studied in depth in this field so far. The main group of works on the subject is that of L Autonne, who in two Memoirs of 1887-88 analytically set out the problem of birational contact transformations of the plane, and studied the most elementary cases; and in subsequent works, culminating in a Memoir of 1905, he began the same research for ordinary space and for spaces with several dimensions.

In this Note, and in some others that will follow, I propose to place on a geometric basis the problem of birational contact transformations of the plane, and to show how this problem, by means of a remarkable representation also due to S Lie, is transformed into an interesting problem of Cremonian transformations of three-dimensional space.
Gino Fano, Sulla rappresentazione di S Lie degli elementi lineari del piano sopra lo spazio punteggiato, Rendiconti Accademia Nazionale Lincei (1928), 529-534.
Comment
The paper begins: S Lie gave a representation of the system of linear elements of a plane on the punctuated space, which is algebraic and biunivocal, hence birational, and can be exploited for the further study of the birational contact transformations of the plane.
Gino Fano, Congruenze _ di curve razionali e trasformazioni cremoniane inerenti a un complesso lineare, Rendiconti Accademia Nazionale Lincei (1928), 623-627.

1929
Gino Fano and Alessandro Terracini, Lezioni di geometria analitica e proiettiva (G B Paravia, Turin, 1929).
Comment
Both authors, Fano and Terracini, had already individually published the Lezioni di Geometria analitica e proiettiva [Analytical and projective geometry lectures], with the A Viretto lithographer in Turin, respectively, in 1926-1927 and in 1927- 1928, and just in 1927-1928, they had both taught at the Polytechnic of Turin, the former responsible for descriptive geometry with applications and the latter for analytic and projective geometry; therefore, their collaboration was greatly facilitated.

1930
Gino Fano, Sulle curve algebriche contenenti serie autoresidue rispetto alla serie canonica, Rendiconti Reale I Istituto Lombardo. Accademia di Scienze e Lettere (2) 63 (1930), 949-967.
Gino Fano, Sulle sezioni spaziali della varietà grassmanniana delle rette dello spazio a cinque dimensioni, Rendiconti Accademia Nazionale Lincei (1930), 329-335.
Gino Fano, Osservazioni sopra una nota del prof H F Baker, Rendiconti Reale I Istituto Lombardo. Accademia di Scienze e Lettere (2) 65 (1930), 93-96.
Comment
Fano's gives the following Abstract: Some results contained in a work by Prof. Baker (Journal London Mathematical Society 6 (1931), 176-185) are completed and shed more light on, relating to the surface $F^{5}$ of $S_{5}$ of Dei Pezzo and to a variety studied by Prof U Perazzo.

Note. Baker's paper which Fano refers to is Segre's Ten Nodal Cubic Primal in Space of Four Dimensions and del Pezzo's Surface in Five Dimensions which appears with date July 1931. Dates can be confusing for although Fano's paper appears in a Volume dated 1930, it was submitted on 21 July 1932.
Gino Fano, Reti di complessi lineari dello spazio S5 aventi una rigata assegnata di rette-centri, Rendiconti Accademia Nazionale Lincei II (1930), 227-232.
Comment
The paper begins: The geometry of lines, in particular of linear complexes of lines and their bundles, in any space has already been studied quite extensively. In 5-dimensional space, a linear complex of lines is in general devoid of singular points (general complex), but it can also have a centre line or a space $S_{3}$ -centre, loci of singular points (singular complexes, or degenerate complexes, respectively of the 1st and 2nd kind).

1931
Gino Fano, Spazi di Riemann e geometrie riemanniane. Loro generalizzazioni, Conferenze di Fisica e di Matematica Università e Politecnico di Torino (1931-32), 17-60.
Comment
From a review: A general overview of Riemannian geometry and its newer generalisations. The author chooses the internal (metric) geometry on a surface according to Gauss as his starting point. From there he goes on to present the ideas on which the n-dimensional geometry according to Riemann can be developed. He then comes to the basic facts for the formal structure of Riemannian geometry and then to tensile algebra and analysis, and then to the concept of parallelism by Levi-Civita. He sticks to the historical development, although this is probably no longer the easiest way of presenting it. At the end he gives an overview of the general theory of relativity according to Einstein and of the theory of spaces with affine (symmetric and non-symmetric) connection in the sense of Weyl and Cartan.
Gino Fano, Sulle varietà algebriche a tre dimensioni aventi tutti i generi nulli, Atti del Congresso Internazionale Dei Matematici Bologna, 3-10 September 1928 Vol 4 (Nicola Zanichelli, Bologna, 1931), 115-121.
Comment
Fano's paper begins: The distinction, which seemed traditional, between sciences of reasoning and experimental sciences is now obsolete. In every science experience and reasoning have a part; the distinction concerns only the reciprocal proportions. In mathematics the part reserved for experience, small and limited to the phase of discovery, consists essentially in the careful examination of some particular case. I propose precisely to set forth here the result of a little experimental work, and of some further conjectures, concerning a difficult and important question, which has long awaited a solution in vain.
Gino Fano, Transformazioni di contatto birazionali del piano, Atti del Congresso Internazionale Dei Matematici Bologna, 3-10 September 1928 Vol 4 (Nicola Zanichelli, Bologna, 1931), 35-42.
Comment
Fano's paper begins: Contact transformations in the plane, in space, and also in several independent variables, are a brilliant creation of S Lie, who studied them exclusively, or almost, from the differential point of view, that is, on the hypothesis that the functions appearing there are any functions, subject only to having regular behaviour in the neighbourhood of a generic point of the region considered. They can also be studied in the algebraic field and in the birational field, in which however they have not been studied in depth up to now. As far as I know, the main works on the subject are those of L Autonne, who, in two Memoirs of 1887-88, analytically set out the problem for the plane, studying some elementary cases; and in subsequent works he also began research for space with three or more dimensions, studying among other things the different types of varieties, called by him "primordiales", which correspond to points and hyperplanes as figures of elements, and which give rise to a greater number of cases as the dimension of the ambient space increases.

1932
Gino Fano, Trasformazioni birazionali sulle varietà algebriche a tre dimensioni di generi nulli, Rendiconti Accademia Nazionale Lincei 15 (1932), 3-5.
Comment
A review states that the paper proves the theorem "A regular algebraic manifold whose genuses are all zero and which admits a continuous group of birational transformations within itself is either rational or birationally equivalent to a cone that projects a regular surface of genus zero and positive bigender. An involution in $S_{3}$ is rational or irrational, depending on whether the $V_{3}$ representing this involution admits a finite continuous group of birational transformations within itself or not. In particular, an irrational involution in $S_{3}$ cannot be mapped onto itself by a finite continuous group of birational transformations."

In the proof, the author relies on a paper published jointly with Enriques and on his work in Atti Congresso Bologna.
Gino Fano, Geometria proiettiva in Enciclopedia italiana di scienze, lettere ed arti XVI (Istituto della Enciclopedia Italiana, Rome, 1932), 630-633.
Gino Fano, Gli indirizzi geometrici moderni in relazione ai gruppi di trasformazioni, in Enciclopedia italiana di scienze, lettere ed arti XVI (Istituto della Enciclopedia Italiana, Rome, 1932), 633-637.
Gino Fano, Nuovi metodi e nuovi indirizzi fino a circa la metà del secolo XIX, in Enciclopedia italiana di scienze, lettere ed arti XVI (Istituto della Enciclopedia Italiana, Rome, 1932), 627-630.

1933
Gino Fano, Prof Enrico D'Ovidio, Annuario della R. Università di Torino 9 (1933).

1934
Gino Fano, Enrico D'Ovidio, Bollettino dell'Unione Matematica Italiana 12 (1934)153-156.
Comment
In this obituary of Enrico D'Ovidio, Fano gives his biography. He was born on 11 August 1843 in Campobasso, received his doctorate in 1868 in Naples, where he was a student of Achille Sannia in particular, taught at high schools until 1872, held the chair for algebra and projective geometry at the University of Turin from 1872 until his retirement in 1918, died in Turin on 21 March 1933. Fano gives an appreciation of D'Ovidio's scientific achievements and his services as an academic teacher. The obituary is accompanied by a list of D'Ovidio's scientific publications.
Gino Fano, Scorrendo il volume di F Klein: "Vorlesungen ueber die Entwicklung der Mathematik im XIX Jahrhundert", Conferenze di Fisica e di Matematica Università e Politecnico di Torino 4 (1934), 151-171.
Comment
Gino Fano reports on the first volume of Klein's lectures (1926) and relates some particularly interesting biographies and remarks on German and French mathematics.

1935
Gino Fano A proposito della nota del prof Majorana: Sull'insegnamento della fisica in Italia, Nuovo Cimento 12 (1935), 49-51.
Gino Fano, Geometria non euclidea: introduzione geometrica alla teoria della relatività) (Nicola Zanichelli, Bologna, 1935).
Comment
Introduction. - The name of non-Euclidean geometry designates some (essentially two) scientific constructions, which originated from critical research on fundamental propositions of ordinary (Euclidean) geometry, and which were first concretised in suppressing or modifying some of these propositions, particularly Euclid's 5th postulate or the postulate of parallels, and in developing the logical consequences of the premises thus modified. These same constructions were then approached also from other points of view, thus managing to frame them in the complex of modern mathematics, to recognise their full importance, and at the same time to throw more light on the critical questions that had given rise to them. The development of non-Euclidean geometry is also connected to the affirmation of new ideas and new scientific views, which have profoundly modified pre-existing ideas and views on geometry, physics, and the whole of scientific philosophy. From these comes the consideration of geometry as an experimental science, part of physics. With them, through the theory of special and general relativity of A. Einstein, the view matures that geometric, mechanical, physical knowledge does not have an existence and a meaning in itself, but only in its synthesis.
Gino Fano, Complementi di geometria (G.U.F., Turin, 1935).
Comment
This work is produced with lithographic printing.

1936
Gino Fano, Superficie algebriche e varietà a tre dimensioni a curve-sezioni canoniche, Rendiconti Accademia Nazionale Lincei 23 (1936), 813-818.
Gino Fano, Su alcune varietà algebriche a tre dimensioni aventi curve sezioni canoniche, in Scritti mat. offerti a L Berzolari (Istituto Mat. R. Univ., Pavia, 1936), 329-349.
Gino Fano, A proposito di un lavoro del sig. Ramamurti (Sulle rigate razionali normali), Atti della Reale Accademia delle Scienze di Torino 71 (1936) 105-109.
Comment
Summary. - Taking advantage of a Note by Mr Ramamurti, some observations are made on the system of normal rational ruled lines $R^{n-1}$ of $S_{n}$ having a curve $C^{n}$ assigned as a directrix, and on the minimum directrixes of these ruled lines.
Gino Fano, Osservazioni su alcune "geometrie finite" I, Rendiconti Accademia Nazionale Lincei 26 (1936) 55-60.
Comment
From a referee's report: Following his earlier work "Sui postulati fondamentali della geometria proiettiva ...", Gino Fano develops in these two notes a general, very transparent geometric method for the systematic construction of the incidence tables of those finite projective geometries of the Veblen systems $S(n^{2}-n+1|2|n)$ , for which the Fano axiom $F_{1}$ on the curvature of the adjacent vertices of a complete quadrilateral is not fulfilled. These are the Veblen systems $S(7|2|3), S(21|2|5), S(73|2|9), ... , S(n^{2}+n+1|2|n+1)$ with $n = 2^{m}, m > 2$ . - This method can also be used analogously for the other Veblen systems $S(n^{2}+n+1|2|n+1)$ with $n = p^{m}$ ( $p$ prime number), as shown by the example of the system $S(91|2|10)$ .

The Fano method is partly similar to the combinatorial method developed earlier by W Heuser in his Heidelberg dissertation inspired by H Liebmann and M Steck (see the term "diagonal combination" there and compare the Fano principal matrix and its submatrices), but is more general, purely geometric and completely transparent. - With the two Fano notes, the problem of the purely formal, explicit arrangement of the incidence tables of the Veblen systems mentioned can be considered as finally solved. The rewriting of the tables thus obtained into involutory-reciprocal ones (mirror-image arrangement of the incidence symbols to a main diagonal) can be done with the help of the "reflection principle" given by M Steck and, geometrically speaking, amounts to subjecting the Veblen system resulting from the Fano method to a polarity or reciprocity. This is the geometric meaning of M Steck's reflection principle, as can be seen immediately.
Gino Fano, Osservazioni su alcune "geometrie finite" II, Rendiconti Accademia Nazionale Lincei 26 (1936), 129-134.
Comment
See referee's report above.

1937
Gino Fano, Sulle varietà algebriche a tre dimensioni a curve-sezioni canoniche, Mem. Accademia d'Italia 8 (1937), 23-64.
Comment
Summary. - The varieties indicated in the title are studied, of the type $M_{3}^{2p-2}$ in the space $S_{p+1}$ , $p$ being the genus of curve-sections. They exist only for $p < 37$ , and for $p > 10$ they are rational, except for a single case $p = 13$ , still of dubious rationality. Among them, those of the higher orders are particularly determined, together with the linear surface systems of genera one that represent them on the space $S_{3}$ .
Gino Fano, Sulle varietà algebriche a tre dimensioni a curve sezioni canoniche, Atti I Congr. U.M.I. Firenze (1937), 245-250.
Comment
Gino Fano begins as follows: I dedicated a preliminary communication to the varieties indicated in the title, of the type $M_{3}^{2p-2}$ of $S_{p+1}$ , not cones, with sectional surfaces having all genera equal to unity, at the International Congress of Bologna in 1928. Today I add some further results, obtained in the last two years.

1938
Gino Fano, Sulle varietà algebriche a tre dimensioni le cui sezioni iperpiane sono superficie di genere zero e bigenere uno, Mem. Soc. It. d. Scienze (3) 24 (1938), 41-66.
Gino Fano, Geometrie non euclidee e non archimedee, in Enciclopedia delle Matematiche elementari II-2 (Hoepli, Milan, 1938), 435-511.

1940
Gino Fano, Quelques remarques à propos d'une note de M Amin Yasin, Comptes Rendus Acad. Sciences 210 (1940), 284-285.
Comment
E G Togliatti writes in a review: A Yasin Amin has constructed 991 tangent conics of a plane C5. The author notes that A Yasin Amin's method can be obtained from a method by F P White simply by projection from the space S_4 into three-dimensional space. The position of the 991 conics thus obtained among all tangent conics of $C^{5}$ has been investigated by the author himself and by W L Edge.
Gino Fano, Sulle curve ovunque tangenti a una quintica piana generale, Commentarii Mathematici Helvetici 12 (1940), 172-190.
Comment
Gino Fano begins the paper as follows: F P White, in a work of about ten years ago, after having considered a general plane quintic and certain pairs of conics tangent to it at 5 points, and having related this figure to a general cubic form of 4-dimensional space, adds: "I have not succeeded in discovering how the remaining (1024) conies arise from the four-dimensional figure." And Prof W P Milne recently repeated the same question to me.

The remaining conics (and pairs of conics) mentioned above appear rather in analogous relation to other cubic forms, and to other figures of 3 or 4 dimensions, as is shown in the present Note (which also treats some other questions, connected with the previous one).
Gino Fano, Su alcune particolari reti di quadriche dello spazio ordinario, Universidad Nacional de Tucumán Revista A 1 (1940), 271-281.
Comment
The paper begins: In a work in press in the Commentarli Mathematici Helvetici I have considered the systems $∞^{4}$ of plane quartics $(C^{4})$ everywhere tangent to a general quintic $C^{5}$ (that is, tangent to this in 10 points, generally distinct), distinguishing those (2015 in number) whose curves have the 10 points of contact on a cubic, and which I have called of the 1st type, from those (2080 in number) which do not enjoy this property, and which I have called of the 2nd type.
Gino Fano and Alessandro Terracini, Lezioni di geometria analitica e proiettiva (G B Paravia, Turin, 1940).
Comment
Gino Fano and Alessandro Terracini were both Jewish mathematicians so it is reasonable to ask how their 1929 book Lezioni di geometria analitica e proiettiva was republished in a new edition in 1940 after the racial laws were passed. In a letter to Terracini, Fano explains why, despite the vetoes placed on books by Jewish mathematicians, this second edition was published:
In 1938 the 2nd edition was started, with slight adjustments on stereotype prints. In September, as soon as the storm broke out, the company wrote to me, asking if it could definitely destroy all the material! I, of course, replied that there was no hurry - [I asked them] to wait in the meantime; and in one of my trips from Lausanne to Turin, I don't remember if in November 39 or 40, I went to talk about it with Comm. Tancredi, and I pointed out that the prohibition on reprinting only concerned textbooks, which at the University do not exist; and that our volume, a scientific treatise, could therefore be reprinted, provided it suited them. He said he agreed.
1941
Gino Fano, Sui cerchi ortogonali a due cerchi dati, Universidad Nacional de Tucumán Revista A 2 (1941), 87-94.
Comment
The paper begins: Bertrand Gambier, in a Note published two years ago in the Bulletin des Sciences Mathématiques, considered an interesting problem, susceptible of some further observation and development. Given two circles in distinct planes, not lying on the same sphere, and not linked ("enlacés"), that is, that do not meet the straight intersection of their planes in pairs of points that are both real and separating, he proposes to construct a further circle orthogonal to the first two (that is, that meets them in two points each, and at a right angle). To this end, considering the orthogonal sphere common to the two given circles (that is, orthogonal to the linear system ∞3 of spheres determined by the two bundles that have such circles as bases), a sphere that in the hypotheses made is real, he transforms this latter sphere into a plane $\pi$ by an inversion; therefore the two given circles into circles with centres in $\pi$ and contained in planes perpendicular to $\pi$ respectively according to their diameters $AA', BB'$ . Every circle perpendicular to these latter will also be orthogonal to the plane π; and it is easily recognized that the extremes $OC'$ of its diameter contained in $x$ will be, in this same plane considered as a real representative of a complex variable, the harmonic pair common to $AA', BB'$ . This being stated, analytically, the solution is immediate, unique, and real (in the plane) with the relative circle.

1942
Gino Fano, Osservazioni sulla rappresentazione di corrispondenze birazionali tra varietà algebriche, Commentarii Mathematici Helvetici 14 (1942), 193-201.
Gino Fano, Su alcune varietà algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche, Commentarii Mathematici Helvetici 14 (1942), 202-211.

1943
Gino Fano, Sulle forme cubiche dello spazio a cinque dimensioni contenenti rigate razionali del $4^{o}$ ordine, Commentarii Mathematici Helvetici 15 (1943), 71-80.
Comment
This paper is concerned with those cubic primals which contain a rational normal ruled surface $R^{4}$ of order 4.
Gino Fano, Superficie del 4o ordine contenenti una rete di curve di genere 2, Pontificiae Accademiae Scientiarum. Commentationes 7 (1943), 185-205.
Comment
Fano gives the following Abstract: The class of birational transformations, which pertain to surfaces of the fourth order containing a net of curves of the second kind, is determined. This class is infinite, however, if the curves of the second kind are of even order and not less than six; but if the curves are of odd order, the class is infinite only when a certain corresponding equation of the second degree between two variables admits of no integral solution.

1944
Gino Fano, Alcune questioni sulla forma cubica dello spazio a cinque dimensioni, Commentarii Mathematici Helvetici 16 (1944), 274-283.
Comment
Fano begins the paper as follows: In a Memoir of mine of 1904 I treated some questions of a projective nature on the general cubic form (or variety) of four-dimensional space. Some of them easily extend to higher spaces; not all. On the other hand, a recent work by E. G. Togliatti considered a particular surface of the 5th order of ordinary space, obtained as the apparent contour of a general cubic form of five-dimensional space with respect to a generic straight line, that is, as the locus of the traces on a fixed $S_{3}$ of the planes passing through this straight line and further meeting the cubic form in pairs of straight lines. And in this research, which essentially starts from the equation of the said surface, no mention is made of the relations between the properties of this surface and those of the cubic form mentioned above; important relations, since the former derive precisely from the latter. It therefore seems appropriate, also in view of possible further developments, to clarify these relations; and this also gives me the opportunity to extend to the cubic form of $S_{5}$ some of the properties set out in my Memoir cited.
Gino Fano, Osservazioni varie sulle sulle superficie regolari di genere zero e bigenere uno, Universidad Nacional de Tucumán Revista A 4 (1944), 69-79.
Comment
The paper begins: The first example of algebraic surfaces of geometric and numerical genus zero and non-rational was given in 1896 by Enriques; and they are the surfaces of the 6th order of three-dimensional space having as double lines the edges of a tetrahedron. Subsequently, such surfaces were characterised by him by means of the values $p_{a} = P_{3}= 0, P_{2} = 1$ of these invariants; which consequently leads to the fact that all odd plurigenera are equal to zero (including therefore the geometric genus $p_{g} = P_{1}$ ), and even plurigenera are equal to unity.

1945
Gino Fano, Nuove ricerche sulle varietà algebriche a tre dimensioni a curvesezioni canoniche, Pontificiae Accademiae Scientiarum Acta 9 (1945), 163-167.
Comment
Fano gives the following Abstract: According to other published studies by the same author, simple complete linear systems of the first genus are determined, which exist in algebraic varieties of three dimensions, whose curve-sections are conic, since such curve-sections belong to the fifth, sixth, or eighth genera: for apart from these cases, the rationality of the varieties is already certain.

All these systems are equivalent to the system of hyperplane sections of the same varieties, or another section of the same genus; but if the section is of the eighth genus, the system is also equivalent to the system of intersections of the general cubic form of space of four dimensions with quadrics.

From which it is clear that all such varieties are irrational, not excepting that cubic form, the question of which has been known to students of algebraic geometry for fifty years now.

1946
Gino Fano, Sulla forma cubica generale dello spazio a 4 dimensioni, Rend. Acc. Lincei (8) 1 (1946), 463-466.
Comment
The paper begins: The question of the rationality or otherwise of the general cubic form of 4-dimensional space, that is, of the 3-dimensional algebraic variety of the 3rd order of this space without double points, has arisen in geometry for over 50 years, arousing great curiosity, without having been resolved so far. As is known, the general plane curve of the 3rd order is not rational, while on the other hand every cubic surface of ordinary space is rational, except for the elliptic cone; for the general cubic forms of spaces of dimension ≥ 4 the question remained uncertain, with a rather negative presumption. For spaces of odd dimension, however, it has been known for some time that rational cubic forms exist, even if they lack a double point; however, these are all (as far as is known so far) particular forms: for example, in 5-dimensional space, cubic forms containing a surface also belonging to this space and whose system of $∞^{4}$ chords is of the 1st order (pair of independent planes, rational normal ruled of the 4th order).

1947
Gino Fano, Le trasformazioni di contatto birazionali del piano, Commentarii Mathematici Helvetici 20 (1947), 181-215.
Comment
The paper begins: Contact transformations in the plane and in three- or multi-dimensional space, although they have already been studied by other authors in particular cases, have been considered in all their generality, both from the analytical and geometrical side, by S Lie, who developed the theory mainly from the differential point of view, on the hypothesis that the functions appearing there are any analytical functions of the different variables; and therefore limited to a suitable neighbourhood of a generic point. The concept of infinitesimal contact transformation and the applications to the integration of differential equations are of particular importance; Lie himself has made of them a powerful instrument for further, fundamental research in analysis and geometry. - Contact transformations can also be considered in the algebraic field and in the birational field, applied in this case to the whole plane or the ambient space; but in these cases they have not been studied in great depth up to now. L Autonne in two Memoirs of 1887-88 has analytically set out the problem for the plane, studying some elementary cases; and in subsequent works) he also started research on three- or more dimensional space, considering the different types of varieties, called by him "primordiales", which correspond to points and hyperplanes as figures of elements, and which give rise to numerous cases as the dimension of the ambient space increases.
Gino Fano, Su alcuni lavori di W L Edge, Rend. Acc. Lincei (8) 3 (1947), 179-185.
Comment
Gini Fano begins the paper as follows: W L Edge in a group of works published in the years 1937-40 determined by geometric means some loci projectively related to a network of quadrics of the space $S_{3}$ , using them to find covariants and combinators of the first-member forms of the equations of those quadrics. Some of his results, however, are valid in more general hypotheses; and others, mixed with questions of lesser importance, do not always appear in full light.

1948
Gino Fano, Nuove ricerche sulle varietà algebriche a tre dimensioni a curve-sezioni canoniche, Pontificiae Accademiae Scientiarum Commentationes 11 (1948) 635-720.
Comment
Gino Fano's Summary reads: According to other published studies by Gino Fano, simple linear systems complete with perfection of the first genus are determined, which exist in algebraic varieties of three dimensions, whose curve-sections belong to the fifth, sixth, or eighth genera: for apart from these cases, the rationality or irrationality of varieties is already known.

All these systems are equivalent to a system of hyperplane sections of the same varieties, or other sections of the same genus; but if the sections are of the eighth genus, the system is also equivalent to a system of intersections of the general cubic form of space of four dimensions with quadrics.

From which it is clear that all such varieties are irrational, not excepting that cubic form, the question of which has been known to students of algebraic geometry for fifty years now.
Gino Fano and Alessandro Terracini, Lezioni di geometria analitica e proiettiva (2nd edition) (G. B. Paravia & C., Turin, 1948).
Comment
Review by Donald Coxeter: The first edition of this substantial treatise appeared in 1930. This second edition contains various small portions of new matter, totalling 12 pages. The most notable addition is a beautifully illustrated account of the curves $y^{2} = f(x)$ , where $f(x)$ is a polynomial of degree 3, 4 or 5.

The book is in five parts. Part I [166 pages] is an introduction to the analytic geometry of the Euclidean plane, including such special curves as epicycloids and the involute of a circle, besides the geometrical interpretation of Fourier series. Part II [129 pages] deals similarly with Euclidean three-space, including vector analysis up to the divergence and curl, as well as a good section on nomography. Part III [164 pages] is on the real projective plane, treated as an extension of the Euclidean plane, with cross ratio used as a link between the synthetic and analytic methods. Collineations (including homologies) and correlations (including polarities) are defined synthetically as by von Staudt, but coordinates are used for the discussion of their invariant and self-conjugate points. Part IV [92 pages] is a thorough treatment of conics, partly synthetic but chiefly analytic. Part V [70 pages] is on real projective space, beginning with collineations and correlations, ordinary polarities and null-polarities, oval and ruled quadrics. The projective theory is followed by metrical considerations, without a clear-cut distinction between the affine and Euclidean geometries. Finally, there is a return to projective geometry for a discussion of line coordinates and linear complexes, with applications to statics.

1949
Gino Fano, Su una particolare varietà a tre dimensioni a curve-sezioni canoniche, Rendiconti Accademia Nazionale Lincei (8) 6 (1949), 151-156.

1950
Gino Fano, Chiarimenti sopra particolari superficie aventi tutti i generi eguali all'unità, Atti Acc. Sci. Torino 84 (1950), 94-96.
Comment
Fano's Abstract is as follows: Some properties of a surface already considered by Mr P du Val are set forth, and observations on other surfaces of genus one are added.
Gino Fano, Nozioni sommarie di geometria sulle curve e superficie algebriche (Gheroni, Turin, 1950).
Comment
This is produced with lithographic printing.
Gino Fano, Irrazionalità della forma cubica generale dello spazio a quattro dimensioni, Rendiconti del Seminario Matematico. Università e Politecnico Torino xb (1950), 21-32.

1953
Gino Fano, Les surfaces du quatriéme ordre, Rendiconti del Seminario Matematico. Università e Politecnico Torino 12 (1953), 301-313.
Gino Fano, G Castelnuovo, Commemorazione, Atti Acc. Lincei (1953).
Comment
This is a special issue of the Atti della Accademia Nazionale dei Lincei. Castelnuovo died on 27 April 1952 and Fano was asked to write a commemoration for his friend and present it at a meeting of the Accademia dei Lincei in December 1952. Although Fano completed the paper, he was taken to hospital in Verona where he died on 8 November 1952.

1958
Gino Fano and Alessandro Terracini, Lezioni di geometria analitica e proiettiva (3rd edition) (G. B. Paravia & C., Torino-Milano-Padova-Firenze-Pescara-Roma-Napoli-C, 1958).
Comment
Review by Saly Ruth Struik: It was L Cremona who, about 1890, inaugurated in Italy that form of instruction in geometry which combines both analytic and projective geometry, and of which the present book is an outstanding example. Originally based on the teaching of G Fano, it was first published by both authors at Turin in 1930. A second edition appeared in 1940, reprinted in 1948. This third edition, somewhat revised, is due to A Terracini. The author uses freely the analytical or synthetic method wherever it is advisable from a didactic standpoint. There are five parts. The first part deals with analytic geometry of the plane, and the second part with analytic geometry of space; both sections concentrate on linear and quadratic curves and surfaces. He inserts a sketch of the nomographic representation of equations in three variables. At the end of the second part is a section on vectors, with some information on "cursori", sliding vectors. The third part is devoted to the elements of projective geometry of the plane, and the fourth to the applications to conic sections. The last part deals with a number of properties of space figures, such as polarity, the metrical classification of quadrics and line complexes. An appendix gives a sketch of the point of view expressed by the Erlangen Program. The text is illuminated by 210 figures.

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