The Rivista di Matematica: Aims, Styles, and Legacy

1. Giuseppe Peano editor of an international journal ? The Rivista di Matematica (1891-1908)Aims, Styles and Legacy C. Silvia Roero MLI Djursholm 20-23 June 2016

2. Aims - with some concluding remarks • Describe the founding and the development of the Rivista di Matematica(RdM) in order to discuss if, and in which sense, it could be considered an international journal, whilst its aims and prospectus were addressed to a national community, one of teachers and professors of mathematics in Italy • Peano’srole in its development as one of the new Italian mathematical journals ‘open to the world’ (Europe, USA): editorial strategies (ICM), public of authors and readers, exchanges, … • Peano’sgoal: the improvement of the methods of teaching Maths and the diffusion not only of his own ideas, methods, programme and results on the foundations of mathematics, but also those of the teamof his coworkers (the importance of Formulaire) • the reasons for delays in the outputs, and the final closing (Form.).

3. Some features or styles of the RdM • (sections: articles, reviews, obituaries, translations, debates) • compared to those of other journals • Support given by Peano and his team of coworkers to editors of new journals with similarobjectives and audiences in Italy and abroad (influences and legacy linked to the textbooks written by members of Peano’s School and reviewed in Italy and abroad) • Map of Italian publishers of textbooks • CHIOSSO (Giorgio) (dir.). – TESEO. Tipografi e editori scolastico-educativi dell’Ottocento Milan, Editrice Bibliografica, 2003 – 760 p. • Dizionario Biografico dell'Educazione (DBE) 1800-2000 • http://dbe.editricebibliografica.it/dbe/ricerche.html

4. G. Peano to David E. Smith, 4 November 1893 In addition to my booklets Arithmetices Principia and Principii di Geometria I have published several educational articles in Rivista di Matematica, which also contains some articles by prof. Burali [Forti] and by other authors. Always in the same journal I have also published some reviews. Instead I scattered my works of scientific character in various scientific journals.

5. Peano’sPapers in international journals of Maths Nouvelles Annales de Mathématiques (1884a, 1884b, 1891r, 1892aa, 1892bb) 5 MathematischeAnnalen(1888b, 1890b, 1890f) 3 Monatshefte für Mathematik und Physik (1894a, 1895p) 2 American Journal ofMathematics(1895c) 1 Mathesis (Mansion, Neuberg)(1889b, 1889c, 1889e, 1890d, 1890e, 1892s, 1914b) 7 CR Acad. des Sciences de Paris (1889g) 1 Rendiconti Circolo Mat. Palermo Peano7 (1888-1921) Rend. Acc. Naz. LinceiPeano11 (1890-1916) L’Intermédiaire des Mathématiciens (1894i, 1894j, 1895s, 1896e, 1896f, 1896g, 1896h, 1896i, 1898i, 1898j, 1898k, 1898l, 1899d, 1903g, 1906h) 15 [Answers] L’Enseignement Mathématique (1906d, 1909c, 1916t) 3 Bull. Sciences Mathématiques et Physiques élémentaires (1902e) 1 [Letter] Société Physique et Mathématique de Kasan(1904b) 1 Archivs de l’Institut de Ciences(Barcelona) (1911d) 1

6. Translations of Peano’s mathematical papers (Italian) published in national and international journalsand books German (2 books, 6 articles) Genocchi-Peano 1899 tr. Bohlmann, Schepp + 5 papers (analysis, series, logic) 1891t (geometrical calculus) tr. Schepp; 1897u (geometrical calculus) tr. A. Lanner Russian(2 books) Genocchi-Peano 1903 tr. NS Sineokov, 1922u tr. KA Posse Portuguese(1) 1889t DefinicçãogeométricadasfuncçoesellipticasJornal de scienciasmathematicas e astronomicas Coimbra G.Teixeira’s journal Polish(1) 1897tZarysRachunkugeometrycznegotr. S. DicksteinWarsawa Spanish (1) Principios de lògicamatemáticaElprogrésomatemático French1915t Importance des symboles en mathématiques. Tr. J. Rouquet Scientia Latin1916t Residuo in formula de quadratura Cavalieri-Simpson L’Enseign. Mat.

7. Rivista di Matematica 1891-1908 Thescope of the Rivista di Matematica is essentially didactic, its principal object being the improvement of the methods of teaching. The Rivistawill contain articles and discussions concerning the fundamental principles of the science and also the history of mathematics. The review of text-books and all publications having reference to the teaching of mathematics will form animportant feature. This journal, whose programme is very different from all journals now published in Italy, has modest aims, but very important, in our opinion. The favor with which it was greeted by friends and acquaintances gives us hope that, with the support and collaboration of the readers, the journal will become a useful guide for teachers and scholars.

8. Rivista di Matematica I 1891, 272 p. II 1892, 215 p.III 1893, 192 p.IV 1894, iv+198 p.V 1895, 195 p.Revue de MathématiquesVI 1896-99, iv+188 p. VII 1900-01, iv+184 p. Revista de MathematicaVIII 1902-06, iv+160 p. IX 1908, 29 p.subscriptionfees6 L. Italy, 7 abroadannual volume8 L. • languages (title-page) Italian, French, latino sine flexione • languages (papers) Italian, French, German, English, Spanish, Latin s. f. • monthly issues (16 p.) & annual volume founders & financing Giuseppe Peano, Francesco Porta, Filiberto Castellano, Francesco Porro professors at the Turin Military School Lists of subscribers Correspondence Peano-Cesàro (from 1894 onward get even) publisher: Libreria Fratelli Bocca, Turin

9. Content and Styles of RdM articles (105) E. Bertini RdM 1891,22-24 published Math. Annalen 44 (1894) reviews of treatises and textbooks (76) Peano Veronese Fondamenti di geometria a più dimensioni RdM 2 Burali Forti Schlömilch Elementi di geometria analitica RdM 2 Lazzeri GremigniElementi di Euclide Giudice Cesàro Corso di Analisi algebrica Loria Vivanti Il concetto di infinitesimo chapters for the Formulaire (44) letters (correspondence) (18) solutions of problems or questions (10) News and report of Conferences, Institutions Gottingen (8) observations (8), eulogies (7), prizes (1),list of bibliographical references (2), translations (2), questions (4)

10. Peano Sopra la raccolta di formule di matematica, RdM, 2, 1892, 76 It would be extremely useful to publish the collections of all the theorems now known referring to given branches of the mathematical sciences in such a way that the scholar need not consult this collection in order to know how much had been done on a given point, and whether his research was new or not. Such a collection, extremely difficult and lengthy in ordinary language, is notably facilitated by the use of the notations of logical mathematics; and the collection of the theorems on a given subject perhaps becomes less long than its bibliography. PeanoSur la définition de la limite d’une fonction. Exercice de logique mathématique, 1895; review C. Arzelà, Lezioni di Calcolo infinitesimale. Confronto col Formulario, 1902 Peano to Vitali 3 April 1905 Per la Rivista di Matematica ogni recensione deve avere il confronto col Formulario e la riduzione in simboli delle proposizioni dell’Autore non ancora scritte nel Formulario. Dal libro del Lebesgue potrà risultare un rigo, o mezza pagina.

11. RdM Foreign Authors Belgium: E. Catalan; Holland: P. Molenbroek; Great Britain: B. Russell, P. Jourdain; Denmark: G. Eneström; Germany: G. Cantor, G. Frege,E. Lampe, O. Stolz, R. Haussner, H. Mehmke; Poland: S. Dickstein; Russia: M. Poretzki, A. Vassilief; Spain: J. Duràn Loriga; Portugal: R. Guimaraes; Greece: C. Stephanos; USA: S. Kimura ItalianAuthors Bertini, Segre, Gerbaldi, Loria, Pincherle, Jadanza, Cesàro, Capelli, Morera, Ascoli, Fano, Favaro, De Amicis, Del Re, Milesi, Testi, Nagy, Amodeo, Sforza, E. Pascal, G. Lazzeri, Mollame, Pirondini, Maggi, Ovazza, Gremigni, Sbrana, Garbasso, Volterra, D’Arcais, Bagnera, Buffa, Chini, Cipolla, Cantoni, Sibirani (Peano’s team) Vailati, Pieri, Novarese, Burali-Forti, Bettazzi, Castellano, Vacca, Vivanti, Giudice, Catania, Porta, Padoa, Nassò, Ramorino, Chionio, Zanotti-Bianco

12. Bulletin of American Mathematical Society A. Ziwet, A New Italian Mathematical Journal: Rivista di Matematica,1891 Almost simultaneously with the Bulletin of the New York Mathematical Society, a new journal of a somewhat similar character has been founded in Italy. Like the Bulletin, the Rivista di Matematica is a monthly…. According to the prospectus “its scope is essentially didactic, its principal object being the improvement of the methods of teaching.” The Rivista will containarticles and discussions concerning the fundamental principles of the science and also the history of mathematics. The review of text-books and all publications having reference to the teaching of mathematics will form an important feature. The editor, Prof. Giuseppe Peano of the University of Turin, is well known through his original investigations in Mathematical Logic and in Grassmann's Geometrical Calculus, as well as through his rigorous and elegant treatment of the Infinitesimal Calculus. His own contributions to the Rivista so far (Jan. 1891) relate mainly to the fundamental logical principles of the science of mathematics.

13. A. Ziwet, A New Italian Mathematical Journal Rivista di Matematica, BAMS 1891 Among the longer articles by other contributors we find an interesting paper by Prof. Segre of Turin, addressed to his students, in which he points out some of the distinctive features of modern mathematics… The author is evidently inspired by what may be called the modern Göttingen school (Riemann, Clebsch, and in particular Felix Klein), insisting as he does on the organic unity of the wholeof mathematics, warning against excessive specialization… It is curious to note that, in the opinion of Segre, there exists a very pronounced preference for the study of pure geometry, to the injury of analytical studies, among the younger generation of Italian mathematicians. Some remarks in this paper as to mathematical rigor and the use of hyperspace gave rise to an interesting discussion between the author and the editor [Peano]. … Among the reviews, the very full account given by Gino Loria of R. de Paolis' theory of geometrical groups is most prominent. Hyde's Directional Calculus finds a competent and appreciative critic in the editor [Peano]. 10 August 1891

14. J. Tannery, Peano G. Applicazioni geometriche del calcolo infinitesimale …BulletindesSciencesMathématiques, 1887, p. 238 les définitions qui se rapportent aux champs de points, aux points extérieurs, intérieurs ou limites par rapport à un champ, aux fonctions distributives (coexistences d’après Cauchy), à la longueur (à l’aire ou au volume) externe, interne ou propre d’un champ, la notion d’intégraleétendue à un champ sont présentées sous une forme abstraite, très précise et très claire. H. Lebesgue,A propos de quelques travaux mathématiques récents, 1917 Les recherches sur la notion d’aire, les recherches de Riemann et de P. du Bois-Reymond sur les conditions d’intégralité, les recherches de MM. Cantor, Peano, Jordan et Borel sur les ensembles de points et les recherches sur les intégrales multiples, que je laisse de coté ainsi que celles sur l’intégration et la différentiation sous le signe d’intégrale et celles sur l’intégration par parties, toutes ces recherches conduisent naturellement à une généralisation des notions de longueur, d’aire et de volume, qui est la notion de mesure d’un ensemble. … C’est ainsi que M. Peano a pu rétablir les définitions adoptées par Archimède: la longueur (ou l’aire) d’une courbe (ou surface) convexe est la limite inférieure de la longueur (ou de l’aire) des lignes polygonales (ou surfaces polyèdres) circonscrites et la limite supérieure des inscrites. On avait perdu l’habitude de procéder avec précautions dans ces questions délicates surtout dans les livres d’enseignement, de sorte que, à l’époque où j’étais au collège, on démontrait victorieusement que =2.

15. G. Vivanti, Peano G., Applicazioni geometriche …, JFM, 19, 1887, p. 248 “Das vorliegende Werk, welches freilich nicht über die Elemente der Infinitesimalgeometrie hinausgeht, verdient dennoch wegen der Eigentümlichkeit der Methoden und wegen der sorgfältigen Genauigkeit der Durchführung ein hohes Interesse. Die fortwährende Anwendung der Möbius-Grassmann’schen Streckentheorie, die eine solche Knappheit und Eleganz mit sich bringt, wie sie kaum bei der Quaternionentheorie zu finden ist, die exakte Begründung des Grenzbegriffes in der Geometrie, endlich die Einführung der Punktgebiete und der Funktionen von Gebieten, wodurch die Infinitesimalgeometrie zu demselben Masse von Strenge und Allgemeinheit gebracht wird, welches die Infinitesimal-rechnung neuerdings erlangt hat, dies alles macht die charakteristische Eigenart des Buches aus.” Relazione della Commissione … cattedra di ...calcolo infinitesimale, BPU 1891 Brioschi, Beltrami, Pincherle, A. Tonelli, Volterra il trattato delle applicazioni geometriche del calcolo infinitesimale è inferiore a molte opere sullo stesso argomento uscite prima e contemporaneamente al lavoro del Peano, avendo l’autore tralasciato molti dei più importanti capitoli della geometria differenziale, forse perché troppo preoccupato del metodo che ha voluto usare(il calcolo dei segmenti) metodo che non sarebbe opportuno introdurre nell’insegnamento in sostituzione di quelli classici. Peano’s treatise on the geometric applications of infinitesimal calculus is less valid than other works on the same subject published before or at the same time, having the author left out many of the most important chapters in differential geometry, perhaps because too worried about the method, a method that would not be appropriate to introduce teaching in substitution of the classic ones.

16. Alexander Ziwet, Two New Works on Grassmann’s Geometrical Calculus, Annals of Mathematics 6 (1891): 14-19 31.1.1891 • Hyde, The DirectionalCalculusbased … methods of H. Grassmann • Peano, Calcolo geometrico … Grassmann1888 Prof. Hyde's remarks on the comparative value of the methods of Hamilton and Grassmannare very just, and particularly valuable… The view taken by Prof. Peanoof his subject is much broader.The geometrical calculus consists of a system of operations to be performed on geometric objects, analogous to those applied to numbers in Algebra. It gives us the means of expressing the results of geometrical constr. by means of formula, of representing geometrical propositions by means of equations, and of substituting transformations of equations to ordinary reasoning. The geometrical calculus has some analogy to analytic geometry; but it differs from it in the fact that, while in analytic geometry all calculations are performed on certain numbers that determine the geometrical objects, in this new science the calculations are performed on the objects themselves."

17. Alexander Ziwet, Two New Works on Grassmann’s Geometrical Calculus, Annals of Mathematics 6 (1891): 14-19 Prof. Peano is perhaps rather too sanguine in his expectations as to the future of the "new science”. But his words are worth quoting: “In my opinion the time is not far when this geometrical calculus, or something analogous to it, will take the place of the methods actually in use in higher mathematical instruction. … it must be taken into account that the student will then be in possession of a method which comprises that of analytic geometry as a special case, is far more powerful, and lends itself admirably to the study of the geometrical applicationsof infinitesimal analysis, of mechanics and graphical statics” … Very likely, Prof. Hyde would be the first to subscribe to this view; and he is certainly to be congratulated upon his persistent efforts in introducing the "new science" to English-speaking mathematicians by repeated courses of lectures on the subject in the University of Cincinnati, and finally by a very useful and (in the best sense of the term) practicaltext-book.

18. Bulletin American Mathematical Society E.H. Moore, IX annual meeting of the AMS, 1902, p. 502-504 A flourishing young school of mathematical logic has recently grown up in Italy under the influence of Peano. They have investigated with marked success the foundations of analysis and geometry, and have in particular endeavored to show the non-contradictoriness of the axioms of our number system by making them depend on the axioms of logic, which axioms we must admit in order to reason at all. ... Geometric intuition has no place in this order of ideas which regards geometry as a mere division of pure logic. The efforts of this school have already been crowned with eminent success, and much may be expected from it in the future. Its leaders are Peano, Veronese, Pieri, Padoa, Burali-Forti, and Levi-Civita in Italy, Hubert in Germany, Moore in America, and Russell in England. J. Pierpont, International Congress of Arts and Science, St. Louis, 1904, 147, 158.

19. Bulletin of American Mathematical Society E.H. Moore, On the foundation of mathematics, 1903, p. 402-424 … the general point of view of what may be called abstract mathematics. One comes in touch with the literature very conveniently by the mediation of Peano's Revue des Mathématiques. The Italian school of Peano and the Formulaire Mathématiquepublished in connection with the Revue, are devoted to the codification in Peano's symbolic language of the principal mathematical theories, and to researches on abstract mathematics. … In his dissertation on euclidean geometry, Mr. Veblen, following the example of Pasch and Peano, … The Italian writers on abstract mathematics for the most part make use of Peano's symbolism. One may be tempted to feel that this symbolism is not an essential part of their work. It is only right to state, however, that the symbolism is not difficult to learn, and there is testimony to the effect that the symbolism is actually of great value to the investigator in removing from attention the concrete connotations of the ordinary terms of general and mathematical language. E.H. Moore, O. Veblen, E. Huntington 1905

20. Peano’s project of the Formulaire de Mathématiques Main areas in which the enterprise was gradually build up: • mathematics • logic • history of mathematics • language, i.e. mathematical vocabulary and how it could be spread Mathematics and Logic dictated the contents and made possible the organization History and Language had the role of creating a context in which these concepts, definitions, theorems, methods, etc. had been conceived, and of communicating them exactly and rigorously to the widestpossiblepublic. The Genesis and Aims of the Formulaire de Mathématiques and the main Sources of inspirations could be found in Rivista di Matematica Stages of realisations: Formulaire 5 editions (1895 up to 1908) see Tables

21. increasing importance of Formulario exchanges Peano to Laisant 18 October 1898 je reçois avec beaucoup de plaisir l’annonce du journal L’Enseignement Mathématique, que vous dirigez avec M. Fehr. Son programme a plusieurs points en commun avec celui de la Rivista di Matematica, lorsqu’elle est née. Mais j’ai du limiter mon journal à la publication du Formulaire de Mathématiques, car je ne pouvais pas attendre à un programme qui surpassait mes forces. Je souhaite donc la meilleure réussite à un journal, qui satisfait à un désir très vifs de nos jours. Je suis un peu accablé par les devoirs de l’enseignement, les heures de loisir sont occupé par la publications du Formulaire. Toutefois j’espère de répondre par quelques travaux à votre adresse. J’enverrai une copie de mon journal et du Formulaire à M. Fehr, qui ne l’a pas encore, si vous acceptez l’échange des journaux.

22. Strategies to promote the Formulario asking for collaboration • Journals • International Conferences (Math., Phil.) • Societies of teachers (Ass. Franç. Avan. Sciences, Mathesis) • Contacts with colleagues, teachers, editors of journals, historians of mathematics, philosophers, assistants, students (Klein, Mittag-Leffler, Jordan, Frege, Cantor, …, Cesàro, Vitali, Levi-Civita, Marcolongo, Pincherle, …, R. de Montessus, …) • Editorial Announces in RdM, other periodicals (for mathematicians, teachers, …), meetings • ‘Payment’ for coworkers consisted in the annual subscription to his Rivista di Matematica

23. C. Angas Scott, The Int. Congress of Math in Paris, AMS 1900 E.B. Wilson, The Foundations of Mathematics, AMS 1904 ... In the fields of arithmetic and algebra, too, Burali-Forti and Padoa, adherents of Peano, had reached a point far beyond the widest view of the chief of the German school that deals with the same subjects. Anyone who is acquainted with the articles presented to the Philosophical Congress at Paris in1900by Peano, Burali-Forti, Padoa, and Pieri, cannot be convinced that these authors had become deadened, and the artificiality of their system is by no means so certain as it might be. ... The articles by Peano, Burali-Forti, Padoa, and Pieri show the point at which the Italian school had arrived in 1900.

24. Revue Semestrielle des publications mathématiquesAmsterdam 1893- Table de journaux nations America, Belgique, Danemark, Deutschland, France, Great Britain, …

25. Team of coworkers Jahrbuchüber dieFortschritte der Mathematik Giulio Vivanti (1859-1949) 1740 reviews 1884-1938 Italian maths 1881Graduated Civil engineering Univ. Turin, 1883 Univ. Bologna (Arzelà, Pincherle) Habilitation Inf. Calculus 1892 G. Vivanti, Peano G., Applicazioni geometriche …, JFM, 19, 1887, p. 248 “Das vorliegende Werk, welches freilich nicht über die Elemente der Infinitesimalgeometrie hinausgeht, verdient dennoch wegen der Eigentümlichkeit der Methoden und wegen der sorgfältigen Genauigkeit der Durchführung ein hohes Interesse. Die fortwährende Anwendung der Möbius-Grassmann’schen Streckentheorie, die eine solche Knappheit und Eleganz mit sich bringt, wie sie kaum bei der Quaternionentheorie zu finden ist, die exakte Begründung des Grenzbegriffes in der Geometrie, endlich die Einführung der Punktgebiete und der Funktionen von Gebieten, wodurch die Infinitesimalgeometrie zu demselben Masse von Strenge und Allgemeinheit gebracht wird, welches die Infinitesimal-rechnung neuerdings erlangt hat, dies alles macht die charakteristische Eigenart des Buches aus.”

26. Aims – correspondence and invitation to collaborate G. Peano to Ernesto Cesàro, 5 February 1891 For some time I wanted to be able to come with you in personal correspondence, and now the new journal entitled the Rivista di Matematica, of which today we published the first issue, give me the occasion. … As you will see from the program, and the first issue, it is absolutely different from current journals in mathematics published in Italy. … Unfortunately, the journals are already many, and lead all difficult life, but we believe that there is a real reason to make it. If it is able to eliminate at least some of the teaching too many inaccurate books , and to make a real improvement in the new it has reached its goal. To bear fruit it needs the cooperation and help of able men, and I strongly recommend to you to help us with work and advices.

27. Peano’s International Network USA G. PeanotoE. Cesàro, 10 December 1899 Quest’autunno, passando per Napoli, parlai con Lei di professori americani. Dal nuovo elenco non risultano persone nuove, oltre quelle di cui si è già parlato. Ad ogni modo eccone alcuni: Hagen, autore della Synopsis höh. Math., Washington. Ci siamo sentiti qualche volta. Hyde Univ. Cincinnati. Si occupa di calcolo geometrico, vettori e quaternioni. Sono in corrispondenza. Peano’sreview (RdM 1, 1891) of Hyde The directionalCalculusbasedupon the methods of H. Grassmann, Boston, 1891 Macfarlane Pennsylvania Martin Editore del Math. Magazine Washington SmithBrockport N. Y. è direttore di una Scuola normale. Ziwet traduttore della Meccanica di Somoff, Prof. all’Università di Michigan. (L’ho conosciuto personalmente). Peanoreview (RdM 3, 1893) ZiwetAn Elementary treatise on theoretical mechanics, New York, Macmillan, 1893

28. G. Peano to E. Cesàro, 20 January1892 … You alluded to some sorrows that I had as director of the Rivista for having refused some papers. But how can we do? Some have sent me treatises to be published and therefore you should keep the Journal came out into sets of 300 pages per month, and then the poor owners! Other items were worthless; others were entirely outside of the journal program, and so on… Apart from that, I have never told anyone, not even in confidence, the name of an author, of which for some reason the paper has not been published; and if you know of some, it is because the author himself has taken care to spread the news.

29. Some concludingremarks Publications of members of Peano’s school Nouvelles Annales de Mathématiques (Peano [12 articoli] 1884-1893; Burali-Forti 1895; Padoa 1903-1908) Revue de Métaphysique et de Morale(Vailati 1899, Pieri 1906, Padoa 1911,1917,1938; Vacca 1911, Cassina 1933) Revue du mois (Vailati 1907) L’Intermédiaire des Mathématiciens (Peano [30 articoli] 1894-1906; Bettazzi 1895, Burali-Forti 1900, Padoa 1900) L’Enseignement mathém. (Peano [6 articoli] 1906-1916; Bettazzi 1900-1905, Burali-Forti 1899-1912, Vailati 1908, Peyroleri 1909, Cassina 1932) L’Enseignement secondaire (Vailati 1907) Bull. Sciences Math. et Astronomiques (Burali-Forti 1899) Bull. Sciences Mathématiques et Physiques élémentaires (Peano 1902) Bull. des Sciences Mathématiques(Boggio 1911, Bottasso 1914, 1915, 1918) CR Acad. des Sciences de Paris (Peano 1899; Boggio 1911) Arch. Int. Hist. des Sciences (Gliozzi 1948-1968)

30. Influences and legacy of RdM in Italy mathematics history of maths F. EnriquesQuestioni riguardanti la geometria elementare, Bologna, 2 vol. 1900Questioni riguardanti la matematica elementare, Bologna 1912, 1914Questioni riguardanti le matematiche elementari, Bologna 1924-1927 L. Berzolari, G. Vivanti, D. Gigli Enciclopedia delle matematiche elementari e complementi, con estensione alle principali teorie analitiche e geometriche, loro applicazioni e notizie storico-bibliografiche Milano (Hoepli) 3 vol. 1930-37 • Journals for teachers • Text-books and treatises written by members of Peano’s School

31. Marco Nassò (1864-1920) teacher in secondary school Algebra elementare ad uso dei Licei e degli Istituti tecnici Torino, Salesiana 1898 (10 reprints) – 1919 (ed. Borio); Aritmetica generale ed Algebra ad uso dei Licei Torino, Salesiana 1898; Elementi di Calcolo Algebrico ad uso delle Scuole Normali Torino, Salesiana 1899 Nassò to G. Mittag-Leffler 22 January 1899 I have now published a treatise on algebra for high schools and I would be highly honored to provide a copy … as humble homage from a schoolboy to a great master … If you were also kind enough to do a review in the renowned journal Acta Mathematica you would do something very grateful. W. J. Greenstreet, M. Nassò …, The Mathematical Gazette, 1899 This is an excellent book for beginners, more after the English style than most Continental textbooks, inasmuch as the sections are followed by examples, in all 2000 in numbers. … An interesting feature of the book is the number of purely historical notes. Professor Nassò is to be congratulated on having produced the best elementary Italian text-book we have seen.

32. Raymond Clare ARCHIBALD 18.9.1922

33. Benjamin Abram BERNSTEIN 8.2.1928 ARCHIBALD 29.7. 1925

34. K. Gödel Collected Works vol. V - Correspondence Ralph Hwastecki to K. Gödel March 17, 1971 I am a special student in Education at Elmhurst College in Elmhurst, Illinois and as part of the curriculum we are required to do a unit on mathematics for the elementary grade level. …. K. Gödel to R. Hwastecki [1971] What should be pointed out to beginning students in math (but of course only after they have learned some math) … is the truly astonishing number of simple & nontrivial theorems and relationships that exist (prevail in math) (see e.g. the laws for the elements & subsidiary lines in a triangle or cnf). Peano's Formul. des math. … many elem. math rel. of this kind.

35. Sources to construct a Map of Italian publishers of textbooks for schools G. CHIOSSO (ed.)TESEO. Tipografi e editori scolastico-educativi dell’Ottocento Milan, Editrice Bibliografica, 2003 – 760 p. G. CHIOSSO (ed.) Dizionario Biografico dell'Educazione (DBE) 1800-2000 http://dbe.editricebibliografica.it/dbe/ricerche.html Thanks

36. Some concludingremarks G. Candido,Il giornalismo matematico in Italia, 1903 Atti Congr. Mathesis NA Nel 1890, per opera dell’illustre Professore Peano, dell’Università di Torino, venne alla luce la Rivista di Matematica con scopo essenzialmente didattico, occupandosi specialmente di perfezionare i metodi di insegnamento, di discutere i principi fondamentali della scienza, pubblicando articoli riferentesi alla storia delle matematiche e recensioni dei trattati e pubblicazioni che riguardano l’insegnamento. L’opera di G. Peano riguarda tutta la matematica epperò riesce utile anche all’insegnante della scuola secondaria, animato dalla convinzione sana che indarno si cerca il rigore nelle parti elementari ove non si riceva l’influsso di una critica severa, ma giusta, di un’analisi rigorosa dei ragionamenti, di un’esposizione concisa e chiarissima.

37. G. Candido,Il giornalismo matematico in Italia, 1903 Proceedings of congress Mathesis in Naples exagerated emphasis Sono discussi nella Rivista di Matematica i problemi fondamentali della scienza, sia della parte elementare che della parte superiore, e vi si trovano recensioni che riescono più utili di tanti trattati. La Rivista di Matematica cominciò le sue pubblicazioni con in testa un articolo del prof. Peano sulla Logica Matematica. … L’insegnante della scuola secondaria ha dato e dà largo contributo allo sviluppo della scuola del prof. Peano. Già la logica matematica viene a formare oggetto di articoli in giornali d’indole affatto elementari, … già il mondo matematico ha riconosciuto l’estrema importanza di questa scuola, e questo a me pare che basti perché l’insegnante della scuola secondaria italiana riconosca doveroso tenersi al corrente della cosa.