Concealed harmonies. Letters of Federigo Enriques to Guido Castelnuovo.
(Riposte armonie. Lettere di Federigo Enriques a Guido Castelnuovo.)

*(Italian)*Zbl 0861.01020
Torino: Bollati Boringhieri. xxxv, 722 p. (1996).

This volume contains the texts of the surviving 670 letters written by Enriques (1871-1946) to Castelnuovo (1865-1952), which had been conserved by Castelnuovo’s daughter Emma. Sadly, Enriques’ descendants seem not to have been so scrupulous (or else Enriques himself was not a saver), for very few traces are left of Castelnuovo’s replies (mainly pp. 593-598). Almost all of the letters belong to the period 1892-1906. They are dominated by the correspondents’ mutual interest in algebraic geometry (the title of the edition alludes to consequences of their classification of algebraic surfaces). Enriques regularly informed his friend of current conjectures and results; his occasional mathematical sketches are reproduced from the original. Their relationship reached even to joint papers (then still not a regular practice among mathematicians); the best known are their pair of pieces on this topic published in German in the Encyclopädie der mathematischen Wissenschaften in 1908 and 1914 but prepared in the late 1890s (pp. 357-364, 403). He also noted newly published literature, including in 1895 a ‘lunga memoria de Poincaré sull’Analysis situs’ (sic on p. 294). From time to time other topics emerge, such as the philosophy of science and mathematics, which became a major concern in Enriques’s later career. The editors have produced a reliable text with excellent apparatus. They begin with short accounts of the work of the correspondents and a chronology, and then indicate their manner of treating the letters; each one carries its own set of editorial footnotes when needed. An appendix contains some closely relevant letters written to Castelnuovo by other mathematicians; a set of plates includes three of them, as well as one from each correspondent and their portraits (that of Enriques seemingly from a later time). The bibliographies cover not only the main writings of the correspondents but also the many works which Enriques mentioned. Unfortunately there is only an index of names. The volume is an important addition to our knowledge of the rich vineyards of Italian geometry during that period.

Reviewer: I.Grattan-Guinness (Bengeo/Herts)