##
**Correspondence with Johann I Bernoulli and Niklaus I Bernoulli. Edited by Emil A. Fellmann and Gleb K. Mikhajlov. With the collaboration of Beatrice Bosshart, Adolf P. Juškevič and Judith Kh. Kopelevič.
(Briefwechsel mit Johann I Bernoulli und Niklaus I Bernoulli.)**
*(German)*
Zbl 0899.01033

Leonhardi Euleri Opera Omnia. Series Quarta A: Commercium Epistolicum 2. Basel: Birkhäuser (ISBN 3-7643-5271-X/hbk). x, 747 S. (1998).

[See also Vol. 1 of the same series (1975; Zbl 0323.01018), Vol. 5 (1980; Zbl 0462.01022) and Vol. 6 (1986; Zbl 0587.01022).] The core of the volume are the 38+11 extant letters along with their translations into German of the correspondences of Euler with Johann I and Niklaus I Bernoulli. At the age of 20 Euler initiated the correspondence with his teacher Johann I Bernoulli, a few months after he had left Basel for St. Petersburg in 1727. The last letter was written in May 1746 by Johann Bernoulli, less than two years before his death. There is an unexplained gap between 1731 and 1736, and 8 letters from Euler (after 1740) have not survived. Niklaus Bernoulli was, at the time of his correspondence with Euler (1742-1745), professor of the laws and solicitor and also occupied with administrational offices at the university of Basel. In both of the correspondences the historian of ‘pure’ mathematics can trace the development of Euler’s algebraic approach to the infinitesimal calculus up to the mid-fourties when he prepared his influential Introductio in analysin infinitorum (publ. 1748). Soon after this the vibrating string affair would force him to revise and broaden his views on functions.

Another major topic of the correspondence with Johann is mechanics. Earlier partial publications were by P. H. Fuss 1843 and G. Eneström 1903-1906. The present edition is based on careful inspections of the original letters, written in Latin (exept two German ones on music). Each original letter is followed here by a German translation (by Fellmann and Bosshart) with extensive annotations, prepared by Fellmann. Moreover, there are introductions to each of the two correspondences, prepared by Fellmann (on mathematics proper) and by Mikhajlov (on mechanics and other applications, in the case of Johann). Infinite series are present in the great majority of letters, including values of the zeta function, the sine product, and (in the Niklaus part) generating functions (partitio numerorum) and debates on divergent series. Euler ows much to the insistence of Niklaus on conceptual clarity. He comes to a distinction of infinite numbers: Definite infinities, as Niklaus is ready to admit them in calculations, had been used by Euler in earlier papers on series. Absolute infinity here appears in the clear description of convergence on 4 February 1744 (Euler: the remainder term becomes as small as one may wish). The famous explanation of the value of a divergent series as the value of the expression from which the series is obtained is another outcome of the debate with Niklaus. Unfortunately this clarity did then not appear explicitly in the Introductio. Both Euler and Johann solve special classes of ordinary differential equations. Methods of lasting value were worked out here for the first time. par In the first letter Euler mentions \(y=(-1)^x\), and Johann continues to insist on his \(\log (-1)=0\), as in his earlier debate with Leibniz. The further development, after Euler’s final solution beginning in 1746, was commented in vol. 5 of the same series par Euler also communicates his first approach to the Gamma function, without arousing enthusiasm on the side of Johann. Geodesics on surfaces are discussed with some success. Mechanics plays a major role in the correspondence with Johann. (There is little of other physics.) Special problems of point mechanics considered here are the motion of bodies on moving orbits under central forces; motion in resisting media; motion of bodies in rotating pipes. The most interesting passages are on the beginnings of a theoretical hydraulics, on the stability of the equilibrium of floating bodies, and on vibrations of such. Some parts of the correspondences were commented elsewhere: E. A. Fellmann, Non-Mathematica im Briefwechsel Leonhard Eulers mit Johann Bernoulli. In: Amphora. Basel: Birkhäuser. 189-228 (1992; Zbl 0788.01026); E. A. Fellmann, Partielle Differentiation im Briefwechsel Eulers mit Niklaus I Bernoulli par eine Miszelle. In: History of mathematics: states of the art. San Diego, CA: Academic Press. 223-236 (1996; Zbl 0855.01020). Further contents: Brief presentations of the Euler and Bernoulli families in Basel (Mikhajlov); the somewhat strange prehistory of the edition (Mikhaijlov); biographies of Johann and Niklaus (Fellmann as printed information on Niklaus was insufficient up to now the latter essay deserves special interest); three appendices, containing an exchange of letters between Johann and the administration of the St. Petersburg Academy, an unpublished text of his Hydraulica, and a treatise by Niklaus on the sum of the reciprocal squares. The bibliography and an index of names are of the excellent quality that distinguishes the whole volume.

Another major topic of the correspondence with Johann is mechanics. Earlier partial publications were by P. H. Fuss 1843 and G. Eneström 1903-1906. The present edition is based on careful inspections of the original letters, written in Latin (exept two German ones on music). Each original letter is followed here by a German translation (by Fellmann and Bosshart) with extensive annotations, prepared by Fellmann. Moreover, there are introductions to each of the two correspondences, prepared by Fellmann (on mathematics proper) and by Mikhajlov (on mechanics and other applications, in the case of Johann). Infinite series are present in the great majority of letters, including values of the zeta function, the sine product, and (in the Niklaus part) generating functions (partitio numerorum) and debates on divergent series. Euler ows much to the insistence of Niklaus on conceptual clarity. He comes to a distinction of infinite numbers: Definite infinities, as Niklaus is ready to admit them in calculations, had been used by Euler in earlier papers on series. Absolute infinity here appears in the clear description of convergence on 4 February 1744 (Euler: the remainder term becomes as small as one may wish). The famous explanation of the value of a divergent series as the value of the expression from which the series is obtained is another outcome of the debate with Niklaus. Unfortunately this clarity did then not appear explicitly in the Introductio. Both Euler and Johann solve special classes of ordinary differential equations. Methods of lasting value were worked out here for the first time. par In the first letter Euler mentions \(y=(-1)^x\), and Johann continues to insist on his \(\log (-1)=0\), as in his earlier debate with Leibniz. The further development, after Euler’s final solution beginning in 1746, was commented in vol. 5 of the same series par Euler also communicates his first approach to the Gamma function, without arousing enthusiasm on the side of Johann. Geodesics on surfaces are discussed with some success. Mechanics plays a major role in the correspondence with Johann. (There is little of other physics.) Special problems of point mechanics considered here are the motion of bodies on moving orbits under central forces; motion in resisting media; motion of bodies in rotating pipes. The most interesting passages are on the beginnings of a theoretical hydraulics, on the stability of the equilibrium of floating bodies, and on vibrations of such. Some parts of the correspondences were commented elsewhere: E. A. Fellmann, Non-Mathematica im Briefwechsel Leonhard Eulers mit Johann Bernoulli. In: Amphora. Basel: Birkhäuser. 189-228 (1992; Zbl 0788.01026); E. A. Fellmann, Partielle Differentiation im Briefwechsel Eulers mit Niklaus I Bernoulli par eine Miszelle. In: History of mathematics: states of the art. San Diego, CA: Academic Press. 223-236 (1996; Zbl 0855.01020). Further contents: Brief presentations of the Euler and Bernoulli families in Basel (Mikhajlov); the somewhat strange prehistory of the edition (Mikhaijlov); biographies of Johann and Niklaus (Fellmann as printed information on Niklaus was insufficient up to now the latter essay deserves special interest); three appendices, containing an exchange of letters between Johann and the administration of the St. Petersburg Academy, an unpublished text of his Hydraulica, and a treatise by Niklaus on the sum of the reciprocal squares. The bibliography and an index of names are of the excellent quality that distinguishes the whole volume.

Reviewer: D.Laugwitz (Darmstadt)

### MSC:

01A75 | Collected or selected works; reprintings or translations of classics |

01A45 | History of mathematics in the 17th century |

26-03 | History of real functions |

34-03 | History of ordinary differential equations |

70-03 | History of mechanics of particles and systems |

76-03 | History of fluid mechanics |

01A70 | Biographies, obituaries, personalia, bibliographies |