Translators’ Notes Citations from original sources –– mostly letters ––writteninGerman(in the orthography of the time) are reproduced here as is if the authors are one of the principal players, Euler and the Bernoullis, with the English translation immediately following within brackets. Otherwise, and also in cases of citations which have previously been translated into German, only the English translation is provided. A few factual changes which have occurred, or come to light, since the publication of the original text have been incorporated in this translation without special mention. Transliteration of Cyrillic characters follow the system used by the Li- brary of Congress (see, e. g., A. J. Lohwater, Russian-English Dictionary of the Mathematical Sciences, American Mathematical Society, Providence, RI, 1961, p. 1). 140 Notes Primary and secondary literature are cited in text and notes according to the following rules: 1. W o r k s of Leonhard Euler are quoted in the text normally with an abbreviated title (mostly translated into English), in the Notes according to the work edition included in the Bibliography of this book, which is arranged in four series, and of which, as of today, there exist 76 quarto volumes. Euler’s works are numbered according to the bibliography of G. Enestr¨om, to which we adhere, and are identified with the respective number, placed in front of the location of a work. For Opera omnia we use O., the series numbers are given in roman, and the volume numbers in arabic nu- merals. Example: E. 65: O.I,24 refers to Euler’s work Methodus inveniendi lineas curvas ...inVolume24ofthefirstseries,whichcarriestheEnestr¨om number 65. In each of the four volumes of the fourth series that have appeared so far, there is a table, with the help of which any work of Euler with known Enestr¨om number can easily be found in the Opera. 2. L e t t e r s of and to Leonhard Euler are identified by their R´esum´enumberinthefirst volume of Series IVA, be they published or not. Example: O.IVA, 1, R 465 refers to Euler’s letter to G. Cramer of July 6, 1745 (on p. 93 of the first volume of the Series IVA). If the letter partner is named in the text, normally only the date and R´esum´e number of the letter is given within round parentheses. Under the R´esum´enumber in O.IVA,1 one can find information about the respective letter as well as a brief indication of its content. 3. S e c o n d a r y l i t e r a t u r about Euler are sometimes linked to the “Burckhardt-Ver- zeichnis” (BV) in the Basel memorial volume (EGB 83, p. 511–552). This contains ca. 800 work titles in altogether fourteen languages. Example: BV Spiess (1929) refers to O. Spiess: Leonhard Euler, Frauenfeld, Leipzig 1929. –– Verdun-Bibliography. 4. Other frequently used abbreviations likeDSB,IMI, Harnack, Euler’s corre- spondence, Euler-Goldbach can easily be inferred via the Bibliography. 1 EGB 83 2 Cf. Bibliography 3 Original in the Archive of the Russian Academy of Sciences in St. Petersburg, where there is also the bulk of Euler’s manuscripts. We thank the directorate of the AAN St. Petersburg for the release of the photographs of the originals and for permission to publish them. 4 K. Euler (1955) 5 Razvitiye idey (1988) 6 Raith (1983) 7 Staehelin (1960) 8 Raith (1983), p. 460 141 Notes 9 Jakob Bernoulli: Opera. Geneva 1744, p. 361–373. Its content, on the whole, is ele- mentary and offers almost no mathematical insights new at the time. 10 More details in Raith (1983), p. 468, Note 5 11 Raith (1983), p. 468, Note 9 12 About life and activity of Paulus Euler, especially about his attitude to pietism, see Raith (1983) and (1979) 13 Cf. Raith (1980), p. 102 14 Herzog (1780), p. 32f. 15 Cf. DSB XI, article by K. Vogel 16 Burckhardt-Biedermann (1889) 17 Magni Euleri praeceptor in mathematicis. Letter from Daniel Bernoulli to Leonhard Euler of Sept. 4, 1743 (O.IVA,1,R 151) 18 Schafheitlin (1922) 19 Schafheitlin (1924) 20 About the role of a “respondent” in such disputes, cf. Staehelin (1957), p. 135f. 21 Cf. Wolf (1862), p. 88, footnote 3 22 Unfortunately, the text of the lecture has not been handed down to us. About this uncertainty, cf. Aiton (1972) 23 B¨orner (1752), p. 99f. 24 J. Bernoulli (1742), Vol. 2, §26, p. 615f. (Transl. from the Latin by O. Spiess) 25 Doctissimo atque ingeniosissimo Viro Juveni Leonhardo Eulero (R 191) 26 Clarissimo atque doctissimo Viro Leonhardo Eulero (R 194) 27 Viro Clarissimo ac Mathematico longe acutissimo Leonhardo Eulero (R 201) 28 Viro incomparabili Leonhardo Eulero Mathematicorum Principi (R 226) 29 Such prize questions were posed by the big academies in the 18th century ––the Paris, Petersburg, and Berlin Academy –– almost regularly and constituted an essential stimulant of research. Their solutions were always submitted anonymously in the form of memoirs or even books, which, if they won a prize, were also printed at the expense of the respective academy. 30 Condorcet (1783), p. 288 (in: O.III,12) 31 The respective repudiation of this statement as a legend in EGB 83, p. 81, Note 11, has turned out to be an error, as was pointed out by K.-R. Biermann (Berlin) in two letters to the author of August 12 and 17, 1988. 32 Transl.fromtheLatinbyO.Spiess 33 All the same, one can prove that Euler has also taken up experiments. Mikhailov (1959), p. 258f. 34 About the procedure of electing a professor in Basel, cf. Staehelin (1957) 35 About the building history of St. Petersburg, of the fortress island Kronstadt, and of the navy and mercantile marine, as also about the sociological and demographic conditions, cf. Donnert, p. 10; Buganov, p. 199, p. 291–338, p. 402f. 36 Finster/van den Heuvel (1990), p. 45f. 37 Buganov, p. 368–373 38 Grau, passim 39 Grau,p.120 40 Ostrovityanov, p. 30f. 41 Spiess (1929), p. 52 142 Notes 42 Euler-Goldbach 43 Fuss, II, p. 409f.; Mikhailov (1957), p. 24–26 44 Euler’s autobiography, here p. 4. –– About currency issues, cf. Fellmann (1992), p. 219–222 45 Mikhailov (1959), p. 275–278 46 The correspondence between Wolff and J. Bernoulli is worked on by F. Nagel in Basel and is to be edited as part of the Bernoulli-edition. 47 Mikhailov (1959), p. 276, Note 12 48 Commentarii Academiae Imperialis scientiarum Petropolitanae, I–XIV (1726–1746) 1728–1751 49 Kopelevich (1973), p. 121–133 50 About the complicated and tragigruesome history of Russia in the post Peter period, cf. for example Gitermann, Vol. 2, p. 143–295; v. Rimscha, p. 322–399 51 Gitermann, Vol. 2, p. 152 52 Lomonosov (1948; 1957), in the Bibliography under Vavilov 53 Kopelevich (1974), p. 176–229 54 Kopelevich (1977), Spiess (1929), p. 52f. 55 Bernoulli-Sutter (1972), p. 62f. 56 Cf. DSB VI; NDB 8. Only since recently has there been in print a complete bibliog- raphy of J. Hermann; see Nagel (1991). 57 Euler’s correspondence, Part III, p. 66, Note 5 58 After Euler’s departure to Berlin (1741), there ensued a correspondence between Euler and Krafft, which, in all respects, is very interesting and continued until 1753. It is published –– for the most part only in the form of extracts or summaries ––in Euler’s correspondence, Part III, p. 134–176. 59 Gmelin (1747); Posselt (1990) 60 Cf. Nevskaya (1983), p. 363–371, especially the literature indicated there on p. 370 in the Notes 6 and 7 61 Cf. Euler’s correspondence,PartI,p.5 62 For a biography of G. F. M¨uller, cf. B¨usching (1785), p. 1–160 63 Euler’s correspondence,PartI 64 Goldbach was second godfather to Euler’s first-born son Johann Albrecht. 65 An excellent biography exists in Yushkevich-Kopelevich (1983). It appeared in Ger- man translation by A. and W. Purkert with Birkh¨auser, Basel 1994. Short expositions in Euler-Goldbach,p.1–16;DSBV,articlebyM.S.Mahoney 66 In a special formulation, this conjecture of Goldbach, stated in the letter to Euler of June 7, 1742, says: Every even natural number not equal to 2 can be represented as sum of two prime numbers (Example: 8 = 5 + 3; 52 = 47 + 5). This statement, to this day, could not be proved in full generality, but it continues to have a stimulating influence on mathematical research into our days. 67 Euler-Goldbach 68 Spiess (1929), p. 63f. 69 Winter (1958) 70 E. 5: O.I,27 71 E. 6: O.II,6 72 E. 7: O.II,31 143 Notes 73 D. Bernoulli (1738). For the early development of hydrodynamics and hydraulics under Daniel Bernoulli and Euler, cf. Mikhailov (1983) 74 Euler’s letter to Daniel Bernoulli of May 25, 1734 (R 98), published in Bibl. Math.(3), 7, p. 139 75 ibid. 76 Spiess (1929), p. 73 77 Registres,p.11 78 Very recently, however, we discovered in the Euler-Archive of Basel among the diary entries of J. A. Euler the copy of a folio, which –– next to geometrical sketches –– shows a doodle of two figures, which could depict the parents of Euler (cf. the figure on p. 118) 79 D. Bernoulli to Euler, March 29, 1738 (R 118). Possibly, the artist was the above- mentioned painter from Holstein, Georg Brucker, art master from 1735 to 1737 at the “Gymnasium” of the Petersburg Academy. The idea that the father-in-law Gsell has made the portraits cannot indeed be dismissed entirely, for the shipment of portraits contained also one of Tsarina Anna, as Bernoulli reports.