Abstract
The historical importance of Lambert’s Mémoire turns out evident as soon as one realizes the issues tackled by the Swiss. There is little doubt that fame goes to the first part of the article, in which Lambert, showing a high level of skill with such then-recent analytic tools like continued fractions, demonstrates with unusual rigour for the 18th century standards the irrationality of \(\pi \). The issue of the nature of this constant had taken a new impulse since the herculean efforts by Ludolph van Ceulen at the end of the 16th century with the use of new analytic tools and their application to some geometric problems. Authors like Gregory, Huygens, Mengoli, Leibniz or Wallis faced these issues, and in particular, the circle-squaring problem, in which \(\pi \) played a central role. Lambert takes up the baton of this analytic tradition —enriched by Euler with his first systematic study of continued fractions— and settles the question of its irrationality.
Therefore the circumference of the circle is not to the diameter as an integer number to an integer number.
—J. H. Lambert Mémoire.
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Notes
- 1.
I only intend to make some comments without going into details, since all the relevant explanations will be included in the part dedicated to the annotated translation.
- 2.
See Baltus (2003).
- 3.
Cantor (1908, p. 447) (translated by José Ferreirós).
- 4.
Hintikka and Spade (2019, p. 12).
- 5.
Euler (1744, p. 325).
- 6.
- 7.
Vorläufige Kenntnisse für die, so die Quadratur und Rektifikation des Cirkuls suchen.
- 8.
I want to thank José Ferreirés for his comments on this respect.
- 9.
(Beckmann 1971, pp. 170, 171).
- 10.
See Chap. 3, §. 15 (words in bold are mine).
- 11.
The irrationality of \(e^{x}\) with x a non zero rational.
- 12.
Concerning the importance of Lambert in the field of logic, see (Hintikka and Spade 2019), where he is claimed to have been without doubt «the greatest 18th-century logician».
- 13.
I thank José Ferreirés for the translation.
- 14.
Legendre (1794, p. 296). In the second edition the demonstration is included in Note V and from the fourth in Note IV (I could not consult the third edition). The comment to Lambert from the fourth edition is reduced to a brief footnote:
This proposition was first demostrated by Lambert, in the Memoirs of Berlin, anno 1761.
- 15.
During the writing of this book, on which we began work in late 2019, a preliminary draft of an English translation of (Serfati 1992) by Denis Roegel was published online (available at: https://hal.archives-ouvertes.fr/hal-02984214) but, to our knowledge, a final version of it has not yet been published.
- 16.
Speiser’s annotations will be indicated throught the translation of the Mémoire by means of footnotes as follows: «See the note by A.S. in Appendix C».
- 17.
I have to say that this summary does not show my journey in chronological order, since there are, as the reader will know, sources that are faster and easier to consult than others. For example the aforementioned work by Adreas Speiser was the last one that I have been able to analyze, long after I had almost completely prepared the translation with the annotations.
- 18.
Lemmermeyer and Mattmuller (2015, p. 55 nota 65).
- 19.
See [Lambert 1770, p. [II]].
- 20.
The interested reader can consult the aforementioned minutes on the Berlin Academy of Science website. The reference to the reading of this work also appears in the lower left corner of the first page of the Mémoire: «Read in 1767». Concerning the aforementioned dating of Lambert’s work, (Rudio 1892) warns us that, although many people repeat it, the data 1761 as the publication date is wrong. The relevant parts of the Monatsbuch in this regard are Petrie (2009, pp. 112 (note 527), 164 (note 733), 169 (note 763), 172 (note 773)). I would like to clarify that I have been able to access this work thanks to the kindness of Armin Emmel, who sent me the parts related to my investigation in a totally disinterested way. Likewise, I thank José Ferreirés for the translation of these parts.
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Dorrego López, E., Fuentes Guillén, E. (2023). Introductory Remarks About the Mémoire (1761/1768). In: Irrationality, Transcendence and the Circle-Squaring Problem. Logic, Epistemology, and the Unity of Science, vol 58. Springer, Cham. https://doi.org/10.1007/978-3-031-24363-9_4
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