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the Marquis de L'Hospital published the first textbook on calculus in 1696. It is based in part on lectures given by Johannes Bernoulli around 1692. L'Hospital also received private tutoring from Bernoulli during that interval. In 1922 a manuscript was discovered written (if I remember correctly by Bernoulli's nephew) in 1705, which was a copy of a manuscript (which was never published and is now lost written by Johannes Bernoulli in 1693 or so. ) These may be lecture notes for Johannes lectures.

In both Bernoulli's notes and in L'Hopital's textbook a curious problem with navigational implications appears. Using the newly invented differential calculus, determine the day of the year with the shortest period of twilight (defined as the time for the sun to rise from 18 degrees below the horizon to the horizon) for any given latitude. The solutions of the two men are not the same ... in fact they are quite different. And neither actually caries the solution out to a particular answer, they only show how to set the problem up.

Here comes the interesting question(s). I have seen several references that state that this very same problem was the topic of a book entitles De Crepusculis [ Concerning the twilight] published by Pedro Nunes, a very prominent Portuguese navigator in 1542 (150 years earlier than either Bernoulli or L'Hospital). It is a substantial book of nearly 150 pages. Certainly, he could not have used calculus, but must have attacked the problem geometrically.

Nunes also includes sections of an 11th century arab work.
Does anyone know of an English translation of any of this? Does anyone know how Nunes approached the problem? Does anyone know why anyone would care about the day of shortest twilight?

Author Nunes, Pedro, 1502-1578
Uniform Ti De crepusculis
Title Petri Nonii Salacie[n]sis, De crepusculis liber unus, / nu[n]c rece[n]s & natus et editus. Item Allacen Arabis uetustissimi, De causis crepuscolorum liber unus, /à Gerardo Cremonensi iam olim Latinitate donatus, nunc uero omniu[m] primum in lucem editus
Note The second work, formerly attributed to Alhazen, is now ascribed to Muḥammad ibn Muʻādh, Abū ʻAbd Allāh, al-Jajjāni. Cf. Sabra, A.I. "Authorship of 'Liber de crepusculis'", in: Isis, v. 58 (1967), p. 77-85
Imprint from colophon

Thanks for any leads or insights or comments.

Joel Silverberg

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