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Joel Silverberg asks some interesting questions about early views about the 
duration of twilight. I have no familiarity with that topic but have a few 
comments to offer.

The moment of twilight being considered, with the Sun18 degrees below the 
horizon, is what we describe as "astronomical twilight", at which no trace 
of light from the Sun remains in the sky. It differs from civil twilight, 
at 6 degrees below, when bright stars start to appear, and nautical 
twilight,12 degees below, when it becomes too dark to make out the horizon. 
Of course, these are no more than nominal values, with a lot of variation 
in practice. Star-navigation takes place between civil and nautical 
twilight. So it seems to me that this question is unrelated to navigation, 
even though the name of Pedro Nunes is associated with a publication about 
it.

At first sight, the question appears to be an entirely trivial one. It 
seems to me (without thinking about it too hard) that there are two days in 
the year when twilight is shortest, which are the days of the solstices, 
and those dates are unrelated to the observer's latitude. Is that too 
superficial a view? Are there fine-points to the question that have quite 
escaped me?

But the matter has rather a long history. Joel tells us "Nunes also 
includes sections of an 11th century arab work." This is presumably the 
text "Liber de crepusculis", a work in Arabic originally attributed to 
Al-Hazen, but it has since been ascribed (in an article in Isis, vol 58 no 
1 (1967), by Sabra) to the work of another author, Abu Abd Allah Muhammad 
ibn Mullah. Sabri desribes it as "a short work containing an estimation of 
the angle of depression of the sun at the beginning of morning twilight and 
the end of evening twilight, and an attempt to calculate on the basis of 
this and other data the height of the atmospheric moisture responsible for 
the refraction of the sun's rays". Sabri adds- "The reasonably accurate 
value of 18º, which it for the angle described has been found remarkable by 
many cmmentators."

I haven't seen this work, or read any more of Sabri's paper than the first 
page, all that's available free on the web, at 
http://www.jstor.org/pss/228388. It seems to me that if we substituted 
"particles" for "moisture", and "scattering" for "refraction" in Sabri's 
description, it would constitute an entirely rational piece of  research 
into atmospheric physics, even today. It had been translated into Latin, in 
Spain, which presumably made it available to Nunez.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

----- Original Message ----- 
From: "Joel Silverberg" 
To: 
Sent: Tuesday, June 29, 2010 12:00 AM
Subject: [NavList] shortest twilight problem...


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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