Dear Frank,
You pose two questions:
1. How do you lay out Babylonian and
Italian hour-lines?
2. Why use dubious definitions of
sunrise and sunset?
I attend to the dubious definitions below
but let's live with them for a moment.
BABYLONIAN AND ITALIAN HOUR-LINES
Let B = Babylonian = Hours since sunrise
Let I = Italian = Hours since sunset
Let F = French = 'normal' local sun time
At an equinox you can trivially convert B
or I to F:
F = B + 6 F = I - 6
Checks:
At sunrise B = 0 F = 6 I = 12
At noon B = 6 F =12 I = 18
At sunset B =12 F =18 I = 24
When the solar declination is not zero the
conversions are modified slightly:
F = B + 6 - xd F = I - 6 + xd
Here xd is the amount of extra dawn or extra
dusk compared with an equinox. If sunrise
is at 5 [local sun time] then xd = 1.
Expressed as an hour-angle:
sin(xd) = tan(dec).tan(latitude)
Of course, xd is negative when dec < 0.
All I did was to set up a spreadsheet and
for each Babylonian hour I chose five
declinations and worked out the equivalent
French hours.
That gave me five hour-angles and declinations
which I translated into (X,Y) points on the
slate.
I checked that the straight line of best fit
through the five points didn't miss any by
more than 0.5mm and drew the line.
Job done.
Well, job nearly done...
When you are cutting slate by hand, you lose
the line the instant you make the first cut!
Accordingly, you actually draw THREE lines:
the middle one and one on either side. You
then make a vee-cut between the two outer
lines and accept that you lose the middle one.
Where possible, the five declinations I choose
were +/- eps0, +/- 12 and 0. Many of the lines
went out of reach at the solstices so I chose
smaller declinations in such cases.
SUNRISE AND SUNSET
You say (correctly):
... the assumption seems to be made that sunrise
and sunset occurs when the altitude of the sun's
centre is zero. This is far from sunset in any
practical sense.
I certainly won't disagree. Like industry standards,
the great thing about definitions of sunrise and
sunset is that there are so many of them!
If you are unfortunate enough to have to measure
the sun's altitude with a sextant when the sun is low,
you have to make all kinds of tedious corrections...
If you are really doing the job properly you have to
allow for your height above sea level and refraction.
You therefore have to allow for temperature, pressure
and humidity and, probably, allow for pollutants too.
All this would make it hard to deal with even a simple
conventional horizontal sundial. The 6-6 lines run
west to east but the sun is (apparently) due east
detectably before 6am on occasions: when the declination
is small and negative.
You can't win!
My understanding is that Babylonian and Italian hours
came in AFTER the advent of mechanical clocks when (of
mechanical necessity) equal hours took over from
unequal hours.
With an equal-hours instrument you needed a reference
point to start and end the 24-hour period. There are
four obvious choices:
Midnight, Noon, Sunrise, Sunset
[Aside: there is a fifth utterly insane choice which
is 'one hour before midnight' and, curiously, that is
the one which is chosen for most of the civilised
world at the moment. Grrrrh!]
If you want to set a clock, midnight is not a good
time to do it by the sun. Noon IS a good time but
it is not easy to estimate noon if it is cloudy
and raining.
Sunrise and sunset can be estimated approximately
(within 30 minutes) even in the foulest of weather
conditions. This made these times attractive as
references on early clocks (which didn't keep very
good time).
It was common in Italy to deem sunset to be half
an hour after dark when the Ave Maria Office would
be said. Some Italian Hours sundials are marked
out in this way with the crossing points on the
equinoctial line displaced half an hour.
With that kind of history, I don't feel unhappy
about using a simple geometric definition of
sunrise and sunset.
You will now make a note to keep me well away
from the Navigation Officer's chart table!
All the best
Frank
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