Dear Noam,
You pose interesting questions...
> I hope this is not off topic...
Certainly on topic for me!!
> Will a south facing vertical dial split
> into 15 degrees using a polar gnomon
> divide the day in the exact same way
> as an ancient Greek or roman sundial
> such as a haemuspherium with a vertical
> or horizontal gnomon?
The key feature of a polar gnomon is that
the direction of the shadow at a given
time of day (hour-angle) is independent
of the time of year (declination).
Your dial with the polar gnomon will
therefore divide the day in the same
way every day of the year.
If you label your 15-degree interval
lines as equal hours, 6am to 6pm,
and hope to read off equal hours,
then the error at a given indicated
hour will be the same each day.
If you label your 15-degree interval
lines as temporary hours, 0 (sunrise)
to 12 (sunset), and hope to read off
temporary hours, then the error at a
given indicated hour will vary from
day to day (because temporary hours
vary from day to day).
You will need an expert on Greek or
Roman sundials to confirm the next
point...
My understanding is that, in Greek
and Roman sundials, the *tip* of the
shadow of the gnomon was used to
determine the time so we are talking
about an indicated point rather than
an indicated direction.
Also, such dials were usually intended
to indicate temporary hours. They
did this erroneously and the error
at a given indicated time changed
as the declination changed.
> If it does divide the day in the same
> fashion, as I think it does,...
I may well have misunderstood your
question but my interpretation is
that the division is different.
> Does anyone know the formula for
> breaking up the day on such a sundial
> with a horizontal gnomon?
Assuming a direct south-facing vertical
dial in the northern hemisphere there
are two formulae to note:
For a polar-oriented gnomon:
tan(ang) = tan(H).cos(phi)
Here ang is the angle of the shadow to
the downward vertical when the hour-angle
is H and the latitude is phi. Note that
there is no term for the declination since
the direction is independent of the time
of year.
For a horizontal gnomon:
sin(H).cos(dec)
tan(ang) = --------------------------------------------
cos(H).cos(dec).cos(phi) - sin(dec).sin(phi)
This is more complicated because the direction
does depend on the solar declination dec.
Notice that if the declination is zero, as
at the equinoxes, the formula reduces to:
tan(ang) = tan(H)/cos(phi)
Karlheinz Schaldach and others have shown that
the lines on the 1334 Braunschweig dial are
close to the angles given by this formula.
Accordingly, this dial would have indicated
the time correctly at the equinoxes. That is
not what you are asking about though!
Try some experiments with the long formula and
you will quickly get a feeling for how matters
change as the declination changes and you will
also, I suspect, be able to answer other questions
that occur to you.
Best wishes
Frank King
Cambridge, U.K.
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