Because in the ancient sundial that need
restoration (very frequent in Italy) it is not always possible to measure the
angles between the hour lines, some time ago I have devised a different method
to find the sundial parameters in
which we need only the knowledge of the equinoctial line and of some points where the hour line
cross it.
Let be:
-
E the point in which the meridian line crosses the equinoctial one;
-
P1 and P2 the points in which two hour lines cross the equinoctial
;
-
(P1-E) and (P2-E) the distances between the points P1 , E and P2 , E (measured on the sundial). These values
are positive if the point P is in the afternoon, are negative if it is
before noon.
-
(N1*15) and (N2*15) the hour
angles of the two hour lines
We get then
:
(P1-E)/(P2-E) =
[tan(N1*15-Ws)+tan(Ws)] / [tan(N2*15-Ws)+tan(Ws)]
where Ws is the hour angle of the sub-style line
.
If k = [(P1-E)*tan(N2*15)] /
[(P2-E)*tan(N1*15)] we
obtain the formula
:
tan(Ws) = (k-1) / [tan(N2*15) -
k*tan(N1*15)]
from which Ws .
If Mu is the angle between the
Equinoctial line and the horizontal
one and
A= tan(Ws) ; B = tan(Mu) we have
cos(Latitude)=[ B/A]*SQR {(1+A*A)
/ 1+ B*B)}
From
sin(d) =tan(Mu)*tan(Latitude) we obtain
the declination d of the wall , and
from
sin(Gam) = cos(Latitude)*cos(d)
the
angle Gam between the style and the
wall.
Finally from
(P1-E) = R*[tan(N1*15-Ws) +
tan(Ws)] / cos(Gam)
we may calculate the length of the
ortho-style R
--------------------------------
I have tried to do some measures
from the photo of the The Oldest Stained Glass Sundial
that is in John Carmichaels site and I have found the followings results
(obviously very approximate):
P1 on the equinoctial line = intersection with the 8h line; N1*15 = -60
P2 on the equinoctial line =
intersection with the 14h line;
N2*15 = 30
From the photo I have found : P1-E = -59 ; P2-E = +38.5 ; Mu = 13d
The results are
:
Latitude = 44.9d
Wall_Decl = -13.3d (East)
Sub-style hour angle = -18.5d that is, sub-style hour = 10h 46m
Angle between the plane and the
style = 43.6d
Ortho-style = 0.593 times the
distance between the points E and P1
It would be interesting to
compare these results with those gotten with other methods.
Note 1 in my opinion the sundial
was probably calculated for a latitude of 45d and not for a particular place in
Germany or in
Switwerland.
Note 2 - This method , that I have published some
years ago on my volume, is useful also for sundials with Italic or Babylonian
hours.
Best
Gianni
Ferrari
