Re: A Tough One?

[fer j. de vries]

Mike Cowham wrote:
 
 Dear Friends,
 I have a vertical east declining sundial that I believe was once
 fixed to a church building.  Its gnomon is missing.
 What I wish to calculate is the latitude of the dial, and its
 declination.  I am sure that it is a very easy problem to solve, but so
 far I have failed.
 When I have this information, I hope to be able to locate its
 original site, (I already have a rough idea of the area of England), and
 maybe find some evidence of where it was fixed to the building.
 The only real clue to its location is given by the angles made
 by its hour lines - assuming them to be accurate.
 Thanks in advance to anyone who may be able to help.
 Regards,
 Mike Cowham.
 Cambridge, England.

Dear Mike,

Here I give just a number of formulae by which the latitude phi and the
declination d of a vertical sundial can be calculated, assuming the
pattern is well drawn.

Measure the angles of the following 2 hourlines :
- for east decliner : hour 6 and 9
- for west decliner : hour 18 and 15
Name these angles t45 and t90 and use positive signs for the angles.

Calculate :
P = cot(t45) - cot(t90)
Q = cot(t90)

X = P*P
Y = Q*Q

a = Y
b = X + Y - 1
c= -1

Z = (-b + sqrt(b.b - 4.a.c)) / (2.a)
or
Z = (-b - sqrt(b.b - 4.a.c)) / (2.a)
Take the positive answer for Z

Then
phi = atn(1/sqrt(Z))

d = asin(Q/tan(phi))

( sqrt is square root out of... )

Example :
t45 = 29 degrees
t90 = 68.78 degrees
X = 2.0044
Y = 0,1508
a = 0,1508
b = 1.1552
c = -1 ( of course )
Z = 0,7852
phi = 48.4552 degrees
d = 20.1244 degrees ( east or west )

I hope I didn't make any typing error.
Otherwise have a look in bulletin of De Zonnewijzerkring, 88.3, page 31.

Best wishes, Fer.

-- 
Fer J. de Vries
[EMAIL PROTECTED]
http://www.iaehv.nl/users/ferdv/
lat. 51:30 Nlong. 5:30 E

Re: A Tough One?

[John Davis]

Mike,

I encountered a similar problem in analysing the stained glass dial from
Wendon Lofts.

Assuming that there are no signs of the gnomon support to define the
sub-style angle on your dial, I would get a value of the declination from
the asymmetry of the earliest and latest hour lines on the dial.

Then, having measured all the hour (and half- and quarter-hours if they
exist), I would plot y = hourline angle - LHA vs x = LHA.  This should give
a curve which starts at the origin, rises to a peak, then returns to zero at
LHA = 90 degrees (this for the case of a direct south dial - the curve will
be offset for the am and pm halves for a decliner).  This experimental plot
can then be compared to a series of theoretical ones calculated for
different latitudes eg by the BSS Sundial Constructor program.  The closest
fit tells you the latitude of your dial.

The advantage of the rather strange-looking x-y parameters that I have
suggested is that it (a) it makes use of all the hour line information you
have, (b) it is most sensitive to lines around 45 degrees, and (c) plotting
the difference between hourline angle and LHA maximises the differences
between curves for different latitudes.

You may have to iterate around the above loop a couple of times if there is
significant uncertainty as to the declination.

I hope this helps - I have the Excel plots from the Wendon Lofts dial if
they are of any use.

John Davis