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    Re: Agnostic gnomon and horary error
    From: Frank Reed
    Date: 2024 Jun 8, 06:08 -0700

    Geoff Hitchcox, you wrote:
    "I assume that if the Sundial was (very carefully) mounted on the surface of Lake Saint-Jean, it would read the correct sundial time.
    So for the 20 June 2024, I tabulated the Solar Azimuth for both locations for every minute of the day - the Sun was above the horizon. I then subtracted the Solar Azimuth at the same UTC minute, (Lake - Treworgy) and plotted this difference (as per attached graph). If this is how one would find the answer Frank - I find the graph quite interesting. The maximum time error of about +40 minutes in the morning, and -40 minutes in the afternoon. Like a broken watch (always accurate once a day), the Treworgy Sundial will be useful each day at Solar Meridian Transit!"

    That's an intriguing idea! On the face of it, it sounds good! But it also sounds a bit 'too good' :). O, if only we had some experimental data! Of course spherical sundials are relatively easy to construct. I have described building one before that we make in my "Sundials" workshop. Maybe we just need to set one up on a clear day and assess the errors for some known offset in the gnomon inclination. :)

    You mentioned that the sundial will be correct each day at solar meridian transit (also known as "local noon"). Of course this has to be true, right? Picture setting up a sundial like this at local noon. The gnomon points to north (or south) in azimuth, or so we assume. The gnomon shadow fall directly on the 12:00, or in the case of an armillary-style spherical sundial, the gnomon shadow falls entirely within the shadow of the meridian ring, which amounts to the same thing. If I rotate the sundial about the east-west axis, the gnomon will continue to be aimed north-south while its inclination changes. The shadow will continue to fall on the 12:00 mark at local noon no matter how much I rotate the sundial around that east-west axis, which is equivalent to changing the "designed" latitude.

    How does the time from a navigator's "time sight" depend on latitude? What error do we find if we use the wrong latitude, and is the error similar for a sundial? For anyone unfamiliar with it, a time sight is not a separate variety of sextant sight. It's just a slightly different way of working with a sight. We take an ordinary afternoon or morning Sun sight (typically a Sun sight but another object works, too) when the Sun is away from the meridian. Rather than getting a line of position from it, we can directly calculate the Sun's dLon (the difference in longitude between the navigator and the Sun's longitude (the Sun's GHA). That dLon is also identical to the Sun's HA or hour angle or "sundial time". The calculation of the dLon or HA from a sextant altitude in a modern, familiar form is simple enough:
       sin(Alt) = sin(Lat)sin(Dec) + cos(Lat)cos(Dec)cos(dLon),
    which we can easily solve for dLon. But what is the sensitivity to the latitude? Is it similar to the sundial case?

    Frank Reed

    PS: Yikes, it's been a week since you wrote that reply. Sorry for the delay in replying!

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