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    Re: In the Wake of Magellan
    From: Frank Reed
    Date: 2019 Oct 10, 11:39 -0700

    Your first two questions address real history, important history. I'll skip those for now and comment on your third question, which is speculative, good fun, and applies just as well today as it would have five hundred years ago. 

    Assuming you know that the Sun's Dec has some value to some accuracy, e.g. +/-23.45° on June 21/December 21, and you also know that the Dec will be zero on March 20/September 23 (different dates on Magellan's calendar, but some specific dates that we can easily specify), how well can we reproduce the Sun's correct Dec values? A straight line approximation would be wrong about halfway between the solstices and equinoxes by as much 5°. That's poor. 

    We can do much better by making a circle diagram, as I have described previously. Draw a quadrant of a circle with a radius of 23.45 cm. Mark 89 to 93 (depending on the season) evenly-spaced dots on the circumference of the quadrant. Label every tenth dot by date. On any date, measure the height of the dot up from the baseline. This is the approximate Dec on that date. This approximation still has issues but it's much better than the linear approach. We can use the same diagram to estimate the daily rate of change of the Sun's Dec by measuring the distance of any daily dot horizontally from the vertical axis. 

    No matter how you handle this interpolation, you'll be limited by the +/-12 hours (or more) uncertainty in the timing of the equinoxes. The Sun's Dec is changing at very nearly one minute of arc per hour (moving N/S at just about one knot) near the equinoxes, so you can expect errors around then on the order of 10 minutes of arc. Can you live with that? In many cases, yes.

    Frank Reed

       
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