Friday, April 8, 2016
I was able to get a good fit for Regiomontanus' positions for the Sun for the year 1495 using Legendre polynomials to "smooth" the data. The difference between the two is consistent with rounding off to the nearest minute.
Again the minimum for daily changes in the positions for the smoothed data were used to determine the position of apogee and an ellipse was then fit to the daily changes in position.
The rms deviation between the fitted ellipse and the smoothed positions of the Sun was about 1.5 minutes so the original positions may not have been solely based on observations. The orbital elements are similar to those for 1494 but with a better value for the length of the year.
I tried to do a set of elements for the year 1500 but the difference between the smoothed and original positions showed relatively large deviations near Feb 28.
Supplemental (Apr 9): The fitted elliptical orbit was chosen so as to minimize the rms deviation for the interval from 20 to 320 days after the beginning of the year to make the fit more precise. The use of an eccentric circle may have may have introduce a systematic error into the Sun's positions which in turn may have affected the eccentricity of the elliptical orbit.
Thursday, April 7, 2016
Christopher Columbus may have had a copy of Regiomontanus' Ephemerides with him in 1504 when he predicted the lunar eclipse for the natives in Jamaica to get them to continue with their provisions while he and his crew were shipwrecked. The Wikipedia article on the eclipse asserts that he had an almanac by the Portuguese astronomer Abraham Zacuto and in another place that he had Regiomontanus' Calendarium. With the rivalry between Portugal and Spain at the time it is unlikely that they would be sharing information concerning navigation. The information in the Calendarium is less detailed than that in the Ephemeris and the position given for the Sun differs somewhat. The first three columns of the Calendarium contain the day of the month, a letter indicating the day of the week and a column with days as designated in a Roman calendar. The table of regions for the Ephemeris and Calendarium both indicate a 0h 0m offset for Nuremberg so that may be the meridian that was used.
It's not clear when Regiomontanus produced his Ephemerides for 1494-1506 or the location for which it was computed. The ephemeris for 1494 was used to compute a set of orbital elements and presumably the positions of the Sun were calculated for noon which was the practice since Ptolemy's time. The ephemerides of the Sun were based on observations but comparisons of the positions for the months December and January with the fit show some distortion. As with the orbital elements for 1672 the daily changes were used to determine the time and position of the Sun's apogee.
The fit of the Sun's positions for 1494 gave the following elements.
The position of perigee is a few degrees earlier than for 1672 but occurs about the same time of year. The eccentricity, e, is also slightly larger too but still less than the value for the eccentricity in Ptolemy's time.
Regiomontanus died while working on calendar reform in Rome in 1476.
Supplemental (Apr 7): I used Legendre polynomials to fit the daily positions of the Sun and then fit a Keplerian orbit to the daily changes in fit's position for the Sun. The fit of the Legendre polynomials is worse at the beginning and end of the year. The central part of the curve is a good match to the elliptical orbit.
Supplemental (Apr 8): I was checking the length of the year 1494 in Regiomontanus' Ephemerides and noticed that there were only 365 days present where it should have been designated as bisextilis, i.e., a leap year. But February, 1494 in the Ephemerides does not have 29 days. The year 1500 is properly designated as bisextilis and February, 1500 does have 29 days.