0; 7,58

Chapter 5 gives an example for using the tables for the mean motion of the Sun for May 12, 1447 at 15;20,30h. The result, after selecting the entries in the various tables, is

2,20;52,1° = 7;20,48° (1000y) + 2;56,20° (400y) + 0;17,38° (40y) + 5,59;33,9° (6y) + 1,38;33,52 (100d) + 29;34,10° (30d) + 1;58,17° (2d) + 0;36,58° (15h) + 0;0,49° (20 min) + 0;0,0° (30s).

The mean longitude of the Sun, 59;9,35°, is obtained by adding the previous value to the radix of the Incarnation (4,38;17,34°).

There were two conventions in the Middle Ages for dates in the Julian calendar. One convention was to use complete years, months, and days, and the other was to use current years, months, and days. The worked example presented in Chapter 5 for May 12, 1447 indicates that Bianchini employs a mixed system: the given date uses a current year and a current month, but complete days (e.g., 'May 12' has to be understood as '132 days have elapsed since the beginning of the year' or '12 days of May have elapsed'). The epoch for this calendar is the Nativity which, in this context, means noon of the day preceding Jan. 1, 1 A.D. In other words, at the given date, only 1446 and 4 months have elapsed. This is not the criterion used nowadays, that is, May 12, 1447 means that 1447 is the current year (1446 complete years have elapsed since epoch), May is the current month (4 complete months have elapsed in the current year), and day 12 means that 11 complete days have elapsed in the current month. Moreover, the Parisian Alfonsine Tables and the editio princeps of the Alfonsine Tables use current years and months and, for both, 'May 12' means that '11 days of May have elapsed'. Ultimately, Bianchini's criterion might derive from the way he constructed his tables, for they are based on adding excesses of days for different dates in order to find the correct time for which a quantity is to be computed. With this criterion, it would seem easier then to work with complete rather than incomplete days.

Table 8. Position of the solar apogee

Na, ff. 16r-17v: Tabula solis in auge Nu, f. 21v: Tabula ingressus solis in augem

Va, ff. 187v-189r: Tabula ad inveniendum introitum solis in augem

Ed. 1495, ff. a8r-b3v: Tabula solis in auge

Ed. 1526, ff. 8r-11v: Tabula solis [symbol for the Sun] in auge

In this table there are five columns: see Table 8. The first is for the argument: collected years from 0y to 2000y at intervals of 4y (at intervals of 20y in Regiomontanus's copy). Thus, all the entries correspond to leap years. The entries in the second column, under the heading 'June', are given in days, hours, and minutes (from 3d 23;38h for year 0 to 13d 1;6h for year 2000). As explained in Chapter 7, the entries show the date and the time the Sun is at its apogee in a given year. The heading of the third column is 'equation' and entries are only given for some selected arguments, in hours, minutes, and seconds; they represent the amount to be added to the corresponding entry in the second column when the given year is one, two, or three years after a leap year. Column 4 displays the longitude of the solar apogee, in physical signs, degrees, minutes, and seconds, whereas column 5, headed motus in anno, represents the yearly progress (in seconds) of the solar apogee. We note that the entries in column 5 are the same, although with a lower precision, as the corresponding ones in Table 1 for the yearly motion of the apogees.

The instructions for using this table are found in Chapter 7 where the worked example consists in determining for year 1446 the position of the solar apogee and the time when the Sun reaches it. The date and time sought are June 13 and 8;50h, and this date was used previously in the discussion of Tables 1 and 2. The resulting position of the solar apogee is said to be 1,30;40,16°. This value appears as 1,30;40,17° in Chapter 6 (see our comments to Table 2).

Table 8: Position of the solar apogee (excerpt)


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