Page:On Chronology and the Construction of the Calendar.pdf/18

days, viz, 30,44, is the mean length of the twelve months in the European Calendar.

The Chinese month is the synodical month, commencing with the New-Moon's day and lasting until the day of the next New-Moon. A synodical month or a lunation being on an average equal to 29,53059 mean solar days, is nearly one day less than two tsie-khi or one European month of 30,44 days on an average. As to the months of every chinese year--the common year consisting of 12 lunations as well as the leap year consisting of 13 lunations--always only 12 names, never 13 names are used, the intercalary lunation bearing the same name (number) as the antecedent lunation, the characters of the sexagesimal cycle Kiah-Tsze, which are added in the chinese Calendar also to the months, turn back after five chinese years.

In the annexed table (7) are inscribed the Chinese and European names of the 28 Moon-stations or Moon-stars, and also their positions on the heavenly sphere on the 1 January 1850 A.D. determined by their rightascension and declination or by their longitude and latitude. Besides these data there are added the annual precession and its secular variation in rightascension and declination, this being necessary for calculating the right ascension and declination at any other given time not very remote from 1850 A.D.

The longitudes of all stars increase each year about 50,2 seconds in arc, but the latitudes scarcely alter.

For the calculation of the exact value of the rightascension $\alpha '$ and declination $\delta '$ of a star at any time $T$ between say 1700 A. D. and 2000 B. C. from its given rightascension $\alpha$ and declination $\delta$ at 1800 A. D. the following formulas, exhibited in the work „Berliner astronomisches Jahrbuch für 1866“ may be used.

${\begin{aligned}\operatorname {tg.} {N}&={\frac {\operatorname {tg.} {\delta }}{\cos .{(\alpha +A)}}}\\\operatorname {tg.} {(\alpha '+A')}&=\operatorname {tg.} {(\alpha +A)}{\frac {\cos .{N}}{\cos .{(N+\theta )}}}\\\operatorname {tg.} {\delta '}&=\operatorname {tg.} {(N+\theta )}.\cos .{(\alpha '+A')}\end{aligned}}$

$A$, $A'$ and $\theta$ to be taken from the table (I).

In the same table is also noted the obliquity of the ecliptic $\epsilon$ for the time 2000 B. C. until 1700 A.D., so that from $\alpha '$ and $\delta '$ can be found also the longitude $\lambda '$ and latitude $\beta '$ of the star corresponding to the time $T$ (to which $\alpha '$ and $\delta '$ belong) by means of the formulas:

${\begin{aligned}\cos .{\beta '}\cos .{\lambda '}&=\cos .{\delta '}\cos .{\alpha '}\\\cos .{\beta '}\sin .{\lambda '}&=\sin .{\delta '}\sin .{\alpha '}\cos .{\epsilon }+\sin .{\delta '}\sin .{\epsilon }\\\sin .{\beta '}&=-\cos .{\delta '}\sin .{\alpha '}\sin .{\epsilon }+\sin .{\delta '}\cos .{\epsilon }\end{aligned}}$