Establishment of tangent function table of Jiufu shadow algorithm

The solar depolarization of various gases in Yangcheng is given in the calendar, so we know the solar zenith distance difference of various gases, which is equal to anywhere. In this way, for any place, as long as we know the solar zenith distance of a solar term (such as summer solstice), we can calculate the solar zenith distance of other gases by adding and subtracting this difference. The remaining two problems need to be solved: first, how to find the solar zenith distance of a certain summer solstice (or winter solstice); Secondly, it is known how zenith distance transforms the shadow length. These two problems can be solved by establishing a table corresponding to the shadow length of the sun and the zenith distance.

If we list a numerical table with zenith distance as the independent variable and the shadow length every other degree, we can solve the above two problems: firstly, measure the shadow length of the measured site from summer solstice to the sun (this measurement is carried out everywhere in the geodetic survey led by a line), get the sun zenith distance from the shadow length look-up table, and then add and subtract a difference ai as mentioned above to get the zenith distance of each gas in the site, and then look up the table again. A line in Da Yan Li's "Step Leakage" established such a corresponding table between the shadow length of each degree from 0 to 80 degrees and the zenith distance of the sun, which is the earliest tangent function table in the history of mathematics in the world. Abroad, around 920, al-Battani (about 858-929), an Arab scholar, compiled the shadow length table of 12 foot bar every 0-90 degrees, which is actually the table of 12ctga. Aboul Waha (940-998), another Arab scholar, compiled a tangent cotangent function table around 980, and gave a value every 15 degree and 10 degree. He also introduced secant and cotangent function for the first time. The method of compiling a line into tangent and cotangent function tables is almost the same as that of Al Batani. A line uses the zenith distance of the sun, and Al Albatani uses the elevation angle of the sun. They are complementary angles, so their findings are the same. The tangent function table of a line is nearly 200 years earlier than that of Al Albatani and 250 years earlier than that of Al Weaver. Although the tangent function table of a row is only from 0 degrees to 80 degrees, and the error is relatively large, it is the earliest tangent function table in the world after all.