《统天历》的岁实消长与气朔算法分析
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摘要
《统天历》是宋代最特殊的历法,它放弃传统的上元时刻选择方法,并首创岁实消长法。详细分析了《统天历》岁实消长与积年算法,指出其回归年常数和消长量比理论值偏大,但其岁实消长算法是合理的。将《统天历》计算定朔和冬至时刻的算法用现代公式表示出来,并给出气朔算法的一个具体算例,计算绍熙二年(1191AD)天正冬至时刻和定朔时刻。利用计算机计算出《统天历》自公元1101—1300年的所有冬至时刻和1192—1209年的所有定朔时刻,与真实值比较,得到了《统天历》气朔推算的精度。《统天历》推算定朔的整体精度是误差21.9min,而冬至的推算精度则是误差29.1min。
Concord with Heaven System is the most special calendar in Song Dynasty. The traditional selection method of epoch was given up, and the method of tropic year growth and decline was initiated in this calendar. Acording to a detailed analysis of the algorithm of epoch and tropic year growth and decline, the paper points out that the tropical year constant and its fluctuation of Concord with Heaven System is large than the theoretical value, but algorithm of tropic year growth and decline is reasonable. The algorithm of computing the real new moon and the winter solstice is represented into modern formulas. The winter solstice and the real new moon of 1191 AD as a concrete example of the algorithm is calculated. All winter solstice between the year 1101 and 1300 and all the real new moon form 1192 to 1209 AD are computed by computer. Compared with the real value, the caculation accuracy Concord with Heaven System is obtained. The accuracy of the real new moon is 21.9 minutes, and the accuracy of winter solstice is 29.1 minutes as a whole.
Concord with Heaven System is the most special calendar in Song Dynasty. The traditional selection method of epoch was given up, and the method of tropic year growth and decline was initiated in this calendar. Acording to a detailed analysis of the algorithm of epoch and tropic year growth and decline, the paper points out that the tropical year constant and its fluctuation of Concord with Heaven System is large than the theoretical value, but algorithm of tropic year growth and decline is reasonable. The algorithm of computing the real new moon and the winter solstice is represented into modern formulas. The winter solstice and the real new moon of 1191 AD as a concrete example of the algorithm is calculated. All winter solstice between the year 1101 and 1300 and all the real new moon form 1192 to 1209 AD are computed by computer. Compared with the real value, the caculation accuracy Concord with Heaven System is obtained. The accuracy of the real new moon is 21.9 minutes, and the accuracy of winter solstice is 29.1 minutes as a whole.
引文
[1] 宋濂,王祎.元史·历志[M].历代天文等志汇编(9) [Z].北京:中华书局,1976:3365.
[2] (元)脱脱,帖睦尔达世,贺惟一等.宋史·律历志[M].历代天文等志汇编(8) [Z].北京:中华书局,1976:2895.
[3] 中山茂.消长法の研究I[J].科学史研究(日本),1963,66:68-84.
[4] 中山茂.消长法の研究II[J].科学史研究(日本),1963,67:128-130.
[5] 中山茂.消长法の研究III[J].科学史研究(日本),1964,69:8-17.
[6] 曲安京.中国数理天文学[M].北京:科学出版社,2008:83-85,90-92.
[7] 滕艳辉,王鹏云.《纪元历》等8部宋代历法的定朔推步及精度分析[J].中国科技史杂志,2009,30(1):55-64.
[8] 滕艳辉.宋代朔闰与交食研究[D].西安:西北大学,2012:68-72.
[9] 滕艳辉,王鹏云.《纪元历》定朔算法模型及分析[J].西北大学学报(自然科学版),2008,38(5):855-858.
[10] 薄树人.《纪元历》解[A].薄树人文集[Z].合肥:中国科学技术大学出版社,2003:369-405.
[11] 滕艳辉,唐泉.《开禧历》定朔计算精度及分析[J].自然科学史研究,2011,30(1):19-27.
[12] 张培瑜.三千五百年历日天象[M].郑州:河南教育出版社,1990.
[13] 李勇.中国古代太阳中天观测及二至点测算精度[J].天文学进展,2005,23(1):70-79.
[14] 关增建.中国古代对回归年长度的测定[J].中国计量,2006,10(1):77-82.
(1)《元史历志》载《开禧历》行用44年,见《历代天文律历等志汇编》(9),3365页,自嘉定戊辰(1208)至淳佑辛亥(1251)应为44年。
(2)关于一般历法冬至时刻求法刻参见文献[9]。
(3)原文为207489,我们重新计算1194年天正冬至,得227489日分,此处当改。
(4)事实上,我们定义的距差是具有正负号的,《统天历》的距差都是正值,因此它要在小字中说明运算时正负号的取法。
(5)每日日躔表是根据《统天历》24气日躔表经过二次插值计算得到,具体算法参考文献[10]。
[2] (元)脱脱,帖睦尔达世,贺惟一等.宋史·律历志[M].历代天文等志汇编(8) [Z].北京:中华书局,1976:2895.
[3] 中山茂.消长法の研究I[J].科学史研究(日本),1963,66:68-84.
[4] 中山茂.消长法の研究II[J].科学史研究(日本),1963,67:128-130.
[5] 中山茂.消长法の研究III[J].科学史研究(日本),1964,69:8-17.
[6] 曲安京.中国数理天文学[M].北京:科学出版社,2008:83-85,90-92.
[7] 滕艳辉,王鹏云.《纪元历》等8部宋代历法的定朔推步及精度分析[J].中国科技史杂志,2009,30(1):55-64.
[8] 滕艳辉.宋代朔闰与交食研究[D].西安:西北大学,2012:68-72.
[9] 滕艳辉,王鹏云.《纪元历》定朔算法模型及分析[J].西北大学学报(自然科学版),2008,38(5):855-858.
[10] 薄树人.《纪元历》解[A].薄树人文集[Z].合肥:中国科学技术大学出版社,2003:369-405.
[11] 滕艳辉,唐泉.《开禧历》定朔计算精度及分析[J].自然科学史研究,2011,30(1):19-27.
[12] 张培瑜.三千五百年历日天象[M].郑州:河南教育出版社,1990.
[13] 李勇.中国古代太阳中天观测及二至点测算精度[J].天文学进展,2005,23(1):70-79.
[14] 关增建.中国古代对回归年长度的测定[J].中国计量,2006,10(1):77-82.
(1)《元史历志》载《开禧历》行用44年,见《历代天文律历等志汇编》(9),3365页,自嘉定戊辰(1208)至淳佑辛亥(1251)应为44年。
(2)关于一般历法冬至时刻求法刻参见文献[9]。
(3)原文为207489,我们重新计算1194年天正冬至,得227489日分,此处当改。
(4)事实上,我们定义的距差是具有正负号的,《统天历》的距差都是正值,因此它要在小字中说明运算时正负号的取法。
(5)每日日躔表是根据《统天历》24气日躔表经过二次插值计算得到,具体算法参考文献[10]。