Mathematical Analysis of the Eight-Year Cycle in the Eight-Year Calculation Method: A Study of the Calendar System in the Book “Ajā’ib al-Makhlūqāt wa Gharā’ib al-Mawjūdāt”

Abstract

This study investigates the origin and validity of the Eight-Year Hisab Method, a calendrical system commonly used in Islamic calendars of the Malay-Indonesian archipelago, such as the Javanese Islamic calendar and regional taqwim traditions. The research focuses on tracing its historical roots through an analysis of Zakariya al-Qazwini’s cosmographical work, ‘Ajā’ib al-Makhlūqāt, and applies a mathematical approach to reconstruct the eight-year cycle it presents. Employing a qualitative library research method, the study draws upon primary texts and secondary astronomical data, particularly the Jean Meus lunar conjunction data. The findings reveal an accumulated time of approximately 2.94168 days in each eight-year cycle, implicitly indicating the presence of three leap years. Statistical analysis shows a significant improvement in correlation—rising from a negative value to 0.64—when a one-year temporal adjustment is applied every ±96 years. Although this model exhibits less precision than the 30-year cycle, it demonstrates sufficient accuracy for practical calendar use. The study concludes that al-Qazwini was among the earliest scholars to formally structure an eight-year lunar cycle, and his model remains relevant for reconstructing lunar-based civil calendars today.