| In recent years, the studies on the intercommunion and comparison of mathematical astronomy between ancient China and India have been generally paid close attention in the circles, either domestic or overseas, history of science. This paper intends to compare some mathematical methods in the field of astronomy in ancient China and India, so as to explore the characteristics of the sciences in ancient China and India, and to attempt to reveal the internal factors of the intercommunion and mutual influence of science and technology in the two countries.When determined the moment of the Winter Solstice, ancient Chinese mathematicians and astronomers often calculate the length of solar shadow, therefore, measuring and calculating the length of solar shadow becomes one of the most fundamental problems in ancient Chinese astronomy. From the initial simple measurement via Pythagorean theorem to the complex calculation in the Song and Yuan Dynasties, Chinese Chouren (mathematicians and astronomers) construct many different methods of calculation, and complete the change from measurement to calculation of the length of solar shadow. Similarly, in ancient Indian calendars, the astronomers also attach great importance to the calculation of the length of solar shadow. In many ancient Indian literatures of mathematics and astronomy, a large number of materials about the length of solar shadow, gain by both measuring via Pythagorean theorem and calculating have been recorded.On the basis of previous researches, this paper straightens out the evolution of the methods of measurement and calculation about the length of solar shadow in both ancient China and India, and analyzes thoroughly and compares systematically those methods. The paper consists of four parts.1. The author of the paper goes over some main Ancient Chinese and Indian mathematical and astronomical books that contain problems about measurement via Pythagorean theorem and studys the essence of their methods and theoretical system. The author thinks that although there are much common ground in the methods of measurements in ancient China and India, they develop from their native land and have not been influenced by external cultures.2. The author of the paper interprets some representative ancient Chinese literatures, which contain the algorithms about the solar shadow, in different historical periods, and divides the evolution of calculation on the length of solar shadow into three stages. The first historical period is from the Han Dynasty to the Sui Dynasty, in which the main method is the proportional algorithm; the second historical period is from the end of the Sui Dynasty to the end of the Tang Dynasty, in which the main method is the quadratic interpolation algorithm; and the third period is from Song to Yuan Dynasties, in which the main method is approximation algorithm of polynomial function. The paper analyzes the characteristics of the algorithm on solar shadow in different stages.3. Based on the historical periods of Indian astronomy that made by David Pingree, an American scholar, the paper deeply analyzes the Indian method on calculating the length of solar shadow, sums up its characteristics, and discusses the effect of foreign culture on it.4. Finally, the paper compares the characteristics and accuracy of the algorithm on solar shadow in ancient China and India and discusses the effect of cosmology on the method of measurement and calculation on the length of solar shadow in the two countries. The author of the paper holds that two countries' algorithms are different, Chinese algorithms, which are numerical methods, are gradually improved, but Indian calculating method is scientific after affected by Greek tradition, and the cosmology has exerted influence on the measurement and calculation about the length of solar shadow. |
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