Xu Guangqi (or Xu Guang-qi) is also known as Hsu Kuang-ch'i. He obtained the highest level in the civil-service examination having been educated in astronomy and calendar computation. He rose in importance to eventually become the leading minister in the Imperial Court of the Ming Dynasty. Before this he had studied Western culture under Matteo Ricci but, before we explain their work together, we should set the scene by quoting from  concerning the background:-
In spite of the dismal political history of the Ming Dynasty (1368-1644), China progressed in many fields including trade and industry, science and technology, philosophy and literature, mainly owing to the wisdom and effort of the people. The economic development was especially marked in the Yangtze Valley. It was therefore a natural consequence that figures like Xu Guang-qi and others appeared during this period of 'Renaissance'. ... [Xu Guang-qi] led a long, but politically rather futile, ministerial life for a quarter of a century in the Imperial Court of the Ming Dynasty. It was also during these last decades of the Ming Dynasty that the Chinese first came into contact with European science through the Jesuits. The Jesuits were intent on spreading the Catholic faith in the old Empire, and in order to win over the people, they endeavoured first to gain the favour and the following of the educated class. As an expedient means they brought in various new technological gadgets and apparatus unknown to China, as well as scientific theories which were, though not all of them up-to-date knowledge at the time in Europe, nonetheless of a sufficient novelty and attraction to some educated Chinese.
In fact Chinese mathematics had been in a period of decline for some time. Xu Guang-qi was well aware of this and attributed the decline to academics neglecting practical learning and also to a confusion between mathematics and numerology. The brilliant "tian yuan" or "coefficient array method" or "method of the celestial unknown" for solving equations which had been expounded with such skill by Li Zhi in the 13th century was no longer understood in China. The remarkable progress which the Chinese had made in algebra had been largely forgotten, and practical problems which had been solved by algebra were by this time solved by ad hoc means. Even the Nine Chapters on the Mathematical Art was almost unknown, and Xu Guang-qi himself had never read the brilliant Chinese classic, while The Ten Classics were thought to have been lost. Into this weak period in Chinese mathematics came European mathematics brought by scholars such as Ricci.
Ricci was a Jesuit who had studied in Rome under Clavius. He arrived at Macau on the east coast of China in 1582. He settled in Chao-ch'ing, Kwangtung Province, and began his study of Chinese. He also worked at acquiring understanding of Chinese culture. In 1589 Ricci moved to Shao-chou and began to teach Chinese scholars the mathematical ideas that he had learnt from his teacher Clavius. Together with one of his students, Qu Rukui, he translated the first book of Euclid's Elements. Ricci attempted to visit Peking in 1595 but, finding the city closed to foreigners, he went instead to Nanking where he lived from 1599. There he presented his essay on the first book of the Elements but this preliminary work is now lost. Ricci was well received in Nanking and this encouraged him to try again to visit Peking which he did in 1601. This time he was allowed to live in the city and he made this his home from that time. He taught mathematics to Chinese students and one of these was the high-ranking public official Xu Guang-qi.
Xu Guang-qi became the first native of China to publish translations of European books into Chinese. Collaborating with Ricci he translated Western books on mathematics, hydraulics, and geography. The first six books of Euclid's Elements were translated into Chinese in 1607 by Ricci and Xu Guang-qi. They took the Latin commentary on Euclid's Elements first published by Clavius in 1574. As an amusing note, we remark that their translation of "Clavius" into Chinese meant "nail" so they referred to Clavius as "Mr Ding"! Their method of translation is described in :-
The translation technique involved Ricci explaining the contents of the original text orally to [Xu Guang-qi] who would then write down what he had understood. ... Ricci and Xu Guang-qi's translation respects the order of Clavius's work completely; however, it is much less verbose ...
The approach to mathematics in these books must have seemed totally alien to Chinese readers whose approach to the subject had been so radically different. Xu Guang-qi wrote in the foreword to the translation:-
Four things in this book are not necessary; it is not necessary to doubt, to assume new conjectures, to put to the test, to modify. In addition four things in this book are impossible. It is impossible to remove any particular passage, to refute it, to shorten it, or to place it before that which precedes it, or vice versa.
Clearly Xu Guang-qi was a total convert to Western thinking but most other Chinese mathematicians stuck to their traditional way of thinking, questioning what to them was absurd such as "a point has no part". The Chinese approach to mathematics had been highly practical and to try to fit the Elements into that tradition Xu Guang-qi explained in his preface how the contents had application to the problem of the calendar, to music and to technology. Most readers remained unconvinced. However the new Chinese terminology which Xu Guang-qi had to invent for point, curve, parallel line, acute angle, obtuse angle etc. (these concepts being alien to Chinese mathematics, there were no Chinese words for them) soon became part of Chinese mathematics, as did the style of the geometric figures, in particular the characters Xu Guang-qi chose to label them.
In one sense Xu Guang-qi did a disservice to Chinese mathematics. He was converted to Christianity by Ricci and adopted the position that Chinese culture was inferior to that of the West, in particular, as we indicated above, in their mathematical tradition. This was a great shame, for although clearly much had to be learnt from the transmission of knowledge, there was no need to talk down the fine achievements of the Chinese through a different approach. Xu Guangqi wrote:-
Rules in the West different from ours we do not have. Rules in the East that are the same as in the west are all right, those different from those in the west are all wrong. ... Therefore though The Ten Mathematical Classics are lost, this is not a pity, for they were nothing but worn-out shoes.
He predicted that soon everyone in China would be studying the Elements and in this he was largely correct since Western schools were set up in China in which the study of the elements was a compulsory topic.
Ricci's main aim in China was to convert the Chinese to Christianity. Many, both in China and in the European Christian Church, felt that he used Western mathematics, science and technology as an unreasonable means to achieve conversions. Ricci himself felt that reforming the Chinese calendar would be the most effective step that could be taken, so "proving" the power of Christianity. Indeed the question of calendar reform had occupied the Chinese for 200 years but, despite various proposals being made, the Bureau of Astronomy had been cautious and done nothing. Xu Guang-qi was already interested in calendar reform before he met Ricci, so he was soon involved in the debate.
The Western approach to astronomy and the calendar scored a major success shortly after Ricci died when it accurately predicted the eclipse of 15 December 1610. Another eclipse was predicted for 1629 and a competition was held by the Chinese government to determine who could give the most accurate prediction of its timing. Three different predictions were made, one by the Da Tong traditional Chinese school, one by the Islamic calendar school, and one by the New Method School led by Xu Guang-qi which used European methods. The most accurate prediction for the eclipse of 21 June 1629 was made by Xu Guang-qi and the emperor then appointed him take charge of calendar reform. Four European Jesuits assisted Xu Guang-qi but the reform process had not been completed on his death in 1633 and was taken over by Li Tang-jing.
During the last few years of his life Xu Guang-qi was an extremely influential figure at the Imperial Court of the Ming Dynasty. The Ming were under attack by from the Manchu who were descendants of the Juchen tribes who had ruled North China as the Chin dynasty in the 12th century. Xu Guang-qi, with his strong belief in the superiority of all things European, persuaded the Ming emperor to have his army adopt advanced European artillery against the Manchu. Initially effective, the strategy collapsed after Xu Guang-qi's death when the Manchu learned European iron-smelting technology and acquired Western arms themselves. The Ming dynasty was defeated by them in 1644.
Article by: J J O'Connor and E F Robertson