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Simon Lhuilier (sometimes written Simon L’Huilier) was the son of Laurent Lhuilier, a jeweller and goldsmith. His mother, Suzanne-Constance Matte, was Laurent Lhuilier’s second wife and there were three older children in the family. The Lhuilier’s were a Huguenot family, originally from Mâcon, but after the Edict of Nantes (which had granted religious liberty to the Huguenots) was revoked by Louis XIV in 1685, they had to flee. They settled in Geneva in 1691.

There was a strange episode in Lhuilier’s life when he was still young. A wealthy relation proposed that he would leave Lhuilier a large fortune if he followed a career in the church. However, Lhuilier had already found the attraction of mathematics and money was not going to make him give up the attractions of the topic so he refused his relative’s offer.

Lhuilier was an exceptional secondary school pupil and he went on to study mathematics at a Calvin Academy where he was taught mathematics by one of Euler’s former pupils, Louis Bertrand, and physics by Georges-Louis Le Sage. It was through Le Sage that Lhuilier obtained his first post as tutor to the Rilliet-Plantamour family, a post he held for two years.

The next career move by Lhuilier was also as a result of him knowing Le Sage. Another of Le Sage’s students, Christoph Pfleiderer, had been appointed to the position of professor of mathematics and physics at the Military Academy in Warsaw. Pfleiderer was put in charge of a competition to find the best authors to write texts for Polish schools and in 1775 he sent details to his old teacher Le Sage. Le Sage tried to persuade Lhuilier to submit an entry to write a physics text but Lhuilier preferred to enter the competition to write a mathematics text. Lhuilier’s proposal won the competition giving him the right to write a mathematics textbook to be used in Polish schools.

Adam Kazimierz Czartoryski was a Polish prince who had been educated in England and prepared to take over the Polish throne but he refused it in 1763. He later became the first minister of education in a European country and his palace at Pulawy became an important centre of culture providing an excellent school for his sons and for the sons of other important people in the neighbourhood. Czartoryski had been involved in the competition to find authors of Polish school texts and he was so impressed by Lhuilier’s entry that he offered him a position as tutor at Pulawy in 1777, in particular as a tutor to his son Adam Jerzy Czartoryski who was seven years old at the time.

Lhuilier spent eleven years at Pulawy. Adam Jerzy Czartoryski proved an extremely bright and gifted pupil and, in addition to his tutoring duties, Lhuilier found time to write his mathematics text, undertake research in mathematics which resulted in several fine publications, and enjoy a busy social life with hunting parties. He also submitted an entry for the prize topic proposed in 1784 by the Berlin Academy.

The Academy sought the best article on the theory of the mathematical infinity and they designed the competition to encourage mathematicians to seek a sound basis for the new differential calculus. Lhuilier submitted the paper Exposition élémentaire des principes des calculs supérieurs and his essay won the prize and was published in Berlin in 1786. The standard concepts and notation for derivatives, and the standard elementary theorems on limits which appear in an undergraduate calculus text today appear in a remarkably similar form in Lhuilier’s prize winning essay. Lhuilier introduced the notation “lim”, and was the first to allow two-sided limits.

The topic of limits was a particularly fortunate one for Lhuilier since he had been thinking about limits before the topic was ever proposed for the prize. In fact his Polish school textbook which was published in 1780 contains a section on limits.

In 1789 Lhuilier returned to Switzerland but the political situation there seemed fragile and he feared that there would soon be a revolution. Pfleiderer, who had become a friend through the Polish episode, was by this time teaching mathematics in Tübingen and Lhuilier went to be with Pfleiderer there. He would stay with Pfleiderer in Tübingen until 1794. In the following year Lhuilier was offered a chair of mathematics in Leiden, but he preferred to compete for the chair in Geneva which had been held by his former teacher Louis Bertrand. Having won the competition, Lhuilier was appointed in 1795 and held this chair until he retired in 1823.

Not only was 1795 the year of his appointment to the Academy in Geneva but it was a year marked by two other important events in Lhuilier’s life. In that year an improved version of his prize-winning essay on limits was published in Latin in Tübingen. Also in that year Lhuilier married Marie Cartier and they would have two children, a son and a daughter.

Lhuilier was also involved in politics in Geneva, being President of the Legislative Council there in 1796. He also achieved a high position in the Academy at Geneva, becoming its rector. He enjoyed many academic honours too, being elected a corresponding member of the Berlin Academy, of the Göttingen Academy, of the St Petersburg Academy, and of the Royal Society of London.

His work on Euler’s polyhedra formula, and exceptions to that formula, were important in the development of topology. Lhuilier also corrected Euler’s solution of the Königsberg bridge problem. He also wrote four important articles on probability during the years 1796 and 1797.

One further work by Lhuilier is worth commenting on. This is the two volume work Eléments raisonnés d’algèbre that he published in 1804 for his students in Geneva. This work was really a sequel to the text which he wrote for Polish schools many years before. Speziali writes in [1]:-

The main value of these two volumes lay in the author’s clear exposition and judicious selection of exercises …

Also in [1] his character is described as follows:-

Whereas the Poles found Lhuilier distinctly puritanical, his fellow citizens of Geneva reproached him for his lack of austerity and his whimsicality, although the latter quality never went beyond putting geometric theorems into verse and writing ballads on the number three and on the square root of minus one.

His most famous pupil was Charles-François Sturm who studied under Lhuilier during the last few years of his career in Geneva.

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 [6] 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 [3]:-

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.

Born on April 24th

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