Born: 26 July 1271 in Poyang, now Jiangxi province, China
Died: about 1335 in Longyou Mountains, Zhejiang province, China
Zhao Youqin was skilled in a large range of topics. He was an expert in astronomy, mathematics and physics, with particular skills in optics. He was also, however, a religious philosopher and a specialist in alchemy. Before he died he gave a copy of the manuscript of his book Ge xiang xin shu, to his disciple Zhu Hui. The manuscript was passed from Zhu Hui to Zhang Jun who published the work.
The Ge xiang xin shu is usually translated as New Elucidation of the Heavenly Bodies but in  Volkov suggests New Writing on the Image of Alteration. He writes:-
The 'alteration' was traditionally related to the changes in numerical data of celestial phenomena and to the establishment of a new calendar. Thus, it appears plausible to suggest New Writing on the Image of Alteration as a tentative rendering of the title. This suggestion is reinforced by Zhao Youqin's remark in the section entitled 'Li fa gai ge' ('On Changes and Alterations (ge) of Calendrical Methods') of the first chapter of the treatise. There, he states that the accumulation of small errors in numerical data on heavenly phenomena led to changes of the values of parameters of calendars used in China from antiquity to his time.In this treatise, Zhao is concerned with the structure of the universe. For him this consists of a flat Earth inside spherical heavens. He gives an explanation of eclipses of both the moon and of the sun. There is also an interesting description of optical experiments he has carried out trying to determine the relationship between the apparent luminosity of a source and its distance from the observer. These experiments are carried out with a pinhole source and so relate to a camera obscura. Zhao also looks at various instruments which can be used for surveying. In particular he describes a gnomon which he uses to determine the distances of the sun, moon and stars from the earth. He also describes an instrument he has designed to measure the difference in right ascension between objects on the celestial sphere. Another of the instruments he has designed allows him to calculate the angle between a given star and the north pole. One of the most interesting features of the book, however, relates to Zhao's calculation of π.
Here is Volkov's translation of a part of Zhao's text (quoted in ):-
Let us take the small square and develop it with calculational procedures in order to obtain the image of a circle and to achieve the circle's perimeter which is to be established. Start from the square which has four angles and widen it in order to obtain a 'circle' of 8 angles and sides, this is the first step. If we perform the second step, then what we research has 16 sides. If we perform the third step, then what we research has 32 sides. If we perform the fourth step, then what we research has 64 sides. In all cases, if the number of steps is increased by one, the number of sides is necessarily doubled. If we reach the 12th step, then what we research will have 16,384 sides. The starting small square is gradually increased and developed, the circle is gradually completed and filled. The greater the number of angles, the more what once was a square is no longer a square but is transformed into a circle!We see that Zhao is describing an iterative procedure, calculating the length of the side of the regular 2n-gon at each step. He ends by saying:-
To summarize, the square is the beginning of the calculations [or 'numbers' (shu)], the circle is the end of the calculations. The circle starts from the square, the square ends in the circle. This is the procedure of the Zhou bi [suan jing], and nothing goes beyond its bounds!The Ge xiang xin shu is not the only book by Zhao to have survived. There is a second work, namely the Xian Fo tongyuan, which is concerned with the teachings of Transcendentals and Buddhas. This is a work on alchemy whose aim was to attain immortality using meditation.
Article by: J J O'Connor and E F Robertson
MacTutor History of Mathematics