Ceyuan (測圓海鏡) and Jiuyong yandai (九容演代)

The book《Ceyuan Haijing》studies inscribed and circumscribed circles in a right triangle and shows equations that give the diameters of the circles. We discuss the development of mathematical contents written by the scholar Yang Zhaoyun in Qing dynasty on the contents of《Ceyuan Haijing》in his book《Jiuyong Yandai》. He derived equations to find the diameters of the circles based on algebraic knowledge known in the Qing dynasty. In this paper, we conclude that Yang's methods in devising the equations include the Gou-Gu Theorem, mathematical expressions derived from Gou-Gu ratio table, and the technique of interchanging triangles and events. We conclude that the Gou-Gu ratio table was a very important tool when Yang devised the equations in《Ceyuan Haijing》. Copyright © 2014 The Korean Society for History of Mathematics.