Hong Gilju's two major works on mathematics are Giha sinseol 幾何新說 and Hogak yeonrye 弧角演例. In Giha sinseol, which consists of three parts, "Ssangchu eoksan" 雙推臆算, "Gaebang monggu" 開方蒙求, and "Japswae sucho" 雜鎖隨 , he suggested many new algorithms. For example, he proposed the method of Ssangchu eoksan; he showed how to infer and calculate with two arbitrary values; and this method employed the counting rod which was very common in Korean mathematics in the early 19th century but totally forgotten in China. Another significant invention of Hong's is a new algorithm of extracting roots. He proposed such a simple method of extracting roots that it needs only simple calculations and the counting rod. Hong's new methods suggest that these originated from a new intellectual milieu in the 19th century. Hogak yeonrye dealt mainly with spherical trigonometry. Every step of calculations inferred from the theorems in Yeoksang goseong 曆象考成 was explained in each case.The background of Hong's mathematical works can be firstly traced to his mother and friends. His mother was well trained in mathematics, and his friends, especially Kim Yeong and Seo Yu-bon, gave much influence on him. His pedigree, which connects his family with Seo family, the best-known mathematician group at that time, provided a strong background of his mathematics. Finally and most importantly, the intellectual milieu of the late 18th and early 19th centuries Korea, where most scholars were interested in mathematics and astronomy, offered Hong fertile soil to make such an impressive achievement.