section_path: Scholarly achievements/Mathematics
section_name: Mathematics
paragraph_text: Mathematics
  
  In the broad field of mathematics, Shen Kuo mastered many practical mathematical problems, including many complex formulas for geometry, circle packing, and chords and arcs problems employing trigonometry. Shen addressed problems of writing out very large numbers, as large as (10). Shen's "technique of small increments" laid the foundation in Chinese mathematics for packing problems involving equal difference series. Sal Restivo writes that Shen used summation of higher series to ascertain the number of kegs which could be piled in layers in a space shaped like the frustum of a rectangular pyramid. In his formula "technique of intersecting circles", he created an approximation of the arc of a circle s given the diameter d, sagitta v, and length of the chord c subtending the arc, the length of which he approximated as s = c + 2v /d. Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by Guo Shoujing (1231–1316). He also simplified the counting rods technique by outlining short cuts in algorithm procedures used on the counting board, an idea expanded on by the mathematician Yang Hui (1238–1298). Victor J. Katz asserts that Shen's method of "dividing by 9, increase by 1; dividing by 8, increase by 2," was a direct forerunner to the rhyme scheme method of repeated addition "9, 1, bottom add 1; 9, 2, bottom add 2".
  
  Shen wrote extensively about what he had learned while working for the state treasury, including mathematical problems posed by computing land tax, estimating requirements, currency issues, metrology, and so forth. Shen once computed the amount of terrain space required for battle formations in military strategy, and also computed the longest possible military campaign given the limits of human carriers who would bring their own food and food for other soldiers. Shen wrote about the earlier Yi Xing (672–717), a Buddhist monk who applied an early escapement mechanism to a water-powered celestial globe. By using mathematical permutations, Shen described Yi Xing's calculation of possible positions on a go board game. Shen calculated the total number for this using up to five rows and twenty five game pieces, which yielded the number 847,288,609,443.