executed man, to examine his arteries and viscera in order to discover cures for illnesses. Shen also took interest in human anatomy, dispelling the long-held Chinese theory that the throat contained three valves, writing, "When liquid and solid are imbibed together, how can it be that in one's mouth they sort themselves into two throat channels?" Shen maintained that the larynx was the beginning of a system that distributed vital qi from the air throughout the body, and that the esophagus was a simple tube that dropped food into the stomach. Following Shen's reasoning and correcting the findings of the dissection of executed bandits in 1045, an early 12th-century Chinese account of a bodily dissection finally supported Shen's belief in two throat valves, not three. Also, the later Song dynasty judge and early forensic expert Song Ci (1186–1249) would promote the use of autopsy in order to solve homicide cases, as written in his Collected Cases of Injustice Rectified. 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 (104)43. 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 + 2v2/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