Ming Antu's infinite series expansion of trigonometric functions
Ming Antu's infinite series expansion of trigonometric functions. Ming Antu, a court mathematician of the Qing dynasty did extensive work on the infinite series expansion of trigonometric functions in his masterpiece Geyuan Milü Jiefa "(Quick Method of Dissecting the Circle and Determination of The Precise Ratio of the Circle)". Ming Antu built geometrical models based on a major arc of a circle and the nth dissection of the major arc. In Fig 1, "AE" is the major chord of arc "ABCDE", and "AB", "BC", "CD", "DE" are its nth equal  segments. If chord "AE" = "y", chord "AB" = "BC" = "CD" = "DE" = "x", the task was to find chord "y" as the infinite series expansion of chord "x". He studied the cases of "n" = 2, 3, 4, 5, 10, 100, 1000 and 10000 in great detail in volumes 3 and 4 of "Geyuan Milü Jiefa". Historical background. In 1701, French Jesuit missionary Pierre Jartoux (1669-1720) came to China, and he brought along three infinite series expansions of trigonometric functions by Isaac Newton and J. Gregory: These infinite series stirred up great interest among Chinese mathematicians , as the calculation of with these "quick methods" involved only multiplication, addition or subtraction, being much faster than classic Liu Hui's π algorithm which involves taking square roots. However, Jartoux did not bring along the method for deriving these infinite series. Ming Antu suspected that the Europeans did not want to share their secrets, and hence he was set to work on it. He worked on and off for thirty years and completed a manuscript called "Geyuan Milü Jiefa". He created geometrical models for obtaining trigonometric infinite series, and not only found the method for deriving the above three  infinite series, but also discovered six more infinite series. In the process, he discovered and applied Catalan numbers. Two-segment chord. Figure 2 is Ming Antu's model of a 2-segment chord. Arc "BCD" is a part of a circle with unit ("r = 1") radius. "AD" is the main chord, arc "BCD" is bisected at "C", draw lines BC, CD, let BC = CD = "x" and let radius AC = 1. Apparently, formula_4 Let EJ = EF, FK = FJ; extend BE straight to L, and let EL = BE; make BF = BE, so F is  inline with AE. Extended BF to M, let BF = MF; connect LM, LM apparently passes point C. The inverted triangle BLM along BM axis into triangle BMN, such that C coincident with G, and point L coincident with point N. The Invert triangle NGB along BN axis into triangle; apparently BI = BC. BM bisects CG and let BM = BC; join GM, CM; draw CO = CM to intercept BM at O; make MP = MO; make NQ = NR, R is the intersection of BN and AC. ∠EBC = 1/2 ∠CAE = 1/2 ∠EAB; formula_6 ∠EBM =  ∠EAB; thus we obtain a series of similar triangles: ABE, BEF, FJK, BLM, CMO, MOP, CGH and triangle CMO = triangle EFJ; So formula_11, and formula_12 Because kite-shaped ABEC and BLIN are similar. Thus formula_22 or formula_23 then And so on Add up the following two equations to eliminate formula_30 items: Expansion coefficients of the numerators: 1, 1, 2, 5, 14, 42, 132 ... (see Figure II Ming Antu original figure bottom line, read from right to left) are the Catalan numbers; Ming Antu discovered the Catalan number. Thus: in which formula_38 is Catalan number. Ming Antu pioneered the use  of recursion relations in Chinese mathematics substituted into formula_4 Finally he obtained In Figure 1 BAE angle = α, BAC angle = 2α × x = BC = sinα × q = BL = 2BE = 4sin (α /2) × BD = 2sin (2α) Ming Antu obtained formula_45 formula_48 Three-segment chord. As shown in Fig 3,BE is a whole arc chord, BC = CE = DE = an are three arcs of equal portions. Radii AB = AC = AD = AE = 1. Draw lines BC, CD, DE, BD, EC; let BG=EH = BC, Bδ =  Eα = BD, then triangle Cαβ = Dδγ; while triangle Cαβ is similar to triangle BδD. As such: formula_54 Eventually, he obtained formula_55 Four-segment chord. Let formula_56 denotes the length of the main chord, and let the length of four equal segment chord =x, formula_57+... formula_58. formula_59 formula_60. Five-segment chord. formula_61 Ten-segment chord. From here on, Ming Antu stop building geometrical model, he carried out his computation by pure algebraic manipulation of infinite series. Apparently ten segments can be considered as a composite 5 segment, with each segment in turn consist of two subsegments. formula_63  formula_64, He computed the third and fifth power of infinite series formula_65 in the above equation, and obtained: formula_66+... Hundred-segment chord. A hundred segment arc's chord can be considered as composite 10 segment-10 subsegments, thus sustutde formula_67 into formula_68, after manipulation with infinite series he obtained: formula_69 formula_70 Thousand-segment chord. formula_71 formula_72... Ten-thousand-segment chord. formula_73... When number of segments approaches infinity. After obtained the infinite series for n=2, 3, 5, 10, 100, 1000, and 10000 segments, Ming Antu went on to handle the case when n approaches infinity. y100, y1000 and y10000 can be  rewritten as: formula_74... formula_75... formula_76... He noted that when n approaches infinity, the denominators 24.000000240000002400, 24.000002400000218400×80 approach 24 and 24×80 respectively, and when n -> infinity, na (100a, 1000a, 1000a) becomes the length of the arc; hence formula_77... formula_78 Ming Antu then performed an infinite series reversion and expressed the arc in terms of its chord formula_79...