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.
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.
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:[1] These infinite series stirred up great interest among Chinese mathematicians, as the calculation of π 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.
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.
[2] 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.
Because kite-shaped ABEC and BLIN are similar,.
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.
Ming Antu pioneered the use of recursion relations in Chinese mathematics[10] substituted into
Finally he obtained[11] In Figure 1 BAE angle = α, BAC angle = 2α × x = BC = sinα × q = BL = 2BE = 4sin (α /2) × BD = 2sin (2α) Ming Antu obtained
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.
From here on, Ming Antu stop building geometrical model, he carried out his computation by pure algebraic manipulation of infinite series.
, He computed the third and fifth power of infinite series
+......[16][17] A hundred segment arc's chord can be considered as composite 10 segment-10 subsegments, thus substituting
, after manipulation with infinite series he obtained:
............[12] 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.
Ming Antu then performed an infinite series reversion and expressed the arc in terms of its chord
