The Accuracy of Seki Kōwa and Tatebe Kenhiro’s Calculations for Arc Lengths Based on Arrow Heights and Chord Lengths in Wasan
Author: tonagaiExecutions: 28
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In the January 2025 issue of Nikkei Science, in the special feature “Rediscovering Wasan,” an article by Kenichi Sato titled “The True Portrait of Seki Kōwa, the ‘Sage of Arithmetic’” presents an approximation formula for the square of an arc. We will compare this formula with the exact value to assess its accuracy. Note that the diameter of the circle is fixed at 1.
Method in Seki and Tatebe’s “Kenki Algorithm”
F(x) = (-9/40 + 6131089x/120 + 752638x²/45 + 340127x³/30 - 5966x⁴/3 + 699904x⁵/45)/113²
Method in Seki and Tatebe’s “Kakuyō Sanpō”
H(x) =(5,107,600x - 23,835,413x² + 43,470,240x³ - 37,997,429x⁴ + 15,047,062x⁵ - 15,010,25x⁶ - 281,290x⁷)
/(1276900*(1-x)⁵)
Exact value
cos⁻¹(1-2x)²
F(x) = (-9/40 + 6131089x/120 + 752638x²/45 + 340127x³/30 - 5966x⁴/3 + 699904x⁵/45)/113²
Method in Seki and Tatebe’s “Kakuyō Sanpō”
H(x) =(5,107,600x - 23,835,413x² + 43,470,240x³ - 37,997,429x⁴ + 15,047,062x⁵ - 15,010,25x⁶ - 281,290x⁷)
/(1276900*(1-x)⁵)
Exact value
cos⁻¹(1-2x)²
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