球面三角學中四鄰公式之推導與其推論

The Derivations of the Four-Part Formulae and Its Inference in Spherical Trigonometry

四鄰公式為環繞球面三角的兩邊及兩角關係式。事實上，在誤差傳播為零的評估準則下，四鄰公式分別為大圈航法計算程序中求解大圈初航向角與天文觀測船位計算程序中求解天體方位角等的最佳公式。四鄰公式在航海學上有其重要性，然相關文獻卻很少論述它。本文從幾何、代數及向量等不同觀點推導四鄰公式，並進而直接推論特殊球面三角中納皮爾法則二的公式。

The four-part formula gives the relation between two angles and the two sides adjacent to them around the spherical triangle. In the fact, the four-part formula, based on the evaluating criteria of zero error propagation in steps of procedures, is optimal formula for solving initial course angle in great circle sailing problem and for solving azimuth angle in astronomical vessel position problem, respectively. Although the four-part formula is very important in application of navigation but litter attention has been given to the point in related literature. In this article, respectively derive the four-part formulae from geometry, algebra and vector viewpoints and then use it to directly infer formulae of Napier's rule 2 of special spherical triangle.