Trigonometry

Trigonometry

Trigonometry

Introduction

Spherical trigonometry as the computational basis for astronomy is built on and utilizes the basic properties of plane trigonometry. Problems in spherical trigonometry are solved by associating arcs in spherical triangles or quadrilaterals with their trigonometric line segments in three-dimensional space, or by first transforming the spherical figures, via projection or three-dimensional solids, to right triangles in the plane and then making use of algorithms developed in plane trigonometry. In this paper, I will examine the evolution of the “media” used for such transformations in China over the period from the early-17th century to the mid-18th century.1 Specifically, I will analyze several trigonometric treatises authored by the Jesuit missionary Giacomo Rho (1593-1638?), by Mei Wending (1633-1721), and by Dai Zhen (1724-1777). For the latter two Chinese scholars, I will also attempt to relate their choices of transformation media to their views on Western learning.Problems of spherical trigonometry in 17th- and 18th-century China were often reduced to problems in plane trigonometry and then solved by means of the proportionality of corresponding sides of similar right triangles. Nevertheless, in the literature on the history of Chinese mathematics, there is not much discussion on the transformation and reduction of spherical problems to the plane, and how the techniques utilized for such transformations evolved over time. In this article, I investigate the evolution of the transformation media involved. I will show that in the trigonometric treatises by Mei Wending (1633-1721) and Dai Zhen (1724-1777), the authors' views on Western learning shaped their choices of transformation media, and conversely their choices of transformation media offered support to their views on trigonometry in the debate of Chinese versus Western methods. Based on my analysis, I also propose a reassessment of Dai's treatise of trigonometry, which was controversial ever since its publication in the 18th century.

Discussion

Throughout Chinese history, astronomy has been closely associated with calendrical science and calendar making.2 The calendar mattered politically to the court in many ways; for example, an unpredicted yet verified solar eclipse could be interpreted as a sign of the emperor's lack of virtue.3 Many historians of mathematics contend that trigonometry was introduced to China by the Jesuits in the 17th century as part of calendar reform efforts.4 Spherical trigonometry in 17th-century China was considered an integral part of astronomy, being the basis for the computations in calendar-making. Problems in astronomy were often of one of two forms: finding the distance between two points on the celestial sphere, and finding the angle between two intersecting arcs. Here distance is defined as the length of the great arc connecting the two points. In general, problems in spherical trigonometry were formulated as finding the length of a side or an angle of a spherical triangle or quadrilateral.

Both Jesuit and Chinese scholars in the beginning of the 17th century employed the following two fundamental properties to solve problems in trigonometry: proportionality of the corresponding sides of similar (right) triangles, and the Pythagorean theorem, known in Chinese as the gouge ...