chapter 3.2 Planar and Spherical Trigonometry Glen Van Brummelen 1 Introduction The motions of the stars influenced the ancients in various ways: they helped them to track times of year; they provided them a means to predict the future on Earth; and, when suitably interpreted, they enabled the ancients to predict the future in the celestial realm. To be able to foretell major events like eclipses as well as the risings, settings, and conjunctions of planets, ancient astronomers were obliged to quantify what they saw. Some cultures (especially that of the Babylonians) employed arithmetic schemes to represent the recurring patterns which they observed and then projected into the future. Hellenistic astron- omy chose a different path: it began geometrically by representing the celestial motions in terms of circles and straight lines linked in various ways. But using these representations to make predictions required the introduction of quan- titative measurements into the representations themselves; and so trigonom- etry was born. Since the object of study was the celestial sphere rather than a flat surface, spherical trigonometry was fundamental to locating objects in the sky. In cases in which all the relevant motions are within a plane (such as the motions of the Sun, the Moon, and the planets along the zodiacal circle), planar trigonometry sufficed. 2 Ancient Trigonometry The word “trigonometry” means the measurement of triangles; and today it conjures up sines, cosines, and tangents. But none of these functions existed in Hellenistic trigonometry. Indeed, the subject itself was not really an inde- pendent discipline but simply the mathematical underpinning needed to con- vert geometrical representations into astronomical predictions—in a sense, a transformation of Euclidean geometry into a scientific tool. However, as is not the case in modern science, the scope of the applications of trigonometry remained strictly astronomical. Since the typical astronomical diagram consisted of circles representing the Sun or the Moon or various components of orbital paths, the heart of the quan- © koninklijke brill nv, leiden, 2020 | doi:10.1163/9789004400566_011 planar and spherical trigonometry 55 figure 1 The definition of the chord titative problem was to convert the lengths of arcs into lengths of related line segments and vice versa. Probably as early as Hipparchus of Rhodes (ca 130 bce), this problem was reduced to its essence. In Figure 1, p. 55, suppose that we have a circle of a given radius R and that we know some arc ϑ on it. (Note: the arc ϑ is equal in value to the angle at the center of the circle.) The goal is to determine from this information the length of segment AB subtending ϑ (liter- ally, to determine the straight line (εὐϑεῖα γραμμή) AB). If this one task can be accomplished, then it can be adapted to extend the knowledge of quantities in astronomical diagrams. Obviously this length (which we call Crd ϑ since it is the chord subtending arc ϑ) depends on the size of the circle. Since Claudius Ptolemy (ca 140 ce) calculated in a sexagesimal number system, he chose the radius R of 60 in the Almagest. Hipparchus’ earlier value is not known since his work is lost but he might have chosen R = 3,438. This apparently peculiar choice has an explanation: if one divides the circle into minutes of arc and considers each of the 21,600 minutes to be a very short straight line of unit length, then the radius of the circle will be approximately 3,438 of these units (21,600/2π ≈ 3,438). 3 Determining Chord-Lengths Some chord-lengths are easy to determine geometrically, for instance, in Fig- ure 1, ϑ = 60°, so the triangle is equilateral and Crd ϑ = R = 60 (using Ptolemy’s value of R). One can also find the chords of arcs with lengths 90° and 120° and, with a little more effort, 36°, 72°, 108°, and 144°. To compose a table of chords of many arcs one at a time would be tedious; instead, theorems were derived that.