Trigonometry and Spherical Astronomy 1. Sines and Chords
CHAPTER TWO TRIGONOMETRY AND SPHERICAL ASTRONOMY 1. Sines and Chords In Almagest I.11 Ptolemy displayed a table of chords, the construction of which is explained in the previous chapter. The chord of a central angle α in a circle is the segment subtended by that angle α. If the radius of the circle (the “norm” of the table) is taken as 60 units, the chord can be expressed by means of the modern sine function: crd α = 120 · sin (α/2). In Ptolemy’s table (Toomer 1984, pp. 57–60), the argument, α, is given at intervals of ½° from ½° to 180°. Columns II and III display the chord, in units, minutes, and seconds of a unit, and the increase, in the same units, in crd α corresponding to an increase of one minute in α, computed by taking 1/30 of the difference between successive entries in column II (Aaboe 1964, pp. 101–125). For an insightful account on Ptolemy’s chord table, see Van Brummelen 2009, where many issues underlying the tables for trigonometry and spherical astronomy are addressed. By contrast, al-Khwārizmī’s zij originally had a table of sines for integer degrees with a norm of 150 (Neugebauer 1962a, p. 104) that has uniquely survived in manuscripts associated with the Toledan Tables (Toomer 1968, p. 27; F. S. Pedersen 2002, pp. 946–952). The corresponding table in al-Battānī’s zij displays the sine of the argu- ment at intervals of ½° with a norm of 60 (Nallino 1903–1907, 2:55– 56). This table is reproduced, essentially unchanged, in the Toledan Tables (Toomer 1968, p.
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