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. 29; F. S. Pedersen 2002, pp. 957–959) and it was rarely modified during the Middle Ages. For instance, it is found in the Alfonsine corpus as the first of a series of tables ascribed to John of Lignères (e.g., Oxford, Bodleian Library, MS Can. Misc. 27, ff. 88v–89r, where this table is found under the title: Incipiunt tabule sinuum et cordarum (. . .) composuit magister iohannes de lineris picardus diocesis ambianensis anno domini nostri ihesu christi 1322), but close inspection reveals that it was just copied from the Toledan 20 chapter two
Table 2.1A: sines (excerpt) Argument Corde mediate (°, ′) (°, ′) (°) 0 30 179 30 0;31,25 1 0 179 0 1; 2,50 ...... 30 0 150 0 30; 0, 0 ...... 60 0 120 0 51;57,42 ...... 89 30 90 30 59;59,52 90 0 90 0 60; 0, 0
Tables. Table 2.1A displays an excerpt of the table ascribed to John of Lignères. The most extensive table of this type in a 14th-century manuscript that we have found is in Paris, Bibliothèque nationale de France, MS lat. 7316A, ff. 89r–111r, where the argument is given at intervals of 1′, thus displaying 2,700 entries for the sine of the argument. At the end of this table there is a trigonometric circle, similar to that displayed in Figure 1, and a small text which includes the name of John of Murs (f. 111r). In some cases, columns for other trigonometric functions are dis- played next to the column for the sine (see Figure 1 for the geometric definitions of these functions). For example, in the Almanac of Azar- quiel columns for cosine and versine appear next to the column for sine. Table 2.1B displays such a table, as presented in Paris, Biblio- thèque de l’Arsenal, MS 8322, f. 127r. Towards the middle of the 15th century substantial progress was made in this field. In his Tabulae primi mobilis Bianchini compiled a table of sines with a norm (i.e., a value of Sin 90°) of 60,000. This table is later found in Regiomontanus’s Tabule directionum, first printed in 1490 in Augsburg, although Regiomontanus composed tables with a norm of 6,000,000 in his De triangulis as well as one with a norm of 10,000,000, in effect displaying decimal values of the sine function to 7 places (Zinner 1990, pp. 55 and 95).