Computing Planetary Positions: User-Friendliness and the Alfonsine Corpus

Abstract

(ProQuest: ... denotes formulae omitted.)Astronomical tables are ways to turn the treatment of complex problems into elemen- tary arithmetic. Since Antiquity astronomers have addressed many problems by means of tables; among them stands out the treatment of planetary motion as well as that for the motions of the Sun and the Moon. It was customary to assign to the planets constant mean velocities to compute their mean longitudes at any given time in the past or the future, and to add to these mean longitudes corrections, called equations, to determine their true longitudes. In this paper we restrict our attention to the five planets,1 with an emphasis on their equations. Section 1 deals with what we call the standard tradition, beginning with Ptolemy's Handy tables, and Section 2 deals with the new presentations that proliferated in Latin Europe in the fourteenth and fifteenth centuries, some of which reflect a high level of competence in mathemati- cal astronomy.21. The Standard TraditionBy the middle of the second century a.D. Ptolemy displayed tables for the equations of the five planets with specific layouts and based on specific models, algorithms, and parameters. We argue that this category of tables, as is the case for many others, provides a clear example of user-friendliness, the driving force that prevailed in the history of table-making.In Almagest XI. 11 Ptolemy presented tables for the planetary equations, one for each of the five planets.3 Each table has eight columns, of which the first two are for the argument (one from 6° to 180° and the other for its complement in 360°). The argument is given at intervals of 6°, from 6° to 90° (and for 270° to 354°), and at intervals of 3°, from 90° to 180° (and for 180° to 270°). According to Toomer, Ptolemy computed the entries at 6°-intervals, even where the function is tabulated at 3°-intervals.4 Columns 3 and 4 are for the equation in longitude and the difference in equation, respectively. Column 3 assumes an eccentric model, which Ptolemy rejected in favour of an equant model. Column 4 displays the difference between the equation for an equant model and the equation for an eccentric model. The sum of corresponding entries in these two columns is the equation of centre, which replaced columns 3 and 4 that appear in Almagest XI. 11 (see Table A, col. 3).5 Columns 5 and 7 give the subtractive and additive differences to be applied to the equation of anomaly (displayed in col. 6), when the planet is at greatest and least distance, respectively. Column 8 is for the minutes of proportion, to seconds, used for interpolation purposes. We note that, in the case of Venus, the entries for the equation in longitude (col. 3) are exactly the same as those for the solar equation, although Ptolemy does not call attention to this fact.6 We display Ptolemy's model for Mars to illustrate how a planet's position can be computed directly from the model: see Figure 1. To do this, one must solve plane triangles by means of trigonometric procedures that were already available in Ptolemy's time. The solution is as follows. Given ic, we wish to compute the correction angle, c3, by solving triangle ECO. But, before we can do this, we have to find the length of EC, where DC, the radius of the deferent, is 60. So first we must solve triangle EDC to find EC, where angle CED is the supplement to angle ic and ED is the eccentricity (a given parameter in the model). With ic, EC and EO (twice the eccentricity), we can solve triangle ECO, which yields the values for c3 and CO. We then have to solve triangle MCO to find c(a). In this triangle two sides and an angle are known: angle MCO is equal to 180° - (cc - c3), CM is the radius of the epicycle (a given parameter in the model), and CO has already been determined. Then...where λ(?), the longitude of the apogee, is a given parameter in the model. Using the planetary equation tables takes trigonometric functions out of the computational scheme. …