RECENT WORKS ON PTOLEMY'S ARGUMENT FOR THE ECCENTRICITY OF THE DEFERENT

In recent decades historians of science have been interested in Ptolemy's second argument. Pedersen, Swerdlow, Evans and Jones,Footnote ⁴ have separately written about the subject, putting forward possible scenarios and motivations for the discovery of the equant point by Ptolemy.

Pedersen analyzes the problem and divides it into two questions. First, why does Ptolemy add the equant point to the simple epicycle-deferent model? What was his observational evidence? Second, why was the eccentricity of the deferent chosen as precisely one half of the eccentricity of the equant? As these two questions are related to each other but not the same, an answer to the second one may be an answer to the first one. Even if we find an argument for the calculation of the eccentricities of the deferent center and equant point, we would not be sure that it is the very argument which Ptolemy used to “discover” the equant point.Footnote ⁵

In Pedersen's view, the Almagest gives no explicit answer to the first question. But if we look for the observable consequences of the choice between the first model without the equant and the second one with the equant, it is obvious that the apparent size of the epicycle would be different in the two models and this difference would influence several phenomena: the length of the retrograde arcs, the latitude of the planet, and the distance of the planet from the observer; the latter affecting the apparent velocity of the planet in different parts of its path, particularly at the oppositions.

Swerdlow and Evans are interested in the first observational phenomenon, i.e., retrograde arcs. To them it seems the most plausible candidate for the basis of Ptolemy's calculation of deferent eccentricity, because Ptolemy himself, in the text already quoted, mentions his use of retrograde arcs to find the eccentricity of the deferent. Swerdlow provides a detailed calculation to find the eccentricity of the deferent given one half of the retrograde arcs and one half of the time of the retrograde motion at both the apogee and the perigee. These data have been extracted from the results obtained by Ptolemy in Almagest 12.6 and have been checked against observations.Footnote ⁶

By contrast, to find the deferent eccentricity, Jones takes into consideration the apparent velocity of the planet. In any retrogradation, the planet passes three times through the middle point of the retrograde arc. Given the time intervals and the longitude of the oppositions, Jones calculates the eccentricities of the deferent and the equant points.

THE MARĀGHA SCHOOL AND THE EQUANT POINT

As it is well-known, the question of the foundations of Ptolemaic models was discussed in Islamic astronomy at least from the 10^th–11^th centuries onwards. These debates resulted in the invention of non-Ptolemaic models in the 13^th century by a group of astronomers known as the School of Marāgha. Al-ʿUrḍī (d. 1266 AD), al-Ṭūsī (d. 1276 AD), al-Shīrāzī (d. 1311 AD) and Ibn al-Shāṭir (d. 1375 AD), among others, are the most famous figures of this school, so called mainly because of the association of the first three with the observatory built by the Īlkhānīd monarch Hulāgū in the city of Marāgha in northwest Iran in 1259 AD.

In these astronomers' critique of Ptolemaic models, the equant point played a crucial role, because its existence seemed to contradict some principles of the Aristotelian physics, especially that of uniform circular motion. Astronomers of Marāgha devised new models to avoid non-uniform motions, without putting into doubt Ptolemy's observational data.Footnote ⁷ In their effort to get rid of all non-uniform motions, while preserving the equant point, it was only too natural for the Astronomers of the Marāgha School to be interested in the origin of this point.

AL-SHĪRĀZĪ AND THE EQUANT POINT

Quṭb al-Dīn al-Shīrāzī presents his astronomical theories in his three main works: Nihāyat al-Idrāk fī dirāyat al-aflāk and al-Tuḥfa al-shāhiyya in Arabic and Ikhtiyārāt-i Muẓaffarī in Persian.

Of course al-Shīrāzī was not a historian of science, but as an astronomer of the Marāgha School, for whom the non-uniform motion of the spheres was a serious problem, he was interested in the empirical origin of the introduction of the equant point in Ptolemaic models and the way through which Ptolemy had devised this theoretical device. In his book Nihāyat al-Idrāk fī dirāyat al-aflāk, he introduces his predecessors' non-Ptolemaic planetary models and remarks his objections. In Book II, Chapter 8 of Nihāya, entitled “About the Spheres of the Superior Planets and Venus and their Motions in Longitude”, he presents a model for the outer planets which he calls “Aṣl al-ḥāfiẓa wa-al-mudīr” (The Principle of the Keeper and the Director) which is, as Saliba has shown, the very model developed by al-ʿUrḍī.Footnote ⁸ Al-Shīrāzī's criticism of this model is based on the idea that the place of the deferent center must coincide with the point already obtained by Ptolemy, whereas in this model the deferent center is displaced to a new point, midway between the old deferent center and the equant point. Like the modern historians mentioned above, al-Shīrāzī believes that the position of the deferent center was not arbitrarily chosen by Ptolemy, so the fact that he does not present an argument for his choice doesn't mean that there is none. Then he goes on to present his suggestion and justify it by an argument.

Of the view of the problem of the origin of the equant prevailing among his contemporaries, al-Shīrāzī writes at the beginning of his text:

This is exactly the view expressed by al-Ṭūsī in the eleventh chapter of his Taḥrīr al-Majisṭī (The Recension of Almagest), entitled “On the Values of the Eccentricities of the Superior Planets and the Positions of their Apogees”:

Against this opinion, al-Shīrāzī suggests a method based on the concept of “retrograde arc”, a concept which designates by the term “the value of retrogradation” (miqdār al-rujūʿ). As far as we know, this term has been introduced in this context by al-Shīrāzī himself. It has no equivalent in the Arabic versions of Almagest,Footnote ¹⁰ and neither al-Ṭūsī in his Taḥrīr, nor his contemporary, al-ʿUrḍī, in his Kitāb al-Hayʾa, mention it. The latter writes:

Again, in this quotation, there is no mention of “the value of retrogradation”. Indeed, the two authors, al-Ṭūsī and al-ʿUrḍī, may refer to Almagest 12.6, in which Ptolemy calculates the retrograde arcs of the planets on the basis of the eccentricities of each of them and states that “the amounts [of the retrogradation] we have demonstrated agree very closely, with those derived from the actual phenomena associated with each planet.”Footnote ¹² Even if Ptolemy does not explicitly present this as a proof of his conjecture for the eccentricity of the deferent, probably al-Ṭūsī and al-ʿUrḍī considered it as a plausible proof of Ptolemy's conjecture.

AL-SHĪRĀZĪ'S ARGUMENTS FOR THE ECCENTRICITIES OF THE DEFERENT AND THE EQUANT POINTS

Eccentricity of the deferent

Al-Shīrāzī not only suggests an empirical origin for Ptolemy's choice of the deferent on the basis of “retrograde arc”, but also provides an argument for the eccentricity of the equant point. His argument for the eccentricity of the deferent is, in this case, similar to the one Swerdlow and Evans suggest. To calculate the size of the epicycle, he uses two separate observations of retrograde arcs at the apogee and the perigee. Then, by comparing the results, he calculates the difference between the distances of apogee and perigee to the center of the universe. Thereupon he obtains the eccentricity of the deferent.

Although the Almagest already contains an argument for the eccentricity of the equant point, al-Shīrāzī continues to develop a different method. It is based on of the eccentricity of the deferent, which has already been calculated through the previous argument, as well as on an observation made of a special opposition. The calculations have not been checked against observational numerical values. Here we first present these arguments and, in the case of the eccentricity of the deferent, we check them against the numerical values of the Almagest.

In Fig. 1 three steps of a retrogradation are represented by the points 1, 2 and 3. These points show the first station, the opposition and the second station of the retrogradation. Following the standard Ptolemaic practice in Almagest 12.2,Footnote ¹³ al-Shīrāzī shows these three steps on one and the same epicycle as shown on Fig. 2.

Fig. 1.

Fig. 2.

In Fig. 2, two epicycles at points C₁ and C₂ can be seen, each of them showing these three steps on one circle. Of these two circles, one is centered on the apogee (C₁) and the other is centered on the perigee (C₂). With O and D as center, we draw two circles. The point O is the center of ecliptic (circle ABG), the point D is the center of the deferent (circle C₁C₂) and E is the equant point. The planet is in the station at P₁ and in the opposition, when the center of the epicycle is at the apogee, at P₂. Moreover, when the center of the epicycle is close to the perigee, the planet is at the station at P′. If the center of the epicycle is considered to be fixed, these two angles P₁OC₁ and P′OC₂ would represent the halves of the apparent distances covered by the planet on its epicycle. Unfortunately al-Shīrāzī presents these two angles as the halves of the retrograde arcs; which is obviously wrong.Footnote ¹⁴ We will discuss the consequences of this mistake in detail later.

Al-Shīrāzī continues: we drop a perpendicular from C₁ to the line OP₁ which crosses it at the midpoint I (Fig. 2). According to the theorem of Apollonius,Footnote ¹⁵ when the planet is at the stationary point, we have:

$$\displaystyle{{{\rm IP}_1 } \over {{\rm P}_1 {\rm O}}}=\displaystyle{{{\rm v}_{\rm c} } \over {{\rm v}_{\rm p} }}$$

where v_p is the angular velocity of the planet on the epicycle and v_c is the uniform angular velocity of the center of the epicycle. Al-Shīrāzī concludes:

$$\displaystyle{{{\rm IP}_1 } \over {{\rm IO}}}=\displaystyle{{{\rm v}_{\rm c} } \over {{\rm v}_{\rm c}+{\rm v}_{\rm p} }}$$

In the next step we have,

$$\eqalign {{\rm IO}&={\rm OC}_1 \cdot \cos {\rm P}_1{\rm OC}_1\cr \quad {\rm IC}_1&={\rm OC}_1 \cdot \sin {\rm P}_1{\rm OC}_1}$$

where OC₁ = 60.Footnote ¹⁶

So, given the retrograde arc at the apogee, al-Shīrāzī concludes that we can calculate the radius of the epicycle:

$${\rm r}_1={\rm C}_1{\rm P}_1=\sqrt {{\rm IP}_1^2+{\rm IC}_1^2}.$$

Similarly, it is possible to calculate the radius of the epicycle on the basis of the retrograde arc at the perigee:

$$\displaystyle{{{\rm NP}^{\prime}} \over {{\rm NO}}}=\displaystyle{{{\rm v}_{\rm c} } \over {{\rm v}_{\rm c}+{\rm v}_{\rm p}}}$$

$$\eqalign {{\rm NO}&={\rm OC}_2 \cdot \cos {\rm P}^{\prime}{\rm OC}_2\cr \quad {\rm NC}_2&={\rm OC}_2 \cdot \sin {\rm P}^{\prime}{\rm OC}_2}$$

where OC₂ = 60. Then

$${\rm r}_2={\rm C}_2{\rm P}^{\prime}=\sqrt {{\rm NP}^{\prime 2}+{\rm NC}_2^2 }.$$

Now we have two values for the radius of the epicycle, the first one is in the units where C₁O = 60 and the second one is in the units where OC₂ = 60. But these two values refer to one and the same thing, i.e. the radius of the epicycle; so we can calculate OC₂ where C₁O = 60:

$${\rm OC}_2=60 \cdot \displaystyle{{r_1 } \over {r_2 }}.$$

Thereupon OD, the eccentricity of the deferent, is known;

$${\rm OD}=1/2\;({\rm OC}_1 - {\rm OC}_2).$$

The eccentricity of the equant point

Now we go on to discuss al-Shīrāzī's own method for the calculation of the eccentricity of the equant point. One can suppose an opposition with the Sun at H, 270 degrees from the planet's apogee, while the center of the epicycle is at C₃ (90° from its apogee) and the planet is at P₃ (Fig. 2). One can find the mean motion (angle C₁EC₃) of the planet from the astronomical tables. Now, if one subtracts the mean motion of the planet from the mean motion of the sun, the remainder is the mean anomaly of the planet (ZC₃P₃)

$${\rm ZC}_3{\rm P}_3={\rm C}_1{\rm OH} - {\rm C}_1{\rm EC}_3$$

and the greatest equation (EC₃P₃) of the planet is:Footnote ¹⁷

$${\rm EC}_3{\rm P}_3=180 - {\rm ZC}_3{\rm P}_3$$

where ZC₃P₃ is the mean anomaly (the distance covered by the planet on the epicycle, measured from the mean apogee), C₁OH is the mean motion of the Sun (= 270°) and UC₃P₃ is the greatest equation of the planet. Thus one can find the line segments EO* and OC*₃ where EC₃ = 60.

$$\eqalign{{\rm EO}^{\ast}&=60.{\rm sin}\,{\rm UC}_3{\rm P}_3\cr \quad {\rm OC}^{\ast}_3&=60.{\rm cos}\,{\rm UC}_3{\rm P}_3.}$$

Then it is possible to obtain OC₃, where OC₁ = 60. OD and DC₃ (DC₁ = OC₁ – OD) are related through the following formula:

$${\rm OC}_3 =\sqrt {{\rm DC}_3^2 - {\rm OD}^2 }.$$

Then one can find OE where OC₁ = 60,

$${\rm OE}=\displaystyle{{{\rm OE}^{\ast} \times {\rm OC}_3 } \over {{\rm OC}_3^{\ast} }}.$$

These two arguments can be summarized as follows:

1. 1. Given the retrograde arcs in the perigee and the apogee, we obtain the epicycle radius and, given the epicycle radius, we obtain the deferent eccentricity.

2. 2. Given the mean motion of the planet in opposition, a quadrant from the apogee, and the eccentricity of the deferent, we obtain the equant eccentricity.

Al-Shīrāzī believes that if one applies these calculations, the numerical value of the equant eccentricity becomes twice as much as the numerical value of the deferent eccentricity.

NUMERICAL CALCULATIONS

In what follows, we use observations of the retrograde arcs of MarsFootnote ¹⁸ to calculate the eccentricity of the deferent by the method al-Shīrāzī suggests. We have:

• Maximum retrograde arc at the apogee: 20° in 80 days;

• Minimum retrograde arc at the perigee: 12° in 64 days.

• Angular Velocity of the center of the epicycle (v_c): 0;31 minutes per day;

• Angular Velocity of the planet (v_p): 0;27 minutes per day.

If one ascribes (as al-Shīrāzī wrongly suggests) halves of the observed retrograde arcs at apogee and perigee (10° and 6°) to P₁OC₁ and P′OC₂, not only the final value for the eccentricity of the deferent (OD) would be negative (!) and far from the expected value (e_Ptolemy = 6) but also the calculated values for the radius of the epicycle (r₁ and r₂) would not be in a good agreement with the Ptolemaic value (r_Ptolemy = 39;30). Table 1 shows the results obtained by calculation according to the method of al-Shīrāzī.

Table 1.

In fact halves of retrograde arcs (λ_a and λ_p) are related to the angles P₁OC₁ and P′OC₂ through the following formulas:

$$\eqalign{\lambda_{\rm a}+{\rm k}_{\rm a} & ={\rm P}_1{\rm OC}_1\cr \quad \lambda_{\rm p} + {\rm k}_{\rm p} & ={\rm P}^{\prime}{\rm OC}_2}$$

where k_a is the eccentric anomaly when the planet is at the station near the apogee, and k_p is the eccentric anomaly when the planet is at the station near the perigee. Angles λ_a and λ_p are halves of the retrograde arcs at the apogee and the perigee; they can be obtained through observation (Fig. 3).Footnote ¹⁹ But, as was mentioned before, al-Shīrāzī doesn't take into account these relations and wrongly presents angles P₁OC₁ and P′OC₂ as the retrograde arcs.

Fig. 3.

To find the true values of P₁OC₁ and P′OC₂, one has to know eccentric anomalies (k_a and k_p) of the planet, when the planet is at the station near the apogee and the perigee. But to know these values, one needs to know the distances of the apogee and the perigee to the observer, i.e. the eccentricity of the deferent.

THE EFFECT OF THE EQUANT POINT ON RETROGRADE ARCS

As Evans has shown,Footnote ²⁰ in a simple eccentric model the apparent size of a retrograde arc at the perigee is larger than the same arc at the apogee. Since the planet is closer to the observer at perigee, it makes a bigger apparent arc in the sky. But the observations show the contrary, namely the retrograde arcs attain their greatest value at the apogee and their least value at the perigee. For example, the retrograde arcs of Mars are as follows:

• Maximum at apogee: 20° in 80 days;

• Minimum at perigee: 12° in 64 days.

So Ptolemy finds a new eccentricity on the basis of these observations and presents it as the eccentricity of the deferent, but keeps the previous eccentricity for the center of the uniform motion (the equant point). In this new model we have two eccentricities, the first for the equant point and the second for the deferent center. Thereupon the equant point affects the velocity of the center of the epicycle, so that it rotates faster at the perigee and slower at the apogee. In this manner the planet retrogrades in a longer path at the apogee and a shorter path at the perigee, and produces larger and smaller arcs in the sky, as the observations show. This variation in the retrograde arcs is the result of the non-uniform motion of the center of the epicycle with respect to the center of the deferent.

Swerdlow considers these parameters and obtains a very good agreement with Ptolemy's values.Footnote ²¹ But in taking the angles P₁OC₁ and P′OC₂ as the retrograde arcs, al-Shīrāzī ignores the main property of the equant point. In fact the equant point affects the velocity of the center of the epicycle which determines the angles k_a and k_p, the angles that is ignored by al-Shīrāzī.

CONCLUDING REMARKS

Ptolemy provides an argument for the eccentricity of the superior planets. He ascribes this eccentricity to the center of the uniform motion (equant point) and, without any argument, considers that the eccentricity of the deferent is half of the former. This argument plays a key role in finding out why Ptolemy displaces the deferent center from the equant point. Although Ptolemy mentions “retrograde arcs” as the empirical origin of this unknown argument, neither the Arabic versions of Almagest, nor the works of al-Shīrāzī's contemporaries mention these arcs explicitly. Al-Shīrāzī in his Nihāya not only suggests the “retrograde arcs” as the empirical origin of the eccentricity of the deferent but also provides an argument based on the observed values of these arcs to calculate this eccentricity.

Although al-Shīrāzī's argument is not without difficulty, his suggestion is in line with the suggestions of the historians of science in recent decades, and shows his interest, as an astronomer, in the empirical origin of this geometric device, i.e. equant point, in the mathematical astronomy of his time.

More research is needed to decide whether al-Shīrāzī is the first scholar, in Islamic period, to have found the empirical origin of the equant point. A point to be noted is that, apart the manuscripts of al-Shīrāzī's Nihāya, this text also exists as a fragment added at the end of the Astan Quds's manuscript of al-Ṭūsī's Taḥrīr al-Majisṭī by an unknown copyist. This fragment is the same as the text we find in Nihāya, but the copyist doesn't present it as such, rather he thinks it may belong to Thābit Ibn Qurra (d. 901). We couldn't find anything related to this subject in the extant works of Thābit Ibn Qurra.Footnote ²² Further studies may reveal the connection between this fragment of Nihāya and Thābit.

This quotation at the end of a copy of al-Ṭūsī's Taḥrīr al-Majisṭī by a probable informed copyist shows that the problem remained important for the scholars from the time of al-Shīrāzī to the time of this copy of Taḥrīr, i.e. 1092 AH/1681 AD.