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In epicycle-deferent astronomy, adding a second ”minor” epicycle to account for observational discrepancies is observationally equivalent to shifting the deferent into a so-called eccentric, or a circle with a center not at the Earth (see e.g. DeWitt 2010, Worldviews, p. 120, and the illustrations I've borrowed from that chapter), as was pointed out already by Hipparchus. Ptolemy famously chose the eccentric approach for the Almagest. I'm trying to wrap my head around how to geometrically prove the equivalence.
More precisely, the idea I am after is that the addition of a second epicycle to the original epicycloid (compounded only of a deferent and a major epicycle) is supposed to have the same effect upon the movement as projected unto the outer circle of fixed stars by a line of sight from Earth as just shifting the original epicycloid by a distance equal to the diameter of the minor epicycle. I think it's the earliest proof of an ”empirical equivalence” of two hypotheses.
The conceptual structure of copernican astronomy
Figure 60 is a partial recursive frame for PATH circa 1543, with activated value nodes representing a Copernican theorica for Saturn. The frame for DAILY MOTION now represents the twenty-four-hour motion of the earth, which for Copernicus includes everything inside the sphere of the moon. Copernicus' treatment of the diurnal motion uses practically the same attribute-value combinations as Ptolemy's. Only the radius of the motion changes from the largest allowed value to the smallest. The most important change, of course, is that this whole motion is now regarded as a real motion of the earth, not the fixed stars. When it comes to making astronomical calculations this makes no difference at all.
The proper motion of each planet is now understood primarily through the motion of a large circle eccentric to the mean sun. Notice that this change requires no addition or deletion of attributes, and no major differences in the activated values. Within the conceptual system of sixteenth-century astronomy, the choice of the mean sun as center for the proper motion is just the choice of a new center for the eccentric circle that differs from the center of the earth. But Ptolemaic astronomers were already using such points in all their models.
Turning to the other two value nodes in the frame for PROPER MOTION: the distances involved remain comparable to the Ptolemaic
figure 60. Partial recursive frame for 'path', circa 1543, with activated value nodes representing a Copernican theorica for Saturn.
ones, although for Copernicus particular values generally become larger than their Ptolemaic equivalents. Although there is now a relation among these distances (changing one requires that you change the rest) this connection has no consequences for calculating planetary positions. The SPEED of these motions also assumes different (but related) values from those in the Ptolemaic frame, but again, these are drawn from the existing set of allowed values. So in the cases of all three attributes we activate values within ranges already admitted in Ptolemaic astronomy.
What has just been said describes the mathematical models for calculating planetary positions presented in the body of DeRevolutionibus, and not the cosmological sketch from Book I. Once again, in the case of Copernicus' account of proper motion, the major difference from
Ptolemy is obscured if we concentrate only on the basis for astronomical calculations. For Ptolemy, the main element in the proper motion is an eccentric circle. But exactly the same results would follow from a concentric circle carrying an epicycle, and the absolute size of these circles is arbitrary. So judging only from its path we cannot discern the real motion of the planet. For Copernicus, however, the motion that results from the eccentric circle and its ancillary minor epicycle is a real motion, with a definite spatial location.
Finally, let us examine the lowest of the three right-hand frames in Figure 60. In Ptolemaic astronomy (Figure 59) this frame represents the properties of the epicycle used to accommodate retrograde motion. For Copernicus, retrograde motion is the result of the annual motion of the earth combined with the proper motion of the planets that has already been introduced. As is well known, Copernicus explains retrogression through change in the line-of-sight as a moving earth overtakes an outer planet or is overtaken by an inner one (Kuhn 1957:166-169). At the same time he can explain why outer planets retrogress while in opposition to the sun (and inner ones in conjunction), and why the retrogressions begin and end where they do (Kuhn 1957:165-167).
To derive actual positions for retrogressions we need a theorica for the earth to replace the theorica for the sun. Copernicus provides this by giving the earth a purely circular path centered on the mean sun. So, just as Ptolemy does, he introduces a circular motion to explain retrogressions. The attributes and values of this circular motion are surprisingly familiar. Of course the center of this motion is the mean sun, that is, a hypothetical point differing from the center of the earth. The speed of this motion is the speed attributed to the sun in the Ptolemaic theorica. And the radius of this motion is the distance attributed to the sun in the Ptolemaic theorica. Again Copernicus' theory introduces no new attributes, and the values he uses for the attributes already introduced fall within the ranges already admitted in Ptolemaic astronomy.
In our reconstruction Copernicus uses the same overall structure as Ptolemy for the key concept PATH, which encompasses the positional data of astronomy. We do not need to add attributes or values, we do not need to delete attributes or values, and we do not need to add new kinds of attributes or values. Not only does Copernicus employ the same attributes, but the values activated in his frame are almost the same pattern as the Ptolemaic ones (in contrast to those activated in an Averroist account) and there is nothing objectionable in the particular values assigned to these attributes. This includes the attribute-value sets used in the treatment of retrogression, in which the proper motion of the earth serves the same function as the epicycle in a Ptolemaic theorica. So if incommensurability is judged by degree ofmismatch between attribute and value nodes, the conceptual structures of Ptolemaic planetary astronomy and Copernican planetary astronomy (Figures 59 and 60) are not incommensurable. Now remember that it is the goal of astronomy in the sixteenth century to calculate planetary positions against the sphere of the heavens as viewed from the earth. The astronomer reading De Revolutionibus is reading it with that goal in mind. And with that goal in mind the sixteenth-century astronomer will find the conceptual structure underlying Copernican calculational techniques to be the same structure that appears in Ptolemaic astronomy. Copernicus' intent is to restore an astronomy that uses only the attributes of circular motion that we have displayed in our frame, and hence to conserve an existing conceptual structure.
Our analysis has concluded that Copernicus' conceptual structure is not incommensurable with Ptolemy's - if anything, it appears to be a variation on it. This is exactly the way Copernicus and Ptolemy were seen during the late sixteenth and early seventeenth centuries. Erasmus Reinhold at the University of Wittenberg adopted Coper-nican calculation techniques to produce a new and improved set of astronomical tables (the Prutenic Tables). Following his lead, a whole host of Ptolemaic astronomers spread Copernican methods through Northern Europe. Farther south, the Jesuit Christopher Clavius, who led the successful reform of the calendar, also counted Copernicus as an intellectual ally of Ptolemy in the common fight against the Aver-roists (Gingerich 1993 Lattis 1994 Barker and Goldstein 1998).
All these words are from astronomy, used to describe the motion of the Sun, Moon and planets. The simplest model is this. Suppose that Earth is the center of a circle. This circle is called deferent. A point moves on the deferent with constant angular velocity, this point is called the center of the epicycle, and epycycle itself is a circle centered at this point. The planet moves on the epicycle with constant angular speed. In another equivalent model, earth is not in the center. But the center of the deferent moves on another circle which is called the eccenter. In more comlicated models the motion of the center of the epicycle is not uniform with respect to the center of the epicycle but uniform as seen from another point which is called equant.
I hope I explained all words. The two simplest models that I described in the beginning were introduced by Apollonius of Perga (262–190 bc) and he proved that they are equivalent. (In modern language this is equivalent to commutativity of addition of vectors). Hipparchus built first models for Sun and Moon, and possibly of planets using this machinery. Equant was introduced by Ptolemy (100-170 ad) who built a satisfactory model for the whole Solar system. This model, with minor variations was used successfully until 17th century, when it was replaces by Kepler's laws, where the motion is on ellipses and the angular velocity is not constant.
However the model with circles and uniform motion is a very good approximation that worked.
For more detail, see Ptolemy's Almagest.
The problem of the equant point
In the previous section we established that the Copernican account of celestial motions and the simplest Ptolemaic account use the same conceptual structure. The same result was established in the previous chapter for the Averroist account and the Ptolemiac account.
figure 61. (a) Ptolemaic eccentric-plus-equant model for an outer planet, compared with (b) Copernican model with 'concealed equant' atE'.
According to the simplified frame introduced in the current chapter, the conceptual structures used by Averroists and followers of Ptolemy differ only in the values assigned to a single attribute (CENTER) and whether certain values are allowed for that attribute. The introduction of the equant changes the situation (Figure 61).
Ptolemy probably recognized that the simple eccentric-plus-epicycle model failed to predict both the direction and the angular width of planetary retrogressions (Evans 1998: 355-359). To correct this he introduced a new device (see Figure 61(a)). Ignoring the epicycle for the present, and considering a diameter of the eccentric (line AB) that passes through the position of the earth O and the eccentric center C, Ptolemy defined a point E at the same distance e as the earth O from the center but on the opposite side. He then used this point E, which he called the equant, to control the motion of the epicycle that carried the planet (compare Figure 54(a)). In his complete model for outer planets, the center of the epicycle moves uniformly along the eccentric not when viewed from the geometrical center of the eccentric C, but when viewed from the equant E. By means of this subsidiary device Ptolemy was able to bring his theory into excellent agreement with observations based on the naked eye. However, from the viewpoint of conceptual structure, and the physical underpinnings of astronomy, this success in calculation is achieved at a very high price.
figure 62. Partial frame for 'circular motion' showing modifications required to accommodate Ptolemaic equant.
In all previous frame diagrams for circular motions it has been taken for granted that the center used to define the radius of a motion and the center used to define the speed of a motion are the same point. In Ptolemy's complete model for the outer planets these are not a single point. In order to accommodate the equant, we therefore need to add a new attribute node to the basic structure for circular motion (Figure 62 compare Figure 57). It is not obvious how this revision should be made in the frames representing astronomical theories that we have considered so far (especially Figure 59). Although Ptolemy makes use of the equant only for a single one of the circular motions making up a planet's path, this change raises the question whether a similar revision is needed in the case of the other motions. Rephrasing this in terms of frame diagrams, the issue is whether to add a new attribute node only in the case of the circular motion corresponding to the eccentric that carries the epicycle (the proper motion) or in all the circular motions needed to specify the planet's path. In these other cases, and especially in the case of the epicycle used for retrogressions, it seems the value of the new attribute happens to coincide with the value for the circle's geometrical center (Figure 63 compare also Figure 59).
Adding an attribute node for MOTION CENTER is not a conservative revision of the prior conceptual structure. We may now recognize two classes of circles required in Ptolemy's theorica for outer planets and generated as subconcepts by the revised frame: circles in which
figure 63. Partial recursive frame for 'path', circa 1500, with activated value nodes representing a Ptolemaic theorica for Saturn, showing modifications required to accommodate Ptolemaic equant.
the center of motion is identical to the geometrical center and circles in which it is not. When the new attribute node is introduced in the frame for CIRCULAR MOTION, an existing entity, the major circle that carries the epicycle, is reclassified from an existing category (circle for which the geometrical center and center of motion coincide) to a new and previously nonexistent category (circle in which the center of motion differs from the geometrical center). Entities of this new sort cannot be accommodated within the old conceptual structure. If these changes had taken place over time, with Figure 62 replacing Figure 57 as the generally accepted frame for circular motion, then it would count as an instance of revolutionary change. According to the standards introduced in Chapter 5, the later conceptual structure is incommensurable with the earlier one.
Ptolemy's account of the motions of the sun and the moon is developed without using equants, and hence using only the conceptual structure of Figure 57, but all of the features of Figure 62 are present in Ptolemy's subsequent account of the motion of the planets. From the viewpoint of later readers all these models date from a single source, the Almagest, so the difficulty of making sense of the equant is perhaps better understood as a conceptual problem within Ptolemaic astronomy. The peculiar status of the equant was a long-standing source of discontent within Ptolemaic astronomy, and the changes that it introduces in the concept of CIRCULAR MOTION, as we have presented it, go a long way toward explaining this phenomenon: resistance to the equant was equivalent to resistance to a revolutionary change in conceptual structure. Ptolemy does not motivate the introduction of the equant by anything like the specification of an anomaly that can be resolved by modifying the frame, so the student of the Almagest is left with two different conceptual structures for CIRCULAR MOTION and no way of reconciling the discrepancies between them.
The equant is embarrassing not only because of the difficulty in understanding how to revise the basic conceptual structure of Ptolemaic astronomy in order to accommodate it, but also because it could not be connected in the usual way with a physical mechanism. As already described, all other circular motions in Ptolemaic astronomy could be imagined as the result of uniform rotations of earth-centered orbs, or spheres carried by such orbs (Figure 54(b)). The equant motion could not be replaced by an earth-centered orb and could not be modeled by a uniform rotation of any of the orbs already accepted (Barker 1990).
Although Averroist natural philosophers objected to eccentrics and epicycles, the main difficulty that concerned Ptolemaic astronomers within their own tradition was the equant. The seeming impossibility of accommodating this necessary technical device within the basic
figure 64. (a) Copernican model for an outer planet, heliocentric arrangement, compared with (b) Copernican model for an outer planet, geocentric arrangement.
conceptual structure of circular motion, or of connecting it with physical models in the usual way, led to the creation of an entire school in Islamic astronomy centered at Maragha in Persia, which developed new mathematical devices and equivalent systems of orbs to avoid it (Ragep 1993). By the fourteenth century this school had found a number of different, calculationally adequate means of avoiding the equant (Pedersen 1993: 241ff.). Although these results remained generally unknown in the West, it is clear that Copernicus encountered some version of them during his education in Italy (di Bono 1995 Barker 1999). When he published his new astronomical models, he was mistakenly given credit for many innovations that had actually occurred in Islam.
Copernicus' model for the outer planets avoids using an equant by adopting a device introduced by Ibn ash-Shatir of Damascus (1304— 1376) (Pedersen 1993: 242-245). No new center of motion for points on the eccentric is introduced. Instead a small subsidiary epicycle is inserted in the model at point D (Figure 61(b) cf. Figure 64(a)). In Ptolemy's original model the distance from the equant to the center of the eccentric and from the center of the eccentric to the observer had been equal (EC = CO, in Figure 61(a)). Take the sum of these two distances to define a unit distance. Copernicus' model in effect retains the same magnitude for this total distance. He then assigns a distance of three-quarters of the unit between the center of the eccentric and the physical center of the system S (formerly the position of the observer on the earth, now the mean sun). Copernicus' model adds a minor epicycle carried by the eccentric at point D, the radius of which is the remaining one-quarter of the unit distance. In Copernicus' presentation this epicycle carries the planet. Its center moves uniformly about the geometrical center of the eccentric. However, the conditions placed on the motion of the minor epicycle (angle BCD = angle CDP) are such that the planet carried by it moves uniformly with respect to a point E' farther along the center line AB (the line of apses) from the eccentric center (shown in Figure 61(b)). So, although the equant point does not appear in Copernicus' diagrams, it is still possible to define an equant point in Copernicus' models, and the planets move injust the way they would if their motion were controlled by an equant in the Ptolemaic manner (for a discussion see Evans 1998: 421-422 Voelkel 2001: 19). The ease with which an equant point can be defined for Copernicus' construction has led some modern commentators to question whether he eliminated the equant at all (Neugebauer 1968). These mathematical considerations should not, however, make us lose sight of more fundamental points about the conceptual structure of Copernican astronomy and its physical interpretation.
The motions described so far represent the main motion of the planet - its proper motion - around the mean sun. For Copernicus, retrogressions are explained by viewing the motion so defined from the moving earth (point O in Figure 64(a)), which is itself in motion around the mean sun S. For Ptolemy, the proper motion is described by the eccentric, while retrogressions are accommodated by the epicycle. In Copernicus' models, the motion of the earth around the sun, which is still treated as a circular motion, replaces the Ptolemaic epicycle. To make a prediction about the angular position of a planet in the sky, however, we still require not only the eccentric, but also this second circle or epicycle, in addition to the new minor epicycle that Copernicus has inserted as part of his mechanism to avoid using an equant point. Copernicus' model, then, can be represented as a double-epicycle system (Figure 64(b)). If the earth is placed at O, this converts the model back to a geocentric system, an option used by the group led by Erasmus Reinhold and now called the Wittenberg astronomers (Westman 1974 Barker and Goldstein 1998).
If the mean sun is placed at O in Figure 64(b), a double-epicycle heliocentric system appears. Kepler, for example, presents Copernicus in this way (Kepler 1609: 14).
In Copernicus' model, all the motions are simple circular motions that can be understood in terms of the original conceptual structure for circular motion presented in Figure 57. No separation of centers of motion from geometric centers is required. Second, because only simple circular motions are used, either Copernicus' original models or their geocentric equivalents can be represented by sets of orbs with centers either at the mean sun, in the case of Copernicus, or at the earth, in the case of Ptolemaic astronomers. In fact, the physical location of the center of the system is irrelevant to the success of the model as a calculating device. (The equivalence is easily seen in vector diagrams-see Figures 64(a) and 64(b).) So although Copernicus' model can be readily reinterpreted in terms of an equant, and although he is describing a motion that is originally defined by means of one, his real achievement is to specify a mechanism that avoids both the deviant conceptual structure required by Ptolemy's complete model and the associated problems of physical interpretation. This was clearly the response of his contemporaries, who regarded Copernicus as amending and improving Ptolemaic astronomy, rather than undermining it. Erasmus Reinhold wrote on the front page of his personal copy of Copernicus' book, "The first axiom of astronomy - all motion is in circles at constant speed" (Gingerich 1993). Georg Rheticus in his preliminary survey of Copernicus' theories simply announced that Copernicus had eliminated the equant (Rheticus 1540/1979: 136137). And later thinkers like Maestlin and Kepler presented Coperni-can models that were consistent with this understanding of his work and that could be interpreted in terms of three-dimensional orbs (Kepler 1596).
If we compare the frame diagrams for the simple Ptolemaic model for the outer planets, Copernicus' model, and Ptolemy's full model including the equant (Figures 59, 60, and 63), it is apparent that it is Ptolemy's full model (Figure 63) that differs most from the other two, because it includes new attribute nodes in all the frames used to recursively expand the attributes of the superordinate concept PATH. If the addition or deletion of attribute nodes leads to the redistribution of entities across categories in ways that are prohibited in the unrevised structure, then the result is incommensurability. On this basis it can be said that Ptolemy's complete model is incommensurable with both the simple model and with Copernicus' model.
Although the introduction of the equant did not cause a failure of communication or lead to the impossibility of comparison between the full Ptolemaic model and alternatives, it can be seen from this reconstruction that the long-standing discomfort with the equant was motivated by a discrepancy in conceptual structures of exactly the same kind that we have already identified in cases of incommensurability between different scientific traditions. It should also be apparent that Copernican astronomy is not incommensurable with either the conceptual structure favored by the Averroists or the Ptolemaic alternative, apart from the difficulties with the equant, which the Ptolemaic astronomers regarded Copernicus as having resolved. But Copernican cosmology, with its central sun and the earth reclassified as a planet, is clearly incommensurable with Ptolemaic cosmology, with its central earth. How was this conflict avoided? The Wittenberg interpretation of Copernicus simply disregarded the cosmology (as obviously wrong on physical and scriptural grounds) and referred the astronomy to a central earth, using a model like that shown in Figure 64(b). This was the most influential interpretation of De Revolutionibus, from the death of Copernicus in 1543 until the appearance of major works by Kepler and Galileo in 1609 and 1610 (Westman 1975 Barker 2002). Kepler insisted on introducing physical considerations based on heliocen-trism that led to a revision in the conceptual structure of astronomy and the first major incommensurability with the structures used by Ptolemy and Copernicus. However, to explain how this change came about we need to consider issues outside positional astronomy, and the conceptual structures we have considered so far.
Equivalence of minor epicycle and eccentric - Astronomy
In the previous article we read about one of the great misconceptions in Indian astronomy, namely, the tilt or obliquity of the earth’s axis, and how the European colonial scholars of yesteryear managed to obfuscate the ancient Indian value as an imprecise 24°, when it was actually 23.975°.
In this article, we move on to the topic of Wonders in Indian astronomy.
There are actually dozens of items in Indian astronomy, big and small, that would easily qualify as a wonder. Of those many marvels, let us examine in this article a wonder, which, in my opinion, is perhaps the most astonishing of all.
Wonder # 1 – The Pulsating Indian Epicycle
Today it is common knowledge that the Sun is the center of the solar system, and that the planets move around it in orbits of elliptical shape.
Ancient man, quite obviously, was unaware of these modern discoveries. When he looked up at the sky, he beheld a vast dome (an inverted bowl, in the words of Khayyam) on which were stuck a fascinating array of objects – the sun, the moon, the planets, myriads of stars, and the occasional comet or meteor whizzing by.
He noticed that every few hours the sky changed its position, and it occurred to him that this enormous dome was actually rotating in a circular fashion around his home (the earth) carrying with it the sun, the moon and the planets. Today, of course, we know that the apparent rotation of the sky is an illusion caused by the rotation of the earth on its axis.
Anyways, sometime later, when our ancient ancestor looked more carefully at the sky, and made some rudimentary measurements, he was puzzled by what he found.
While the stars appeared to have a perfectly uniform circular motion around the earth, other objects in the sky, like the sun, the moon and the planets, did not. These latter bodies seemed to have an independent and strange motion of their own.
At times these bodies appeared to be nearer (they looked a little larger), and at other times farther away (they looked smaller). Sometimes they were moving slowly, and at other times faster. And, most perplexing of all, at times some planets were seen to stop moving, and go in reverse (backwards)!
What could possibly be causing such an irregular motion of these bodies? Why were they not moving in nice, uniform, circular orbits like the stars? His curiosity, no doubt, impelled him to investigate and speculate on the reasons for this. But, for a long time, the problem was beyond his understanding.
Eventually though, when his mathematics and geometry had advanced sufficiently, he developed an ingenious geometric model, the so-called EPICYCLE-SYSTEM, which, believe it or not, very neatly explains all these variations in speed, proximity and direction of these heavenly bodies.
We will examine the epicycle-system shortly, but, before we step into that, it must be mentioned that the subject of epicycles is today a contentious area, with plenty of accusations and acrimony. European scholars have been claiming for the past 200 years that it was the Greeks who invented the epicycle-model, and that the Indians copied it from them. Indian scholars are now beginning to challenge that. Hopefully, by the end of this article, readers will have enough information to form a knowledgeable opinion of their own.
Before we take up epicycles, an understanding of the elliptical orbit is essential.
The ellipse is a geometric figure that looks like a squished circle, which mathematicians find utterly fascinating.
Fig. 1A shows a circle and its center. Now imagine that you grabbed hold of that circle by its left and right sides, and pulled it apart a little. You would end up with something like Fig. 1B.
Fig 1: The Ellipse
Notice that not only is the circle now an oval, but your pulling also split the center into two centers. An ellipse is an oval with two centers, also called two focuses, or foci (plural). If you kept pulling, the ellipse would become more eccentric, and the foci more apart, as seen in Fig. 1C.
As an aside, note that the ellipse holds a special pride of place for India. Great mathematicians of the past, including the likes of Kepler and Euler, have struggled to find a formula that would give the exact boundary length (perimeter) of the ellipse. Ramanujan in 1914 provided the most accurate approximation till date, and shortly thereafter improved it further with a second approximation. Today, there are exact formulas to calculate the perimeter, but these are all based on infinite series, which are computationally cumbersome, and end up being approximate anyway, because it is practically impossible to compute an infinite number of terms. Ramanujan’s two formulas are elegant, concise, and used ubiquitously in most modern computations of the ellipse.
To give you an idea of the accuracy of these formulas, consider this. If Pluto’s orbit was exactly elliptical, you could use Ramanujan’s second formula to find the perimeter of its orbit (of billions of kilometers) with an error of less than one-thousandth of a millimeter 1 . Incredible!
The Elliptical Orbit
In the field of astronomy too the ellipse holds a special place of honor. Any object gravitationally bound to another (i.e. in orbit around it) will always move in an elliptical orbit.
The earth, the moon, the planets, the binary-stars, and even the satellite which an ISRO rocket hurls into space, all move in elliptical orbits.
Let us briefly examine the elliptical orbit, taking the Moon as an example.
Fig 2: Energy Transformation and the Moon’s Orbit
The left side of Fig. 2 shows a child on a swing. There is an interesting energy transformation taking place here. At points A and C, the swing is at its highest, or farthest, from the earth. At these points its kinetic energy (speed) is the lowest, while its gravitational potential energy (height) is maximum. At point B, the reverse is true there the swing is moving at its maximum speed (highest kinetic energy) while its gravitational potential energy (height) is the lowest. As the swing moves to and from, there is a constant energy transformation taking place, from kinetic to potential (speed to height), and vice-versa.
A very similar thing occurs in an elliptical orbit, as shown on the right side of the figure. Observe that the earth is at one focus of the Moon’s elliptic orbit, while the other focus is empty. The Moon comes closest to the earth at the Perigee (P), and is farthest at the Apogee (A). At P, its distance from the earth (or height) is minimum and its speed is maximum. At A, the opposite is true it has a maximum height and a minimum speed. Like the swing, the Moon’s kinetic and potential energy (speed and height) are constantly being transformed from one to the other as it moves around its elliptical orbit.
As an aside, note that the eccentricity of the Moon’s orbit has been exaggerated in Fig. 2 for instructional purposes. In reality, the Moon’s orbit is quite close to a circle, with only a small eccentricity.
Anyhow, now that we have examined the elliptical orbit, let us return back to our ancient ancestor, who was puzzled by the movement of the Moon.
Obviously (he reasoned), the Moon was not moving in a uniform circular orbit, because it was clearly closer at times, and farther away at other times. Also, when it was closer its speed was higher, and when it was farther away its speed was slower.
Now, what kind of a geometric model could possibly account for these variations?
After a long struggle, he hit upon a brilliant solution!
What exactly is an epicycle? In simple terms, an epicycle is a circle moving on another circle.
Fig 3: The Epicycle Model
As shown in Fig. 3, the bigger (blue) circle, with the earth at its center, is called the Deferent. The smaller (orange) circle is the Epicycle. The planet moves around the epicycle, while the epicycle itself travels around the deferent.
Now, at first sight, you might not be too impressed by this but wait till you see it in action!
Try it yourself
To give you an idea of the power of the epicycle, I created a couple of interactable movies using Adobe Flash.
To try the epicycle yourself, please go to this link and download two movies: Epicycle.swf and Pulsation.swf. Clicking on the files will take you to a download page. Click the download button and the file will be downloaded to your computer.
These movies need Adobe Flash Player to run. There are two ways to handle this. The easiest way is to use your browser. Open your browser and drag-and-drop the .SWF movie into it. You may be asked if you wish to activate Flash Player, or download an extension. Say yes. In my experience Firefox handles this better than Google Chrome.
The other (harder) way is to install the free Adobe Flash Player on your computer. You can get it here. Remember to uncheck any ‘special offers’ from Abobe before downloading and installing.
When you are finally able to run the Epicycle.swf movie, you should see the below:
Fig 4: Opening Screen of Epicycle.swf
There are 4 parameters in the epicycle-model: 1) Deferent radius, 2) Epicycle radius, 3) Epicycle speed, 4) Planet speed.
Of these four, we will keep two fixed (Deferent radius=200 and Epicycle speed=30 deg/second), and play with the other two (Planet Speed and Epicycle radius).
Click the ‘SETUP’ button, and you should see the model show up, ready to run, as below:
Fig 5: Epicycle Model Ready to Run after clicking SETUP
Click the ‘TRACE’ checkbox on the right, to have the planet leave a green trail as it moves, so you can see the orbit shape.
Now click the ‘Start/Stop’ button. You should see the epicycle in action, as below:
Fig 6: Epicycle in Action
To try out various combinations, follow these steps:
- Open (double-click) Epicycle.swf
- Set the Epicycle Radius and Planet Speed to some values, as you like (note that Planet Speed can be negative as well, going counter-clockwise).
- Click the ‘TRACE’ checkbox
- Click ‘SETUP’ to create the Model
- Click the Start/Stop button to start the epicycle and see the planetary orbit you created with your setting.
You can click the Start/Stop button anytime to pause/resume the action.
Unfortunately, there is no easy way to restart the movie in Flash with a different setting. So, for each setting that you want to try, you have to close the file, and reopen it.
I was able to obtain some interesting orbits, as shown below. The settings are embedded in each figure (PS=Planet Speed, ER=Epicycle Radius).
Fig 7: Some Orbital Shapes created using Epicycle.swf
Surprisingly, it is even possible to obtain an orbit that is a perfect straight line!
Try out some settings of your own, and let us know if you created any interesting orbits, and what settings you used!
Hopefully, the reader is now sufficiently impressed by the power of the epicycle, and by the ingenuity of our ancient ancestors who created it (remember, they had no Adobe Flash!). Also note the fact that we have employed only a single epicycle so far, which is merely scratching the surface of this technique. Using multiple epicycles, almost any conceivable shape can be traced. Check out this link to explore deeper.
Moving on, let us see how the power of the epicycle was applied to astronomy, starting with the Greeks.
The Greek Epicycle
The Greek epicycle is very simple.
The two speeds, namely, that of the epicycle on the deferent, and of the planet on the epicycle, are set equal (but in opposite directions, one clockwise, the other counter-clockwise).
As seen in the first item in Fig. 7 above, such a setting results in a circular orbit. This circular orbit is shifted rightward of the deferent, so that the earth is not its center. Such a shifted circle is also called an ‘eccentric’ (an off-center circle), as shown in Fig. 8.
Fig 8: The Greek Epicycle
Does this eccentric orbit exhibit the same features as the elliptical orbit? Let us see.
At point P, we note that the eccentric orbit is closest to the earth, and at point A it is farthest. Well, that at least explains the variation in proximity (planet is sometimes near and sometimes far).
Further, we observe that the orbital fragment XPY is clearly smaller than the fragment XAY. But note that both these arcs are equal to 180 degrees (half the sky) as seen from the earth. That is, the planet takes lesser time to cover arc XPY, of 180 degrees, and more time to cover arc XAY, also of 180 degrees. In other words, as seen from the earth, the planet appears to move faster near the Perigee P, and slower at the Apogee A, which is exactly how an elliptical orbit behaves.
In summary, the Greek epicycle produces only a shifted (eccentric) circle, and not an ellipse or other complex shapes. While this is not great, it is not bad either, as the eccentric does simulate the main features of the elliptical orbit, namely, the proximity and speed variations.
The Indian Epicycle
The Indian epicycle shares the basic feature of the Greek epicycle, namely, the speed of the epicycle on the deferent is equal to the speed of the planet on the epicycle.
The similarities, however, end there. The Indian epicycle adds three levels of complexity on top of this, which puts it on a different plane altogether. Let us examine these added complexities.
Firstly, the epicycle radius is not a constant. So, the epicycle enlarges and shrinks (pulsates) as it moves on the deferent. Check out the Pulsation.swf movie to see the pulsating epicycle in action.
Secondly, the details of pulsation are surprisingly complex. All said and done, the epicycle is an ancient technique, created by ancient people. How intricate could such an ancient technique be? A linear variation of radius with angle would have been complex enough for this ancient technology, and yet, the actual variation of pulsation is far more complex – it is sinusoidal! Though we know the ancient Indians were advanced in mathematics and geometry, still, this extra bit of complexity comes as a surprise.
Thirdly, the direction of pulsation is not the same for all planets. This is an intricacy that is not easily explained in a non-technical forum, so I will not attempt it.
Apart from these complexities, another striking fact of the Indian epicycle system is the fine-ness of pulsation. Fig. 9 shows a Table of the maximum and minimum circumferences of the epicycle for each planet during pulsation. Manda and Sheegra are two kinds of Indian epicycles, which we won’t go into here.
From the Table we note that all of these pulsations are exceedingly fine – the epicycle size changes only a little, making minor adjustments to the final orbit. We are filled with wonder for whoever crafted such fine control of the planetary orbits!
Fig 9: Pulsation Details
To summarize, the pulsating Indian epicycle is laden with technical complexities that are far beyond the simple Greek epicycle. While the Greek epicycle creates only an eccentric circle, the Indian epicycle produces a very complex orbit that defies simple description. For small pulsations, the Indian orbit is close to an ellipse.
The Sun’s Orbit – Comparing Greek and Indian Epicycles
The proof of the pudding, it is said, is in the eating. To round out this discussion, let us compare Indian and Greek epicycle techniques by applying them to the simplest possible orbit (that of the Sun) and examine how well they predict its orbit.
Fig 10: Comparing Indian and Greek Epicycles of the Sun
Fig. 10 shows the error in arc-minutes of Indian and Greek epicycles for the Sun, for the years 2000 AD and 100 AD 2 . It is seen that the Indian error for both dates is half that of the Greek. In other words, the Indian epicycle of the Sun is twice as accurate as the Greek.
Fig. 10 also shows another interesting fact. The Indian Epicycle of the Sun was more accurate 2000 years ago than it is today! This naturally raises the question – what if we go back further in time?
The results are shown in Fig. 11.
Fig 11: Error of the Indian Epicycle of the Sun in Time
This figure highlights the interesting fact that the pulsating Indian epicycle of the Sun grows progressively more accurate (has less error) as we go back in time, reaching peak accuracy (minimum error) at about 5000 BC (red dashed line). Very curious!
Does that mean the Indian epicycle of the Sun was created in that timeframe (5000 BC)? Dare we make that conclusion? Is Indian astronomy that old?
Well, looking at the intricate design of the pulsating Epicycle, we can say for certain that the ancient Indian astronomer spent a lot of time tinkering and fiddling with his model. Why would anyone do that? Unless it was to make his model as close to reality as possible. If that line of reasoning is any good, then, applying a cushion on observational accuracy of say 5 minutes of arc, we can state that the pulsating Indian Epicycle of the Sun was developed somewhere in the period (8000 BC – 3000 BC).
Colonial attempts to downplay the Pulsating Indian Epicycle
As can be expected, the British colonial rulers and their evangelical brethren were anxious to keep a lid on this astounding feature of Indian astronomy.
The western translators of the Surya-Siddhanta quickly glossed over the matter like this …
A remarkable peculiarity of the Hindu system is that the epicycles are supposed to contract their dimensions as they leave the apsis or the conjunction, becoming smallest at the quadrature, then again expanding till the lower apsis, or opposition is reached.
Move along folks, nothing much here. Only a little Hindu peculiarity, that’s all.
And a few pages later, they would be writing paragraphs upon paragraphs proving that the Indians borrowed their epicycle from the Greeks!
The Situation Today
The situation today is little better than what it was 150 years ago, thanks to the complete hijacking of Indian astronomy by western historians during the 19 th and 20 th centuries. Here, for example, is what wiki (Elliptic Orbit) says on the Sun’s orbit …
The Babylonians (3000 BC) were the first to realize that the Sun’s motion along the ecliptic was not uniform…
The Babylonians never used any epicycles, much less a complex one like the pulsating Indian epicycle.
It would be nice if Indian astronomy got at least an ‘honorable mention’ in regards to the Sun’s orbit.
The Pulsating Indian Epicycle is one of the great wonders of ancient science. The very fine pulsations that are set into the Indian models for each planet indicate that the Indian epicycle system is the result of a long and sustained tinkering over time. It also implies that the Indian astronomer had access to a vast collection of very accurate data.
We salute the ingenuity of the ancient Indian astronomer, not only in observational technique, but also in devising the elegantly conceived pulsating epicycle.
Before we close, we must briefly discuss the thought unsaid, the elephant in the room. Was there any borrowing between the Indians and the Greeks?
After all, the epicycle is an idea, and two people, far away from each other, can have the same idea.
This is because epicycles can be represented as a complex Fourier series so, with a large number epicycles, very complicated paths can be represented in the complex plane. [ 23 ]
where a_0 and k_0 are constants, i=sqrt <-1>is an imaginary number, and t is time, correspond to a deferent centered on the origin of the complex plane and revolving with a radius a_0 and angular velocity
If z_1 is the path of an epicycle, then the deferent plus epicycle is represented as the sum
Generalizing to N epicycles yields
which is a particular type of complex Fourier series known as a Besicovitch almost periodic function. Finding the coefficients a_j to represent a time-dependent path in the complex plane, z=f(t) , is the goal of reproducing an orbit with deferent and epicycles, and this is a way of "saving the phenomena" (σώζειν τα φαινόμενα). [ 24 ]
This parallel was noted by Giovanni Schiaparelli. [ 25 ] [ 26 ] Pertinent to the Copernican Revolution debate of "saving the phenomena" versus offering explanations, one can understand why Thomas Aquinas, in the 13th century, wrote:
Equivalence of minor epicycle and eccentric - Astronomy
Ancient Theories of the Sun:
1. Eccentric Model Applet
Select from the Details menu.
There are two mathematically equivalent models of ancient Greek astronomy explaining the unequal motion of the Cun:
Lenghts of the seasons:
The rounded number of days in zodiac signs agree with those of Geminus:
| ||Ari||Tau||Gem||Can||Leo ||Vir||Lib||Sco||Sag||Cap||Aqu||Pis|| |
The equation of the center (EoC) is the difference between the actual position of the Sun and the position it would have if its angular motion were uniform.
From apogee to perigee the actual Sun is behind the mean Sun ( EoC negative), f rom perigee to apogee the apparent Sun is in advance ( EoC positive).
The maximum value of the equation of the center is at 90° from apogee:
e=1/24.04 max EoC = 2° 23'
In the appendix 2 "Calculation of the Eccentric-Quotient for the Sun" of Thurston's book, e is computed to be 143/3438 = 24.04, using the lengths of the seasons and 365 d 14 /60 h 48 /3600 min = 365.2467 d for the length of the tropical year given by Ptolemy.
Hugh Thurston : Early Astronomy, Springer, Berlin, New York 1994.
Jean Meeus: Astronomical Tables of the Sun, Moon and Planets. 2nd ed., Willmann-Bell, Richmond 1995.
Galileos discoveries and the conceptual structure of astronomy
At the time of Copernicus all natural philosophers in the Latin West agreed that the earth was the center of the cosmos, and that celestial objects somehow moved around it. The main physical constituents were a series of spherical shells, centered on the earth. Celestial objects like planets and stars were minor imperfections in these shells. They were carried around the heavens by the spheres as they moved (Swerdlow 1976 Van Helden 1985). How planets moved was a matter of bitter dispute. Averroist natural philosophers believed that the heavens consisted of a series of shells of the element ether, all concentric to the earth (Barker 1999). Ptolemaic astronomers agreed that overall the planets moved within a series of nesting concentric shells, but they gave a detailed account of the shells for each planet that included some parts generating circular motions not centered on the earth. In both cases the overall construction of the heavens was intended to conform to the principles of Aristotle's physics, and it was generally agreed that all celestial motions were compounded from motions that were circular and performed at constant speed (Barker and Goldstein 1998).
Although the fixed stars actually appear to follow paths across the sky that are circles traversed at constant speed, it is well known that the sun, moon, and planets do not. The planets are the most complex case, possessing both a proper motion in the opposite direction from
figure 50. Partial frame for 'circular motion' as applied to astronomy, circa 1500.
the twenty-four-hour daily motion of the stars and a regularly repeated reversal of this motion called retrogression. Averroists and Ptolemaic astronomers differed radically in the explanations they gave for these two aspects of a planet's motion. Let us consider each of these positions in turn.
Figure 50 shows a partial frame for circular motion as it applies in understanding the motion of celestial objects. The concept has four important attributes for Averroists. First, all circular motions take place about some definable center, and for an Averroist this center must be the center of the cosmos (which is also the center of the earth). Although other centers of circular motion are geometrically possible, for physical and metaphysical reasons only one value of this attribute is allowed in any Averroist account of the heavens. Second, all circular motions must have a definite radius, although in practice Averroists were unable to specify precise values. It was generally recognized that for the heavens, the minimum radius was that of the motion of the moon - the nearest object - and the maximum was that of the fixed stars - assumed to be at equal distances and forming a boundary to the cosmos. In principle planets could move on circles at any radius between these boundaries. An array of boxes appears in the frame between MOON and FIXED STARS to indicate an indefinitely large range of intermediate values. The possible values for speed, or angular velocity, range from 24 hours - the speed of the daily rotation - through
figure 51. Geocentric system of the world. Reproduced from Peter Apian, Cosmographia, Antwerp (1540), fol 6 R. Copyright the History of Science Collections, the University of Oklahoma Libraries, and reproduced by permission.
the slowest proper motion, that of the planet Saturn, which returns to its original position in the sky in slightly less than 30 years. Again, these values were seldom specified with any precision by Averroists, but the speeds of rotation for the proper motions of other planets must fall in the range between 24 hours and 30 years, also indicated by an array of intermediate boxes.
For the Averroists and their rivals, the physical constitution of the heavens consisted not so much of planets moving in circles as of planets carried by spheres. The celestial spheres were hollow shells that fitted perfectly inside one another (Figure 51). The sixteenth-century name for a spherical shell bounded by two spherical surfaces is an 'orb'. Although astronomers made this technical distinction, they often spoke of 'spheres', expecting their audience to understand that they were referring to orbs.
In all cases the circles used in describing the motions of particular planets are believed to result from the uniform rotation of an orb. For an Averroist the direction of the orb's axis is an important variable -here displayed as the fourth attribute node. While the axis for the orb creating the daily rotation passes through the celestial poles, the axis for the orb producing the proper motion coincided with the axis of the ecliptic. Most importantly, retrogressions are created by the combined effects of at least two concentric orbs with offset axes, carried within the orbs for the daily and proper motion (Pedersen 1993: 63-70, 235-236).
The Averroist account of the path for a planet was built from a minimum of four circular motions, corresponding to four concentric spheres (Figure 52). The spheres fit together perfectly, one inside another, with no empty space between. It is assumed that the axis of an inner sphere is carried by fixed points on the next sphere out. Consequently, a planet carried on an inner sphere does not perform a simple circle when viewed from the central earth. It follows a path that is the resultant of the motions of the sphere that carries it and all the spheres to which that one is attached, directly or indirectly.
The Averroist conceptual structure for the path of a planet may be presented as a recursive frame diagram (Figure 53), using the frame for circular motion (Figure 50) to specify the attributes and values of each circular motion involved in the frame for PATH. Averroists and Ptolemaic astronomers give identical accounts of the daily motion (the fixed stars rotate about an axis through the poles once in twenty-four hours, carrying everything else with them). To simplify our diagrams the corresponding branch will not be included in the next few figures. Omitting the daily motion for simplicity, the Averroist conceptual structure will have one frame corresponding to the proper motion and two corresponding to retrograde motions (as indicated previously, one circular motion accounts for the proper motion, while two circular motions account for retrograde motion). The recursive frame diagram shown in Figure 53 incorporates a corresponding number of iterations of the frame for circular motion (Figure 50). Since
figure 52. Averroist orb cluster, showing concentric orbs for daily motion (a), proper motion (b), and retrogression (c) and (d).
Averroists allow only a single value for the attribute CENTER, an identical node (CENTER OF EARTH) is activated in each circular motion considered here. Under PROPER MOTION, the value for AXIS ORIENTATION will be ECLIPTIC. Under RETROGRADE, the values will be neither POLAR nor ECLIPTIC, but specific to individual planets, here shown as OTHER.
figure 53. Partial recursive frame for 'path' version.
of a celestial object, Averroist figure 53. Partial recursive frame for 'path' version.
of a celestial object, Averroist
The Ptolemaic account of a planet's path requires a simpler recursive frame and the activation of a different set of value nodes. The basic explanation for a planet's path, in addition to its daily motion, makes use of two mechanisms: an eccentric deferent and an epicycle
figure 54. (a) Ptolemaic eccentric-plus-epicycle model for the proper motion and retrogressions of an outer planet. (b) Cross section through a set of Ptolemaic spherical shells that reproduce the circles of (a) as they rotate.
(Figure 54(a)). The complete model for planets like Mars, Jupiter, and Saturn makes use of an additional feature called the equant, which will be discussed in the next chapter. The two main features of the planet's motion, its proper motion and retrogressions, are explained primarily by the separate circular motions of the deferent and the epicycle, respectively (Pedersen 1993: 81-87).
There will be an obvious difference between the recursive frame for this part of Ptolemaic astronomy (Figure 55) and the corresponding Averroist frame (Figure 53). The lower part of the diagram, corresponding to RETROGRADE MOTION, will consist of two frames for CIRCULAR MOTION in the Averroist case but only one in the Ptolemaic case. This is not the kind of difference that creates incommensurability. As in the case of the comets discussed in Section 5.3, the greater complexity of the Averroist recursive frame is again created without introducing any new kinds of attributes, or new ranges of values. The two recursive frames do not, therefore, allow the appearance, in one frame, of objects that violate the no-overlap principle in the other.
In the Ptolemaic frame (Figure 55), while both the deferent and the epicycle correspond to circular motions, neither has the same
figure 55. Partial recursive frame for 'path' of a celestial object, simple Ptolemaic version.
values assigned to attributes as in the Averroist case. The deferent is not centered on the earth but at a point some distance away and is therefore an eccentric circle. The epicycle center is carried on the eccentric deferent as it rotates, and its center is therefore remote from the center of the earth.
As did the Averroists, Ptolemaic astronomers believed that the circles in their planetary models were generated by the uniform rotation of earth-centered spheres. The eccentric deferent is generated by the rotation of two nonuniform spherical shells, or 'orbs', which appear as crescent shapes when displayed in cross section (see Figure 54(b)). Although the uniform gap between these shells is usually displayed in a contrasting color, it is itself a further solid object in which the small sphere representing the epicycle is physically embedded. The planet in turn is physically embedded in this minor sphere. With the exception of the epicycle sphere, all these shells rotate about axes that pass through the center of the earth, so that even the motion of the epicycle can be seen to be constrained by earth-centered spheres, which move in conformity with Aristotle's physics. This did not prevent Averroists from objecting to both eccentrics and epicycles on the grounds that the individual circular motions had impermissible centers. To decide between the Averroist position and the Ptolemaic position, the best evidence would be an example of a celestial motion that was inarguably centered at some other point than the center of the earth. This is exactly what Galileo provided.
Two pieces of telescopic evidence collected by Galileo between 1610 and 1613 could be used as decisive arguments against the Averroist conceptual structure. First, the observation of the phases of Venus seemed to require that Venus travel on a circle centered on the sun (Drake 1990: pt. 3). It is important to note that the pattern of the phases obliges this conclusion and not the mere observation of the phases themselves. Both Averroist and Ptolemaic accounts of the motion of Venus predict phases that appear as Venus moves away from the direct line between the earth and sun (Ariew 1987). However, in the Averroist account the fact that Venus never moves farther than about forty-six degrees away from this line would limit the observable phases to crescents, and the requirement that Venus be carried on a sphere concentric with the earth would make all phases the same apparent size. Galileo actually observed a full range of phases with widely varying sizes. In particular the (nearly) full phases were small, suggesting they took place on the far side of the sun, while the crescent phases were large, suggesting they took place nearer the earth. Although inconsistent with the original Ptolemaic account of the location of Venus, Galileo's results could be accommodated by the simple expedient of moving the center of Venus' epicycle from its original position on the earth-sun line, to coincide with the position of the sun. A Ptolemaic astronomer might well have said that Galileo's observations of phases for Venus confirmed the Ptolemaic account of its motion using an epicycle and accurately located the center of the epicycle for the first time (Ariew 1999: 97-119).
An even clearer case for non-earth-centered motion could be made from the discovery of Jupiter's satellites. In the very first book on his telescopic discoveries, Galileo (1610/1989) argued persuasively that
Jupiter was accompanied by four satellites moving on circles of different sizes around the planet as it traveled through the sky. The general acceptance of Galileo's discovery of these new objects made it impossible to maintain the Averroist prohibition on centers of motion other than the center of the earth. (Note, however, that neither of these pieces of evidence in itself establishes whether the earth is in motion around some external center, or vice versa.) Returning to the frame diagram for circular motion (Figure 50), we may now summarize the dispute between the Averroists and the followers of Ptolemy as follows: because the Averroists insisted that only one center was allowed for celestial motions, they not only denied the possibility of values other than their preferred value for the attribute CENTER, but can be seen as rejecting the inclusion in the recursive frame of any attribute-value pairs other than their preferred one. The Ptolemaic astronomers, on the other hand, insisted that it was at least a legitimate question to inquire, for any particular motion used in astronomy, whether the center was identical to the center of the earth or some other point, and their conceptual structure made use of some of these additional value nodes. Galileo's telescopic discoveries vindicated the Ptolemaic insistence on the inclusion of these nodes, by showing that several celestial motions could not be accommodated without them.
When Galileo's telescopic discoveries are analyzed in this way we can see why the initial response to them did not lead to major changes in the conceptual structure of astronomy. Many Ptolemaic astronomers, for example, the Jesuits trained by Christopher Clavius at the Colle-gio Romano, rapidly endorsed the telescopic discoveries (Lattis 1994). Although the phases of Venus and the satellites of Jupiter require the recognition that some value nodes for 'center of motion' must be accepted beyond the Averroist choice, the corresponding attribute node was not yet identical to the node appearing in the seventeenth-century structure we examined earlier. Figure 46 contains nodes for ORBIT CENTER and SHAPE. An orbit is a continuous track in space traced by a planet, and it defines both the direction from the observer to a planet and its distance. In all astronomical theories before Kepler predictions were confined to directions, that is, angular positions of planets with respect to a fixed reference line in space (Barker and
Goldstein 1994). The node for CENTER in the conceptual structure of Ptolemaic astronomy designates a center of angular motion and not an orbit center. After Kepler introduced the concept of an orbit in his 1609 Astronomia Nova, anyone accepting the new conceptual structure for astronomy presented there would be obliged to substitute a node that did represent ORBIT CENTER, together with a variety of choices for the shape of an orbit. Initially the two most important choices are the circle and the ellipse. Newton demonstrated that motions subject to an inverse-square law created orbits that were conic sections and may be seen as adding a new set of value nodes to an existing structure in which the attribute nodes were provided by Kepler and Galileo.
From the viewpoint of Kuhn's account of conceptual change in science, two points about this reconstruction deserve special mention. First, it can be seen that the transition from the Ptolemaic conceptual structure to the Newtonian one was not a process that took place instantaneously, but rather one in which an existing structure was successively modified. Kepler's theoretical work and Galileo's telescopic discoveries happened at almost the same moment. Kepler could argue, in favor of the new structure that he proposed, that by means of his new style of calculations he was able to specify the position of the planet Mars with an unprecedented accuracy. But the existence of the separate set of arguments, based on Galileo's discoveries of the phases of Venus and moons of Jupiter and supporting a conceptual structure diverging from the Averroist one in the same way as Kepler's, meant that his work and Galileo's rapidly became mutually supportive in the emergence of what was ultimately Newton's conceptual structure.
The same considerations also allow us to locate and appraise some of the most important incommensurabilities between pre-Newtonian and post-Newtonian astronomy. The first serious incommensurability appears with the replacement of the attribute nodes for PATH with those for ORBIT CENTER and ORBIT SHAPE. But it is important to recognize that other parts of the frame remained constant despite this change. Consequently the many astronomical questions that drew primarily on the attributes of celestial objects that remained unaffected by the change were uncontroversial, and supporters of both pre- and post-Newtonian astronomy could agree on their solution. As late as 1728 Ephraim Chambers found it useful to present the elements of both conceptual structures in a single work. The idea that the introduction of any change in a conceptual structure leads to total communication failure between supporters of the new structure and supporters of the old is therefore seen to be completely unfounded.
In Section 4.5 we suggested that conceptual changes vary in degree, and that the frame account explains why some changes are more severe than others. By the same token, the degree of severity of incommensurability may also be appraised by analyzing where replacements are made in the frame. Roughly speaking, as described in Section 4.5, the higher in a kind hierarchy the replacement of attributes appears, the more acute the problem will be. Phrasing the point in terms of frames, if we consider the frame for a multiple-level kind hierarchy (Figure 16) we can say that the more general the attribute that is replaced, the greater the incommensurability. The mark of incommensurability between two conceptual structures is therefore not a total failure of correspondence between them, but rather the appearance of two or more attributes that differ and that introduce different sets of values. In general, merely introducing a new set of values for an existing attribute will not generate incommensurability. Averroist astronomy and the simple version of Ptolemaic astronomy we have discussed so far are not incommensurable, although the full version may be (as we will see in the next chapter). The addition or deletion of an attribute will create incommensurability only if the new attribute-value sets violate the no-overlap principle (or another of the hierarchical principles introduced in Section 4.2) as applied to the attribute-value sets of the previous frame. Thus, the incommensurability between Keplerian astronomy and Ptolemaic astronomy created by the deletion of the attribute PATH in favor of the attributes ORBIT CENTER and ORBIT SHAPE is significant (Figures 46 and 49), but the incommensurability between the concept of PHYSICAL OBJECT in post-Newtonian physics and in pre-Newtonian physics will be considerably more severe, as that concept is superordinate to the concept of an astronomical object which we have been considering (Figures 43 and 45).
The difficulties labeled incommensurability have so far appeared when two or more conceptual structures from different scientific traditions have been compared. We will see in the next chapter that individual traditions may suffer from similar difficulties. Copernicus' main announced objection to Ptolemaic astronomy may be seen as a problem of just this kind.
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This is because epicycles can be represented as a complex Fourier series so, with a large number of epicycles, very complicated paths can be represented in the complex plane. 
where a_0 and k_0 are constants, i=sqrt <-1>is an imaginary number, and t is time, correspond to a deferent centered on the origin of the complex plane and revolving with a radius a_0 and angular velocity
If z_1 is the path of an epicycle, then the deferent plus epicycle is represented as the sum
This is an almost periodic function, and is a periodic function just when the ratio of the k_j 's is rational. Generalizing to N epicycles yields the almost periodic function
which is periodic just when every pair of k_j 's is rationally related. Finding the coefficients a_j to represent a time-dependent path in the complex plane, z=f(t) , is the goal of reproducing an orbit with deferent and epicycles, and this is a way of "saving the phenomena" (σώζειν τα φαινόμενα). 
This parallel was noted by Giovanni Schiaparelli.   Pertinent to the Copernican Revolution's debate about "saving the phenomena" versus offering explanations, one can understand why Thomas Aquinas, in the 13th century, wrote: