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81-402: Ptolemy does not have a word for the equant – he used expressions such as "the eccentre producing the mean motion". The equant point (shown in the diagram by the large • ), is placed so that it is directly opposite to Earth from the deferent 's center, known as the eccentric (represented by the × ). A planet or the center of an epicycle (a smaller circle carrying

162-646: A Fourier expansion : with Bessel functions J n {\displaystyle J_{n}} and parameter β = 1 − 1 − e 2 e {\displaystyle \beta ={\frac {1-{\sqrt {1-e^{2}}}}{e}}} . Omitting all terms of order e 4 {\displaystyle e^{4}} or higher (indicated by O ⁡ ( e 4 ) {\displaystyle \operatorname {\mathcal {O}} \left(e^{4}\right)} ), it can be written as Note that for reasons of accuracy this approximation

243-422: A complex Fourier series ; therefore, with a large number of epicycles, very complex paths can be represented in the complex plane . Let the complex number where a 0 and k 0 are constants, i = √ −1 is the imaginary unit , 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 where T

324-466: A circular path not centered on the Earth. The moving object's speed will vary during its orbit around the outer circle (dashed line), faster in the bottom half and slower in the top half, but the motion is considered uniform because the planet goes through equal angles in equal times from the perspective of the equant point. The angular speed of the object is non-uniform when viewed from any other point within

405-464: A constant speed. According to the astronomer Hipparchos, moving the center of the Sun's path slightly away from Earth would satisfy the observed motion of the Sun rather painlessly, thus making the Sun's orbit eccentric. Most of what we know about Hipparchus comes to us through citations of his works by Ptolemy. Hipparchus' models' features explained differences in the length of the seasons on Earth (known as

486-478: A planet. The location was determined by the deferent and epicycle, while the duration was determined by uniform motion around the equant. He did this without much explanation or justification for how he arrived at the point of its creation, deciding only to present it formally and concisely with proofs as with any scientific publication. Even in his later works where he recognized the lack of explanation, he made no effort to explain further. Ptolemy's model of astronomy

567-498: A preliminary unpublished sketch called the Commentariolus . By the time he published De revolutionibus orbium coelestium , he had added more circles. Counting the total number is difficult, but estimates are that he created a system just as complicated, or even more so. Koestler, in his history of man's vision of the universe, equates the number of epicycles used by Copernicus at 48. The popular total of about 80 circles for

648-514: A semblance of constant circular motion of celestial bodies , a long-standing article of faith originated by Aristotle for philosophical reasons, while also allowing for the best match of the computations of the observed movements of the bodies, particularly in the size of the apparent retrograde motion of all Solar System bodies except the Sun and the Moon . The equant model has a body in motion on

729-415: A solar eclipse (585 BC), or Heraclides Ponticus . They also saw the "wanderers" or "planetai" (our planets ). The regularity in the motions of the wandering bodies suggested that their positions might be predictable. The most obvious approach to the problem of predicting the motions of the heavenly bodies was simply to map their positions against the star field and then to fit mathematical functions to

810-588: A system that employs elliptical rather than circular orbits. Kepler's three laws are still taught today in university physics and astronomy classes, and the wording of these laws has not changed since Kepler first formulated them four hundred years ago. The apparent motion of the heavenly bodies with respect to time is cyclical in nature. Apollonius of Perga (3rd century BC) realized that this cyclical variation could be represented visually by small circular orbits, or epicycles , revolving on larger circular orbits, or deferents . Hipparchus (2nd century BC) calculated

891-465: A theory to make its predictions match the facts. There is a generally accepted idea that extra epicycles were invented to alleviate the growing errors that the Ptolemaic system noted as measurements became more accurate, particularly for Mars. According to this notion, epicycles are regarded by some as the paradigmatic example of bad science. Copernicus added an extra epicycle to his planets, but that

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972-415: A thousand years after Ptolemy's original work was published. When Copernicus transformed Earth-based observations to heliocentric coordinates, he was confronted with an entirely new problem. The Sun-centered positions displayed a cyclical motion with respect to time but without retrograde loops in the case of the outer planets. In principle, the heliocentric motion was simpler but with new subtleties due to

1053-528: A time. This is not to say that he believed the planets were all equidistant, but he had no basis on which to measure distances, except for the Moon. He generally ordered the planets outward from the Earth based on their orbit periods. Later he calculated their distances in the Planetary Hypotheses and summarized them in the first column of this table: Had his values for deferent radii relative to

1134-491: Is a function of time t as follows: where Ω is the constant angular speed seen from the equant which is situated at a distance E when the radius of the deferent is R . Ptolemy introduced the equant in " Almagest ". The evidence that the equant was a required adjustment to Aristotelian physics relied on observations made by himself and a certain "Theon" (perhaps, Theon of Smyrna ). In models of planetary motion that precede Ptolemy , generally attributed to Hipparchus ,

1215-460: Is considered as established, because thereby the sensible appearances of the heavenly movements can be explained; not, however, as if this proof were sufficient, forasmuch as some other theory might explain them. Being a system that was for the most part used to justify the geocentric model, with the exception of Copernicus' cosmos, the deferent and epicycle model was favored over the heliocentric ideas that Kepler and Galileo proposed. Later adopters of

1296-441: Is no bilaterally-symmetrical, nor eccentrically-periodic curve used in any branch of astrophysics or observational astronomy which could not be smoothly plotted as the resultant motion of a point turning within a constellation of epicycles, finite in number, revolving around a fixed deferent. Any path—periodic or not, closed or open—can be represented with an infinite number of epicycles. This is because epicycles can be represented as

1377-465: Is no uniquely determined line of nodes. One uses the true longitude instead: where: The relation between the true anomaly ν and the eccentric anomaly E {\displaystyle E} is: or using the sine and tangent : or equivalently: so Alternatively, a form of this equation was derived by that avoids numerical issues when the arguments are near ± π {\displaystyle \pm \pi } , as

1458-524: Is that historians examining books on Ptolemaic astronomy from the Middle Ages and the Renaissance have found absolutely no trace of multiple epicycles being used for each planet. The Alfonsine Tables, for instance, were apparently computed using Ptolemy's original unadorned methods. Another problem is that the models themselves discouraged tinkering. In a deferent-and-epicycle model, the parts of

1539-405: Is the period . 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 constants k j is rational . Generalizing to N epicycles yields the almost periodic function which is periodic just when every pair of k j is rationally related. Finding

1620-405: Is usually limited to orbits where the eccentricity e {\displaystyle e} is small. The expression ν − M {\displaystyle \nu -M} is known as the equation of the center , where more details about the expansion are given. The radius (distance between the focus of attraction and the orbiting body) is related to the true anomaly by

1701-447: Is usually very difficult to find any details of previously used models, except from writings by Ptolemy himself. For many centuries rectifying these violations was a preoccupation among scholars, culminating in the solutions of Ibn al-Shatir and Copernicus . Ptolemy's predictions, which required constant review and corrections by concerned scholars over those centuries, culminated in the observations of Tycho Brahe at Uraniborg . It

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1782-589: The Keplerian orbit , the equant method causes the body to spend too little time far from the Earth and too much close to the Earth. For example, when the eccentric anomaly is π/2, the Keplerian model says that an amount of time of π / 2 − e {\displaystyle \pi /2-e} will have elapsed since perigee (where the period is 2 π {\displaystyle 2\pi } , see Kepler equation ), whereas

1863-593: The Tusi couple as an alternative explanation, and Nicolaus Copernicus , whose alternative was a new pair of small epicycles for each deferent. Dislike of the equant was a major motivation for Copernicus to construct his heliocentric system. The violation of uniform circular motion around the center of the deferent bothered many thinkers, especially Copernicus, who mentions the equant as a "monstrous construction" in De Revolutionibus . Copernicus' displacement of

1944-477: The heliocentric model did not exist in Ptolemy 's time and would not come around for over fifteen hundred years after his time. Furthermore, Aristotelian physics was not designed with these sorts of calculations in mind, and Aristotle 's philosophy regarding the heavens was entirely at odds with the concept of heliocentrism. It was not until Galileo Galilei observed the moons of Jupiter on 7 January 1610, and

2025-600: The lengths of the seasons . This can be observed in the lengths of seasons, given by equinoxes and solstices that indicate when the Sun traveled 90 degrees along its path. Though others tried, Hipparchos calculated and presented the most exact lengths of seasons around 130 BCE. According to these calculations, Spring lasted about ⁠94 + 1 / 2  ⁠ days , Summer about ⁠92 + 1 / 2  ⁠ , Fall about ⁠88 + 1 / 8  ⁠ , and Winter about ⁠90 + 1 / 8  ⁠ , showing that seasons did indeed have differences in lengths. This

2106-411: The mean anomaly . For elliptic orbits, the true anomaly ν can be calculated from orbital state vectors as: where: For circular orbits the true anomaly is undefined, because circular orbits do not have a uniquely determined periapsis. Instead the argument of latitude u is used: where: For circular orbits with zero inclination the argument of latitude is also undefined, because there

2187-410: The "first anomaly"), and the appearance of retrograde motion in the planets (known as the "second anomaly"). But Hipparchus was unable to make the predictions about the location and duration of retrograde motions of the planets match observations; he could match location, or he could match duration, but not both simultaneously. Between Hipparchus's model and Ptolemy's there was an intermediate model that

2268-400: The 13th century, wrote: Reason may be employed in two ways to establish a point: firstly, for the purpose of furnishing sufficient proof of some principle [...]. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results, as in astronomy the theory of eccentrics and epicycles

2349-578: The 13th century. (Alfonso is credited with commissioning the Alfonsine Tables .) By this time each planet had been provided with from 40 to 60 epicycles to represent after a fashion its complex movement among the stars. Amazed at the difficulty of the project, Alfonso is credited with the remark that had he been present at the Creation he might have given excellent advice. As it turns out, a major difficulty with this epicycles-on-epicycles theory

2430-457: The Earth and the Sun. When ancient astronomers viewed the sky, they saw the Sun, Moon, and stars moving overhead in a regular fashion. Babylonians did celestial observations, mainly of the Sun and Moon as a means of recalibrating and preserving timekeeping for religious ceremonies. Other early civilizations such as the Greeks had thinkers like Thales of Miletus , the first to document and predict

2511-404: The Earth from the center of the cosmos obviated the primary need for Ptolemy's epicycles: It explained retrograde movement as an effect of perspective, due to the relative motion of the earth and the planets. However, it did not explain non-uniform motion of the Sun and Moon, whose relative motions Copernicus did not change (even though he did recast the Sun orbiting the Earth as the Earth orbiting

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2592-417: The Earth was where they stood and observed the sky, and it is the sky which appears to move while the ground seems still and steady underfoot. Some Greek astronomers (e.g., Aristarchus of Samos ) speculated that the planets (Earth included) orbited the Sun, but the optics (and the specific mathematics – Isaac Newton 's law of gravitation for example) necessary to provide data that would convincingly support

2673-557: The Earth. The uniformity was generally assumed to be observed from the center of the deferent, and since that happens at only one point, only non-uniform motion is observed from any other point. Ptolemy displaced the observation point from the center of the deferent to the equant point. This can be seen as violating the axiom of uniform circular motion. Noted critics of the equant include the Persian astronomer Nasir al-Din Tusi who developed

2754-472: The Earth–Sun distance been more accurate, the epicycle sizes would have all approached the Earth–Sun distance. Although all the planets are considered separately, in one peculiar way they were all linked: the lines drawn from the body through the epicentric center of all the planets were all parallel, along with the line drawn from the Sun to the Earth along which Mercury and Venus were situated. That means that all

2835-419: The Keplerian model it is π / 2 + arcsin ⁡ ( e ) , {\displaystyle \pi /2+\arcsin(e),} which is more. However, for small eccentricity the error is very small, being asymptotic to the eccentricity to the third power. The angle α whose vertex is at the center of the deferent, and whose sides intersect the planet and the equant, respectively,

2916-518: The Moon's Motion which employed an epicycle and remained in use in China into the nineteenth century. Subsequent tables based on Newton's Theory could have approached arcminute accuracy. According to one school of thought in the history of astronomy, minor imperfections in the original Ptolemaic system were discovered through observations accumulated over time. It was mistakenly believed that more levels of epicycles (circles within circles) were added to

2997-431: The Moon, moving faster at perigee and slower at apogee than circular orbits would, using four gears, two of them engaged in an eccentric way that quite closely approximates Kepler's second law . Epicycles worked very well and were highly accurate, because, as Fourier analysis later showed, any smooth curve can be approximated to arbitrary accuracy with a sufficient number of epicycles. However, they fell out of favor with

3078-427: The Ptolemaic system seems to have appeared in 1898. It may have been inspired by the non-Ptolemaic system of Girolamo Fracastoro , who used either 77 or 79 orbs in his system inspired by Eudoxus of Cnidus . Copernicus in his works exaggerated the number of epicycles used in the Ptolemaic system; although original counts ranged to 80 circles, by Copernicus's time the Ptolemaic system had been updated by Peurbach toward

3159-548: The Sun, the two are geometrically equivalent). Moving the center of planetary motion from the Earth to the Sun did not remove the need for something to explain the non-uniform motion of the Sun, for which Copernicus substituted two (or several) smaller epicycles instead of an equant. Deferent and epicycle In the Hipparchian , Ptolemaic , and Copernican systems of astronomy , the epicycle (from Ancient Greek ἐπίκυκλος ( epíkuklos )  'upon

3240-408: The actual spacing and widths of retrograde arcs, which could be seen later according to Ptolemy's model and compared. Ptolemy himself rectified this contradiction by introducing the equant in his writing when he separated it from the center of the deferent, making both it and the deferent's center their own distinct parts of the model and making the deferent's center stationary throughout the motion of

3321-534: The angle of the epicycle is not a linear function of the angle of the deferent. In the Hipparchian system the epicycle rotated and revolved along the deferent with uniform motion. However, Ptolemy found that he could not reconcile that with the Babylonian observational data available to him; in particular, the shape and size of the apparent retrogrades differed. The angular rate at which the epicycle traveled

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3402-435: The bodies revolve in their epicycles in lockstep with Ptolemy's Sun (that is, they all have exactly a one-year period). Babylonian observations showed that for superior planets the planet would typically move through in the night sky slower than the stars. Each night the planet appeared to lag a little behind the stars, in what is called prograde motion . Near opposition , the planet would appear to reverse and move through

3483-518: The body, as seen from the main focus of the ellipse (the point around which the object orbits). The true anomaly is usually denoted by the Greek letters ν or θ , or the Latin letter f , and is usually restricted to the range 0–360° (0–2π rad). The true anomaly f is one of three angular parameters ( anomalies ) that defines a position along an orbit, the other two being the eccentric anomaly and

3564-465: The center of the circular deferents that distinguished the Ptolemaic system. For the outer planets, the angle between the center of the epicycle and the planet was the same as the angle between the Earth and the Sun. Ptolemy did not predict the relative sizes of the planetary deferents in the Almagest . All of his calculations were done with respect to a normalized deferent, considering a single case at

3645-417: The changing positions. The introduction of better celestial measurement instruments, such as the introduction of the gnomon by Anaximander, allowed the Greeks to have a better understanding of the passage of time, such as the number of days in a year and the length of seasons, which are indispensable for astronomic measurements. The ancients worked from a geocentric perspective for the simple reason that

3726-439: The circle', meaning "circle moving on another circle") was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon , Sun , and planets . In particular it explained the apparent retrograde motion of the five planets known at the time. Secondarily, it also explained changes in the apparent distances of the planets from the Earth. It was first proposed by Apollonius of Perga at

3807-522: 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

3888-432: The discovery that planetary motions were largely elliptical from a heliocentric frame of reference , which led to the discovery that gravity obeying a simple inverse square law could better explain all planetary motions. In both Hipparchian and Ptolemaic systems, the planets are assumed to move in a small circle called an epicycle , which in turn moves along a larger circle called a deferent (Ptolemy himself described

3969-477: The eccentric and epicycles were already a feature. The Roman writer Pliny in the 1st century CE, who apparently had access to writings of late Greek astronomers, and not being an astronomer himself, still correctly identified the lines of apsides for the five known planets and where they pointed in the zodiac. Such data requires the concept of eccentric centers of motion. Before around the year 430 BCE, Meton and Euktemon of Athens observed differences in

4050-521: The end of the 3rd century BC. It was developed by Apollonius of Perga and Hipparchus of Rhodes, who used it extensively, during the 2nd century BC, then formalized and extensively used by Ptolemy in his 2nd century AD astronomical treatise the Almagest . Epicyclical motion is used in the Antikythera mechanism , an ancient Greek astronomical device, for compensating for the elliptical orbit of

4131-459: The epicyclic model such as Tycho Brahe , who considered the Church's scriptures when creating his model, were seen even more favorably. The Tychonic model was a hybrid model that blended the geocentric and heliocentric characteristics, with a still Earth that has the sun and moon surrounding it, and the planets orbiting the Sun. To Brahe, the idea of a revolving and moving Earth was impossible, and

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4212-422: The equant model gives π / 2 − arctan ⁡ ( e ) , {\displaystyle \pi /2-\arctan(e),} which is a little more. Furthermore, the true anomaly at this point, according to the equant model, will be only π / 2 + arctan ⁡ ( e ) , {\displaystyle \pi /2+\arctan(e),} whereas in

4293-436: The models to match more accurately the observed planetary motions. The multiplication of epicycles is believed to have led to a nearly unworkable system by the 16th century, and that Copernicus created his heliocentric system in order to simplify the Ptolemaic astronomy of his day, thus succeeding in drastically reducing the number of circles. With better observations additional epicycles and eccentrics were used to represent

4374-466: The more realistic n-body problem required numerical methods for solution. The power of Newtonian mechanics to solve problems in orbital mechanics is illustrated by the discovery of Neptune . Analysis of observed perturbations in the orbit of Uranus produced estimates of the suspected planet's position within a degree of where it was found. This could not have been accomplished with deferent/epicycle methods. Still, Newton in 1702 published Theory of

4455-442: The motions of the planets. The empirical methodology he developed proved to be extraordinarily accurate for its day and was still in use at the time of Copernicus and Kepler. A heliocentric model is not necessarily more accurate as a system to track and predict the movements of celestial bodies than a geocentric one when considering strictly circular orbits. A heliocentric system would require more intricate systems to compensate for

4536-427: The need for deferent/epicycle methods altogether and produced more accurate theories. By treating the Sun and planets as point masses and using Newton's law of universal gravitation , equations of motion were derived that could be solved by various means to compute predictions of planetary orbital velocities and positions. If approximated as simple two-body problems , for example, they could be solved analytically, while

4617-572: The newly observed phenomena till in the later Middle Ages the universe became a 'Sphere/With Centric and Eccentric scribbled o'er,/Cycle and Epicycle, Orb in Orb'. As a measure of complexity, the number of circles is given as 80 for Ptolemy, versus a mere 34 for Copernicus. The highest number appeared in the Encyclopædia Britannica on Astronomy during the 1960s, in a discussion of King Alfonso X of Castile 's interest in astronomy during

4698-427: The night sky faster than the stars for a time in retrograde motion before reversing again and resuming prograde. Epicyclic theory, in part, sought to explain this behavior. The inferior planets were always observed to be near the Sun, appearing only shortly before sunrise or shortly after sunset. Their apparent retrograde motion occurs during the transition between evening star into morning star, as they pass between

4779-469: The now-lost astronomical system of Ibn Bajjah in 12th century Andalusian Spain lacked epicycles. Gersonides of 14th century France also eliminated epicycles, arguing that they did not align with his observations. Despite these alternative models, epicycles were not eliminated until the 17th century, when Johannes Kepler's model of elliptical orbits gradually replaced Copernicus' model based on perfect circles. Newtonian or classical mechanics eliminated

4860-513: The numbers by more than two degrees. Saturn is surpassed by the numbers by one and a half degrees." Using modern computer programs, Gingerich discovered that, at the time of the conjunction, Saturn indeed lagged behind the tables by a degree and a half and Mars led the predictions by nearly two degrees. Moreover, he found that Ptolemy's predictions for Jupiter at the same time were quite accurate. Copernicus and his contemporaries were therefore using Ptolemy's methods and finding them trustworthy well over

4941-460: The orbit. Applied without an epicycle (as for the Sun), using an equant allows for the angular speed to be correct at perigee and apogee, with a ratio of ( 1 + e ) 2 / ( 1 − e ) 2 {\displaystyle (1+e)^{2}/(1-e)^{2}} (where e {\displaystyle e} is the orbital eccentricity ). But compared with

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5022-545: The phases of Venus in September 1610, that the heliocentric model began to receive broad support among astronomers, who also came to accept the notion that the planets are individual worlds orbiting the Sun (that is, that the Earth is a planet, too). Johannes Kepler formulated his three laws of planetary motion , which describe the orbits of the planets in the Solar System to a remarkable degree of accuracy utilizing

5103-405: The planet) was conceived to move at a constant angular speed with respect to the equant. To a hypothetical observer placed at the equant point, the epicycle's center (indicated by the small · ) would appear to move at a steady angular speed. However, the epicycle's center will not move at a constant speed along its deferent. The reason for the implementation of the equant was to maintain

5184-442: The planets actually orbited the Sun. Ptolemy's and Copernicus' theories proved the durability and adaptability of the deferent/epicycle device for representing planetary motion. The deferent/epicycle models worked as well as they did because of the extraordinary orbital stability of the solar system. Either theory could be used today had Gottfried Wilhelm Leibniz and Isaac Newton not invented calculus . According to Maimonides ,

5265-422: The planets in his model moved in perfect circles. Johannes Kepler would later show that the planets move in ellipses, which removed the need for Copernicus' epicycles as well. True anomaly In celestial mechanics , true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit . It is the angle between the direction of periapsis and the current position of

5346-403: The planets were different, and so it was with Copernicus' initial models. As he worked through the mathematics, however, Copernicus discovered that his models could be combined in a unified system. Furthermore, if they were scaled so that the Earth's orbit was the same in all of them, the ordering of the planets we recognize today easily followed from the math. Mercury orbited closest to the Sun and

5427-512: The point but did not give it a name ). Both circles rotate eastward and are roughly parallel to the plane of the Sun's apparent orbit under those systems ( ecliptic ). Despite the fact that the system is considered geocentric , neither of the circles were centered on the earth, rather each planet's motion was centered at a planet-specific point slightly away from the Earth called the eccentric . The orbits of planets in this system are similar to epitrochoids , but are not exactly epitrochoids because

5508-422: The required orbits. Deferents and epicycles in the ancient models did not represent orbits in the modern sense, but rather a complex set of circular paths whose centers are separated by a specific distance in order to approximate the observed movement of the celestial bodies. Claudius Ptolemy refined the deferent-and-epicycle concept and introduced the equant as a mechanism that accounts for velocity variations in

5589-533: The rest of the planets fell into place in order outward, arranged in distance by their periods of revolution. Although Copernicus' models reduced the magnitude of the epicycles considerably, whether they were simpler than Ptolemy's is moot. Copernicus eliminated Ptolemy's somewhat-maligned equant but at a cost of additional epicycles. Various 16th-century books based on Ptolemy and Copernicus use about equal numbers of epicycles. The idea that Copernicus used only 34 circles in his system comes from his own statement in

5670-408: The scripture should be always paramount and respected. When Galileo tried to challenge Tycho Brahe's system, the church was dissatisfied with their views being challenged. Galileo's publication did not aid his case in his trial . "Adding epicycles" has come to be used as a derogatory comment in modern scientific discussion. The term might be used, for example, to describe continuing to try to adjust

5751-465: The shift in reference point. It was not until Kepler's proposal of elliptical orbits that such a system became increasingly more accurate than a mere epicyclical geocentric model. Owen Gingerich describes a planetary conjunction that occurred in 1504 and was apparently observed by Copernicus. In notes bound with his copy of the Alfonsine Tables , Copernicus commented that "Mars surpasses

5832-437: The similar number of 40; hence Copernicus effectively replaced the problem of retrograde with further epicycles. Copernicus' theory was at least as accurate as Ptolemy's but never achieved the stature and recognition of Ptolemy's theory. What was needed was Kepler's elliptical-orbit theory, not published until 1609 and 1619. Copernicus' work provided explanations for phenomena like retrograde motion, but really did not prove that

5913-408: The two tangents become infinite. Additionally, since E 2 {\displaystyle {\frac {E}{2}}} and ν 2 {\displaystyle {\frac {\nu }{2}}} are always in the same quadrant, there will not be any sign problems. so The true anomaly can be calculated directly from the mean anomaly M {\displaystyle M} via

5994-408: The whole are interrelated. A change in a parameter to improve the fit in one place would throw off the fit somewhere else. Ptolemy's model is probably optimal in this regard. On the whole it gave good results but missed a little here and there. Experienced astronomers would have recognized these shortcomings and allowed for them. According to the historian of science Norwood Russell Hanson : There

6075-413: The yet-to-be-discovered elliptical shape of the orbits. Another complication was caused by a problem that Copernicus never solved: correctly accounting for the motion of the Earth in the coordinate transformation. In keeping with past practice, Copernicus used the deferent/epicycle model in his theory but his epicycles were small and were called "epicyclets". In the Ptolemaic system the models for each of

6156-406: Was later used as evidence for the zodiacal inequality, or the appearance of the Sun to move at a rate that is not constant, with some parts of its orbit including it moving faster or slower. The Sun's annual motion as understood by Greek astronomy up to this point did not account for this, as it assumed the Sun had a perfectly circular orbit that was centered around the Earth that it traveled around at

6237-422: Was not constant unless he measured it from another point which is now called the equant (Ptolemy did not give it a name). It was the angular rate at which the deferent moved around the point midway between the equant and the Earth (the eccentric) that was constant; the epicycle center swept out equal angles over equal times only when viewed from the equant. It was the use of equants to decouple uniform motion from

6318-434: Was not until Johannes Kepler published his Astronomia Nova , based on the data he and Tycho collected at Uraniborg, that Ptolemy's model of the heavens was entirely supplanted by a new geometrical model. The equant solved the last major problem of accounting for the anomalistic motion of the planets but was believed by some to compromise the principles of the ancient Greek philosophers, namely uniform circular motion about

6399-434: Was only in an effort to eliminate Ptolemy's equant, which he considered a philosophical break away from Aristotle's perfection of the heavens. Mathematically, the second epicycle and the equant produce nearly the same results, and many Copernican astronomers before Kepler continued using the equant, as the mathematical calculations were easier. Copernicus' epicycles were also much smaller than Ptolemy's, and were required because

6480-411: Was proposed to account for the motion of planets in general based on the observed motion of Mars. In this model, the deferent had a center that was also the equant, that could be moved along the deferent's line of symmetry in order to match to a planet's retrograde motion. This model, however, still did not align with the actual motion of planets, as noted by Hipparchos. This was true specifically regarding

6561-424: Was used as a technical method that could answer questions regarding astrology and predicting planets positions for almost 1,500 years, even though the equant and eccentric were regarded by many later astronomers as violations of pure Aristotelian physics which presumed all motion to be centered on the Earth. It has been reported that Ptolemy's model of the cosmos was so popular and revolutionary, in fact, that it

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