Deferent and epicycle

720:, Copernicus commented that "Mars surpasses 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 a thousand years after Ptolemy's original work was published. 210:(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 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. 36: 157: 575: 2255: 2244: 762: 700: 724:

motion was simpler but with new subtleties due to 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".

742:, 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 the Ptolemaic system seems to have appeared in 1898. It may have been inspired by the 217:. All of his calculations were done with respect to a normalized deferent, considering a single case at 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 754:. 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 the similar number of 40; hence Copernicus effectively replaced the problem of retrograde with further epicycles. 680:(2nd century BC) calculated 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. 1552:

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

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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

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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

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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

1502:

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

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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

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to a remarkable degree of accuracy utilizing 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.

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Another problem is that the models themselves discouraged tinkering. In a deferent-and-epicycle model, the parts of 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

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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

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In the Ptolemaic system the models for each of 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

554:

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

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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 the models to match more

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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 the shift in reference

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Copernicus added an extra epicycle to his planets, but that 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

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As it turns out, a major difficulty with this epicycles-on-epicycles theory 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

813:

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

519:

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 the bodies revolve in their epicycles in lockstep with Ptolemy's Sun (that is, they all have exactly a one-year period).

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Had his values for deferent radii relative to 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

1535:

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 the scripture should be always paramount and

955:

There 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

917:

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

2060:"The popular belief that Copernicus's heliocentric system constitutes a significant simplification of the Ptolemaic system is obviously wrong ... he Copernican models themselves require about twice as many circles as the Ptolemaic models and are far less elegant and adaptable." 1503:

its results, as in astronomy the theory of eccentrics and epicycles 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.

1526:

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 the epicyclic model such as

728:

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 the rest of the planets fell into place in order outward, arranged in distance by their periods of revolution.

128:, an ancient Greek astronomical device, for compensating for the elliptical orbit of 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 545:

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 the Earth and the Sun.

1557:

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 the planets in his model moved in perfect circles.

881:

With better observations additional epicycles and eccentrics were used to represent 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

643:

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).

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later showed, any smooth curve can be approximated to arbitrary accuracy with a sufficient number of epicycles. However, they fell out of favor with the discovery that planetary motions were largely elliptical from a

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perspective for the simple reason that 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.,

590:

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 epicycles of the planets in orbit around Earth (Earth at the center). The path-line is the combined motion of the planet's orbit (deferent) around Earth and within the orbit itself (epicycle).

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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

2318: 1160: 1050: 2247: 180:(Ptolemy himself described 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 ( 1536:

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

84: 'upon 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 825:, 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 582:

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

526:

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

188:, 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 160:

The basic elements of Ptolemaic astronomy, showing a planet on an epicycle (smaller dashed circle), a deferent (larger dashed circle), the eccentric (×) and an

1548:"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 869:

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

219: 732:

numbers of epicycles. The idea that Copernicus used only 34 circles in his system comes from his own statement in a preliminary unpublished sketch called the

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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.

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A deferent/epicycle model is in fact used to compute Lunar positions needed to define modern Hindu calendars. See Nachum Dershovitz and Edward M. Reingold:

821:

eliminated the need for deferent/epicycle methods altogether and produced more accurate theories. By treating the Sun and planets as point masses and using

2365: 2454: 936:

whole it gave good results but missed a little here and there. Experienced astronomers would have recognized these shortcomings and allowed for them.

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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 a

899:

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

1191: 71: 714:

describes a planetary conjunction that occurred in 1504 and was apparently observed by Copernicus. In notes bound with his copy of the

100:

of the five planets known at the time. Secondarily, it also explained changes in the apparent distances of the planets from the Earth.

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about the number of epicycles. Their response was that the original author of the entry had died and its source couldn't be verified.

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was devoted to a description of the trigonometry used to make the transformation between geocentric and heliocentric coordinates.

1351: 2689: 2559: 2527: 2080: 1913: 1845: 1655: 200:, but are not exactly epitrochoids because the angle of the epicycle is not a linear function of the angle of the deferent. 649: 3161: 2957: 856:

which employed an epicycle and remained in use in China into the nineteenth century. Subsequent tables based on Newton's

738: 586:

to the changing positions. The introduction of better celestial measurement instruments, such as the introduction of the

877:

in order to simplify the Ptolemaic astronomy of his day, thus succeeding in drastically reducing the number of circles.

1989: 1946: 1820: 2662: 2398: 2713: 2591: 2475: 2223: 2166: 1878: 3192: 2682: 668:(3rd century BC) realized that this cyclical variation could be represented visually by small circular orbits, or 1116: 204:

at which the epicycle traveled was not constant unless he measured it from another point which is now called the

19:"Deferent" redirects here. For the acknowledgement of the legitimacy of the power of superior or superiors, see 988: 691:

methodology he developed proved to be extraordinarily accurate for its day and was still in use at the time of

1561:

would later show that the planets move in ellipses, which removed the need for Copernicus' epicycles as well.

571:). The regularity in the motions of the wandering bodies suggested that their positions might be predictable. 3094: 631:'s philosophy regarding the heavens was entirely at odds with the concept of heliocentrism. It was not until 1631: 538:

before reversing again and resuming prograde. Epicyclic theory, in part, sought to explain this behavior.

2238: 1615: 887: 534:, the planet would appear to reverse and move through the night sky faster than the stars for a time in 535: 111:

of Rhodes, who used it extensively, during the 2nd century BC, then formalized and extensively used by

97: 2650: 2393:

made by Christián Carman and Ramiro Serra, which uses 1000 epicycles to retrace the cartoon character

901: 2891: 2656: 1531:, who considered the Church's scriptures when creating his model, were seen even more favorably. The 787: 2157: 1320: 52: 48: 2886: 2705: 3033: 2906: 2841: 961: 948: 583: 129: 64: 2433: 909:'s interest in astronomy during the 13th century. (Alfonso is credited with commissioning the 623:'s time and would not come around for over fifteen hundred years after his time. Furthermore, 2998: 2736: 2631: 1487: 1483: 125: 2671:

Reconstruction of a planet's bizarre orbit with Ptolemy's system of epicycles and deferents.

2507: 976:; therefore, with a large number of epicycles, very complex paths can be represented in the 3140: 2881: 2831: 2751: 2726: 1667: 1491: 1476: 906: 845: 624: 600: 531: 1725: 8: 2911: 2816: 2766: 2741: 2446: 1475:, is the goal of reproducing an orbit with deferent and epicycles, and this is a way of " 874: 818: 747: 665: 560: 104: 1671: 687:

as a mechanism that accounts for velocity variations in the motions of the planets. The

3145: 3059: 2637: 2357: 2349: 2196: 2072: 2062: 1792: 1745: 1737: 1683: 1495: 945: 1186:

is the path of an epicycle, then the deferent plus epicycle is represented as the sum

3054: 2901: 2806: 2761: 2587: 2565: 2555: 2523: 2471: 2468:

The forgotten revolution. How science was born in 300 BC and why it had to be reborn.

2423: 2341: 2219: 2200: 2162: 2076: 2018: 1985: 1942: 1919: 1909: 1884: 1874: 1851: 1841: 1816: 1784: 1749: 1729: 1687: 1590: 1549: 1324: 841: 834: 751: 556: 2420:

To save the phenomena, an essay on the idea of physical theory from Plato to Galileo

2361: 2295:, 1968, vol. 2, p. 645. This is identified as the highest number in Owen Gingerich, 3166: 3119: 3089: 3079: 3028: 3003: 2515: 2493: 2488: 2333: 2188: 2014: 1776: 1721: 1675: 1640: 1580: 1513: 1171: 1109: 910: 826: 716: 136: 2508:"Tycho the Prophet: History, Astrology and the Apocalypse in Early Modern Science" 849: 3114: 3049: 3018: 2916: 2846: 2133: 2097: 2066: 1656:"The Structure and Function of Ptolemy's Physical Hypotheses of Planetary Motion" 1558: 1532: 1337: 645: 632: 542: 527: 523: 2977: 2856: 2791: 2771: 2299:. Gingerich also expressed doubt about the quotation attributed to Alfonso. In 2031: 1679: 1537: 1508: 1086: 973: 970: 830: 711: 246: 2668: 2402: 2390: 2179:

Goldstein, Bernard R. (1972). "Theory and Observation in Medieval Astronomy".

683:

Claudius Ptolemy refined the deferent-and-epicycle concept and introduced the

107:

at the end of the 3rd century BC. It was developed by Apollonius of Perga and

3186: 3099: 3084: 3064: 2569: 2427: 2394: 2345: 2267: 2141: 2105: 1923: 1855: 1788: 1733: 1457: 1096: 977: 704: 213:

Ptolemy did not predict the relative sizes of the planetary deferents in the

141: 2519: 1888: 1610: 2947: 2796: 2415: 829:, for example, they could be solved analytically, while the more realistic 653: 616: 615:

for example) necessary to provide data that would convincingly support the

608: 2674: 1903: 2786: 2731: 2262: 1585: 1528: 197: 603:) speculated that the planets (Earth included) orbited the Sun, but the 35: 3104: 2921: 2866: 2826: 2756: 1741: 1709: 1575: 870: 810: 806: 802: 798: 692: 677: 595: 185: 108: 44: 27: 2353: 2047:

Gingerich, "Crisis versus Aesthetic in the Copernican Revolution", in

1796: 1764: 1310:{\displaystyle z_{2}=z_{0}+z_{1}=a_{0}e^{ik_{0}t}+a_{1}e^{ik_{1}t}\,.} 3008: 2861: 2851: 2836: 2781: 2776: 2721: 688: 661: 628: 156: 77: 56: 20: 2403:"La refutabilidad del Sistema de Epiciclos y Deferentes de Ptolomeo" 3109: 2937: 2801: 2337: 2192: 1780: 1570: 791: 761: 660:

The apparent motion of the heavenly bodies with respect to time is

574: 181: 117: 26:"Epicycle" redirects here. For the similar mathematical curve, see 2254: 2243: 699: 3013: 2876: 2871: 2821: 2811: 636: 620: 559:, the first to document and predict a solar eclipse (585 BC), or 135:

Epicycles worked very well and were highly accurate, because, as

112: 1095:

is time, correspond to a deferent centered on the origin of the

932:

apparently computed using Ptolemy's original unadorned methods.

3124: 3074: 3023: 2896: 2746: 2249:

The Gradual Acceptance of the Copernican Theory of the Universe

2005:

Palter, Robert (1970). "Approach to the History of Astronomy".

892:

The Gradual Acceptance of the Copernican Theory of the Universe

684: 627:

was not designed with these sorts of calculations in mind, and

604: 587: 568: 206: 169: 161: 93: 640: 193: 1873:(Rev. ed.). Cambridge, UK: Cambridge University Press. 85: 2087:, p. 204. This is an extreme estimate in favor of Ptolemy. 750:, who used either 77 or 79 orbs in his system inspired by 1714:

The American Journal of Semitic Languages and Literatures

964:, "The Mathematical Power of Epicyclical Astronomy", 1960 758:

did not prove that the planets actually orbited the Sun.

89: 2038:(New York: American Institute of Physics, 1993), p. 125. 1429:{\displaystyle z_{N}=\sum _{j=0}^{N}a_{j}e^{ik_{j}t}\,,} 2552:

Copernicus' secret: how the scientific revolution began

2303:(p. 56), however, Gingerich relates that he challenged 1494:" versus offering explanations, one can understand why 905:

on Astronomy during the 1960s, in a discussion of King

223:

and summarized them in the first column of this table:

16:

Planetary motions in archaic models of the Solar System

2665:, interactive JavaScript coding example, Khan Academy. 2447:"Quasi periodic motions from Hipparchus to Kolmogorov" 1871:

Early physics and astronomy: a historical introduction

840:

The power of Newtonian mechanics to solve problems in

703:

The basic simplicity of the Copernican universe, from

578:

The complexity to be described by the geocentric model

1809:

For an example of the complexity of the problem, see

1769:

Transactions of the American Philological Association

1354: 1194: 1119: 991: 176:, which in turn moves along a larger circle called a 1521: 2514:, London: Palgrave Macmillan UK, pp. 137–156, 1428: 1309: 1154: 1044: 652:, which describe the orbits of the planets in the 184:). Despite the fact that the system is considered 2319:"The Mathematical Power of Epicyclical Astronomy" 2161:, Cambridge University Press, 1997, Chapter 14. ( 1835: 3184: 2007:Studies in the History and Philosophy of Science 1447:is rationally related. Finding the coefficients 172:are assumed to move in a small circle called an 115:in his 2nd century AD astronomical treatise the 168:In both Hipparchian and Ptolemaic systems, the 2451:Rendiconti Lincei – Matematica e Applicazioni. 2036:The Eye of Heaven: Ptolemy, Copernicus, Kepler 1979: 1936: 1838:The lives and opinions of eminent philosophers 1810: 1346:epicycles yields the almost periodic function 2690: 2638:Eccentrics, Deferents, Epicycles, and Equants 2554:. New York: Simon & Schuster Paperbacks. 2453:Series 9, Band 12, No. 2, 2001, p. 125–152. ( 2282:An Approach to the History of Early Astronomy 1905:The history and practice of ancient astronomy 1553:as the paradigmatic example of bad science. 848:. Analysis of observed perturbations in the 148:could better explain all planetary motions. 2704: 2213: 2034:, "Alfonso X as a Patron of Astronomy", in 1155:{\displaystyle k_{0}={\frac {2\pi }{T}}\,,} 146:gravity obeying a simple inverse square law 2697: 2683: 2061: 1762: 1456:to represent a time-dependent path in the 1438:which is periodic just when every pair of 863: 860:could have approached arcminute accuracy. 672:, revolving on larger circular orbits, or 2644: 2505: 2178: 1975: 1973: 1971: 1710:"Babylonian Astronomy: Historical Sketch" 1422: 1303: 1148: 1045:{\displaystyle z_{0}=a_{0}e^{ik_{0}t}\,,} 1038: 939: 777:is the center of the epicycle of the Sun 196:of planets in this system are similar to 2549: 2422:. Chicago: University of Chicago Press. 2132: 2096: 1868: 1707: 760: 698: 573: 522:Babylonian observations showed that for 155: 34: 2317:Hanson, Norwood Russell (1 June 1960). 3185: 2659:, interactive, Wolfram Demonstrations. 2634:– at Rice University's Galileo Project 2316: 2310: 2004: 1968: 1653: 801:, the now-lost astronomical system of 2678: 2414: 1908:. New York: Oxford University Press. 1901: 1726:10.1086/amerjsemilanglit.55.2.3088090 1327:just when the ratio of the constants 823:Newton's law of universal gravitation 639:on 7 January 1610, and the phases of 2609:Library of Congress, Washington, D.C 2399:Deferentes, epiciclos y adaptaciones 2121:Approach to the History of Astronomy 1836:Diogenes Laertius (September 2013). 1660:Journal for the History of Astronomy 2657:Orbits with Epicycles on a Deferent 1629:See page 21 of the Introduction in 739:De revolutionibus orbium coelestium 563:. They also saw the "wanderers" or 227:Ptolemy's estimates of orbit sizes 13: 1608: 144:, which led to the discovery that 124:Epicyclical motion is used in the 14: 3204: 2625: 2586:, Addison–Wesley, 1996. p. 299. ( 1522:Epicycles and the Catholic Church 96:. In particular it explained the 2653:, interactive, Foothill College. 2253: 2242: 607:(and the specific mathematics – 511: 508: 505: 2597: 2576: 2543: 2499: 2481: 2460: 2439: 2408: 2397:; see also Christián Carman's " 2383: 2286: 2274: 2232: 2216:Newton's Forgotten Lunar Theory 2207: 2172: 2149: 2126: 2113: 2090: 2068:The Exact Sciences in Antiquity 2054: 2041: 2025: 1998: 1955: 1930: 151: 142:heliocentric frame of reference 2214:Kollerstrom, Nicholas (2000). 1895: 1862: 1829: 1803: 1756: 1701: 1647: 1623: 1602: 1543: 1498:, in the 13th century, wrote: 790:and Isaac Newton not invented 1: 3095:Inferior and superior planets 1763:Mosshammer, Alden A. (1981). 2019:10.1016/0039-3681(70)90001-4 1937:Owen Gingerich (2004). "4". 1099:and revolving with a radius 769:) is offset from the Earth ( 7: 2669:Ptolemy and Homer (Simpson) 1616:Online Etymology Dictionary 1564: 1482:This parallel was noted by 854:Theory of the Moon's Motion 736:. By the time he published 594:The ancients worked from a 10: 3209: 3172:Medieval Islamic astronomy 2969:On the Sizes and Distances 2651:Ptolemaic System Simulator 1680:10.1177/002182869502600102 549: 98:apparent retrograde motion 70: 25: 18: 3162:Medieval European science 3154: 3133: 3042: 2991: 2930: 2892:Sosigenes the Peripatetic 2712: 2584:Blind Watchers of the Sky 2506:Håkansson, Håkan (2007), 2261:.. The quotation is from 1654:Andrea, Murschel (1995). 1479:" (σώζειν τα φαινόμενα). 788:Gottfried Wilhelm Leibniz 555:Greeks had thinkers like 103:It was first proposed by 2470:Springer, Berlin. 2004, 2158:Calendrical Calculations 1708:Olmstead, A. T. (1938). 1596: 1321:almost periodic function 650:laws of planetary motion 3193:Ancient Greek astronomy 2887:Sosigenes of Alexandria 2706:Ancient Greek astronomy 2582:See e.g., Kolb, Rocky, 2550:Repcheck, Jack (2008). 2520:10.1057/9780230206472_8 2305:Encyclopædia Britannica 2293:Encyclopædia Britannica 1980:Owen Gingerich (2004). 1869:Pedersen, Olaf (1993). 1811:Owen Gingerich (2004). 983:Let the complex number 924:Encyclopædia Britannica 902:Encyclopædia Britannica 864:The number of epicycles 619:model did not exist in 2959:On Sizes and Distances 2645:Animated illustrations 2512:The Word and the World 1982:'The Book Nobody Read' 1939:'The Book Nobody Read' 1813:'The Book Nobody Read' 1519: 1430: 1388: 1311: 1156: 1046: 967: 962:Norwood Russell Hanson 949:Norwood Russell Hanson 940:Mathematical formalism 929: 897: 844:is illustrated by the 782: 708: 635:observed the moons of 584:mathematical functions 579: 267:normalized to Sun = 1) 165: 40: 3070:Deferent and epicycle 2999:Antikythera mechanism 2445:Giovanni Gallavotti: 1902:Evans, James (1998). 1815:. Arrow. p. 50. 1500: 1488:Copernican Revolution 1484:Giovanni Schiaparelli 1431: 1368: 1312: 1157: 1047: 953: 915: 879: 764: 702: 648:formulated his three 577: 159: 126:Antikythera mechanism 38: 3141:Babylonian astronomy 2832:Hippocrates of Chios 2301:The Book Nobody Read 2252:. (New York, 1917), 2218:. Green Lion Press. 1492:saving the phenomena 1477:saving the phenomena 1352: 1192: 1117: 989: 946:historian of science 907:Alfonso X of Castile 846:discovery of Neptune 625:Aristotelian physics 601:Aristarchus of Samos 220:Planetary Hypotheses 2912:Theon of Alexandria 2271:, Book 8, 11.82–85. 2123:, pp. 113–114. 2051:, pp. 193–204. 1672:1995JHA....26...33M 1486:. Pertinent to the 875:heliocentric system 819:classical mechanics 748:Girolamo Fracastoro 666:Apollonius of Perga 561:Heraclides Ponticus 228: 130:Kepler's second law 105:Apollonius of Perga 3146:Egyptian astronomy 3060:Circle of latitude 2663:ANIMATE: Epicycles 2371:on 1 November 2020 2073:Dover Publications 1633:Ptolemy's Almagest 1426: 1340:. Generalizing to 1307: 1152: 1042: 809:lacked epicycles. 783: 709: 613:law of gravitation 580: 226: 166: 53:Copernican systems 41: 3180: 3179: 3055:Celestial spheres 2561:978-0-7432-8952-8 2529:978-1-349-35338-5 2082:978-0-486-22332-2 1963:De Revolutionibus 1915:978-0-19-987445-3 1847:978-1-230-21699-7 1840:. General Books. 1765:"Thales' Eclipse" 1641:Toomer, Gerald J. 1609:Harper, Douglas. 1591:Scientific method 1490:'s debate about " 1325:periodic function 1146: 944:According to the 842:orbital mechanics 835:numerical methods 827:two-body problems 752:Eudoxus of Cnidus 557:Thales of Miletus 536:retrograde motion 516: 515: 268: 265:(modern/Ptolemy, 259: 252: 239: 3200: 3167:Indian astronomy 3120:Sublunary sphere 3090:Hipparchic cycle 3029:Mural instrument 3004:Armillary sphere 2983: 2973: 2963: 2953: 2943: 2699: 2692: 2685: 2676: 2675: 2632:Ptolemaic System 2620: 2619: 2617: 2615: 2601: 2595: 2580: 2574: 2573: 2547: 2541: 2539: 2538: 2536: 2503: 2497: 2489:Summa Theologica 2485: 2479: 2464: 2458: 2443: 2437: 2431: 2412: 2406: 2387: 2381: 2380: 2378: 2376: 2370: 2364:. Archived from 2323: 2314: 2308: 2290: 2284: 2278: 2272: 2257: 2246: 2236: 2230: 2229: 2211: 2205: 2204: 2176: 2170: 2153: 2147: 2145: 2138:The Sleepwalkers 2134:Koestler, Arthur 2130: 2124: 2117: 2111: 2109: 2102:The Sleepwalkers 2098:Koestler, Arthur 2094: 2088: 2086: 2063:Neugebauer, Otto 2058: 2052: 2045: 2039: 2029: 2023: 2022: 2002: 1996: 1995: 1977: 1966: 1959: 1953: 1952: 1934: 1928: 1927: 1899: 1893: 1892: 1866: 1860: 1859: 1833: 1827: 1826: 1807: 1801: 1800: 1760: 1754: 1753: 1705: 1699: 1698: 1696: 1694: 1666:(xxvii): 33–61. 1651: 1645: 1644: 1639:. Translated by 1638: 1627: 1621: 1620: 1606: 1517: 1514:Summa Theologica 1474: 1455: 1446: 1435: 1433: 1432: 1427: 1421: 1420: 1416: 1415: 1398: 1397: 1387: 1382: 1364: 1363: 1345: 1335: 1316: 1314: 1313: 1308: 1302: 1301: 1297: 1296: 1279: 1278: 1266: 1265: 1261: 1260: 1243: 1242: 1230: 1229: 1217: 1216: 1204: 1203: 1185: 1169: 1161: 1159: 1158: 1153: 1147: 1142: 1134: 1129: 1128: 1110:angular velocity 1107: 1094: 1084: 1083: 1082: 1071: 1062: 1051: 1049: 1048: 1043: 1037: 1036: 1032: 1031: 1014: 1013: 1001: 1000: 965: 927: 911:Alfonsine Tables 895: 807:Andalusian Spain 805:in 12th century 717:Alfonsine Tables 543:inferior planets 524:superior planets 503: 492: 486: 480: 474: 463: 457: 451: 445: 434: 427: 422: 418: 413: 409: 399: 393: 387: 383: 378: 374: 364: 358: 352: 348: 342: 332: 326: 320: 316: 311: 307: 293: 287: 282: 278: 264: 258:(modern/Ptolemy) 257: 244: 238:(in Earth radii) 237: 229: 225: 137:Fourier analysis 81: 74: 3208: 3207: 3203: 3202: 3201: 3199: 3198: 3197: 3183: 3182: 3181: 3176: 3150: 3129: 3115:Spherical Earth 3050:Callippic cycle 3038: 3019:Equatorial ring 2987: 2981: 2971: 2961: 2951: 2941: 2926: 2917:Theon of Smyrna 2708: 2703: 2647: 2628: 2623: 2613: 2611: 2603: 2602: 2598: 2581: 2577: 2562: 2548: 2544: 2534: 2532: 2530: 2504: 2500: 2486: 2482: 2465: 2461: 2444: 2440: 2413: 2409: 2388: 2384: 2374: 2372: 2368: 2321: 2315: 2311: 2291: 2287: 2280:Robert Palter, 2279: 2275: 2239:Dorothy Stimson 2237: 2233: 2226: 2212: 2208: 2177: 2173: 2154: 2150: 2131: 2127: 2118: 2114: 2095: 2091: 2083: 2059: 2055: 2046: 2042: 2030: 2026: 2003: 1999: 1992: 1978: 1969: 1960: 1956: 1949: 1935: 1931: 1916: 1900: 1896: 1881: 1867: 1863: 1848: 1834: 1830: 1823: 1808: 1804: 1761: 1757: 1706: 1702: 1692: 1690: 1652: 1648: 1636: 1630: 1628: 1624: 1607: 1603: 1599: 1567: 1559:Johannes Kepler 1546: 1524: 1518: 1507: 1461: 1453: 1448: 1444: 1439: 1436: 1411: 1407: 1403: 1399: 1393: 1389: 1383: 1372: 1359: 1355: 1353: 1350: 1349: 1341: 1333: 1328: 1317: 1292: 1288: 1284: 1280: 1274: 1270: 1256: 1252: 1248: 1244: 1238: 1234: 1225: 1221: 1212: 1208: 1199: 1195: 1193: 1190: 1189: 1184: 1178: 1165: 1162: 1135: 1133: 1124: 1120: 1118: 1115: 1114: 1106: 1100: 1090: 1080: 1078: 1073: 1072:are constants, 1070: 1064: 1061: 1055: 1052: 1027: 1023: 1019: 1015: 1009: 1005: 996: 992: 990: 987: 986: 966: 960: 942: 928: 922: 896: 888:Dorothy Stimson 886: 866: 850:orbit of Uranus 646:Johannes Kepler 633:Galileo Galilei 552: 528:prograde motion 501: 490: 484: 478: 472: 461: 455: 449: 443: 432: 425: 420: 416: 411: 407: 397: 391: 385: 381: 376: 372: 362: 356: 350: 346: 340: 330: 324: 318: 314: 309: 305: 291: 285: 280: 276: 266: 263: 256: 251:in Earth radii) 250: 243: 236: 154: 31: 24: 17: 12: 11: 5: 3206: 3196: 3195: 3178: 3177: 3175: 3174: 3169: 3164: 3158: 3156: 3152: 3151: 3149: 3148: 3143: 3137: 3135: 3131: 3130: 3128: 3127: 3122: 3117: 3112: 3107: 3102: 3097: 3092: 3087: 3082: 3077: 3072: 3067: 3062: 3057: 3052: 3046: 3044: 3040: 3039: 3037: 3036: 3031: 3026: 3021: 3016: 3011: 3006: 3001: 2995: 2993: 2989: 2988: 2986: 2985: 2979:On the Heavens 2975: 2965: 2955: 2952:(Eratosthenes) 2945: 2934: 2932: 2928: 2927: 2925: 2924: 2919: 2914: 2909: 2904: 2899: 2894: 2889: 2884: 2879: 2874: 2869: 2864: 2859: 2857:Philip of Opus 2854: 2849: 2844: 2839: 2834: 2829: 2824: 2819: 2814: 2809: 2804: 2799: 2794: 2789: 2784: 2779: 2774: 2769: 2764: 2759: 2754: 2749: 2744: 2739: 2734: 2729: 2724: 2718: 2716: 2710: 2709: 2702: 2701: 2694: 2687: 2679: 2673: 2672: 2666: 2660: 2654: 2646: 2643: 2642: 2641: 2635: 2627: 2626:External links 2624: 2622: 2621: 2596: 2575: 2560: 2542: 2528: 2498: 2480: 2459: 2438: 2407: 2391:this animation 2382: 2338:10.1086/348869 2332:(2): 150–158. 2309: 2285: 2273: 2231: 2224: 2206: 2193:10.1086/350839 2171: 2148: 2146:, pp. 194–195. 2125: 2112: 2089: 2081: 2071:(2 ed.). 2053: 2040: 2032:Owen Gingerich 2024: 1997: 1991:978-0099476443 1990: 1967: 1961:One volume of 1954: 1948:978-0099476443 1947: 1929: 1914: 1894: 1879: 1861: 1846: 1828: 1822:978-0099476443 1821: 1802: 1781:10.2307/284125 1755: 1720:(2): 113–129. 1700: 1646: 1622: 1600: 1598: 1595: 1594: 1593: 1588: 1583: 1578: 1573: 1566: 1563: 1545: 1542: 1533:Tychonic model 1523: 1520: 1509:Thomas Aquinas 1505: 1496:Thomas Aquinas 1451: 1442: 1425: 1419: 1414: 1410: 1406: 1402: 1396: 1392: 1386: 1381: 1378: 1375: 1371: 1367: 1362: 1358: 1348: 1331: 1306: 1300: 1295: 1291: 1287: 1283: 1277: 1273: 1269: 1264: 1259: 1255: 1251: 1247: 1241: 1237: 1233: 1228: 1224: 1220: 1215: 1211: 1207: 1202: 1198: 1188: 1182: 1151: 1145: 1141: 1138: 1132: 1127: 1123: 1113: 1104: 1087:imaginary unit 1068: 1059: 1041: 1035: 1030: 1026: 1022: 1018: 1012: 1008: 1004: 999: 995: 985: 974:Fourier series 958: 941: 938: 920: 884: 865: 862: 837:for solution. 831:n-body problem 765:The deferent ( 734:Commentariolus 712:Owen Gingerich 551: 548: 514: 513: 510: 507: 504: 498: 494: 493: 487: 481: 475: 469: 465: 464: 458: 452: 446: 440: 436: 435: 429: 423: 414: 405: 401: 400: 394: 388: 379: 370: 366: 365: 359: 353: 344: 338: 334: 333: 327: 321: 312: 303: 299: 298: 295: 289: 283: 274: 270: 269: 260: 253: 247:semimajor axis 240: 233: 153: 150: 15: 9: 6: 4: 3: 2: 3205: 3194: 3191: 3190: 3188: 3173: 3170: 3168: 3165: 3163: 3160: 3159: 3157: 3153: 3147: 3144: 3142: 3139: 3138: 3136: 3132: 3126: 3123: 3121: 3118: 3116: 3113: 3111: 3108: 3106: 3103: 3101: 3100:Metonic cycle 3098: 3096: 3093: 3091: 3088: 3086: 3085:Heliocentrism 3083: 3081: 3078: 3076: 3073: 3071: 3068: 3066: 3065:Counter-Earth 3063: 3061: 3058: 3056: 3053: 3051: 3048: 3047: 3045: 3041: 3035: 3032: 3030: 3027: 3025: 3022: 3020: 3017: 3015: 3012: 3010: 3007: 3005: 3002: 3000: 2997: 2996: 2994: 2990: 2984: 2980: 2976: 2974: 2972:(Aristarchus) 2970: 2966: 2964: 2960: 2956: 2954: 2950: 2946: 2944: 2940: 2936: 2935: 2933: 2929: 2923: 2920: 2918: 2915: 2913: 2910: 2908: 2905: 2903: 2900: 2898: 2895: 2893: 2890: 2888: 2885: 2883: 2880: 2878: 2875: 2873: 2870: 2868: 2865: 2863: 2860: 2858: 2855: 2853: 2850: 2848: 2845: 2843: 2840: 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2227: 2225:1-888009-08-X 2221: 2217: 2210: 2202: 2198: 2194: 2190: 2187:(1): 39–47 . 2186: 2182: 2175: 2168: 2167:0-521-56474-3 2164: 2160: 2159: 2152: 2143: 2142:Penguin Books 2139: 2135: 2129: 2122: 2116: 2107: 2106:Penguin Books 2103: 2099: 2093: 2084: 2078: 2074: 2070: 2069: 2064: 2057: 2050: 2049:Eye of Heaven 2044: 2037: 2033: 2028: 2020: 2016: 2012: 2008: 2001: 1993: 1987: 1983: 1976: 1974: 1972: 1964: 1958: 1950: 1944: 1940: 1933: 1925: 1921: 1917: 1911: 1907: 1906: 1898: 1890: 1886: 1882: 1880:0-521-40340-5 1876: 1872: 1865: 1857: 1853: 1849: 1843: 1839: 1832: 1824: 1818: 1814: 1806: 1798: 1794: 1790: 1786: 1782: 1778: 1774: 1770: 1766: 1759: 1751: 1747: 1743: 1739: 1735: 1731: 1727: 1723: 1719: 1715: 1711: 1704: 1689: 1685: 1681: 1677: 1673: 1669: 1665: 1661: 1657: 1650: 1642: 1635: 1634: 1626: 1618: 1617: 1612: 1605: 1601: 1592: 1589: 1587: 1584: 1582: 1581:Occam's razor 1579: 1577: 1574: 1572: 1569: 1568: 1562: 1560: 1554: 1551: 1550:try to adjust 1541: 1539: 1534: 1530: 1516: 1515: 1510: 1504: 1499: 1497: 1493: 1489: 1485: 1480: 1478: 1472: 1468: 1464: 1459: 1458:complex plane 1454: 1445: 1423: 1417: 1412: 1408: 1404: 1400: 1394: 1390: 1384: 1379: 1376: 1373: 1369: 1365: 1360: 1356: 1347: 1344: 1339: 1334: 1326: 1322: 1304: 1298: 1293: 1289: 1285: 1281: 1275: 1271: 1267: 1262: 1257: 1253: 1249: 1245: 1239: 1235: 1231: 1226: 1222: 1218: 1213: 1209: 1205: 1200: 1196: 1187: 1181: 1175: 1173: 1168: 1149: 1143: 1139: 1136: 1130: 1125: 1121: 1112: 1111: 1103: 1098: 1097:complex plane 1093: 1088: 1076: 1067: 1058: 1039: 1033: 1028: 1024: 1020: 1016: 1010: 1006: 1002: 997: 993: 984: 981: 979: 978:complex plane 975: 972: 963: 957: 952: 950: 947: 937: 933: 925: 919: 914: 912: 908: 904: 903: 893: 889: 883: 878: 876: 872: 861: 859: 855: 851: 847: 843: 838: 836: 832: 828: 824: 820: 817:Newtonian or 815: 812: 808: 804: 800: 797:According to 795: 793: 789: 780: 776: 772: 768: 763: 759: 755: 753: 749: 745: 744:non-Ptolemaic 741: 740: 735: 729: 725: 721: 719: 718: 713: 706: 705:Thomas Digges 701: 697: 694: 690: 686: 681: 679: 675: 671: 667: 663: 658: 655: 651: 647: 642: 638: 634: 630: 626: 622: 618: 614: 610: 606: 602: 597: 592: 589: 585: 576: 572: 570: 566: 562: 558: 547: 544: 539: 537: 533: 529: 525: 520: 499: 496: 495: 488: 482: 476: 470: 467: 466: 459: 453: 447: 441: 438: 437: 430: 424: 415: 406: 403: 402: 395: 389: 380: 371: 368: 367: 360: 354: 345: 339: 336: 335: 328: 322: 313: 304: 301: 300: 296: 290: 284: 275: 272: 271: 261: 254: 248: 242:Modern value 241: 234: 231: 230: 224: 222: 221: 216: 211: 209: 208: 201: 199: 195: 191: 187: 183: 179: 175: 171: 163: 158: 149: 147: 143: 138: 133: 131: 127: 122: 120: 119: 114: 110: 106: 101: 99: 95: 91: 87: 83: 80: 73: 69: 66: 65:Ancient Greek 62: 58: 54: 50: 46: 37: 33: 29: 22: 3069: 2978: 2968: 2962:(Hipparchus) 2958: 2949:Catasterismi 2948: 2938: 2797:Eratosthenes 2640:at MathPages 2612:. Retrieved 2608: 2599: 2583: 2578: 2551: 2545: 2533:, retrieved 2511: 2501: 2494:I q. 32 a. 1 2487: 2483: 2467: 2462: 2450: 2441: 2419: 2410: 2385: 2373:. Retrieved 2366:the original 2329: 2325: 2312: 2304: 2300: 2296: 2292: 2288: 2281: 2276: 2266: 2248: 2234: 2215: 2209: 2184: 2180: 2174: 2156: 2151: 2137: 2128: 2120: 2115: 2101: 2092: 2067: 2056: 2048: 2043: 2035: 2027: 2010: 2006: 2000: 1981: 1962: 1957: 1938: 1932: 1904: 1897: 1870: 1864: 1837: 1831: 1812: 1805: 1772: 1768: 1758: 1717: 1713: 1703: 1691:. Retrieved 1663: 1659: 1649: 1632: 1625: 1614: 1604: 1555: 1547: 1525: 1512: 1501: 1481: 1470: 1466: 1462: 1449: 1440: 1437: 1342: 1329: 1318: 1179: 1176: 1166: 1163: 1101: 1091: 1074: 1065: 1056: 1053: 982: 968: 954: 943: 934: 930: 923: 916: 900: 898: 891: 880: 873:created his 867: 857: 853: 839: 816: 796: 784: 778: 774: 770: 766: 756: 743: 737: 733: 730: 726: 722: 715: 710: 682: 673: 669: 659: 654:Solar System 617:heliocentric 609:Isaac Newton 593: 581: 564: 553: 540: 521: 517: 218: 214: 212: 205: 202: 198:epitrochoids 189: 177: 173: 167: 152:Introduction 134: 123: 116: 102: 78: 75: 68: 60: 42: 32: 3080:Geocentrism 2992:Instruments 2982:(Aristotle) 2787:Cleostratus 2752:Aristarchus 2732:Anaximander 2714:Astronomers 2455:PDF; 205 KB 2389:See, e.g., 2263:John Milton 1775:: 145–155. 1586:Overfitting 1544:Bad science 1529:Tycho Brahe 1323:, and is a 1319:This is an 664:in nature. 497:Star shell 45:Hipparchian 3155:Influenced 3134:Influences 3105:Octaeteris 3034:Triquetrum 2922:Timocharis 2907:Theodosius 2867:Posidonius 2827:Hipparchus 2817:Heraclides 2757:Aristyllus 2742:Apollonius 2737:Andronicus 2614:6 December 2535:6 December 2375:21 October 2140:. Arkana, 2104:. Arkana, 1611:"epicycle" 1576:Epicycloid 871:Copernicus 811:Gersonides 803:Ibn Bajjah 799:Maimonides 746:system of 693:Copernicus 678:Hipparchus 596:geocentric 565:"planetai" 532:opposition 235:Mean size 186:geocentric 109:Hipparchus 63:(from 28:Epicycloid 3009:Astrolabe 2942:(Ptolemy) 2862:Philolaus 2852:Oenopides 2837:Hypsicles 2782:Cleomedes 2777:Callippus 2767:Autolycus 2722:Aglaonice 2570:209693599 2428:681213472 2346:0021-1753 2297:Alfonso X 2201:120700705 2136:(1989) . 2100:(1989) . 2065:(1969) . 1984:. Arrow. 1941:. 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