Equant

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

Equant (or punctum aequans) is a mathematical concept developed by Claudius Ptolemy in the 2nd century AD to account for the observed motion of the planets. The equant is used to explain the observed speed change in different stages of the planetary orbit. This planetary concept allowed Ptolemy to keep the theory of uniform circular motion alive by stating that the path of heavenly bodies was uniform around one point and circular around another point.

Contents

Placement

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 the planet) was conceived to move at a constant angular speed with respect to the equant. In other words, 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. [1]

The reason for the implementation of the equant was to maintain 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.

Equation

The angle α whose vertex is at the center of the deferent and whose sides intersect the planet and the equant respectively is a function of time t:

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. [2]

The equant model has a body in motion on a circular path that does not share a center with Earth. The moving object's speed will actually vary during its orbit around the outer circle (dashed line), faster in the bottom half and slower in the top half. The motion is considered uniform only because the planet sweeps around equal angles in equal times from the equant point. The speed of the object is non-uniform when viewed from any other point within the orbit.

Discovery and use

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). [1]

In models of the universe that precede Ptolemy, generally attributed to Hipparchus, the eccentric and epicycles were already a feature. The Roman 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. [3] Such data requires the concept of eccentric centers of motion.

Before around The year 430 B.C., Meton and Euktemon of Athens observed differences in the lengths of seasons. [1] 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 B.C.. According to these calculations, Spring lasted about 94.5 days, Summer about 92.5, Fall about 88.125, and Winter about 90.125, showing that seasons did indeed exist differences in lengths of seasons. This 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 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. [1]

Most of what we know about Hipparchus comes to us through mentions of his works by Ptolemy in the Almagest. Hipparchus' models' features explained differences in the length of the seasons on Earth (known as 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. [4] Between Hipparchus’s model and Ptolemy’s there was an intermediate model that 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 may 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 the actual spacing and widths of retrograde arcs, which could be seen later according to Ptolemy’s model and compared. [1]

Ptolemy himself rectified this contradiction by introducing the equant in his writing Almagest IX, 5, 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 a planet. [1] 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. [1]

Ptolemy's model of astronomy was used as a technical method that could answer questions regarding astrology and predicting planets positions for almost 1500 years, even though the equant and eccentric were violations of pure Aristotelian physics which required 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 is usually very difficult to find any details of previously used models, except from writings by Ptolemy himself. [1] 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 oversight and corrections by concerned scholars over those centuries, culminated in the observations of Tycho Brahe at Uraniborg.

It wasn't 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. [5] [6]

Criticism

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 philosopher/astronomers, namely uniform circular motion about the Earth. [7] 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 moved the observation point explicitly off the center of the deferent to the equant. This can be seen as breaking part of the uniform circular motion rules. Noted critics of the equant include the Persian astronomer Nasir al-Din Tusi who developed the Tusi couple as an alternative explanation, [8] and Nicolaus Copernicus, whose alternative was a new pair of epicycles for each deferent. Dislike of the equant was a major motivation for Copernicus to construct his heliocentric system. [9] [10] This violation of perfect 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' movement of the Earth away from the center of the universe obviated the primary need for Ptolemy's epicycles by explaining retrograde movement as an optical illusion, but he re-instituted two smaller epicycles into each planet's motion in order to replace the equant.

See also

Related Research Articles

Hipparchus Ancient Greek scholar

Hipparchus of Nicaea was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry but is most famous for his incidental discovery of precession of the equinoxes.

Apparent retrograde motion

Apparent retrograde motion is the apparent motion of a planet in a direction opposite to that of other bodies within its system, as observed from a particular vantage point. Direct motion or prograde motion is motion in the same direction as other bodies.

Axial precession Gravity-induced, slow, and continuous change in the orientation of an astronomical bodys rotational axis

In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. In particular, it can refer to the gradual shift in the orientation of Earth's axis of rotation in a cycle of approximately 26,000 years. This is similar to the precession of a spinning top, with the axis tracing out a pair of cones joined at their apices. The term "precession" typically refers only to this largest part of the motion; other changes in the alignment of Earth's axis—nutation and polar motion—are much smaller in magnitude.

Geocentric model a superseded description of the Universe with Earth at the center

In astronomy, the geocentric model is a superseded description of the Universe with Earth at the center. Under the geocentric model, the Sun, Moon, stars, and planets all orbited Earth. The geocentric model was the predominant description of the cosmos in many ancient civilizations, such as those of Aristotle in Classical Greece and Ptolemy in Roman Egypt.

In the Hipparchian, Ptolemaic, and Copernican systems of astronomy, the epicycle 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.

<i>Almagest</i> Astronomical treatise

The Almagest is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy. One of the most influential scientific texts of all time, it canonized a geocentric model of the Universe that was accepted for more than 1200 years from its origin in Hellenistic Alexandria, in the medieval Byzantine and Islamic worlds, and in Western Europe through the Middle Ages and early Renaissance until Copernicus. It is also a key source of information about ancient Greek astronomy.

Heliocentrism Astronomical model where the Earth and planets revolve around the Sun

Heliocentrism is the astronomical model in which the Earth and planets revolve around the Sun at the center of the Solar System. Historically, heliocentrism was opposed to geocentrism, which placed the Earth at the center. The notion that the Earth revolves around the Sun had been proposed as early as the 3rd century BC by Aristarchus of Samos, but at least in the medieval world, Aristarchus' heliocentrism attracted little attention—possibly because of the loss of scientific works of the Hellenistic period.

Tychonic system Model of the Solar System proposed in 1588 by the Danish astronomer Tycho Brahe

The Tychonic system is a model of the Universe published by Tycho Brahe in the late 16th century, which combines what he saw as the mathematical benefits of the Copernican system with the philosophical and "physical" benefits of the Ptolemaic system. The model may have been inspired by Valentin Naboth and Paul Wittich, a Silesian mathematician and astronomer. A similar model was implicit in the calculations a century earlier by Nilakantha Somayaji of the Kerala school of astronomy and mathematics.

Celestial spheres Term in ancient times for the heavens

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<i>De revolutionibus orbium coelestium</i> Book by Copernicus

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Ibn al-Shatir Arab astronomer and clockmaker

ʿAbu al-Ḥasan Alāʾ al‐Dīn ʿAlī ibn Ibrāhīm al-Ansari known as Ibn al-Shatir or Ibn ash-Shatir was an Arab astronomer, mathematician and engineer. He worked as muwaqqit in the Umayyad Mosque in Damascus and constructed a sundial for its minaret in 1371/72.

Copernican Revolution 16th to 17th century intellectual revolution

The Copernican Revolution was the paradigm shift from the Ptolemaic model of the heavens, which described the cosmos as having Earth stationary at the center of the universe, to the heliocentric model with the Sun at the center of the Solar System. This revolution consisted of two phases; the first being extremely mathematical in nature and the second phase starting in 1610 with the publication of a pamphlet by Galileo. Beginning with the publication of Nicolaus Copernicus’s De revolutionibus orbium coelestium, contributions to the “revolution” continued until finally ending with Isaac Newton’s work over a century later.

Ancient Greek astronomy

Greek astronomy is astronomy written in the Greek language in classical antiquity. Greek astronomy is understood to include the ancient Greek, Hellenistic, Greco-Roman, and Late Antiquity eras. It is not limited geographically to Greece or to ethnic Greeks, as the Greek language had become the language of scholarship throughout the Hellenistic world following the conquests of Alexander. This phase of Greek astronomy is also known as Hellenistic astronomy, while the pre-Hellenistic phase is known as Classical Greek astronomy. During the Hellenistic and Roman periods, much of the Greek and non-Greek astronomers working in the Greek tradition studied at the Musaeum and the Library of Alexandria in Ptolemaic Egypt.

Tusi couple

The Tusi couple is a mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller circle to oscillate back and forth in linear motion along a diameter of the larger circle. The Tusi couple is a 2-cusped hypocycloid.

<i>Commentariolus</i> Work by Copernicus

The Commentariolus is Nicolaus Copernicus's brief outline of an early version of his revolutionary heliocentric theory of the universe. After further long development of his theory, Copernicus published the mature version in 1543 in his landmark work, De revolutionibus orbium coelestium.

Equatorium

An equatorium is an astronomical calculating instrument. It can be used for finding the positions of the Moon, Sun, and planets without calculation, using a geometrical model to represent the position of a given celestial body.

Copernican heliocentrism Concept that the Earth rotates around the Sun

Copernican heliocentrism is the name given to the astronomical model developed by Nicolaus Copernicus and published in 1543. This model positioned the Sun at the center of the Universe, motionless, with Earth and the other planets orbiting around it in circular paths, modified by epicycles, and at uniform speeds. The Copernican model displaced the geocentric model of Ptolemy that had prevailed for centuries, which had placed Earth at the center of the Universe. Copernican heliocentrism is often regarded as the launching point to modern astronomy and the Scientific Revolution.

Jacques du Chevreul (1595-1649). Jacques du Chevreul was born in Coutances, France and died in Paris, France. Du Chevreul grew up in an educated household and was the son of a magistrate. In 1616, he received a Master of Arts for studying humanities and philosophy at the University of Paris. Du Chevreul continued education at a higher level and received a Bachelor of Divinity for theology in 1619. He did not start teaching until 1620 where he remained associated with College Harcourt and University of Paris, up until two years before his death when he taught philosophy at the College Royal. Throughout his lifetime Jacques du Chevreul held various teaching and administrative positions including principal and rector. Little is known about his later life. Although he studied subjects such as philosophy, logic, ethics, metaphysics, and physics, he published his two popular books on mathematics. Arithmetica (1622) and Sphaera were both published in Paris, France. Sphaera, du Chevreul's most popular book was about his view of the world and the universe. He used references from the Bible, Aristotle, and Plato to reject the Copernican model and instead created his own eccentric-epicycle geocentric model of the universe. Du Chevreul believed that the earth was the center of the universe, but that the major planets Venus and Mercury orbited around the sun. He theorized that there were wandering and fixed stars in the heavens and there were a total of thirteen planets in his model. The heavens were in the order of the Moon, the Sun, Mars, Jupiter surrounded by four Medicean stars, Saturn with two satellites, and above all these levels resided God. Du Chevreul's cosmic scheme is a highly original attempt to resist Copernicanism and accommodate Galieleo's telescopic discoveries in an Aristotelian cosmos.

Historical models of the Solar System

The historical models of the Solar System began during prehistoric periods and is updated to this day. The models of the Solar System throughout history were first represented in the early form of cave markings and drawings, calendars and astronomical symbols. Then books and written records then became the main source of information that expressed the way the people of the time thought of the Solar System.

The Vicarious Hypothesis, or hypothesis vicaria, was a planetary hypothesis proposed by Johannes Kepler to describe the motion of Mars. The hypothesis adopted the circular orbit and equant of Ptolemy's planetary model as well as the heliocentrism of the Copernican model. Calculations using the Vicarious Hypothesis did not support a circular orbit for Mars, leading Kepler to propose elliptical orbits as one of three laws of planetary motion in Astronomia Nova.

References

  1. 1 2 3 4 5 6 7 8 Evans, James (April 18, 1984). "On the function and probable origin of Ptolemy's equant" (PDF). American Journal of Physics . 52 (12): 1080–89. Bibcode:1984AmJPh..52.1080E. doi:10.1119/1.13764 . Retrieved August 29, 2014.
  2. Eccentrics, deferents, epicycles and equants (Mathpages)
  3. Pliny the Elder. The Natural History, Book 2: An account of the world and the elements, Chapter 13: Why the same stars appear at some times more lofty and some times more near . Retrieved August 7, 2014.
  4. "The New Astronomy - Equants, from Part 1 of Kepler's Astronomia Nova". science.larouchepac.com. Retrieved August 1, 2014. An excellent video on the effects of the equant
  5. Perryman, Michael (2012-09-17). "History of Astrometry". European Physical Journal H. 37 (5): 745–792. arXiv: 1209.3563 . Bibcode:2012EPJH...37..745P. doi:10.1140/epjh/e2012-30039-4.
  6. Bracco; Provost (2009). "Had the planet Mars not existed: Kepler's equant model and its physical consequences". European Journal of Physics. 30: 1085–92. arXiv: 0906.0484 . Bibcode:2009EJPh...30.1085B. doi:10.1088/0143-0807/30/5/015.
  7. Van Helden. "Ptolemaic System" . Retrieved 20 March 2014.
  8. Craig G. Fraser (2006). The Cosmos: A Historical Perspective. Greenwood Publishing Group. p. 39. ISBN   978-0-313-33218-0.
  9. Kuhn, Thomas (1957). The Copernican Revolution. Harvard University Press. pp.  70–71. ISBN   978-0-674-17103-9. (copyright renewed 1985)
  10. Koestler A. (1959), The Sleepwalkers , Harmondsworth: Penguin Books, p. 322; see also p. 206 and refs therein.