Epicycle in the context of "Kepler's laws of planetary motion"

Claudius Ptolemy (/ˈtɒləmi/; Ancient Greek: Πτολεμαῖος, Ptolemaios; Latin: Claudius Ptolemaeus; c. 100 – 160s/170s AD), better known mononymously as Ptolemy, was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine, Islamic, and Western European science. The first was his astronomical treatise now known as the Almagest, originally entitled Mathēmatikḗ Syntaxis (Μαθηματικὴ Σύνταξις, Mathēmatikḗ Syntaxis, lit. 'Mathematical Treatise'). The second is the Geography, which is a thorough discussion on maps and the geographic knowledge of the Greco-Roman world. The third is the astrological treatise in which he attempted to adapt horoscopic astrology to the Aristotelian natural philosophy of his day. This is sometimes known as the Apotelesmatika (Αποτελεσματικά, 'On the Effects') but more commonly known as the Tetrábiblos (from the Koine Greek meaning 'four books'; Latin: Quadripartitum).

The Catholic Church promoted his work, which included the only mathematically sound geocentric model of the Solar System, and unlike most Greek mathematicians, Ptolemy's writings (foremost the Almagest) never ceased to be copied or commented upon, both in late antiquity and in the Middle Ages. However, it is likely that only a few truly mastered the mathematics necessary to understand his works, as evidenced particularly by the many abridged and watered-down introductions to Ptolemy's astronomy that were popular among the Arabs and Byzantines. His work on epicycles is now seen as a very complex theoretical model built in order to explain a false tenet based on faith.

Copernican heliocentrism is the astronomical model developed by Nicolaus Copernicus and published in 1543. This model positioned the Sun near 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 challenged the geocentric model of Ptolemy that had prevailed for centuries, which had placed Earth at the center of the Universe.

Although Copernicus had circulated an outline of his own theory to colleagues sometime before 1514, he did not decide to publish it until he was urged to do so later by his pupil Rheticus. His model was an alternative to the longstanding Ptolemaic model that purged astronomy of the equant in order to satisfy the theological and philosophical ideal that all celestial motion must be perfect and uniform, preserving the metaphysical implications of a mathematically ordered cosmos. His heliostatic model retained several false Ptolemaic assumptions such as the planets' circular orbits, epicycles, and uniform speeds, while at the same time using accurate ideas such as:

In astronomy, Kepler's laws of planetary motion give a good approximations for the orbits of planets around the Sun. They were published by Johannes Kepler from 1608-1621 in three works Astronomia nova, Harmonice Mundi and Epitome Astronomiae Copernicanae. The laws were based Kepler's concept of solar fibrils adapted to the accurate astronomical data of Tycho Brahe. These laws replaced the circular orbits and epicycles of Copernicus's heliostatic model of the planets with a heliocentric model that described elliptical orbits with planetary velocities that vary accordingly. The three laws state that:

1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

The elliptical orbits of planets were indicated by calculations of the orbit of Mars. From this, Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. The second law establishes that when a planet is closer to the Sun, it travels faster. The third law expresses that the farther a planet is from the Sun, the longer its orbital period.

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.

Ptolemy does not have a word for the equant – he used expressions such as "the eccentre producing the mean motion".