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Ptolemy’s system.

The formation of astronomy as an exact science began thanks to the work of the outstanding Greek scientist Hipparchus. He first began systematic astronomical observations and their comprehensive mathematical analysis, laid the foundations for spherical astronomy and trigonometry, developed the theory of the motion of the Sun and the Moon, and on its basis – methods for predicting eclipses.
Hipparchus discovered that the apparent movement of the sun and moon in the sky is uneven. Therefore, he came to the point of view that these luminaries move uniformly in circular orbits, but the center of the circle is shifted relative to the center of the Earth. Such orbits were called eccentrics. Hipparchus compiled tables by which it was possible to determine the position of the sun and moon in the sky on any day of the year. As for the planets, according to Ptolemy, he “made no other attempt to explain the motion of the planets, but was content to tidy up the observations made before him, adding to them even more of his own. He limited himself to pointing out to his contemporaries the unsatisfactoryness of all the hypotheses by which some astronomers thought to explain the movement of the celestial bodies. ”
Thanks to the work of Hipparchus, astronomers abandoned the imaginary crystal spheres proposed by Eudoxus and moved on to more complex constructions with the help of epicycles and deferents proposed even before Hipparchus by Apollon Pergsky. The classical form of the theory of epicyclic movements was given by Claudius Ptolemy.
Ptolemy’s main work “Mathematical Syntax in 13 Books” or, as the Arabs later called it, “Almagest” (“The Greatest”) became known in medieval Europe only in the 12th century. In 1515 it was printed in Latin translated from Arabic, and in 1528 translated from Greek. Three times “Almagest” was published in Greek, in 1912 it was published in German.
Almagest is a real encyclopedia of ancient astronomy. In this book, Ptolemy did what none of his predecessors could do. He developed a method, using which it was possible to calculate the position of one or another planet at any predetermined point in time. This was not easy for him, and in one place he remarked:
“It seems easier to move the planets themselves than to comprehend their complex movement …”
Having “set” the Earth in the center of the world, Ptolemy presented the visible complex and uneven movement of each planet as the sum of several simple uniform circular motions.
According to Ptolemy, each planet moves uniformly in a small circle – an epicycle. The center of the epicycle, in turn, glides evenly around the circumference of a large circle, called the deferent. For a better agreement between the theory and the observational data, it was necessary to assume that the center of the defender is shifted relative to the center of the Earth. But that was not enough. Ptolemy was forced to assume that the motion of the center of the epicyclic along the descent is uniform (that is, its angular velocity of motion is constant), if we consider this motion not from the center of the descent O and not from the center of the Earth T, but from some “leveling point” E, called later equant.
Combining the observations with the calculations, Ptolemy obtained the method of successive approximations that the ratios of the radii of epicycles to the radii of the referents for Mercury, Venus, Mars, Jupiter, and Saturn are 0.376, 0.720, 0.658, 0.192, and 0.103, respectively. It is curious that in order to calculate the position of the planet in the sky, it was not necessary to know the distance to the planet, but only the ratio of the radii of epicycles and defenders mentioned above.
When constructing his geometric model of the world, Ptolemy took into account the fact that during their movement the planets deviate somewhat from the ecliptic. Therefore, for Mars, Jupiter and Saturn, he “tilted” the plane of the defenders to the ecliptic and the plane of the epicycles to the planes of the defenders. For Mercury and Venus, he introduced oscillations up and down with the help of small vertical circles. In general, to explain all the features observed at that time in the motion of the planets, Ptolemy introduced 40 epicycles. The system of the world of Ptolemy, in the center of which the Earth is located, is called geocentric.
In addition to the ratio of the radii of epicycles and deferents, to compare the theory with observations, it was necessary to specify periods of revolution in these circles. According to Ptolemy, all the upper planets complete a complete revolution around the circumference of epicycles in the same period of time as the Sun in the ecliptic, i.e., for a year. Therefore, the radii of the epicycles of these planets directed to the planets are always parallel to the direction from the Earth to the Sun. In the lower planets – Mercury and Venus – the period of revolution along the epicycle is equal to the time interval, and during which the planet returns to its starting point in the sky. For periods of inversion of the center of the epicycle along the circumference of the deferent, the picture is opposite. At Mercury and Venus they are equal to a year. Therefore, the centers of their epicycles always lie on a straight line connecting the sun and the Earth. For outer planets, they are determined by the time during which the planet, having described the full circle in the sky, returns to the same stars.

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