Early civilizations of india, Greece, Babylon, China and Egypt have progressed in astronomy. Is there any shared knowledge among them and if any how can we find out?
Much of the ancient transmission went through Greece. Astronomical (and other) knowledge was transmitted from Egypt to Greece at least by the time of early Pythagoreans (c. 500 BC), likely earlier. According to some sources, Pythagoras and Democritus visited Egypt personally to study with the priests. There might have been some early transmission from Babylon, such as of the Metonic and Saros cycles. After the Alexander's conquests (c. 320 BC), Babylonian astronomical observations and computational methods (including sexagesimal arithmetic) came to be widely used by Hellenistic astronomers, starting with Hipparchus (c.250 BC), see Babylonian sources. Recently, evidence of Babylon to Hellenistic Egypt transmission came to light, see Ancient Knowledge Transfer: Egyptian Astronomical Instructions based on Babylonian Methods:
"Egyptian astronomers computed the position of the planet Mercury with methods originating from Babylonia. This is proven by a study of two unpublished Egyptian instructional texts from the Ashmolean Museum (Oxford)... The texts date to 1-50 AD and are written in the Demotic language, a late stage of ancient Egyptian, on two ostraca (potsherds). They are the only known texts from Greco-Roman Egypt with instructions for computing astronomical phenomena with Babylonian methods. The instructions correspond exactly to methods invented in Babylonia several centuries earlier (400-300 BC). Surprisingly, the ostraca employ a mathematical formulation not found in Babylonian texts but whose existence has long been suspected by historians of astronomy."
Ptolemy employed Babylonian methods, inherited from Hipparchus, extensively in the Almagest (c.150 AD), the opus magnum of geocentric astronomy. At least parts of the Almagest were known in India by the time of Aryabhata (c.500 AD), see Duke's The Equant in India. China remained isolated until much later, flat Earth beliefs were still current by the time of Jesuit arrival in 1600-s, see Why was China slow to recognise the sphericity of Earth?
The Circulation of Astronomical Knowledge in the Ancient World is a comprehensive scholarly collection, highlights are given in Wikipedia's History of Astronomy.
To quote Subash Kak, “ The Astronomical Frame Myths from diverse cultures have an astronomical basis. Varuna, Ouranos (Greece), Tammuz (Sumer and Babylon), Adon (Canaan, Cyprus), Osiris (Egypt), Telipinus-Sharruma (Hittite) may be seen as being fundamentally the same. These gods were born to the great mother goddesses of their cultural fields: Aditi, Ishtar, Astarte, Isis, the Sun goddess of Arinna, Kybele (Attis). These gods and goddesses were often represented by the night sky. They encode the constellation of Orion and their myths refer to the vernal equinox in it (7th-5th millennium BCE). The precession of the vernal equinox was noted by transfer- ring Orion’s mythology to Taurus and Aries. Thus Tammuz was killed by his hostile brother Sirius, Osiris by his hostile brother Seth (Ursa Major), Tvastr, Dyauspita, Prajaapati were killed by their youngest son Indra or Rudra (Sirius). Terrible events in ancient myths usually refer to dramatic celestial phenomena, thus providing chronological markers. But to use them as such, without supporting evidence, can mislead for they could have been adopted at a late stage from another culture. The parallels in the myths suggest that interaction between civilizations goes back very far. In the Puranas, the myth of the churning of the ocean, amrta-manthana, represents, at one level, the shifting of the astronomical frame. Figure 2.1 shows a traditional representation of this story with the gods and the demons, symbolizing the upper and the lower hemispheres of the sky, respectively, at the opposite ends of the earth. Lakshmi rising from the oceans is Ceres of the Greeko-Roman pantheon.
When myths represent astronomical knowledge, the gods symbolize stars or planets. The word deva for god comes from the root div, to shine. The dramatic incidents of the myths refer to departure from an expected clockwork of the stars. This departure is a result of the precession of the earth that changes the orientation of the poles slowly. Other layers of stories added over time may have nothing to do with astronomical phenomena but rather express other dimensions of human experience.
Greek and Indian Zodiacs are almost the same, the Babylonian one slightly different having a few different images.
“Astronomy as Crowning Science:
Although the focus of the book is the astronomical code of the Rgveda, the evidence presented here allows us to sketch an account of the rise of astronomy in ancient India that, in turn, provides us insights into many different aspects of early Indian culture and civilization. The Puraanas credit the legendary figure of Pururavas with the division of the one fire, of which he had learnt from the gandharvas, into three that are the basis of the Vedic fire ritual. One may conclude that the fire ritual, together with its astronomical basis, was a part of the Vedic religion for as long as the bards of the Puraanas could remember. Corroboration of this view is obtained from the fact that the Greeks and the Romans also had fire ritual.
One may assume that either some fire ritual existed before the dispersal of the Indo-Europeans or having arisen somewhere it spread into the web of different Indo-European language speaking peoples through the process of migration and diffusion. Since current archaeological evidence suggests that such a dispersal occurred as early as 8000 BCE, astronomical concerns that were to be eventually reflected in the design of the three altars by Pururavas are very old. Conversely, if fire ritual spread from one Indo-European region to others, it is still likely to be very old because fire altars have been found in the Sindhu-Sarasvati sites. Such an old tradition implies observations over millennia, which helped find several fundamental facts about the lunar and the solar years as well as the motions of the planets.
The Brahmanas speak of ritual that parallels the passage of the year. Monthly rites like the darshapurnamaasa and seasonal rites like chaturmaasya required careful observation of the movements of the sun and the moon across months. Such rites necessitated the definition of the tithi, the division of the lunar year into 360 parts.
The key that opened the knowledge of the astronomy of the Rgveda and the Shatapatha Braahmana was the idea of numerical equivalences. The areas of the fire altars correspond to the broad astronomical facts about the year. This key is inherent in the Upanishadic equivalence between the outer and the inner. The fact that the altar increases by one unit area in each new construction indicates the intercalation that is necessary to bring the lunar year in line with the solar year. This increase goes on until the ninety fifth year when an additional correction is made to remove this error. We have sketched broad aspects of Rgvedic astronomy but its details remain to be deciphered.
The main elements of the astronomy of the Vedaanga Jyotisha are already contained in the Shatapatha Braahmana. Specifically, we find clear references to the nominal year of 372 tithis, the nakshatra year of 324 tithis, and a solar year of 371 tithis. The choice of 371 tithis for the solar year corresponds to 365.1949 days. But the fact that a further correction was required in 95 years indicates that these figures were known to be approximate.
The existence of an independent tradition of observation of planets and a theory thereof as suggested by our analysis of the Rgvedic code helps explain the puzzle why classical Indian astronomy of the Siddhaanta period uses many constants that are different from that of the Greeks. This confirms the thesis that although Siddhantic astronomy from the time of Aryabhata developed in some knowledge of Greek methods, the reason why it retained its characteristic form was because it was based on an independent, old tradition.
Continuing interaction in subsequent centuries is mirrored in parallels between Indian and Babylonian astronomies. We saw that the ratio of longest to shortest daylight changed from 1.29 to 1.5 as the focus of Vedic astronomy shifts from Sarasvati valleys to Northwest India. The latter value is to be found in Vedaanga Jyotisa of the latter half of the second millennium. One finds the same ratio of 1.5 in the Babylonian texts of the first millennium, although the earliest Babylonian texts spoke of a ratio of 2.0. "
by Subash Kak “The Astronomical code of the Rgveda (3rd edition)