In this historical review, the opinions of Ancient Greece philosophers, astronomers, and poets such as Thales Milesian, Pythagoras, Plato, Eudoxus, Aristotle, Archimedes, Cicero, Diogenes Laertius, Iamblichus, Plutarch, Homer, and Aratus about the planet position calculations and about the possibility of predictions of natural phenomena are analyzed. The planet positions were predicted before Eudoxus (probably before Philolaus) by a spindle of Ananke and after Eudoxus by Antikythera mechanism. Following Pythagoras and Plato, it is established that the regular seismoacoustic observations were performed. In the Ancient World in the Mediterranean area, there was an extensive network of acoustic stations (~10 pcs), which were located in close proximity to the geologic faults. Also, it is shown that the ship that was carrying Antikythera mechanism (A-Ship) was built in 244 BC in Syracuse with direct participation of Archimedes and Archias from Corinthian. Later, the A-Ship was a part of the Roman Republic safety system. The grain volumes, which were delivered to Rome city by large grain vessels, and the population of Rome city in the period 74–71 BC were estimated. Planetary calculator might be used for the chronology of the historical events as a backward prediction in addition to present Radiocarbon dating and Dendrochronology methods.