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Discourse on the Tides (Discorso Sul Flusso E Il Reflusso Del Mare) is an essay written by Galileo Galilei in 1616 as a letter to Alessandro Orsini that explains the motion of Earth's tides as a consequence of Earth's rotation and revolution around the sun.

Background
Galileo's intrigue for tidal theory was motivated by a boat trip in 1595. While aboard a ferry carrying freshwater to Venice, Galileo noticed that the ship's cargo would undulate in accordance with the rocking of the ship. Galileo composed Discourse on the Tides while in Rome and appealing for papal acceptance of the teaching of Copernican theory. The letter is thus not just an explanation of tidal phenomenon but also a private confirmation and defense of Galileo's ideas on heliocentrism, a complete explanation of which is the subject of Dialogue Concerning the Two Chief World Systems.

Summary
The letter compares the ocean's waves to the disturbances in a vase of water, which move for three reasons: the slope of the vase, external forces exerted on the vase-water system, and the possible acceleration of the vase itself. Comparably, the ocean's tides are due to Earth's terrain, wind currents, and circular accelerations. In Galileo's understanding, Earth's rotation and simultaneous orbital revolution dictate that half of the Earth's rotational arc is in concordance with the direction of revolution and the other half is opposed to the direction of revolution. He reasoned that by sometimes matching and sometimes counteracting the motion of orbit, positive and negative acceleration is generated that influences bodies of water to rock back and forth, creating the tides. Though these opposing acceleration cannot be observed, large bodies of water containing points of location far away enough to experience significantly different vectors of acceleration would necessarily contour into waves.

Criticism
Galileo's theory of the tides could not account for one of the two daily occurrences of high tide observed along most coastlines. He countered that the totality of Earth's water system contains an extenuating amount of variables, such as Earth's tilt, uneven surface and coastline, oceanic depth, and inclement winds, that can explain the second high tide. Historical writer E. J. Aiton states that the discourse "is among the least successful of [Galileo's] investigations and completely misrepresents the phenomena it is supposed to explain," and served mainly as an impetus for continued research by Galileo's followers.

Retrospect
The discourse does not include gravitational forces in its theory to explain the Earth's orbit and does not consider the relation between the ocean and cosmic gravitational forces, like that of the moon. Occurring invisibly, gravity was far too mystic for Galileo's consideration. Galileo did end the Discourse on the Tides with reservations that his theory may be incorrect and the hope that further scientific investigation will confirm his proposal. Still, his 1632 Dialogue Concerning the Two Chief World Systems omitted gravitational forces and included a reprisal of his tidal theory in its fourth section.