## E57 -- Inquisitio physica in causam fluxus ac refluxus maris

(A physical inquiry into the cause of the ebb and flow of the sea)

Summary:
##### (based on Eric J. Aiton's introduction (written in English) to Opera Omnia Series 2, Volume 31 and on Clifford A. Truesdell's introduction to Opera Omnia Series 2, Volume 12)
In this essay, Euler explains the tides based on the Newtonian principle of universal gravitation. He first reviews the various explanations of tides that have already been proposed, making criticisms along the way:
• he claims to have refuted Galileo's theory that the tides are the result of a combination of the diurnal and annual motions of the earth since this theory is unable to produce the motion of the reciprocation of the seas;
• he dismisses Descartes' theory because this theory predicts low water when people generally observe high water; Euler also says that Descartes fails to explain how the moon hidden under the earth has the same effect as if it moved over the horizon.
Euler then goes on to calculate the magnitudes of the tide-generating forces and arrives at the following conclusions:
• the lunar tide is greater than the solar tide;
• a sphere is attracted as if its entire mass is concentrated at its center.
He also determines the equilibrium figure of the ocean that the disturbing forces are tending to produce at all times. It is in this exploration that Euler comes across his principal contribution to the theory of tides: The inclination of the surface of the water to the level depends only on the horizontal component of the disturbing force. He also establishes a prediction of the equilibrium theory in the absence of friction. As a result, he is able to tabulate the height of the tide for selected regions (places on the equator and in latitudes 30° and 60°); in each case, he provides the values for the moon on the equator and also when the moon has declination 20° north and south. He also shows that the force of the moon is four times that of the sun. Furthermore, he calculates the rate of change of the disturbing forces as the luminaries move towards or away from the horizon; from this, Euler gives the time of high water after the transit of the moon across the meridian. He is also able to give the moon's hour angle at the time of high water in terms of the angular separation of the sun and moon. Moreover, he shows that the restoring force of water is proportional to the elevation of the free surface, thus reducing the theory of the tides to a non-linear oscillation problem in which the sun and moon attract an elastic rather than a fluid ocean. Realizing the defectiveness of this procedure, Euler tries to introduce corrections a posteriori.

In addition, Euler offers a theory based on geometrical and analytical considerations by assuming that the vertical motion of the water is a forced harmonic oscillation with amplitude and phase influenced by the inertia of the water. He then offers a brief description of the retardation of the tides and explains that they are the result of the effect of the inertia of the water. Finally, Euler brings together the explanations of the principal phenomena of the tides in open oceans and near islands and arranges these systematically and compares them with observations.
###### a.s.
Motto:
Cur nunc declivi nudentur littora Ponto,
Adversis tumeat nunc maris unda fretis;
Dum vestro monitu naturam consulo rerum:
Quam procul a terris abdita causa latet!
In solem lunamque feror. Si plauditis auso;
Sidera sublimi vertice summa petam.

According to the records, it was presented to the St. Petersburg Academy on June 15, 1739.

Publication:
• Originally published in Pièces qui ont remporté le prix de l'académie royale des sciences de Paris in 1740, pp. 235-350
• Republished in Recueil des pièces qui ont remporté les prix de l’académie royale des sciences 4, 1752, pp. 235-350 + 4 diagrams. [E57a]
• Reprinted in I. Newton, Philosophiae naturalis principia mathematica, ed. Leseur and Jaquier, 3, Geneva 1742, pp. 283-374 [E57b]
• Reprinted in I. Newton, Philosophiae naturalis principia mathematica, ed. Leseur and Jaquier, 3, editio nova, Coloniae Allobrogum 1760, pp. 283-374 [E57c]
• Opera Omnia: Series 2, Volume 31, pp. 19 - 124
Documents Available:
• Original publication: E57