Chapter 4, page 9.
4.9 4.1 Action by the Moon and by the Sun.
4.2 Modality of action: attraction.
4.3 The ocean tides observed from the space.
4.4 Values of attraction.
4.5 The direction of the tide waves.
4.6 The continents and the flowing of the tide waves.
4.7 Number of the tide waves.
4.8 Tide waves and sublunar points.
4.9 The physical equation for the ocean tides.
4.10 When Earth, Moon and Sun are aligned.
4.11 Tide cadences.
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The physical equation to be applied in the ocean tides.
On the physical equation, the two approaches agree, though for different and incompatible reasons.
The attraction on the waters of the oceans.
.1 The attractive force between two bodies decreases proportionally to the square of the distance.
.2 Applying this law, the attractive forces exerted on the same water by the Sun and by the Moon are, on average, in the ratio of 1 (that of the Sun) to 0,0056 (that of the Moon).
.3 In other words, the attractive forces exerted on the water of our oceans by the mass of the Sun are around, and on average, 178 times those exerted by the mass of the Moon.
#04 - The physical equation for the ocean tides.
.4 However, it is the Moon that in reality has the larger effect on the tides (Sun 1; Moon over 2,19).
.5

Considering the values of the masses, we may infer that the forces bringing about the ocean tides decrease their action roughly in proportion to the cube of the distance between the masses that generate them.
.6 The two ways of explaining the ocean tides agree on the text of the five paragraphs given up to this point.
The current approach on this issue.
.7

The ocean tides are due to the varying attraction produced by the two masses of the Moon and Sun on the water of the oceans of the Earth, as this revolves on its axis.
.8 However, in the case of the tides, another formula is to be applied: the action of the attracting bodies on the water of our Earth decreases in proportion to the third power of the distance.
.9 That's because, in the case of the ocean tides, and only in that case, the important is not the force in itself, but its variation, the gradient.
.10 Indeed, the distance of the Sun from the Earth is so huge, that the difference of its attraction on the various spots on our planet is very small.
.11 Instead, the attraction of the Moon on the different places on Earth varies a lot.
.12

Quantitative analysis shows that the differences of the gravitational forces across the Earth's surface are proportional to the cube of the distances Sun - Earth and Earth - Moon. (M. Tomczak, 1996).
Comments.
.13 (alternative approach) What has been asserted at the paragraph 9 is arbitrary, because of at least three considerations.
.14 First of all, because that would infringe the principle of universal uniformity in the physical laws.
.15 That would be like going back to the times before Galileo, when one could use an ad hoc explanation, such as that of the epicycles.
.16

Even if you admit, provisionally, that it is the gradient of the forces, and not the forces by themselves, to determine the ocean tides, the analysis referred to at the §12 of this page, would be useless, because the space unit of each tide - where it takes place, from its generation, until its final phase - is not the Earth.
.17 Each basin has its tide, autonomous with regard to the other ones, in process on the other basins, at the same time.
.18 That keeps its validity, does not matter whether the alternative theory, on the ocean tides, presented on the next chapter, proves to be true, or not.
.19 If the assertion of the paragraph 9 were true, one could devise useful and exceptional experiments.
.20 One could even make a low consumption engine, taking advantage of the fact, that the force of attraction could be overcome by its variation, by a good deal.
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